Simulation

DISCRETE-EVENT

http://tolerance.ajou.ac.kr

SYSTEM
SIMULATION
Third Edition

Jerry Banks • John S. Carson II
Barry L. Nelson • David M. Nicol


Part I. Introduction to Discrete-Event
System Simulation
Ch.1 Introduction to Simulation
Ch.2 Simulation Examples
Ch.3 General Principles
Ch.4 Simulation Software

Ch. 1 Introduction to Simulation
Real-world
process

A set of assumptions
concerning the behavior of a system


Modeling
& Analysis

Simulation





the imitation of the operation of a real-world process or system over time
to develop a set of assumptions of mathematical, logical, and symbolic
relationship between the entities of interest, of the system.
to estimate the measures of performance of the system with the simulationgenerated data

Simulation modeling can be used



as an analysis tool for predicting the effect of changes to existing systems
as a design tool to predict the performance of new systems

1.1 When Simulation is the Appropriate Tool (1)


Simulation enables the study of, and experimentation with, the internal
interactions of a complex system, or of a subsystem within a complex
system.
Informational, organizational, and environmental changes can be
simulated, and the effect of these alterations on the model’s behavior
can be observed.
The knowledge gained in designing a simulation model may be of great
value toward suggesting improvement in the system under
investigation.
By changing simulation inputs and observing the resulting outputs,
valuable insight may be obtained into which variables are most
important and how variables interact.
Simulation can be used as a pedagogical device to reinforce analytic
solution methodologies.

1.1 When Simulation is the Appropriate Tool (2)


Simulation can be used to experiment with new designs or policies prior
to implementation, so as to prepare for what may happen.
Simulation can be used to verify analytic solutions.
By simulating different capabilities for a machine, requirements can be
determined.
Simulation models designed for training allow learning without the cost
and disruption of on-the-job learning.
Animation shows a system in simulated operation so that the plan can
be visualized.
The modern system (factory, wafer fabrication plant, service
organization, etc.) is so complex that the interactions can be treated
only through simulation.

1.2 When Simulation is not Appropriate
When the problem can be solved using common sense.
When the problem can be solved analytically.
When it is easier to perform direct experiments.
When the simulation costs exceed the savings.
When the resources or time are not available.
When system behavior is too complex or can’t be defined.
When there isn’t the ability to verify and validate the model.


1.3 Advantages and Disadvantages of Simulation (1)
Advantages
















New polices, operating procedures, decision rules, information flows,
organizational procedures, and so on can be explored without disrupting
ongoing operations of the real system.
New hardware designs, physical layouts, transportation systems, and so on,
can be tested without committing resources for their acquisition.
Hypotheses about how or why certain phenomena occur can be tested for
feasibility.
Insight can be obtained about the interaction of variables.
Insight can be obtained about the importance of variables to the performance
of the system.
Bottleneck analysis can be performed indicating where work-in-process,
information, materials, and so on are being excessively delayed.
A simulation study can help in understanding how the system operates rather
than how individuals think the system operates.
“What-if” questions can be answered. This is particularly useful in the design
of new system.

1.3 Advantages and Disadvantages of Simulation (2)


Disadvantages








Model building requires special training. It is an art that is learned over time
and through experience. Furthermore, if two models are constructed by two
competent individuals, they may have similarities, but it is highly unlikely
that they will be the same.
Simulation results may be difficult to interpret. Since most simulation
outputs are essentially random variables (they are usually based on random
inputs), it may be hard to determine whether an observation is a result of
system interrelationships or randomness.
Simulation modeling and analysis can be time consuming and expensive.
Skimping on resources for modeling and analysis may result in a simulation
model or analysis that is not sufficient for the task.
Simulation is used in some cases when an analytical solution is possible, or
even preferable, as discussed in Section 1.2. This might be particularly true
in the simulation of some waiting lines where closed-form queueing models
are available.

1.4 Areas of Application (1)


WSC(Winter Simulation Conference) : http://www.wintersim.org


Manufacturing Applications

 Analysis of electronics assembly operations
 Design and evaluation of a selective assembly station for high-precision scroll
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compressor shells
Comparison of dispatching rules for semiconductor manufacturing using largefacility models
Evaluation of cluster tool throughput for thin-film head production
Determining optimal lot size for a semiconductor back-end factory
Optimization of cycle time and utilization in semiconductor test manufacturing
Analysis of storage and retrieval strategies in a warehouse
Investigation of dynamics in a service-oriented supply chain
Model for an Army chemical munitions disposal facility

Semiconductor Manufacturing

 Comparison of dispatching rules using large-facility models
 The corrupting influence of variability
 A new lot-release rule for wafer fabs

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Assessment of potential gains in productivity due to proactive reticle management
Comparison of a 200-mm and 300-mm X-ray lithography cell
Capacity planning with time constraints between operations
300-mm logistic system risk reduction

Construction Engineering

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Construction of a dam embankment
Trenchless renewal of underground urban infrastructures
Activity scheduling in a dynamic, multiproject setting
Investigation of the structural steel erection process
Special-purpose template for utility tunnel construction

Military Application
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Modeling leadership effects and recruit type in an Army recruiting station
Design and test of an intelligent controller for autonomous underwater vehicles
Modeling military requirements for nonwarfighting operations
Multitrajectory performance for varying scenario sizes
Using adaptive agent in U.S Air Force pilot retention




Logistics, Transportation, and Distribution Applications
 Evaluating the potential benefits of a rail-traffic planning algorithm
 Evaluating strategies to improve railroad performance
 Parametric modeling in rail-capacity planning
 Analysis of passenger flows in an airport terminal
 Proactive flight-schedule evaluation
 Logistics issues in autonomous food production systems for extended-duration
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space exploration
Sizing industrial rail-car fleets
Product distribution in the newspaper industry
Design of a toll plaza
Choosing between rental-car locations
Quick-response replenishment



Business Process Simulation
 Impact of connection bank redesign on airport gate assignment
 Product development program planning
 Reconciliation of business and systems modeling
 Personnel forecasting and strategic workforce planning



Human Systems
 Modeling human performance in complex systems
 Studying the human element in air traffic control


1.5 Systems and System Environment


System


defined as a group of objects that are joined together in some
regular interaction or interdependence toward the accomplishment
of some purpose.

System Environment


changes occurring outside the system.

The decision on the boundary between the system and its
environment may depend on the purpose of the study.

1.6 Components of a System (1)


Entity : an object of interest in the system.
Attribute : a property of an entity.
Activity : a time period of specified length.
State : the collection of variables necessary to describe the
system at any time, relative to the objectives of the
study.
Event : an instantaneous occurrence that may change the
state of the system.
Endogenous : to describe activities and events occurring
within a system.
Exogenous : to describe activities and events in an
environment that affect the system.

1.6 Components of a System (2)


1.7 Discrete and Continuous Systems
Systems can be categorized as discrete or continuous.



Bank : a discrete system
The head of water behind a dam : a continuous system


1.8 Model of a System


Model




a representation of a system for the purpose of studying the system
a simplification of the system
sufficiently detailed to permit valid conclusions to be drawn about
the real system

1.9 Types of Models


Static or Dynamic Simulation Models




Static simulation model (called Monte Carlo simulation) represents
a system at a particular point in time.
Dynamic simulation model represents systems as they change over
time

Deterministic or Stochastic Simulation Models




Deterministic simulation models contain no random variables and
have a known set of inputs which will result in a unique set of
outputs
Stochastic simulation model has one or more random variables as
inputs. Random inputs lead to random outputs.

The model of interest in this class is discrete, dynamic, and
stochastic.

1.10 Discrete-Event System Simulation


The simulation models are analyzed by numerical rather than by
analytical methods




Analytical methods employ the deductive reasoning of mathematics
to solve the model.
Numerical methods employ computational procedures to solve
mathematical models.

1.11 Steps in a Simulation Study (1)


Problem formulation


Policy maker/Analyst understand and agree with the formulation.

Setting of objectives and overall project plan
Model conceptualization


The art of modeling is enhanced by an ability to abstract the
essential features of a problem, to select and modify basic
assumptions that characterize the system, and then to enrich and
elaborate the model until a useful approximation results.

Data collection


As the complexity of the model changes, the required data elements
may also change.

Model translation


GPSS/HTM or special-purpose simulation software



Verified?



Is the computer program performing properly?
Debugging for correct input parameters and logical structure

Validated?




The determination that a model is an accurate representation of the
real system.
Validation is achieved through the calibration of the model

Experimental design


The decision on the length of the initialization period, the length of
simulation runs, and the number of replications to be made of each
run.

Production runs and analysis


To estimate measures of performances



More runs?
Documentation and reporting




Program documentation : for the relationships between input
parameters and output measures of performance, and for a
modification
Progress documentation : the history of a simulation, a chronology
of work done and decision made.

Implementation

Four phases according to Figure 1.3






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First phase : a period of discovery or orientation
(step 1, step2)
Second phase : a model building and data collection
(step 3, step 4, step 5, step 6, step 7)
Third phase : running the model
(step 8, step 9, step 10)
Fourth phase : an implementation
(step 11, step 12)


Ch2. Simulation Examples


Three steps of the simulations






Determine the characteristics of each of the inputs to the
simulation. Quite often, these may be modeled as probability
distributions, either continuous or discrete.
Construct a simulation table. Each simulation table is different, for
each is developed for the problem at hand.
For each repetition i, generate a value for each of the p inputs, and
evaluate the function, calculating a value of the response yi. The
input values may be computed by sampling values from the
distributions determined in step 1. A response typically depends on
the inputs and one or more previous responses.


The simulation table provides a systematic method for tracking
system state over time.
Inputs
Repetition
s
1
2
·
·
·
n

Xi1

Xi2

…

Xij

Response
…

Xip

yi

2.1 Simulation of Queueing Systems (1)

Calling population

Waiting Line


Server

Fig. 2.1 Queueing System

A queueing system is described by its calling population, the
nature of the arrivals, the service mechanism, the system
capacity, and the queueing discipline.



In the single-channel queue, the calling population is infinite.


If a unit leaves the calling population and joins the waiting line or
enters service, there is no change in the arrival rate of other units
that may need service.

Arrivals for service occur one at a time in a random fashion.


Once they join the waiting line, they are eventually served.

Service times are of some random length according to a
probability distribution which does not change over time.
The system capacity has no limit, meaning that any number of
units can wait in line.
Finally, units are served in the order of their arrival (often called
FIFO: First In, First out) by a single server or channel.



Arrivals and services are defined by the distribution of the time
between arrivals and the distribution of service times,
respectively.
For any simple single- or multi-channel queue, the overall
effective arrival rate must be less than the total service rate, or
the waiting line will grow without bound.


In some systems, the condition about arrival rate being less than
service rate may not guarantee stability



System state : the number of units in the system and the status
of the server(busy or idle).
Event : a set of circumstances that cause an instantaneous
change in the state of the system.


In a single-channel queueing system there are only two possible
events that can affect the state of the system.
 the arrival event : the entry of a unit into the system
 the departure event : the completion of service on a unit.

Simulation clock : used to track simulated time.



If a unit has just completed service, the simulation proceeds in
the manner shown in the flow diagram of Figure 2.2.


Note that the server has only two possible states : it is either busy
or idle.
Departure
Event

Begin server
idle time

No

Another unit
waiting?

Yes

Remove the waiting unit
from the queue
Begin servicing the unit

Fig. 2.2 Service-just-completed flow diagram



The arrival event occurs when a unit enters the system.


The unit may find the server either idle or busy.
 Idle : the unit begins service immediately
 Busy : the unit enters the queue for the server.

Arrival
Event

Unit enters
service

No

Server
busy?

Yes

Unit enters queue
for service

Fig. 2.3 Unit-entering-system flow diagram


Fig. 2.4 Potential unit actions upon arrival

Fig. 2.5 Server outcomes after service completion




Simulations of queueing systems generally require the
maintenance of an event list for determining what happens next.
Simulation clock times for arrivals and departures are computed
in a simulation table customized for each problem.
In simulation, events usually occur at random times, the
randomness imitating uncertainty in real life.
Random numbers are distributed uniformly and independently
on the interval (0, 1).
Random digits are uniformly distributed on the set {0, 1, 2, … ,
9}.
The proper number of digits is dictated by the accuracy of the
data being used for input purposes.



Pseudo-random numbers : the numbers are generated using a
procedure → detailed in Chapter 7.
Table 2.2. Interarrival and Clock Times


Assume that the times between arrivals were generated by rolling
a die five times and recording the up face.

Table 2.3. Service Times




Assuming that all four values
are equally likely to occur,
these values could have
been generated by placing
the numbers one through
four on chips and drawing
the chips from a hat with
replacement, being sure to
record the numbers selected.
The only possible service
times are one, two, three,
and four time units.




The interarrival times and service times must be meshed to simulate
the single-channel queueing system.
Table 2.4 was designed specifically for a single-channel queue which
serves customers on a first-in, first-out (FIFO) basis.



Table 2.4 keeps track of the clock time
at which each event occurs.
The occurrence of the two types of
events(arrival and departure event) in
chronological order is shown in Table
2.5 and Figure 2.6.
Figure 2.6 is a visual image of the
event listing of Table 2.5.
The chronological ordering of events
is the basis of the approach to
discrete-event simulation described in
Chapter 3.



Figure 2.6 depicts the number of customers in the system at the
various clock times.



Example 2.1 Single-Channel Queue
Arrival

Departure

Checkout Counter

 Assumptions
• Only one checkout counter.
• Customers arrive at this checkout counter at random from 1 to 8

minutes apart. Each possible value of interarrival time has the same
probability of occurrence, as shown in Table 2.6.
• The service times vary from 1 to 6 minutes with the probabilities shown
in Table 2.7.
• The problem is to analyze the system by simulating the arrival and
service of 20 customers.



Example 2.1 (Cont.)




A simulation of a grocery store that starts with an empty system is
not realistic unless the intention is to model the system from startup
or to model until steady-state operation is reached.
A set of uniformly distributed random numbers is needed to generate
the arrivals at the checkout counter. Random numbers have the
following properties:
 The set of random numbers is uniformly distributed between 0 and 1.
 Successive random numbers are independent.



Random digits are converted to random numbers by placing a
decimal point appropriately.
 Table A.1 in Appendix or RAND() in Excel.



The rightmost two columns of Tables 2.6 and 2.7 are used to
generate random arrivals and random service times.



Example 2.1 (Cont.) Table 2.8


The first random digits are 913. To obtain the corresponding time between
arrivals, enter the fourth column of Table 2.6 and read 8 minutes from the
first column of the table.



Example 2.1 (Cont.) Table 2.9


The first customer's service time is 4 minutes because the random digits 84
fall in the bracket 61-85



Example 2.1 (Cont.)

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

The essence of a manual simulation is the simulation table.
The simulation table for the single-channel queue, shown in Table
2.10, is an extension of the type of table already seen in Table 2.4.
Statistical measures of performance can be obtained form the
simulation table such as Table 2.10.
Statistical measures of performance in this example.
 Each customer's time in the system
 The server's idle time



In order to compute summary statistics, totals are formed as shown
for service times, time customers spend in the system, idle time of
the server, and time the customers wait in the queue.



Example 2.1 (Cont.)


The average waiting time for a customer : 2.8 minutes
average waitng time =



The probability that a customer has to wait in the queue : 0.65
probability ( wait ) =



total time customers wait in queue 56
=
= 2.8 (min)
total numbers of customers
20

number of customers who wait 13
=
= 0.65
20

The fraction of idle time of the server : 0.21
probability of idle server =



total idle time of server
18
=
= 0.21
total run time of simulation 86

The probability of the server being busy: 0.79 (=1-0.21)



Example 2.1 (Cont.)


The average service time : 3.4 minutes
average service time =

total service time
68
=
= 3.4 (min)
total numbers of customers 20

This result can be compared with the expected service time by finding the
mean of the service-time distribution using the equation in table 2.7.
∞

E ( S ) = ∑sp ( s )
s =0

E ( S ) = 1(0.10) + 2(0.20) + 3(0.30) + 4(0.25) + 5(1.10) + 6(0.05) = 3.2 (min)
The expected service time is slightly lower than the average service time in
the simulation. The longer the simulation, the closer the average will be to

E (S )



Example 2.1 (Cont.)


The average time between arrivals : 4.3 minutes

average time between arrivals =


sum of all times between arrivals 82
=
= 4.3 (min)
numbers of arrivals − 1
19

This result can be compared to the expected time between arrivals by
finding the mean of the discrete uniform distribution whose endpoints are
a=1 and b=8.
E ( A) =

a + b 1+ 8
=
= 4.5 (min)
2
2

The longer the simulation, the closer the average will be to


E ( A)

The average waiting time of those who wait : 4.3 minutes

average waiting time of those who wait =

total time customers wait in queue
56
=
= 4.3 (min)
total numbers of customers who wiat 13



Example 2.1 (Cont.)


The average time a customer spends in the system : 6.2 minutes

average time customer spends in the system =
average time
customer spends
in the system

=

total time customers spend in system 124
=
= 6.2 (min)
20

average time
customer spends
waiting in the queue

+

average time
customer spends
in service

∴ average time customer spends in the system = 2.8 + 3.4 = 6.2 (min)



Example 2.2 The Able Baker Carhop Problem
Able

Baker

 A drive-in restaurant where carhops take orders and bring food to the car.
 Assumptions
• Cars arrive in the manner shown in Table 2.11.
• Two carhops Able and Baker - Able is better able to do the job and works a
bit faster than Baker.
• The distribution of their service times is shown in Tables 2.12 and 2.13.

Example 2.2 (Cont.)








A simplifying rule is that Able
gets the customer if both
carhops are idle.
If both are busy, the customer
begins service with the first
server to become free.
To estimate the system
measures of performance, a
simulation of 1 hour of
operation is made.
The problem is to find how
well the current arrangement
is working.




Example 2.2 (cont.)






The row for the first customer is filled in manually, with the random-number
function RAND() in case of Excel or another random function replacing the
random digits.
After the first customer, the cells for the other customers must be based on
logic and formulas. For example, the “Clock Time of Arrival” (column D) in
the row for the second customer is computed as follows:
D2 = D1 + C2
The logic to computer who gets a given customer can use the Excel macro
function IF(), which returns one of two values depending on whether a
condition is true or false.
IF( condition, value if true, value if false)

Nothing

N
o

Is Baker idle?

N
o

Is Able i dle?

Yes

Is it time of arriv al?

clock = 0


N
o

Yes

Yes

completed?

Is there the service

Yes

N
o

Store clock time (col umn H or K)

service (colum n E)

Generate random di git for

Convert random digit to random
num ber for service time
(column G)

Able service begi n (col um n F )

Generate random digit for
servi ce (col um n E)

num ber for service time

Convert random digit to random
(col um n J)

Baker servi ce begin (c olumn I)

Increm ent clock



Example 2.2 (cont.)


The logic requires that we compute when Able and Baker will become free,
for which we use the built-in Excel function for maximum over a range,
MAX().

F10 = IF ( D10 > MAX ( H $1 : H 9), D10, IF ( D10 > MAX ( K $1 : K 9), "" ,
MIN ( MAX ( H $1 : H 9), MAX ( K $1 : K 9))))




If the first condition (Able idle when customer 10 arrives) is true, then the
customer begins immediately at the arrival time in D10. Otherwise, a second
IF() function is evaluated, which says if Baker is idle, put nothing (..) in the
cell. Otherwise, the function returns the time that Able or Baker becomes
idle, whichever is first [the minimum or MIN() of their respective completion
times].
A similar formula applies to cell I10 for “Time Service Begins” for Baker.



Example 2.2 (Cont.)


For service times for Able, you could use another IF() function to make the
cell blank or have a value:
G10 = IF(F10 > 0,new service time, "")
H10 = IF(F10 > 0, F10+G10, "")



The analysis of Table 2.14 results in the following:
 Over the 62-minute period Able was busy 90% of the time.
 Baker was busy only 69% of the time. The seniority rule keeps
Baker less busy (and gives Able more tips).
 Nine of the 26 arrivals (about 35%) had to wait. The average
waiting time for all customers was only about 0.42 minute (25
seconds), which is very small.
 Those nine who did have to wait only waited an average of 1.22
minutes, which is quite low.
 In summary, this system seems well balanced. One server cannot
handle all the diners, and three servers would probably be too
many. Adding an additional server would surely reduce the waiting
time to nearly zero. However, the cost of waiting would have to be
quite high to justify an additional server.

2.2 Simulation of Inventory Systems (1)


This inventory system has a
periodic review of length N, at
which time the inventory level is
checked.
An order is made to bring the
inventory up to the level M.
In this inventory system the lead
time (i.e., the length of time
between the placement and
receipt of an order) is zero.
Demand is shown as being
uniform over the time period



Notice that in the second cycle, the amount in inventory drops below
zero, indicating a shortage.
Two way to avoid shortages




Carrying stock in inventory
: cost - the interest paid on the funds borrowed to buy the items, renting of
storage space, hiring guards, and so on.
Making more frequent reviews, and consequently, more frequent purchases
or replenishments
: the ordering cost

The total cost of an inventory system is the measure of performance.




The decision maker can control the maximum inventory level, M, and the
length of the cycle, N.
In an (M,N) inventory system, the events that may occur are: the demand for
items in the inventory, the review of the inventory position, and the receipt of
an order at the end of each review period.



Example 2.3 The Newspaper Seller’s Problem










A classical inventory problem concerns the purchase and sale of
newspapers.
The paper seller buys the papers for 33 cents each and sells them
for 50 cents each. (The lost profit from excess demand is 17 cents
for each paper demanded that could not be provided.)
Newspapers not sold at the end of the day are sold as scrap for 5
cents each. (the salvage value of scrap papers)
Newspapers can be purchased in bundles of 10. Thus, the paper
seller can buy 50, 60, and so on.
There are three types of newsdays, “good,” “fair,” and “poor,” with
probabilities of 0.35, 0.45, and 0.20, respectively.



Example 2.3 (Cont.)






The problem is to determine the optimal number of papers the
newspaper seller should purchase.
This will be accomplished by simulating demands for 20 days and
recording profits from sales each day.
The profits are given by the following relationship:
 revenue   cost of   lost profit from   salvage from sale  
Pofit = 
 from sales  −  newspapers  −  excess demand  +  of scrap papers  
 
 
 

 
 
 







The distribution of papers demanded on each of these days is given
in Table 2.15.
Tables 2.16 and 2.17 provide the random-digit assignments for the
types of newsdays and the demands for those newsdays.



Example 2.3 (Cont.)




The simulation table for the decision to purchase 70 newspapers is shown in
Table 2.18.
The profit for the first day is determined as follows:
Profit = $30.00 - $23.10 - 0 + $.50 = $7.40
 On day 1 the demand is for 60 newspapers. The revenue from the sale of 60

newspapers is $30.00.
 Ten newspapers are left over at the end of the day.
 The salvage value at 5 cents each is 50 cents.





The profit for the 20-day period is the sum of the daily profits, $174.90. It can
also be computed from the totals for the 20 days of the simulation as follows:
Total profit = $645.00 - $462.00 - $13.60 + $5.50 = $174.90
The policy (number of newspapers purchased) is changed to other values
and the simulation repeated until the best value is found.



Example 2.4 Simulation of an (M,N) Inventory System










This example follows the pattern of the probabilistic order-level
inventory system shown in Figure 2.7.
Suppose that the maximum inventory level, M, is11 units and the
review period, N, is 5 days. The problem is to estimate, by simulation,
the average ending units in inventory and the number of days when a
shortage condition occurs.
The distribution of the number of units demanded per day is shown in
Table 2.19.
In this example, lead time is a random variable, as shown in Table
2.20.
Assume that orders are placed at the close of business and are
received for inventory at the beginning of business as determined by
the lead time.



Example 2.4 (Cont.)



For purposes of this example, only five cycles will be shown.
The random-digit assignments for daily demand and lead time are
shown in the rightmost columns of Tables 2.19 and 2.20.



Example 2.4 (Cont.)


The simulation has been started with the inventory level at 3 units
and an order of 8 units scheduled to arrive in 2 days' time.
Beginning Inventory of
=
Third day






Ending Inventory of
2 day in first cycle

+

new order

The lead time for this order was 1 day.
Notice that the beginning inventory on the second day of the third cycle was
zero. An order for 2 units on that day led to a shortage condition. The units
were backordered on that day and the next day also. On the morning of day
4 of cycle 3 there was a beginning inventory of 9 units. The 4 units that were
backordered and the 1 unit demanded that day reduced the ending
inventory to 4 units.
Based on five cycles of simulation, the average ending inventory is
approximately 3.5 (88 ÷ 25) units. On 2 of 25 days a shortage condition
existed.

2.3 Other Examples of Simulation (1)


Example 2.5 A Reliability Problem
Milling Machine

Bearing

Bearing

Bearing

Repairperson








Downtime for the mill is estimated at $5 per minute.
The direct on-site cost of the repairperson is $15 per hour.
It takes 20 minutes to change one bearing, 30 minutes to change two
bearings, and 40 minutes to change three bearings.
The bearings cost $16 each.
A proposal has been made to replace all three bearings whenever a
bearing fails.



Example 2.5 (Cont.)





The cumulative distribution function of
the life of each bearing is identical, as
shown in Table 2.22.

The delay time of the
repairperson's arriving at the
milling machine is also a
random variable, with the
distribution given in Table 2.23.



Example 2.5 (Cont.)








Table 2.24 represents a simulation of 20,000 hours of operation under
the current method of operation.
Note that there are instances where more than one bearing fails at the
same time.
This is unlikely to occur in practice and is due to using a rather coarse
grid of 100 hours.
It will be assumed in this example that the times are never exactly the
same, and thus no more than one bearing is changed at any
breakdown. Sixteen bearing changes were made for bearings 1 and
2, but only 14 bearing changes were required for bearing 3.



Example 2.5 (Cont.)


The cost of the current system is estimated as follows:
 Cost of bearings = 46 bearings × $16/bearing = $736
 Cost of delay time = (110 + 125 + 95) minutes × $5/minute = $1650
 Cost of downtime during repair =

46 bearings × 20 minutes/bearing × $5/minute = $4600
 Cost of repairpersons =
46 bearings × 20 minutes/bearing × $15/60 minutes = $230
 Total cost = $736 + $1650 + $4600 + $230 = $7216


Table 2.25 is a simulation using the proposed method. Notice that
bearing life is taken from Table 2.24, so that for as many bearings
as were used in the current method, the bearing life is identical for
both methods.



Example 2.5 (Cont.)






Since the proposed method uses more bearings than the current method,
the second simulation uses new random digits for generating the additional
lifetimes.
The random digits that lead to the lives of the additional bearings are shown
above the slashed line beginning with the 15th replacement of bearing 3.
The total cost of the new policy :








Cost of bearings = 54 bearings × $16/bearing = $864
Cost of delay time = 125 minutes × $5/minute = $625
Cost of downtime during repairs = 18 sets × 40 minutes/set × $5/minute = $3600
Cost of repairpersons = 18 sets × 40 minutes/set × $15/60 minutes = $180
Total cost = $864 + $625 + $3600 + $180 = $5269

The new policy generates a savings of $1947 over a 20,000-hour
simulation. If the machine runs continuously, the simulated time is about 2
1/4 years. Thus, the savings are about $865 per year.



Example 2.6 Random Normal Numbers



A classic simulation
problem is that of a
squadron of bombers
attempting to destroy an
ammunition depot
shaped as shown in
Figure 2.8.



Example 2.6 (Cont.)










If a bomb lands anywhere on the depot, a hit is scored. Otherwise,
the bomb is a miss.
The aircraft fly in the horizontal direction.
Ten bombers are in each squadron.
The aiming point is the dot located in the heart of the ammunition
dump.
The point of impact is assumed to be normally distributed around the
aiming point with a standard deviation of 600 meters in the
horizontal direction and 300 meters in the vertical direction.
The problem is to simulate the operation and make statements
about the number of bombs on target.



Example 2.6 (Cont.)


The standardized normal variate, Z, with mean 0 and standard deviation 1,
is distributed as

Z=





X −µ

σ

X = Zσ + µ

µ
where X is a normal random variable, is the true mean of the distribution of
σ
X, and is the standard deviation of X.
In this example the aiming point can be considered as (0, 0); that is, the
µ
µ
value in the horizontal direction is 0, and similarly for the value in the
vertical direction.
X = Zσ X
Y = Zσ Y
where (X,Y) are the simulated coordinates of the bomb after it has fallen
and σ Y = 300
σ X = 600
Y = 300 Z i
X = 600 Z
i



Example 2.6 (Cont.)


The values of Z are random normal numbers.
 These can be generated from uniformly distributed random numbers,

as discussed in Chapter 7.
 Alternatively, tables of random normal numbers have been generated.
A small sample of random normal numbers is given in Table A.2.
 For Excel, use the Random Number Generation tool in the Analysis
TookPak Add-In to generate any number of normal random values in a
range of cells.




The table of random normal numbers is used in the same way as
the table of random numbers.
Table 2.26 shows the results of a simulated run.

Example 2.6 (Cont.)


Example 2.6 (Cont.)















The mnemonic RNN x
stands for .random normal number to compute
the x coordinate. and corresponds to above.
Zi
The first random normal number used was –0.84, generating an x
coordinate 600(-0.84) = -504.
The random normal number to generate the y coordinate was 0.66,
resulting in a y coordinate of 198.
Taken together, (-504, 198) is a miss, for it is off the target.
The resulting point and that of the third bomber are plotted on
Figure 2.8.
The 10 bombers had 3 hits and 7 misses.
Many more runs are needed to assess the potential for destroying
the dump.
This is an example of a Monte Carlo, or static, simulation, since
time is not an element of the solution.



Example 2.7 Lead-Time Demand






Lead-time demand may occur in an inventory system.
The lead time is the time from placement of an order until the order
is received.
In a realistic situation, lead time is a random variable.
During the lead time, demands also occur at random. Lead-time
demand is thus a random variable defined as the sum of the
demands over the lead time, or

∑

T

i =0



Di

where i is the time period of the lead time, i = 0, 1, 2, … , Di is the
demand during the ith time period; and T is the lead time.
The distribution of lead-time demand is determined by simulating
many cycles of lead time and building a histogram based on the
results.

Example 2.7 (Cont.)
The daily demand is given by the
following probability distribution:





The lead time is a random
variable given by the following
distribution:




Example 2.7 (Cont.)







The incomplete simulation
table is shown in Table 2.29.
The random digits for the first
cycle were 57. This generates
a lead time of 2 days.
Thus, two pairs of random
digits must be generated for
the daily demand.



Example 2.7 (Cont.)





The histogram might appear as
shown in Figure 2.9.
This example illustrates how
simulation can be used to study an
unknown distribution by generating
a random sample from the
distribution.

2.4 Summary


This chapter introduced simulation concepts via examples in order to
illustrate general areas of application and to motivate the remaining
chapters.
The next chapter gives a more systematic presentation of the basic
concepts. A more systematic methodology, such as the eventscheduling approach described in Chapter 3, is needed.
Ad hoc simulation tables were used in completing each example.
Events in the tables were generated using uniformly distributed random
numbers and, in one case, random normal numbers.
The examples illustrate the need for determining the characteristics of
the input data, generating random variables from the input models, and
analyzing the resulting response.

Ch. 3 General Principles


Discrete-event simulation
The basic building blocks of all discrete-event simulation models
: entities and attributes, activities and events.
A system is modeled in terms of





its state at each point in time
the entities that pass through the system and the entities that represent
system resources
the activities and events that cause system state to change.

Discrete-event models are appropriate for those systems for which
changes in system state occur only at discrete points in time.
This chapter deals exclusively with dynamic, stochastic systems (i.e.,
involving time and containing random elements) which change in a
discrete manner.

3.1Concepts in Discrete-Event Simulation (1)


System : A collection of entities (e.g., people and machines) that interact
together over time to accomplish one or more goals.
Model : An abstract representation of a system, usually containing
structural, logical, or mathematical relationships which describe a
system in terms of state, entities and their attributes, sets, processes,
events, activities, and delays.
System state : A collection of variables that contain all the information
necessary to describe the system at any time.
Entity : Any object or component in the system which requires explicit
representation in the model (e.g., a server, a customer, a machine).
Attributes : The properties of a given entity (e.g., the priority of a waiting
customer, the routing of a job through a job shop).



List : A collection of (permanently or temporarily) associated entities, ordered
in some logical fashion (such as all customers currently in a waiting line,
ordered by first come, first served, or by priority).
Event : An instantaneous occurrence that changes the state of a system
(such as an arrival of a new customer).
Event notice : A record of an event to occur at the current or some future
time, along with any associated data necessary to execute the
event; at a minimum, the record includes the event type and
the event time.
Event list : A list of event notices for future events, ordered by time of
occurrence also known as the future event list (FEL).
Activity : A duration of time of specified length (e.g., a service time or
interarrival time), which is known when it begins (although it may be
defined in terms of a statistical distribution).



Delay : A duration of time of unspecified indefinite length, which is not
known until it ends (e.g., a customer's delay in a last-in, first-out
waiting line which, when it begins, depends on future arrivals).
Clock : A variable representing simulated time, called CLOCK in the
examples to follow.
An activity typically represents a service time, an interarrival time, or any other
processing time whose duration has been characterized and defined by the
modeler.
An activity's duration may be specified in a number of ways:





1. Deterministic-for example, always exactly 5 minutes;
2. Statistical-for example, as a random draw from among 2, 5, 7 with equal
probabilities;
3. A function depending on system variables and/or entity attributes-for example,
loading time for an iron ore ship as a function of the ship's allowed cargo
weight and the loading rate in tons per hour.



an end of inspection event
event time = 105
Event notice
100

current simulated time

105

time

Inspection time (=5)

The duration of an activity is computable from its specification at the
instant it begins.
To keep track of activities and their expected completion time, at the
simulated instant that an activity duration begins, an event notice is
created having an event time equal to the activity's completion time.



A delay's duration




Not specified by the modeler ahead of time, But rather determined by
system conditions.
Quite often, a delay's duration is measured and is one of the desired
outputs of a model run.
How long to wait?

A customer's delay in a waiting line may be dependent on the number
and duration of service of other customers ahead in line as well as the
availability of servers and equipment.

Delay


Activity

What so called

a conditional wait

an unconditional wait

A completion

a secondary event

a primary event

by placing an event
notice on the FEL

by placing the associated
entity on another list, not the
FEL, perhaps repre-senting
a waiting line

A management

 System state, entity attributes and the number of active entities, the
contents of sets, and the activities and delays currently in progress are all
functions of time and are constantly changing over time.
 Time itself is represented by a variable called CLOCK.



EXAMPLE 3.1 (Able and Baker, Revisited)







Consider the Able-Baker carhop system of Example 2.2.
System state
 LQ (t ) : the number of cars waiting to be served at time t
 LA (t ) : 0 or 1 to indicate Able being idle or busy at time t
 LB (t ) : 0 or 1 to indicate Baker being idle or busy at time t
Entities : Neither the customers (i.e., cars) nor the servers need
to be explicitly represented, except in terms of the
state variables, unless certain customer averages are
desired (compare Examples 3.4 and 3.5)
Events
 Arrival event
 Service completion by Able
 Service completion by Baker



EXAMPLE 3.1 (Cont.)


Activities
 Interarrival time, defined in Table 2.11
 Service time by Able, defined in Table 2.12
 Service time by Baker, defined in Table 2.13



Delay : A customer's wait in queue until Able or Baker becomes free

The definition of the model components provides a static description of
the model.
A description of the dynamic relationships and interactions between the
components is also needed.



A discrete-event simulation
: the modeling over time of a system all of whose state changes occur
at discrete points in time-those points when an event occurs.
A discrete-event simulation proceeds by producing a sequence of system
snapshots (or system images) which represent the evolution of the system
through time.

Figure 3.1 Prototype system snapshot at simulation time t

3.1.1. The Event-Scheduling/Time-Advanced Algorithm (1)


The mechanism for advancing simulation time and guaranteeing that all
events occur in correct chronological order is based on the future event
list (FEL).
Future Event List (FEL)




to contain all event notices for events that have been scheduled to occur at
a future time.
to be ordered by event time, meaning that the events are arranged
chronologically; that is, the event times satisfy

t <t1 ≤t 2 ≤ 
current value of
simulated time

≤t n

Imminent event

Scheduling a future event means that at the instant an activity begins,
its duration is computed or drawn as a sample from a statistical
distribution and the end-activity event, together with its event time, is
placed on the future event list.



List processing : the management of a list .




the removal of the imminent event
: As the imminent event is usually at the top of the list, its removal is as
efficient as possible.
the addition of a new event to the list, and occasionally removal of some
event (called cancellation of an event)
: Addition of a new event (and cancellation of an old event) requires a
search of the list.

The efficiency of this search depends on the logical organization of the
list and on how the search is conducted.
The removal and addition of events from the FEL is illustrated in Figure
3.2.



The system snapshot at time 0 is defined by the initial conditions and
the generation of the so-called exogenous events.
An exogenous event : a happening “outside the system” which
impinges on the system.
The specified initial conditions define the system state at time 0.


In Figure 3.2, if t = 0, then the state (5, 1, 6) might represent the initial
number of customers at three different points in the system.

How future events are generated?




to generate an arrival to a queueing system
by a service-completion event in a queueing simulation
to generate runtimes and downtimes for a machine subject to breakdowns

To generate an arrival to a queueing system
- The end of an interarrival interval is an example of a primary event.




By a service-completion event in a queueing simulation


A new service time, s*, will be generated for the next customer.
 When one customer completes service, at current time CLOCK = t
 If the next customer is present

⇒ The next service-completion event will be scheduled to occur at future
time t* = t + s* by placing onto the FEL a new event notice of type service
completion.







A service-completion event will be generated and scheduled at the
time of an arrival event, provided that, upon arrival, there is at least one
idle server in the server group.
Beginning service : a conditional event triggered only on the condition
that a customer is present and a server is free.
Service completion : a primary event.
Service time : an activity



By a service-completion event in a queueing simulation (Cont.)



A conditional event is triggered by a primary event occurring
Only primary events appear on the FEL.

To generate runtimes and downtimes for a machine subject to
breakdowns









At time 0, the first runtime will be generated and an end-of-runtime
event scheduled.
Whenever an end-of-runtime event occurs, a downtime will be
generated and an end-of-downtime event scheduled on the FEL.
When the CLOCK is eventually advanced to the time of this end-ofdowntime event, a runtime is generated and an end-of-runtime event
scheduled on the FEL.
An end of runtime and an end of downtime : primary events.
A runtime and a downtime : activities



Every simulation must have a stopping event, here called E, which defines
how long the simulation will run.
There are generally two ways to stop a simulation:


1. At time 0, schedule a stop simulation event at a specified future time T E.
Ex) Simulate a job shop for TE = 40 hours,that is,over the time interval [0, 40].



2. Run length TE is determined by the simulation itself. Generally, T E is the time of
occurrence of some specified event E.
Ex) the time of the 100th service completion at a certain service center.
the time of breakdown of a complex system.
the time of disengagement or total kill in a combat simulation.
the time at which a distribution center ships the last carton in a day's orders.

In case 2, TE is not known ahead of time. Indeed, it may be one of the
statistics of primary interest to be produced by the simulation.

3.1.2. World Views (1)


World views
: the event-scheduling world view, the process-interaction world view, and the
activity-scanning world view.

The process-interaction approach






To focus on entities and their life cycle
Process : the life cycle of one entity
: a time-sequenced list of events, activities, and delays, including
demands for resources, that define the life cycle of one entity
as it moves through a system.
The life cycle consists of various events and activities.
Some activities may require the use of one or more resources whose
capacities are limited (queueing).



The process-interaction approach (Cont.)


Figure 3.4 shows the interaction between two customer processes as
customer n+1 is delayed until the previous customer's “end-service event”
occurs.



The activity-scanning approach







Simple in concept, but slow runtime on computers
: Both the event-scheduling and the process-interaction approaches
use a variable time advance.
: The activity-scanning approach uses a fixed time increment and
a rule-based approach to decide whether any activities can begin
at each point in simulated time.
To focus on the activities and those conditions
At each clock advance, the conditions for each activity are checked and,
if the conditions are true, then the corresponding activity begins.
Three-phase approach
: to combine pure activity-scanning approach with the features of event
scheduling, variable time advance.
: events are considered to be activities of duration-zero time units.



The activity-scanning approach (Cont.)







In the three-phase approach, activities are divided into two categories.
- B activities : activities bound to occur; all primary events and
unconditional activities.
- C activities : activities or events that are conditional upon certain
conditions being true.
Phase A : Remove the imminent event from the FEL and advance the clock
to its event time. Remove any other events from the FEL that
have the same event time.
Phase B : Execute all B-type events that were removed from the FEL.
Phase C : Scan the conditions that trigger each C-type activity and
activate any whose conditions are met. Rescan until no
additional C-type activities can begin or events occur.



EXAMPLE 3.2 (Able and Baker, Back Again)



The events and activities were identified in Example 3.1.
Using the three-phase approach, the conditions for beginning each activity
in Phase C are:

Activity

Condition

Service time by Able
Service time by Baker


A customer is in queue and Able is idle
A customer is in queue, Baker is idle, and
Able is busy

Using the process-interaction approach, we view the model from the
viewpoint of a customer and its “life cycle.” Considering a life cycle
beginning upon arrival, a customer process is pictured in Figure 3.4

3.1.3. Manual Simulation Using Event Scheduling (1)


Example 3.3 (Single-Channel Queue)


Reconsider Example 2.1



System state (LQ(t), LS(t)) :
 LQ(t) is the number of customers in the waiting line
 LS(t) is the number being served (0 or 1) at time t





Entities : The server and customers are not explicitly modeled,
except in terms of the state variables above.
Events :
 Arrival (A)
 Departure (D)
 Stopping event (E), scheduled to occur at time 60.



Example 3.3 (Cont.)


Event notices (event type, event time) :
 (A, t ), representing an arrival event to occur at future time t
 (D, t ), representing a customer departure at future time t
 (E, 60), representing the simulation-stop event at future time 60.



Activities :
 Interarrival time, defined in Table 2.6
 Service time, defined in Table 2.7




Delay : Customer time spent in waiting line.
The effect of the arrival and departure events was first shown in
Figures 2.2 and 2.3 and is shown in more detail in Figures 3.5 and
3.6.



Example 3.3 (Cont.)




The interarrival times and service times will be identical to those
used in Table 2.10

Initial conditions

 the system snapshot at time zero (CLOCK = 0)
 LQ(0) = 0, LS(0) = 1
 both a departure event and arrival event on the FEL.







The simulation is scheduled to stop at time 60.
Server utilization : total server busy time (B) / total time (TE).
a* : the generated interarrival time
s* : the generated service times
The simulation in Table 3.1 covers the time interval [0, 21].



Example 3.4 (The Checkout-Counter Simulation, Continued)


In Example 3.3, to estimate :

 mean response time : the average length of time a customer spends

in the system
 mean proportion of customers who spend 4 or more minutes in the
system.




Entities (Ci, t ) : representing customer Ci who arrived at time t
Event notices :
 (A, t, Ci), the arrival of customer Ci at future time t
 (D, t, Cj), the departure of customer Cj at future time t





Set : “CHECKOUTLINE,” the set of all customers currently
at the checkout counter (being served or waiting to be
served), ordered by time of arrival
A customer entity with arrival time as an attribute is added in order to
estimate mean response time.



Example 3.4 (Cont.)


Three new cumulative statistics will be collected :
 S : the sum of customer response times for all customers who have

departed by the current time
 F : the total number of customers who spend 4 or more minutes at
the checkout counter
 ND : the total number of departures up to the current simulation time.


These three cumulative statistics will be updated whenever the
departure event occurs.



The simulation table for Example 3.4 is shown in Table 3.2.



The response time for customer is computed by
Response time = CLOCK TIME - attribute “time of arrival”



Example 3.4 (Cont.)


For a simulation run length of 21 minutes
 the average response time was S/ND = 15/4 = 3.75 minutes
 the observed proportion of customers who spent 4 or more minutes in

the system was F/ND = 0.75.



Example 3.5 (The Dump Truck Problem, Figure 3.7)
Traveling

Loading
Loader
queue

Weighing
queue
First-Come
First-Served





Scale

First-Come
First-Served

The distributions of loading time, weighing time, and travel time are given in
Tables 3.3, 3.4, and 3.5, respectively, from Table A.1.
The purpose of the simulation is to estimate the loader and scale utilizations
(percentage of time busy).





The activity times are taken from the
following list as needed:



Example 3.5 (Cont.)


System state [LQ(t), L(t), WQ(t), W(t)]
 LQ(t) = number of trucks in loader queue
 L(t) = number of trucks (0, 1, or 2) being loaded
 WQ(t) = number of trucks in weigh queue
 W(t) = number of trucks (0 or 1) being weighed, all at simulation

time t


Event notices :
 (ALQ, t, DTi ), dump truck i arrives at loader queue (ALQ) at time t
 (EL, t, DTi), dump truck i ends loading (EL) at time t
 (EW, t, DTi), dump truck i ends weighing (EW) at time t



Entities : The six dump trucks (DT 1, … , DT 6)



Example 3.5 (Cont.)


Lists :
 Loader queue : all trucks waiting to begin loading, ordered on

a first come, first served basis
 Weigh queue : all trucks waiting to be weighed, ordered on a first
come, first served basis






Activities : Loading time, weighing time, and travel time
Delays : Delay at loader queue, and delay at scale
It has been assumed that five of the trucks are at the loaders and
one is at the scale at time 0.
The simulation table is given in Table 3.6.



Example 3.5 (Cont.)


This logic for the occurrence of the end-loading event
 When an end-loading (EL) event occurs, say for truck j at time t , other

events may be triggered.
 If the scale is idle [W(t)=0], truck j begins weighing and an endweighing event (EW) is scheduled on the FEL.
 Otherwise, truck j joins the weigh queue.
 If at this time there is another truck waiting for a loader, it will be
removed from the loader queue and will begin loading by the
scheduling of an end-loading event (EL) on the FEL.


In order to estimate the loader and scale utilizations, two cumulative
statistics are maintained:
 BL = total busy time of both loaders from time 0 to time t
 BS = total busy time of the scale from time 0 to time t



Example 3.5 (Cont.)


The utilizations are estimated as follows:
 average loader utilization
 average scale utilization





49 / 2
= 0.32
76
76
= = 1.00
76

=

These estimates cannot be regarded as accurate estimates of the
long-run “steady-state” utilizations of the loader and scale.
A considerably longer simulation would be needed to reduce the
effect of the assumed conditions at time 0 (five of the six trucks at
the loaders) and to realize accurate estimates.



Example 3.6 (The Dump Truck Problem Revisited)



The events and activities were identified in Example 3.5.
Using the activity scanning approach
Activity

Condition

Loading time

Truck is at front of loader queue, and at least one loader is idle.

Weighing time

Truck is at front of weigh queue and weigh scale is idle.

Travel time

Truck has just completed weighing.



Using the process-interaction approach

3.2 List Processing
3.2.1 List : Basic Properties and Operations (1)


Head Pointer
Event type

Event type

Event type

Event type

Event time

Event time

Event time

Event time

Any data

Any data

Any data

Any data

Next pointer

Next pointer

Next pointer

Next pointer

Record

Record

Record

List : a set of ordered or ranked records.
Record : one entity or one event notice.
Field : an entity identifier and its attributes
: the event type, event time, and any other event
related data

Tail Pointer



How to store record in a physical location in computer memory



in arrays : successive records in contiguous locations
by pointers to a record : structures in C, classes in C++

The main operations on a list :


Removing a record from the top of the list.
 when time is advanced and the imminent event is due to be executed.
 by adjusting the head pointer on the FEL ⇔ by removing the event at the

top of the FEL.



Removing a record from any location on the list.
 If an arbitrary event is being canceled, or an entity is removed from a list

based on some of its attributes (say, for example, its priority and due date)
to begin an activity.
 by making a partial search through the list.



The main operations on a list (Cont.)


Adding an entity record to the top or bottom of the list.
 when an entity joins the back of a first-in first-out queue.
 by adjusting the tail pointer on the FEL ⇔ by adding an entity to the

bottom of the FEL



Adding a record to an arbitrary position on the list, determined
by the ranking rule.
 if a queue has a ranking rule of earliest due date first (EDF).
 by making a partial search through the list.

The goal of list-processing techniques
: to make second and fourth operations efficient

3.2.2 Using Arrays for List Processing (1)


The notation R(i) : the ith record in the array
Advantage


Any specified record, say the ith, can be retrieved quickly without
searching, merely by referencing R(i ).

Disadvantage






When items are added to the middle of a list or the list must be
rearranged.
Arrays typically have a fixed size, determined at compile time or
upon initial allocation when a program first begins to execute.
In simulation, the maximum number of records for any list may be
difficult or impossible to determine ahead of time, while the current
number in a list may vary widely over the course of the simulation
run.

Memory address

100

101

1

102

2

103

3

104

4


105

5

106

7

adding

100

101

1

102

2

103

3

104

4

8

108

9

109

110

109

110

10

6
105

5

107

106

6

107

7

108

8

9

10

move move move move

Two methods for keeping track of the ranking of records in a list




to store the first record in R(1), the second in R(2), and so on, and the last in
R(tailptr), where tailptr is used to refer to the last item in the list.
a variable called a head pointer, with name headptr, points to the record at
the top of the list.



Example 3.7 (A List for the Dump Trucks at the Weigh Queue)






In Example 3.5, suppose that a waiting line of three dump trucks
occurred at the weigh queue, at CLOCK time 10 in Table 3.6.

Suppose further that the model is tracking one attribute of each
dump truck, its arrival time at the weigh queue, updated each time it
arrives.
Suppose that the entities are stored in records in an array
dimensioned from 1 to 6, one record for each dump truck.



Example 3.7 (Cont.)




Each entity is represented by a record with 3 fields, the first an
entity identifier, the second the arrival time at the weigh queue, and
the last a pointer field to “point to” the next record, if any, in the list
representing the weigh queue, as follows:
[ DTi , arrival time at weigh queue, next index ]
At CLOCK time 10, the list of entities in the weigh queue would be
defined by:
headptr = 3

R(4) = [DT4, 10.0, 0]

R(1) = [DT1, 0.0, 0]

R(5) = [DT5, 0.0, 0]

R(2) = [DT2, 10.0, 4]

R(6) = [DT6, 0.0, 0]

R(3) = [DT3, 5.0, 2]

tailptr = 4



Example 3.7 (Cont.)


To traverse the list, start with the head pointer, go to that record,
retrieve that record's next pointer, and proceed, to create the list in
its logical order, as for example:
headptr = 3
R(3) = [DT3, 5.0, 2]
R(2) = [DT2, 10.0, 4]
R(4) = [DT4, 10.0, 0]
tailptr = 4



Example 3.7 (Cont.)





At CLOCK time 12, dump truck DT 3 begins weighing and thus
leaves the weigh queue.
headptr = 2

At CLOCK time 20, dump truck DT 5 arrives to the weigh queue
and joins the rear of the queue.
tailptr = 5

3.2.3 Using Dynamic Allocation and Linked Lists (1)


In procedural languages such as C and C++, and in most
simulation languages, entity records are dynamically created
when an entity is created and event notice records are
dynamically created whenever an event is scheduled on the
future event list.
The languages themselves, or the operating systems on which
they are running, maintain a linked list of free chunks of
computer memory and allocate a chunk of desired size upon
request to running programs.
With dynamic allocation, a record is referenced by a pointer
instead of an array index. A pointer to a record can be thought
of as the physical or logical address in computer memory of the
record.



In our example, we will use a notation for records identical to
that in the previous section (3.2.2):
Entities: [ ID, attributes, next pointer ]
Event notices: [ event type, event time, other data, next pointer ]

If for some reason we wanted the third item on the list, we would
have to traverse the list, counting items until we reached the
third record.
Unlike arrays, there is no way to retrieve directly the ith record in
a linked list, as the actual records may be stored at any arbitrary
location in computer memory and are not stored contiguously as
are arrays.



Example 3.8 (The Future Event List and the Dump Truck
Problem)








Based on Table 3.6, event notices in the dump truck problem of
Example 3.5 are expanded to include a pointer to the next event
notice on the future event list and can be represented by:
[ event type, event time, DT i , nextptr ]
as, for example,
[ EL, 10, DT 3, nextptr ]
where EL is the end loading event to occur at future time 10 for
dump truck DT 3, and the _eld nextptr points to the next record on
the FEL.
Figure 3.9 represents the future event list at CLOCK time 10 taken
from Table 3.6.


Example 3.8 (Cont.)




Example 3.8 (Cont.)



For example, if R is set equal to the head pointer for the FEL at
CLOCK time 10, then
 R->eventtype = EW
 R->eventtime = 12
 R->next : the pointer for the second event notice on the FEL



so that

 R->next->eventtype = EL
 R->next->eventtime = 20
 R->next->next : the pointer to the third event notice on the FEL





What we have described are called singly-linked lists, because
there is a one-way linkage from the head of the list to its tail.
For some purposes, it is desirable to traverse or search a list
starting at the tail as well as from the head. For such purposes, a
doubly-linked list can be used.

3.2.4 Advanced Techniques


One idea to speed up processing doubly-linked lists
: to use a middle pointer in addition to a head and tail pointer.
With special techniques, the mid pointer will always point to the
approximate middle of the list.
When a new record is being added to the list, the algorithm first
examines the middle record to decide whether to begin searching
at the head of the list or the middle of the list.
Theoretically, except for some overhead due to maintenance of the
mid pointer, this technique should cut search times in half.


headptr
1

51

2

52

…

…

49

99
searching

middleptr
80

where to add?

50

100

tailptr


Chapter 4. Simulation Software

Preliminary


Software that is used to develop simulation models
can be divided into three categories.


General-purpose programming languages
 FORTRAN, C, C++



Simulation programming languages
 GPSS/HTM, SIMAN V®



Simulation Environments
 This category includes many products that are distinguished

one way or another (by, for example, cost, application area, or
type of animation) but have common characteristics such as a
graphical user interface and an environment that supports all
(or most) aspects of a simulation study.

4.1 History of Simulation Software


Historical period








1955 – 60
The Period of Search
1961- 65 The Advent
1966 – 70
The formative Period
1971 – 78
The Expansion Period
1979 – 86
The Period of Consolidation and
Regeneration
1987 The Period of Integrated Environments



The Period of Search (1955 – 60)




In the early years, simulation was conducted in FORTRAN
or other general purpose programming language without the
support of simulation-specific routines.
In the first period, much effort was expended in the search
for unifying concepts and the development of reusable
routines to facilitate simulation



The Advent (1961 - 65)


The forerunner of the simulation programming language
(SPLs) in use today appeared in the period 1961-65.
 FORTRAN-based packages such as SIMSCRIPT and GASP,

the ALGOL descendant SIMULA, and GPSS



The first process-interaction SPL, GPSS was developed by
Geoffrey Gordon at IBM and appeared about 1961.
 Quick simulations of communications and computer systems,

but its ease of use quickly spread its popularity to other
application areas.
 GPSS is based on a block-diagram representation and is suited
for queuing models of all kinds.





Harry Markowitz provided the major conceptual guidance for
SIMSCRIPT, first appearing in 1963.
 SIMSCRIPT originally was heavily influenced by FORTRAN,

but in later versions its developers broke from its FORTRAN
base and created its own SPL.
 The initial versions were based on event scheduling.


Philip J. Kiviat began the development of GASP (General
Activity Simulation Program) in 1961.
 Originally it was based on the general-purpose programming

language ALGOL, but later a decision was made to base it on
FORTRAN.
 GASP, like GPSS, used flow-chart symbols familiar to
engineers.





Numerous other SPLs were developed during this time
period.
 Notably, they included SIMULA, an extension of ALGOL and

The Control and Simulation Language (CSL) that took an
activity-scanning approach.



The Formative Period (1966 – 70)




During this period, concepts were reviewed and refined to
promote a more consistent representation of each
language’s world view. The major SPLs matured and gained
wider usage.
Rapid hardware advancements and user demands forced
some languages, notably GPSS, to undergo major revisions.
 GPSS/360, with its extensions to earlier versions of GPSS,

emerged for the IBM 360 computer.



SIMSCRIPT II represented a major advancement in SPLs.
 With its freeform English-like language and “forgiving” compiler,

an attempt was made to give the user major consideration in
the language design.



The Formative Period (1966 – 70)




ECSL, a descendant of CSL, was developed and became
popular in the UK.
In Europe, SIMULA added the concept of classes and
inheritance, thus becoming a precursor of the modern
object-oriented programming language.



The Expansion Period (1971 – 78)


Major advances in GPSS during this period came from
outside IBM.
 Norden Systems headed the development of GPSS/NORDEN,

a pioneering effort that offered an interactive, visual online
environment.
 Wolverine Software developed GPSS/H, released in 1977 for
IBM mainframes, later for minicomputers and the PC.


With the addition of new features including an interactive
debugger, it has become the principal version of GPS in use
today.



The Expansion Period (1971 – 78)


Purdue made major changes to GASP, with GASP IV
appearing in 1974.

 It incorporated state events in addition to time events, thus

adding support for the activity-scanning world view in addition
to the event-scheduling world view.



Efforts were made during this period to attempt to simplify
the modeling process.
 Using SIMULA, an attempt was made to develop a system

definition from a high-level user perspective that could be
translated automatically into an executable model.
 Similar efforts included interactive program generators, the
“Programming by Questionnaire,” and natural-language
interfaces, together with automatic mappings to the language
choice.



Consolidation and Regeneration (1979 – 86)




During this period, the predominant SPLs extended their
implementation to many computers and microprocessors
while maintaining their basic structure.
Two major descendants of GASP appeared: SLAM II and
SIMAN.
 SLAM sought to provide multiple modeling perspectives and

combined modeling capabilities.


That is, it had an event-scheduling perspective based on GASP, a
network world view, and a continuous component.

 SIMAN possessed a general modeling capability found in SPLs

such as GASP IV, but also had block-diagram component
similar in some respects to SLAM and GPSS.



Consolidation and Regeneration (1979 – 86)
 As did SLAM II, SIMAN allowed an event-scheduling approach

by programming in FORTRAN with a supplied collection of
GORTRAN routines, a block-diagram approach analogous in
some ways to that of GPSS and SLAM, and a continuous
component.



The Present Period (1987 – present)


The most recent period is notable for the growth of SPLs on
the personal computer and the emergence of simulation
environments with graphical user interfaces, animation and
other visualization tools.
 Some packages attempt to simplify the modeling process by

the use of process flow or block diagramming and “fill-in-theblank” windows that avoid the need to learn programming
syntax.



Some of the more predominant simulation environments
introduced since the mid-eighties, such as Arena and
AutoMod.

4.2 Selection of Simulation Software




Evaluating and selecting simulation software:


Do not focus on a single issue such as ease of use.
 Consider the accuracy and level of detail obtainable, ease of

learning, vendor support, and applicability to your problem.



Execution speed is important.
 Do not think exclusively in terms of experimental runs that take

place at night and over the weekend.



Beware of advertising claims and demonstrations.
 Many advertisements exploit positive features of the software

only.






Ask the vendor to solve a small version of your problem.
Beware of “checklists” with “yes” and “no” as the entries.
 For example, many packages claim to have a conveyor entity.

However, implementations have considerable variation and
level of fidelity. Implementation and capability are what is
important.



Simulation users ask if the simulation model can link to and
use code or routines written in external languages such as
C, C++, or FORTRAN.
 This is good feature, especially when the external routines

already exist and are suitable for the purpose at hand.





There may be a significant trade-off between the graphical
model-building environments and ones based on a
simulation language.
 Beware of “no programming required” unless either the

package is a near-perfect fit to your problem domain, or
programming (customized procedural logic) is possible with
the supplied blocks, nodes, or process flow diagram, in which
case “no programming required” refers to syntax only and not
the development of procedural logic.

4.3 An Example Simulation


Example 4.1 (The Checkout Counter: Typical SingleServer Queue)






The system, a grocery checkout counter, is modeled as a
single-server queue.
The simulation will run until 1000 customers have been
served.
Interarrival time of customers
 Exponentially distributed with a mean of 4.5 minutes



Service time
 Normally distributed with a mean of 3.2 minutes and a standard

deviation of 0.6 minutes

4.3 An Example Simulation


Example 4.1 (The Checkout Counter: Typical SingleServer Queue)


When the cashier is busy, a queue forms with no customers
turned away.


THE ART OF
COMPUTER
SYSTEMS
PERFORMANCE
ANALYSIS
Raj Jain


Part 1 An Overview
of Performance Evaluation
Ch. 1 Introduction
Ch. 2 Common Mistakes and How to Avoid Them
Ch. 3 Selection of Techniques and Metrics

CH. 1 INTRODUCTION


Performance is a key criterion in the design, procurement,
and use of computer systems.
The goal is to get the highest performance for a given cost.
A basic knowledge of performance evaluation terminology
and techniques.

1.1 Outline of Topics (1)


Performance Evaluation on system design alternatives
System Tuning : determining the optimal value
Bottleneck Identification : finding the performance bottleneck
Workload Characterization
Capacity Planning : determining the number/size of components
Forecasting : predicting the performance at future loads

Six Examples of the types of problems



1. Select appropriate evaluation techniques, performance metrics,
and workloads for a system.






The techniques for performance evaluation
: Measurement, Simulation, and Analytical modeling
The metric : the criteria used to evaluate the performance
(ex) Response time – the time to service a request
(ex) Throughput – transactions per second
The workload : the requests made by the users of the system

Ex. (1.1) What performance metrics should be used to compare the
performance of the following systems?
(a) Two disk drives
(b) Two transaction processing systems
(c) Two packet retransmission algorithms



2. Conduct performance measurements correctly.




Load Generator : a tool to load the system
(ex) Remote Terminal Emulator for a timesharing system
Monitor : a tool to measure the results

Ex. (1.2) Which type of monitor (software or hardware) would be more
suitable for measuring each of the following quantities?
(a) Number of instructions execute by a processor
(b) Degree of multiprogramming on a timesharing system
(c) Response time of packets on a network



3. Use proper statistical techniques to compare several alternatives.




Most performance evaluation problems basically consist of finding the best
among a number of alternatives.
Simply comparing the average result of a number of repeated trials does
not lead to correct conclusions, particularly if the variability of the result is
high.

Ex. (1.3) The number of packets lost on two links was measured for four
file sizes as shown in Table 1.1. Which link is better?
TABLE 1.1 Packets Lost on Two Links
File Size

Link A

Link B

1000
1200
1300
50

5
7
3
0

10
3
0
1



4. Design measurement and simulation experiments to provide the
most information with the least effort.


Given a number of factors that affect the system performance, it is
useful to separate out the effects of individual factors.

Ex. (1.4) The performance of a system depends on the following three factors
(a) Garbage collection technique used: G1, G2, or none.
(b) Type of workload: editing, computing, or artificial intelligence (AI).
(c) Type of CPU: C1, C2, or C3
How many experiments are needed? How does one estimate the
performance impact of each factor?



5. Performance simulations correctly.


In designing a simulation model, one has to select a language for
simulation, select seeds and algorithms for random-number
generation, decide the length of simulation run, and analyze the
simulation results.

Ex. (1.5) In order to compare the performance of two cache replacement
algorithms:
(a) What type of simulation model should be used?
(b) How long should the simulation be run?
(c) What can be done to get the same accuracy with a shorter run?
(d) How can one decide if the random-number generator in the
simulation is a good generator?



6. Use simple queueing models to analyze the performance of systems.


Queueing models are commonly used for analytical modeling of
computer systems.

Ex. (1.6) The average response time of a database system is 3 seconds.
During a 1-minute observation interval, the idle time on the
system was 10 seconds. Using a queueing model for the system,
determine the following:
(a) System Utilization
(b) Average service time per query
(c) Number of queries completed during the observation interval
(d) Average number of jobs in the system
(e) Probability of number of jobs in the system being greater than 10
(f) 90-percentile response time
(g) 90-percentile waiting time

1.2 The Art of Performance Evaluation(1)


Some requirements for performance
evaluation
What a
performanc
e metric?





An intimate knowledge of the system
being modeled
A careful selection of the methodology,
workload, and tools

Given the same problem, two analysts
may choose different performance
metrics and evaluation methodologies.
Given the same data, two analysts may
interpret them differently.



Example 1.7




The throughputs of two systems A and B were measured in
transactions per second.
The results are shown in Table 1.2
TABLE 1.2 Throughput in Transactions per Second
System

Workload 2

A

20

10

B


Workload 1

10

20

There are three ways to compare the performance of the two
systems.



Example 1.7 (Cont.)


The first way is to take the average of the performance on the
two workloads.
System

Workload 2

Average

A
B


Workload 1
20
10

10
20

15
15

The second way is to consider the ratio of the performances
with system B as the base.
System

Workload 1

Workload 2

Average

A
B

2
1

0.5
1

1.25
1



Example 1.7 (Cont.)


The third way is to consider the performance ratio with system
A as the base.
System

Workload 1

Workload 2

Average

A
B

1
0.5

1
2

1
1.25

Example 1.7 illustrates a technique known as the ratio game.

1.3 Professional Organizations,
Journals, and Conferences (1)


ACM SIGMETRICS
: for researchers engaged in developing methodologies and user
seeking new or improved techniques for analysis of computer
systems

IEEE Computer Society
: a number of technical committees – the technical committee on
simulation may of interest to performance analysts

ACM SIGSIM
: Special Interest Group on SIMulation – Simulation Digest

CMG
: Computer Measurement Group, Inc. – CMG Transactions



IFIP Working Group 7.3
: AFIPS(American Federation of Information Processing Societies)
- ACM, IEEE, etc.

The Society for Computer Simulation
: Simulation(monthly), Transactions of the Society for Computer
Simulation(quarterly)

SIAM
: SIAM Review, SIAM Journal on Control &Optimization, SIAM Journal
on Numerical Analysis, SIAM Journal on Computing, SIAM Journal
on Scientific and Statistical Computing, and Theory of Probability &
Its Applications



ORSA
: Operations Research, ORSA Journal on Computing, Mathematics
of Operations Research, Operations Research Letters, and
Stochastic Models

Each of the organizations organizes annual conferences.
Students interested in taking additional courses on
performance evaluation techniques may consider courses
on statistical inference, operations research, stochastic
processes, decision theory, time series analysis, design of
experiments, system simulation, queueing theory, and other
related subjects.

1.4 Performance Projects


Select a computer subsystem, for example, a network mail
program, an operation system, a language complier, a text
editor, a processor, or a database.
Perform some measurements.
Analyze the collected data.
Simulate or analytically model the subsystem.
Predict its performance.
Validate the model.


Chapter. 2 Common Mistakes and How to
Avoid Them

2.1 Common Mistakes in
Performance Evaluation (1)


No goals




Any endeavor without goals is bound to fail.
Each model must be developed with a particular goal in mind.
The metrics, workloads, and methodology all depend upon the goal.

General- purpose
model
Particular model

What goals?



Biased Goals


The stating the goals becomes that of finding the right metrics
and workloads for comparing the two systems, not that of
finding the metrics and workloads such that our system turns
out better.
I’m a jury.Your statement is wrong. Be
unbiased.

Our system
is better.

Our system
is better.



Unsystematic Approach (Section 2.2)


Often analysts adopt an unsystematic approach whereby they
select system parameters, factors, metrics, and workloads
arbitrarily.

Pick up as
my likes

Parameter A

Metric B

Workload C

Factor D



Analysis without Understanding the Problem




Defining a problem often takes up to 40% of the total effort.
A problem well stated is half solved.
Of the remaining 60%, a large share goes into designing
alternatives, interpretation of the results, and presentation of
conclusions.
Model A
Final
results
Model B



Incorrect Performance Metrics




A metric refers to the criterion used to quantify the performance of
the system.
The choice of correct performance metrics depends upon the
services provided by the system being modeled.

Compare MIPS

RISC

Meaningless

CISC



Unrepresentative Workload




The workload used to compare two systems should be
representative of the actual usage of the systems in the field.
The choice of the workload has a significant impact on the
results of a performance study.

Network
Short Packet Sizes
Long Packet Sizes

Network



Wrong Evaluation Technique






There are three evaluation technique: measurement, simulation,
and analytical modeling.
Analysts often have a preference for one evaluation technique
that they use for every performance evaluation problem.
An analyst should have a basic knowledge of all three
techniques.
Measurement

Simulation

Analytical
Modeling



Overlooking Important Parameters






It is good idea to make a complete list of system and workload
characteristics that affect the performance of the system.
System parameters
- quantum size : CPU allocation
- working set size : memory allocation
Workload parameters
- the number of users
- request arrival patterns
- priority

Ignoring Significant Factors












Parameters that are varied in the study are called factors.
Not all parameters have an equal effect on the performance.
: if packet arrival rate rather than packet size affects the response time
of a network gateway, it would be better to use several different
arrival rates in studying its performance.
It is important to identify those parameters, which, if varied, will
make a significant impact on the performance.
It is important to understand the randomness of various system and
workload parameters that affect the performance.
The choice of factors should be based on their relevance and not on
the analyst’s knowledge of the factors.
For unknown parameters, a sensitivity analysis, which shows the effect
of changing those parameters form their assumed values, should be
done to quantify the impact of the uncertainty.



Inappropriate Experimental Design






Experimental design relates to the number of measurement or
simulation experiments to be conducted and the parameter
values used in each experiment.
The simple design may lead to wrong conclusions if the parameters
interact such that the effect of one parameter depends upon the
values of other parameters.
Better alternatives are the use of the full factorial experimental
designs and fractional factorial designs.



Inappropriate Level of Detail







The level of detail used in modeling a system has a significant
impact on the problem formulation.
Avoid formulations that are either too narrow or too broad.
A common mistake is to take the detailed approach when a highlevel model will do and vice versa.
It is clear that the goals of a study have a significant impact on
what is modeled and how it is analyzed.



No Analysis




One of the common problems with measurement projects is
that they are often run by performance analysts who are good
in measurement techniques but lack data analysis expertise.
They collect enormous amounts of data but do not know to
analyze or interpret it.

Let’s explain how
one can use the
results

1

2

3

4

5



Erroneous Analysis


There are a number of mistakes analysts commonly make in
measurement, simulation, and analytical modeling, for example,
taking the average of ratios and too short simulations.

Simulation time



No Sensitivity Analysis






Often analysts put too much emphasis on the results of their
analysis, presenting it as fact rather than evidence.
Without a sensitivity analysis, one cannot be sure if the
conclusions would change if the analysis was done in a slightly
different setting.
Without a sensitivity analysis, it is difficult to access the relative
importance of various parameters.



Ignoring Errors in Input





Often the parameters of interest cannot be measured.
The analyst needs to adjust the level of confidence on the
model output obtained from input data.
Input errors are not always equally distributed about the mean.

512
octects

Packet

Transmit
buffer

Receive
buffer



Improper Treatment of Outliers







Values that are too high or too low compared to a majority of values
in a set are called outliers.
Outliers in the input or model output present a problem.
If an outlier is not caused by a real system phenomenon, it
should be ignored.
Deciding which outliers should be ignored and which should be
included is part of the art of performance evaluation and
requires careful understanding of the system being modeled.



Assuming No Change in the Future







It is often assumed that the future will be the same as the past.
A model based on the workload and performance observed in
the past is used to predict performance in the future.
The future workload and system behavior is assumed to be the
same as that already measured.
The analyst and the decision makers should discuss this
assumption and limit the amount of time into the future that
predictions are made.



Ignoring Variability




It is common to analyze only the mean performance since
determining variability is often difficult, if not impossible.
If the variability is high, the mean alone may be misleading to
the decision makers.

Load
demand

Weekly
Mean = 80

Not useful
MON

TUE WED

THU

FRI

SAT

SUN



Too Complex Analysis






Performance analysts should convey final conclusions in as
simple a manner as possible.
It is better to start with simple models or experiments, get
some results or insights, and then introduce the complications.
The decision deadlines often lead to choosing simple models.
Thus, a majority of day-to-day performance problems in the
real world are solved by simple models.
My model is simple and
easier to explain it

I’m easily
understood

Decision
maker

Analyst



Improper Presentation of Results




The eventual aim of every performance study is to help in decision
making.
The right metric to measure the performance of an analyst is
not the number of analyses performed but the number of
analyses that helped the decision makers.

I’m analyst.
Let’s explain the
results of the
analysis

Words, pictures, and graphs



Ignoring Social Aspects






Successful presentation of the analysis results requires two
types of skills: social and substantive.
- Writing and speaking : Social skills
- Modeling and data analysis : Substantive skills.
Acceptance of the analysis results requires developing a trust
between the decision makers and the analyst and presentation
of the results to the decision makers in a manner
understandable to them.
Social skills are particularly important in presenting results that
are counter to the decision maker’s beliefs and values or that
require a substantial change in the design.



Ignoring Social Aspects (cont.)




The presentation to the decision makers should have minimal
analysis jargon and emphasize the final results, while the
presentation to other analysts should include all the details of
the analysis techniques.
Combining these two presentations into one could make it
meaningless for both audiences.



Omitting Assumptions and Limitations




Assumptions and limitations of the analysis are often omitted
from the final report.
This may lead the user to apply the analysis to another context
where the assumptions will not be valid.

Assumption(A)

Analysis results

Final report

Assumption(B)

Other context
Is the result right?

2.2 A Systematic Approach to


State Goals and Define the System




Given the same set of hardware and software, the definition of
the system may vary depending upon the goals of the study.
The choice of system boundaries affects the performance metrics
as well as workloads used to compare the systems.

Dual CPU
System
Timesharing system
System : Timesharing system
Part : external components to CPU

Different ALU system
System : CPU
Part : internal components in CPU



List Service and Outcomes


Each system provides a set of services.

1. Request the service

6. Response

2. Send the packets

3. Perform a number of
different instructions
4. Request queries

5. Answer queries



Select Metrics








Select criteria to compare the performance.
Choose the metrics(criteria).
In general, the metrics are related to the speed, accuracy, and
availability of services.
The performance of a network
: the speed(throughput, delay), accuracy(error rate), and
availability of the packets sent.
The performance of a processor
: the speed of (time taken to execute) various instructions



List Parameters







Make a list of all the parameters that affect performance.
The list can be divided into system parameters and workload
parameters.
System parameters
: Hardware/Software parameters
: These generally do not vary among various installations of the
system.
Workload parameters
: Characteristics of user’s requests
: These vary form one installation to the next.



Select Factors to Study








The list of parameters can be divided into two parts
: those that will be varied during the evaluation
and those that will not.
The parameters to be varied are called factors and their values
are called levels.
It is better to start with a short list of factors and a small
number of levels for each factor and to extend the list in the
next phase of the project if the resource permit.
It is important to consider the economic, political, and
technological constraints that exist as well as including the
limitations imposed by the decision makers’ control and the
time available for the decision.



Select Evaluation Technique


The right selection among analytical modeling, simulation, and
measurement depends upon the time and resources available
to solve the problem and the desired level of accuracy.



Select Workload









The workload consists of a list of service requests to the system.
For analytical modeling, the workload is usually expressed as a
probability of various requests.
For simulation, one could use a trace of requests measured on
a real system.
For measurement, the workload may consist of user scripts to
be executed on the systems.
To produce representative workloads, one needs to measure
and characterize the workload on existing systems.



Design Experiments






Once you have a list of factors and their levels, you need to
decide on a sequence of experiments that offer maximum
information with minimal effort.
In first phase, the number of factors may be large but the
number of levels is small. The goal is to determine the relative
effect of various factors.
In second phase, the number of factors is reduced and the
number of levels of those factors that have significant impact
is increased.



Analyze and Interpret Data







It is important to recognize that the outcomes of measurements
and simulations are random quantities in that the outcome
would be different each time the experiment is repeated.
In comparing two alternatives, it is necessary to take into
account the variability of the results.
The analysis only produces results and not conclusions.
The results provide the basis on which the analysts or decision
makers can draw conclusions.



Present Results








It is important that the results be presented in a manner that is
easily understood.
This usually requires presenting the results in graphic form and
without statistical jargon.
The knowledge gained by the study may require the analysts to
go back and reconsider some of the decisions made in the
previous steps.
The complete project consists of several cycles through the
steps rather than a single sequential pass.

Case Study 2.1 (1)


Consider the problem of comparing remote pipes with
remote procedure calls.
Procedure calls


The calling program is blocked, control is passed to the called
procedure along with a few parameters, and when the
procedure is complete, the results as well as the control return
to the calling program.

Remote pipes



When called, the caller is not blocked.
The execution of the pipe occurs concurrently with the
continued execution of the caller. The results, if any, are later
returned asynchronously.

Case Study 2.1 (2)


System Definition





Goal : to compare the performance of applications using
remote pipes to those of similar applications using
remote procedure calls.
Key component : Channel (either a procedure or a pipe)
System

Network
Clinet
System

Server

Case Study 2.1 (3)


Services








Two types of channel calls
: remoter procedure call and remote pipe
The resources used by the channel calls depend upon the
number of parameters passed and the action required on those
parameters.
Data transfer is chosen as the application and the calls will be
classified simply as small or large depending upon the amount
of data to be transferred to the remote machine.
The system offers only two services
: small data transfer or large data transfer

Case Study 2.1 (4)


Metrics






Due to resource limitations, the errors and failures will not be
studied. Thus, the study will be limited to correct operation only.
Resources : local computer(client), the remote computer(server),
and the network link
Performance Metrics
- Elapsed time per call
- Maximum call rate per unit of time or equivalently, the time
required to complete a block of n successive calls
- Local CPU time per call
- Remote CPU time per call
- Number of bytes sent on the link per call

Case Study 2.1 (5)


Parameters


System Parameter
 Speed of the local CPU, the remote CPU, and the network
 Operating system overhead for interfacing with the channels
 Operating system overhead for interfacing with the networks
 Reliability of the network affecting the number of retransmissions

required



Workload Parameters
 Time between successive calls
 Number and sizes of the call parameters
 Number and sizes of the results
 Type of channel
 Other loads on the local and remote CPUs
 Other loads on the network

Case Study 2.1 (6)


Factors












Type of channel
: Two type – remote pipes and remote procedure calls
Speed of the network
: Two locations of the remote hosts will be used – short distance(in the
campus) and long distance(across the country)
Sizes of the call parameters to be transferred
: Two levels will be used – small and large
Number n of consecutive calls
: Eleven different values of n – 1,2,4,8,16,32,…,512,1024
All other parameters will be fixed.
The retransmissions due to network errors will be ignored.
Experiments will be conducted when there is very little other load on
the hosts and the network.

Case Study 2.1 (7)


Evaluation Technique




Since prototypes of both types of channels have already been
implemented, measurements will be used for evaluation.
Analytical modeling will be used to justify the consistency of
measured values for different parameters.

Workload




A synthetic program generating the specified types of channel
requests
This program will also monitor the resources consumed and log
the measured results(using Null channel requests).

Case Study 2.1 (8)


Experimental Design


A full factorial experimental design with 23×11=88 experiments will
be used for the initial study.

Data Analysis


Analysis of variance will be used to quantify the effects of the first
three factors and regression will be used to quantify the effects of
the number n of successive calls.

Data Presentation


The final results will be plotted as a function of the block size n.


Chapter. 3 Selection of Techniques and Metrics

3.1 Selecting an Evaluation Technique (1)


Table 3.1 Criteria for Selecting an Evaluation Technique
Analytical
Modeling

Simulation

Measurement

1. Stage

Any

Any

Postprototype

2. Time Required

Small

Medium

Varies

3. Tools

Analysts

Computer language

Instrumentation

4. Accuracy

Low

Moderate

Varies

5. Trade-off
evaluation

Easy

Moderate

Difficult

6. Cost

Small

Medium

High

7. Saleability

Low

Medium

High

Criterion



Life-cycle stage




Measurement : only if something similar to the proposed
system already exists
Analytical modeling and Simulation : if it is a new concept

The time available for evaluation


Measurements generally take longer than analytical modeling
but shorter than simulations.

The availability of tools


Modeling skills, Simulation languages, and Measurement
instruments



Level of accuracy






Analytical modeling requires so many simplifications and
assumptions that if the results turn out be accurate.
Simulations can incorporate more details and require less
assumptions than analytical modeling, and thus more often are
closer to reality.
Measurements may not give accurate results simply because
many of the environmental parameters, such as system
configuration, type of workload, and time of the measurement,
may be unique to the experiment. Thus, the accuracy of results
can vary from very high to none.



Trade-off evaluation








The goal of every performance study is either to compare
different alternatives or to find the optimal parameter value.
Analytical models provide the best insight into the effects of
various parameters and their interactions.
With simulations, it may be possible to search the space of
parameter values for the optimal combination, but often it is
not clear what the trade-off is among different parameters.
Measurement is the least desirable technique in this respect. It
is not easy to tell if the improved performance is a result of
some random change in environment or due to the particular
parameter setting.



Cost






Measurement requires real equipment, instruments, and time. It
is the most costly of the three techniques.
Cost, along with the ease of being able to change configurations, is
often the reason for developing simulations for expensive systems.
Analytical modeling requires only paper and pencils. Thus, It is the
cheapest alternative.

Saleability of results




The key justification when considering the expense and the
labor of measurements
Most people are skeptical of analytical results simply because
they do not understand the technique or the final result.



Three rules of validation






Do not trust the results of a simulation model until they have
been validated by analytical modeling or measurements.
Do not trust the results of an analytical model until they have
been validated by a simulation model or measurements.
Do not trust the results of a measurement until they have been
validated by simulation or analytical modeling.

Two or more techniques can also be used sequentially or
simultaneously.


For example, a simple analytical model was used to find the
appropriate range for system parameters and a simulation was
used later to study the performance in that range.

3.2 Selecting performance Metrics (1)


One way to prepare a set of performance criteria or metrics
: to list the services offered by the system
The outcomes can be classified into three categories, as
shown in Figure 3.1.
: The system may perform the service correctly, incorrectly,
or refuse to perform the service.

System

Done

Error j

Event k

inc orrectly

cannot do

Done

correc tly

Done

Request f or service i

Time

events

Time betwee

of the event

Duration

errors

Time betwee

Probability

(Utilization)

Resourc e

(Throughput

Rate

(Response ti




If the system performs the service correctly








Performance is measured by time-rate-resources.
(responsiveness, productivity, and utilization)
The responsiveness of a network gateway
: response time (the time interval between arrival of a packet
and its successful delivery)
The gateway’s productivity
: throughput (the number of packets forwarded per unit of time)
The utilization gives an indication of the percentage of time the
resources of the gateway are busy for the given load level.
- The resource with the highest utilization is called the
bottleneck.



If the system performs the service incorrectly



An error is said to have occurred.
Classify errors and to determine the probabilities of each class
of errors. Ex) the probability of single-bit errors for the gateway

If the system does not perform the service



It is said to be down, failed, or unavailable
Classify the failure modes and to determine the probabilities of
each class. Ex) The gateway may be unavailable 0.01% of the
time due to processor failure and 0.03% due to software failure.



The metrics associated with the three outcomes, namely
successful service, error, and unavailability, are so called
speed, reliability, and availability.
For many metrics, the mean value is all that is important.
However, do not overlook the effect of variability.
In computer systems shared by many users, two types of
performance metrics need to be considered : individual and global.
Individual metrics reflect the utility of each user
- Response time and Throughput
Global metrics reflect the systemwide utility.
- Response time and Throughput
- Resource utilization, Reliability, and Availability



Given a number of metrics, use the following considerations to
select a subset: low variability, nonredundancy, and completeness.
Low variability helps reduce the number of repetitions required to
obtain a given level of statistical confidence.
If two metrics give essentially the same information, it is less
confusing to study only one.
The set of metrics included in the study should be complete. All
possible outcomes should be reflected in the set of performance
metrics.

Case Study 3.1 (1)

End system

Intermedate
systems

End system
Intermedate
systems

End system

Intermedate
systems

Intermedate
systems
End system


Consider the problem of
comparing two different
congestion control
algorithms for computer
networks.

End system

Intermedate
systems

Intermedate
systems

End system

End system

End system

The problem of congestion
occurs when the number of
packets waiting at an
intermediate system
exceed the system’s
buffering capacity and
some of the packets have
to be dropped.

Case Study 3.1 (2)


Four possible outcomes






Some packets are delivered in order to the correct destination.
Some packets are delivered out of order to the destination.
Some packets are delivered more than once to the destination (duplicate
packets).
Some packets are dropped on the way (lost packets).

Time-rate-resource metrics






Response time: the delay inside the network for individual packets.
Throughput: the number of packets per unit of time.
Processor time per packet on the source end system.
Processor time per packet on the destination end systems.
Processor time per packet on the intermediate systems.

Case Study 3.1 (3)


The variability of the response time is important since a highly
variant response results in unnecessary retransmissions. Thus, the
variance of the response time became the sixth metric.
In many systems, the out-of-order packets are discarded at the
destination end systems. In others, they are stored in system
buffers awaiting arrival of intervening packets. Thus, the probability
of out-of-order arrivals was the seventh metric.
Duplicate packets consume the network resources without any use.
The probability of duplicate packets was the eighth metric.
Lost packets are undesirable for obvious reasons. The probability
of lost packets is the ninth metric.
Excessive losses could cause some user connections to be broken
prematurely. The probability of disconnect is the tenth metric.

Case Study 3.1 (4)


It is necessary that all users be treated fairly in the network. Thus,
fairness was added as the eleventh metric. It is defined as a
function of variability of throughput across users.
For any given set of user throughputs (x1,x2, … ,xn), the following
function can be used to assign a fairness index to the set:

(∑i =1 xi ) 2
n

f ( x1 , x2 ,  , xn ) =

n∑i =1 xi2
n

For all nonnegative values of xi’s, the fairness index always lies
between 0 and 1.
If only k of the n users receive equal throughput and the remaining n-k
users receive zero throughput, the fairness index is k/n.

Case Study 3.1 (5)


After a few experiments, it was clear that throughput and delay
were really redundant metrics. ⇔ All schemes that resulted in
higher throughput also resulted in higher delay.
The variance in response time was dropped since it was redundant
with the probability of duplication and the probability of
disconnection.

3.3 Commonly Used Performance
Metrics (1)


Response time : the interval between a user’s request and the
system response, as shown in Figure 3.2a.
- This definition is simplistic since the requests as well as the
responses are not instantaneous.
The user spend time typing the request and the system takes time
outputting the response, as show in Figure 3.2b.
- It can be defined as either the interval between the end of a
request submission and the beginning of the corresponding
response from the system or as the interval between the end of a
request submission and the end of the corresponding response
form the systems.

User's request

System's response
Time

Response time
(a) Instantaneous request and response

User
starts
request

User
finishs
request

System
starts
execution

System
System
starts
completes
response response

User starts
next
request
Time

Reaction
time

Think
time

Response
time
(Definition 1)
Response
time
(Definition 2)
(b) Realistic request and response

Metrics (2)


Turnaround time : the time between the submission of a batch job
and the completion of its output.
- Notice that the time to read the input is included in the
turnaround time.
Reaction time : the time between submission of a request and the
beginning of its execution by the system
- To measure the reaction time, one has to able to monitor the
actions inside a system since the beginning of the execution
may not correspond to any externally visible event.
Stretch factor : the ratio of response time at a particular load to
that at the minimum load
- The response time of a system generally increases as the load
on the system increases.

Metrics (3)


Throughput is defined as the rate (requests per unit of time) at
which the requests can be serviced by the system.
- For batch systems, jobs per second.
- For interactive systems, requests per second.
- For CPU, MIPS(Millions of Instructions Per Second), or MFLOPS
(Millions of Floating-Point Operations Per Second)
- For networks, packets per second(pps) or bits per second(bps)
- For transactions processing system, TPS(Transactions Per
Second)
After a certain load, the throughput stops increasing; in most
cases, it may event start decreasing, as shown in Figure 3.3.

Knee
Nominal
capacity

`

Throughput

Usable
capacity

Knee
capacity
Load

`

Response
time

Load


Metrics (4)


Nominal capacity : the maximum achievable throughput under ideal
workload conditions
Usable capacity : It is more interesting to know the maximum
throughput achievable without exceeding a
prespecified response time limit.
Knee capacity : the throughput at the knee
- In many applications, the knee of the throughput or the response
time curve is considered the optimal operating point.
Efficiency : the ratio of maximum achievable throughput (usable
capacity) to nominal capacity
The utilization of a resource is measured as the function of time
the resource is busy servicing requests. ⇔ the ratio of busy time
and total elapsed time over a given period.

Metrics (5)


Idle time : the period during which a resource is not being used.
Reliability : the probability of errors or by the mean time between
errors.
Availability : the fraction of the time the system is available to
service user’s requests.
Downtime : the time during which the system is not available.
Uptime : the time during which the system is available(MTTF-Mean
Time To Failure).
Cost/performance ratio : a metric for comparing two or more
systems.

3.4 Utility Classification of
Performance Metrics


Higher is Better or HB.
: System users and system managers prefer higher values of such
metrics. Ex) System throughput
Lower is Better or LB.
: System users and system managers prefer smaller values of such
metrics. Ex) Response time
Nominal is Best or NB.
: Both high and low values are undesirable. Ex) Utilization
Figure 3.5 shows hypothetical graphs of utility of the three classes
of metrics.

(a) Lower is better

(b) Higher is better

Better
Utility

Better
Utility

Metric

Metric

(c) Nominal is best

Utility

Best

Metric


3.5 Setting Performance
Requirements (1)


Typical requirement statements




The system should be both processing and memory efficient. It should
not create excessive overhead.
There should be an extremely low probability that the network will
duplicate a packet, deliver a packet to the wrong destination, or
change the data in a packet.

These requirement statements are unacceptable since they suffer
from one or more of the following problems.






Nonspecific : No clear numbers are specified.
Nonmeasurable
Nonacceptable
Nonrealizable
Nonthroughput

3.5 Setting Performance
Requirements (2)


What all these problems lack can be summarized in one word
: SMART(Specific, Measurable, Acceptable, Realizable, Thorough)







Specificity precludes the use of words like “low probability” and “rate”.
Measurability requires verification that a given system meets the
requirement.
Acceptability and Realizability demand new configuration limits or
architectural decisions so that the requirements are high enough to be
acceptable and low enough to be achievable.
Thoroughness includes all possible outcomes and failure modes.

Case Study 3.2 (1)


Consider the problem of specifying the performance requirements for a
high-speed LAN system.






The performance requirements for three categories of outcomes were
specified as follows:
Speed : If the packet is correctly delivered, the time taken to deliver it
and the rate at which it is delivered are important. This leads
to the following two requirements:
(a) The access delay at any station should be less than 1 second.
(b) Sustained throughput must be at least 80 Mbits/sec.
Reliability : Five different error modes were considered important. Each
of these error modes causes a different amount of damage
and, hence, has a different level of acceptability. The
probability requirements for each of these error modes and
their combined effect are specified as follows

Case Study 3.2 (2)


(a) The probability of any bit being in error must be less than 10-7.
(b) The probability of any frame being in error (with error indication
set) must be less than 1%.
(c) The probability of a frame in error being delivered without error
indication must be less than 10-15.
(d) The probability of a frame being misdelivered due to an
undetected error in the destination address must be less than
10-18.
(e) The probability of a frame being delivered more than once
(duplicate) must be less than 10-5.
(f) The probability of losing a frame on the LAN (due to all sorts of
errors) must be less than 1%.

Case Study 3.2 (3)



Availability : Two fault modes were considered significant. The first was
the time lost due to the network reinitializations, and the
second was time lost due to permanent failures requiring
field service calls. The requirements for frequency and
duration of these fault modes were specified as follow:
(a) The mean time to initialize the LAN must be less than 15
milliseconds.
(b) The mean time between LAN initializations must be at least 1
minute.
(c) The mean time to repair a LAN must be less than 1 hour. (LAN
partitions may be operational during this period.)
(d) The mean time between LAN partitioning must be at least half a
week.


정 지 영

목차
1. 개요
2. 다중프로세서 시스템
3. 근사 분석 모델
4. 시뮬레이션 모델
5. 분석 모델과 시뮬레이션 모델 결과
6. 다중프로세서 시스템 모델의 확장


1. 개요


다중프로세서 시스템에서의 메모리 , 버스 경쟁 모델
시스템의 분석적 모델 개발
분석 결과를 검사하기 위한 시뮬레이션 모델 개발



기억장치
1

2

M
버스 1
2
B

1

2

N
처리장치

멀티프로세서 시스템 요소



N 개의 프로세서가 물리적으로나 기능적으로 동일하다고 가정
프로세서는 자신의 지역 메모리를 가지고 있으며 이들 메모리는
캐쉬이거나 자료 레지스터와 명령 버퍼의 형태를 취할 수 있다 .
실행 시 프로세서는 각 머신 싸이클 동안에 명령어 인출이나 연산
자 인출 또는 저장 요청 발생
요청확률 h 는 프로세서의 지역 메모리에서 응해지고 , 확률
p=1-h 로 메모리로의 접근 요청
h: 적중률
p: 비 적중률



메모리 요청 처리
1. 프로세서는 머신 싸이클이 시작될 때 메모리 요청을 초기화하는
동시에 실행을 일시 정지 시킨다 . 동일 모듈에 대한 다중 요청
은 중재 메커니즘에 의해 해결한다 .

2. 요청이 성공하면 해당 모듈은 프로세서에 대해 점유되고 , 그렇

지 않으면 다음 싸이클의 시작에서 다시 발생한다 . 두번째 중재
메커니즘은 M 개까지의 성공적 요청의 집합으로부터 B 개까지
의 요청을 선택하여 이들 요청에 대한 버스들을 점유한다 . 여기
서 버스가 점유되는 순서는 모듈이 점유되었던 순서와 동일하다
고 가정



3. 성공적인 요청에 대해서 그 요청이 저장이라면 , 주소와 자료가
버스를 통해 메모리로 전송되고 만일 그 요청이 인출이라면 해
당 싸이클의 종료 시 버스를 통하여 자료가 반환된다 .

4.

버스 싸이클이 끝날 때 , 요청들이 성공적으로 완료된 프로세서
들은 실행으로 되돌아가고 , 이들 요청에 의해 점유된 버스와 모
듈은 해제된다 . 성공하지 못한 요청은 다음 싸이클의 시작에서
새로운 요청과 함께 재발생된다 .
목적 : 프로세서 성능이 메모리 모듈과 버스에 대한 경쟁에 의해
서 얼마나 영향을 받는가를 결정하는 것



시스템 대역폭 (BW)




프로세서와 메모리 사이의 전체 전송률로 단위 시간당 전송
의 관점으로 표현
전송 시간이 1 싸이클이기 때문에 전체 버스 활용은 BW 와
같고 BW 는 종종 사용중인 버스의 평균 수로서 정의

어떤 한 싸이클 동안 발생하는 요청의 확률이 다른 싸이클에서 발
생하는 확률과 같고 또 독립적이라 가정하면 , 프로세서의 실행
간격은 Bernoulli 실행열에 해당
실행간격 평균 : (1-p)/p
메모리 요청은 메모리에 대해 독립 , 일양 분포를 한다고 가정



프로세서 i 가 메모리 j 를 요청할 확률 : p/M
프로세서 i 가 메모리 j 를 요청하지 않을 확률 : 1- p/M
메모리 j 에 적어도 하나의 요청이 있을 확률

q = 1 − (1 − p / M ) N

( 식 5.1)

메모리 요청확률이 동일하고 독립적이라고 가정하면 , M 개의 메
모리 중 i 번째를 요청할 확률 fi 는 이항분포이다 .
M  i
fi =  q (1 − q) M −i
 i 
 

( 식 5.2)

한 싸이클에서 승인되는 버스 요청의 기대값
M

B −1

i=B

i =1

BW = B ∑ fi + ∑ i × fi

( 식 5.3)



추정한 대역폭은 요청이 재발생되지 않는 싸이클에만 적용
프로세서당 평균 요청률 ( 비 적즁률 ): p, 전체 요청률 : Np
일반적인 경우에 모든 싸이클을 고려해 보면 프로세서당 요청률
r 은 비 적중률보다 크다 .
실행

블럭된 요청
b

x

승인된 요청
1

T
프로세서 상호요청 간격 타이밍

단일 프로세서 요청률









r=(b+1)/T = (b+1)/(x+b+1)
분모와 분자를 b+1 로 나누면
r=1/[1+x/(b+1)]
b+1=rT : T 동안에 발생된 요청의 전체 횟수
프로세서당 요청완료율 : BW/N
T=N/BW
b+1=Nr/BW
r= 1/[1+xBW/Nr]




BW 를 추정하기 위한 단순 고정 소수점 반복 알고리즘

1. 식 (5.1) ~(5.3) 을 사용하여 초기 대역폭 BW0 의 추정값을 계산
한다 .
2. 다음에서 r 의 개선된 추정값을 계산한다 .

ri= 1/[1+xBWi-1/Nri-1]
1. q=1-(1-ri/M) N 을 계산하고 식 (5.2) 와 (5.3) 을 사용하여 새로
운 추정값 BWi 를 계산한다 .
2. |Bwi-Bwi-1| < e 라면 종료하고 그렇지 않으면 단계 2 로 돌아간
다.

e=0.005 인 경우 C 알고리즘
real BW(p,B,M,n)
real p; intB, M, N;
{
real bw0, bw1=p*N, r=p, x=1.0/p-1.0, Bwi();
do
{
bw0 = bw1; r=1.0/(1.0+x*bw0/(N*r));
bw1=BWi(r,B,M,N);
}
while (fabs(bw1-bw0) > 0.005);
return(bw1);
}


real Bwi (r,B,M,N)
real r; intB, M, N;
{ /* compute bandwidth for request rate r */
int I; real q, bw=0.0, f();
q=1.0-pow(1.0-r/M, (real)N);
for(i=1; i<B; i++) bw += i*f(i,M,q);
for(i=B; i<=M; i++) bw += B*f(i,M,q);
return (bw);
}
real Fact(n)
int n;
{ /* compute n factorial */
real z=1.0;
while (n) {z*=n; n--;}
return (z);
}


real C(n,k)
int n,k;
{ /* compute binomial coefficient */
return (Fact (n)/Fact(k) * Fact(n-k)));
}
real f(i,M,q)
int i, M; real q;
{ /* compute binomial probability */
real z;
z=C(M,i)*pow(q,(real)i)*pow(1.0-q,(real)(M-i));
return(z);
}


3.1 활용과 대기시간


단일 버스 , 단일 메모리 모듈 그리고 단일 프로세서의 평균 활용
은 각각 Ub=BW/B, Um 그리고 Up=xBW/N 이다 .
요청당 평균 대기 시간은 b 이고 b 는 T-x-1 이다 .
T=N/BW 이고 x+1=1/p 이므로 b 는 다음과 같다 .
b=(N/BW)-(1/p)

3.2 평균 큐 길이


Lb 를 프로세서당 블럭된 요청의 평균 숫자라고 하자 .
리틀의 법칙으로부터
L b =bBW / N
얻어지며 , 이것으로부터 다음을 구할 수 있다 .
L b =1-BW / Np

3.3 시스템 처리량


시스템에서 작업의 단위를 태스크라고 하고 각 태스크는 평균 n
실행시간 간격을 요구한다고 하면 , 태스크당 평균 서비스 시간
은 nx 이다 .
활용 법칙으로부터 시스템 처리량은 Up 태스크임을 알 수 있다 .
설계 비교를 위하여 nx=1 을 취함으로써 얻어진 정규화된 시스
템 처리량 XP 를 이용하는 것이 유용하다 .
XP = NU p = N[xBW/N] = BW[(1/p)-1]

4. 시뮬레이션 모델


분석 모델 검증에 대해 개발하고자 하는 시뮬레이션
모델의 종류는 목적에 따라 달라진다 .
다중 프로세서 시스템에 있어서 목적은 분석 모델이
시스템의 행위를 적절하게 표현하는가를 검증하는 것
이다 .

4.1 시뮬레이션 모델 1


#include <smpl.h>
#define busy 1

real
p=0.250,
/* local memory miss rate
*/
treq[17]
/* next request time for processor
*/
tn=1.0E6;
/ * earliest-occurring request time
*/
int
N=8, M=4, nB=2, /* no. processors, memories, & buses */
modole[17],bus,
/* memory & bus facility descriptors
*/
nbs=0,
/ * no. busy buses current cycle
*/
req[17],
/* currently-requested memory module */
next=1,
/* arbitration scan starting point */

/*----------- MEMOR Y-BUS BANDWIDTH MODEL----- */
main() {
int event, i,n;
smpl (0, “bandwidth Model”);
for (i=1; i<=M, i++) module [i]=facility(“module”,1);
for (n=1; n<=N; n++) {req[n++] {req[n]=0; next_access (n) ; }
schedule(1,tn,0);
while (time() < 10000.0)
{
cause (&event,&n) ;
switch (event) {
case 1: begin_cycle() ; break;
case 2: req_module(n) : break;
case 3: end cycle(n); break;
}
}
printf(“BW=%.3f n”, U(bus));




/-----------COMPUTE NE XT ACCESS TIME---------*/
nest_access(n)
int n;
{
real t;
t=floor(log(ranf())/log(1.0-p))+time();
treq[n]=t; if(t<tn) then tn=t;
}

next_access() 함수가 각 프로세서에 대한 초기 접근 시간을 결정하고
tn 을 계산하기 위하여 호출
treq[n] 은 사건 발생 시간이지 사건간 시간은 아님 , 프로세서 n 의 다음
요청 발생 시간
tn 은 최초로 발생한 요청의 발생시간



이 모델에서는 시간의 단위가 버스 싸이클이므로 , 시간은 이 단
위의 정수배로 진행하고 , 사건은 싸이클의 시작이나 끝에서 발
생한다 .
begin_cycle() 은 시간 tn 에 발생하는 요청에 대해 N 개 프로세
서들을 스캔하여 이러한 각 프로세서에 대해서 임의 메모리 모듈
을 요청 목적지로서 할당하고 메모리 ( 그리고 버스 ) 요청 즉 ,
사건 2 를 스케줄한다 .



req_module() 은 요청된 메모리 모듈이 사용 가능하고 , 버스가
사용 가능한가를 보기 위해 검사한다 .
이들 조건이 만족되면 모듈과 버스는 점유되고 이 싸이클에서의
사용중인 bus 의 숫자 , 즉 nbs 가 증가된다 . 요청의 완료는 그
싸이클의 끝에서 발생되도록 스케줄 된다 .
요청된 모듈이 사용 중이거나 버스가 얻어질 수 없다면 , 요청은
다음 싸이클의 시작에서 재발생되기 위해 블럭된다 .
사건 3 은 한 버스 싸이클의 끝에서 요청의 완료를 표시한다 . 버
스와 메모리 모듈이 해제되고 , req[n] 은 0 으로 설정되며
next_access() 는 프로세서에 대한 다음 요청 발생 시간을 계산하
기 위하여 호출된다 .



/*----EVENT 1: BEGIN CYCLE-------*/
begin_cycle() {
int i,n=next: real t, tmin=1.0E6;
for (i=0; i<N; i++) {
if (!req[n]) then {/* in this version, req[n] always is 0 here */
if ((t=treq[n])==tn)
then
{req[n]=random(1,M); schedule(2,0.0n);}
else if (t<tmin) then tmin=t;
}
n=(n%N)+1;
}
next=(next%N)+1; tn=tmin;
}

/*------EVENT 2: REQUEST MEMORY AND BUS----------*/
req_module(n)
int n;
{
if (status (module[req[n]]!=busy&&status(bus)!=busy)
then {
request(module[req[n]],n,0); request(bus,n,0);
nbs++; schedule(3,1.0,n);
}
else
{req[n]=0; if (++treq[n]<tn) then tn=treq[n];}
}


/*---------EVENT 3: END CYCLE---------*/
end-cycle(n)
{
release(bus,n);
release(module[req[n]].n);
req[n]=0;
next_access(n);
if (--nbs==0) then schedule(1, tn-time(),0);
}




분석 모델에서 블럭된 요청은 재발생될 때 M 개 모듈중의 어느
하나에 임으로 재할당된다고 가정하고 시뮬레이션 모델 1 에 반
영하였으나 쉽게 제거될 수 있다 .
실제 시스템은 요청 큐잉의 몇몇 타입을 제공할 수 있는데 반하
여 , 결과 모델은 큐와 버스에 대한 요청에 대해 다소 임의적인
선택을 제공한다 .
이것이 성능에 얼마나 영향을 주는가를 보기 위하여 , 모듈과 버
스 요청이 선입선출에 기반해서 큐잉되는 시스템의 모델을 살펴
본다 .



프로세서는 원의 집합인 지연서버에 의해 표현된다 . 지연 서버
는 항상 요청을 지원하기 위한 충분한 서버를 가지고 있어서 큐가
필요 없다 .
프로세서가 메모리 요청을 발생시키면 , 요청은 선택된 메모리
모듈에 대해 큐잉되고 , 그것을 점유하고 , 버스에 대해서 큐잉되
고 , 버스를 점유한다 .
메모리 모듈과 버스는 전송을 수행하기 위하여 동시에 점유되어
야 한다 .



메모리
프로세서

메모리큐

버스

1

1

1

2

2

2
해제

N

M
예약

N

다중프로세서 시스템 큐잉 모델 다이어그램



이 모델의 장점은 요청당 평균 지연이 메모리 지연과 버스 지연으로 분
리될 수 있다는 것이다 .

/*-------EVENT2: REQUEST MEMORY------------*/
req_module(n)
int n;
{
if (request (module[req[n]],n,0)==0) then
schedule”(3,0.0,n);
}

/*---------EVENT 3: REQUEST BUS------------------ */
req_bus(n)
int n;
{
if (request (bus,n)==0) then
{nbs++; schedule(4,1.0,n);}
}


/*------------EVENT 4: END CYCLE---------------*/
end_cycle(n)
int n;
{
req[n]=-req[n]; nbs--;
if (nbs==0) then {
for (n=1; n<=N; n++)
if (req[n]<0) then {
release(bus,n);
release(module[-req[n]],n);
req[n]=0; next access(n);
}
schedule(1,tn-time(),0);
}
}




시뮬레이션 모델 1 과 2 는 실 시스템의 동기화 작동을 표현하기
위하여 구성되었기 때문에 고정된 단위 시간 진행을 사용하였고
싸이클 끝점에서 요청 초기화와 완료를 정렬하였다 .
큐잉 네트워크 모델은 다중프로세서 시스템을 모델화하는데 있
어서 성공적으로 사용되어 왔고 , 또 이들은 동기화 행위를 표현
하도록 제한되어 오지도 않았다 .
비동기화 큐잉 모델로부터의 결과와 이러한 모델이 어떻게 비교
되는가 본다 .

#include <smpl.h>
#define queued 1
real p=0.250;
int N=8, M=4, nB=2,
module[17],
bus,
req[17];


/* local memory 비적중율
*/
/ * no. processors, memories, & buses */
/* facility descroptors for modules
*/
/ * focility descriptors for buses
*/
/* currently-requested memory module */

main()
{
int event, I, n; real x=1.0/p-1.0;
smpl(0,”Bandwidth Model”) ;
bus=facility(“bus”,nB) ;
for(i=1; i<=M, i++) module[i]=facility(“module”,1);
for(n=1; n<=N; n++) {
req[n]=random(1,M); schedule(1, expntl(x),n; )
}


while (time()<10000.0) {
cause(&event,&n);
switch(event) {
case 1:
if (request(module [req[n]], n, 0)!=queued)
then schedule(2, 0.0, n);
break;
case 2: /* reserve bus & initiate transfer */
if (request(bus, n, 0) !=queued) then
schedule(3, 1.0, n);
break;
case 3: /* complete: schedule next request */
release(bus, n);
release(module [req[n]], n);
req[n]=random(1, M);
schedule(1,espntl(x), n);
break;
}
}/* end-while */
report();
}/* end-main */


5. 분석 모델과 시뮬레이션 모델 결과


N

M

B

P

ana

sim1

sim2

sim3

4

4

4

1.000

2.734

2.739

2.619

2.613

4

4

2

.500

1.583

1.668

1.664

1.665

4

4

1

.250

.807

.327

.927

.339

4

2

1

.250

.818

.327

.927

.339

4

2

1

.251

.481

.487

.137

.484

8

8

8

1.000

5.251

5.253

4.984

4.934

8

8

4

.500

3.273

3.379

3.334

3.352

8

8

2

.250

1.706

1.774

1.718

1.739

8

4

2

.250

1.890

1.711

1.713

1.709

8

4

1

.251

.860

.866

.993

.861



일정하지 않은 요청률


중앙 및 입출력 프로세서의 혼합과 같은 다른 요청률을 가진
서로 다른 프로세서 타입으로 구성된 시스템을 모델링하기를
원할 수 있다 .

비임의적이거나 비일양적인 메모리 주소지정


특정 모듈에 대한 요청의 쇄도나 블록 전송에 의해 생성된 순
차 메모리 주소에 대한 참조의 형태를 표현하고자 할 수 있다
.



다양한 전송 길이


요청당 하나의 단위의 전송을 가정했었다 . 이것을 요청당 여
러 단위가 전송될 수 있도록 확장시키길 원할 수 있다 .

서로 다른 버스와 메모리 서비스 시간


하나의 전송이 한 싸이클 동안 메모리 모듈과 버스 모두를 사
용중 상태로 유지한다고 가정했으나 메모리 모듈 사용중인
시간이 자료전송 시간보다 긴 메모리 서브 시스템의 모형을
만들기를 원할 수 있다 .

1. Introduction


1. Introduction to SimScript II.5


SimScript II.5 는 CACI Products Company 에서 만든 범용 시
뮬레이션 언어로서 프로세스 지향 , 사건 지향으로 만들어
지는 모델을 구현하는데 적합하다

1.1 변수 (Variable)


문자 (letter), 숫자 (digit), 마침표 (period) 를 조합해 만든다 .
대소문자의 구별은 없다 .

1.2 입력데이터 읽기 (Reading Input Data)


데이터를 읽는 문은 READ 이다 .

1. Introduction
1.3 수식 표현 (Arithmetic Expression)
수식 연산자는 다음과 같다 .
+ (add), - (subtract), * (multiply), / (divide), ** (exponentiate)
example :
read x and y
add x to y
print 1 line with y thus
The sum is : ***


1. Introduction


1.4 변수값 계산 (Computing Variable Values)


변수에 값을 할당하는 문은 LET 이다 . 그러나 굳이 이
문을 쓸 필요는 없다 .

example : let x = x + 1
1.5 특별한 계산문 (Special Computation Statements)
Add / Subtract
example : add 1 to counter

1. Introduction


1.6 계산 결과 보여주기 (Displaying the Results of Computation)
example : print 1 line with PRICE/ITEMS thus
PRICE/ITEM = $*.***
1.7 반복문 (Repetition)
for 문을 사용할 수 있다 . ( do loop 문으로 반복 구간을 정할 수
있다 .)
example :
for i=1 to 5 by 1
do
read X
read Y
loop

1. Introduction


1.8 프로그램 종료



stop 문은 프로그램을 논리적으로 끝내는 명령문이고 ,
end 문은 물리적으로 끝내는 명령문이다 .

1.9 Variable Modes





SimScript II.5 의 numerical variable 은 REAL / INTEGER 2 가
지밖에 없다 .
표현 가능한 숫자의 크기는 Computer 에 dependent 하다 .
variable type definition 은 Preamble 에서 이루어진다 .

1. Introduction


1.10 Routines

CALL routine name : routine 을 호출할 때

RETURN 은 call 된 routine 을 종료할 때 쓴다 .

argument passing
- routine <name> given <argument> yielding <argument>


function 을 사용하여 routine 을 표현할 수가 있다 .
preamble 에서 "DEFINE name AS mode function" 으로 정
의하고 return value 는 function 내에서 "RETURN WITH
arithmetic expression" 으로 한다 .
example : function Absolute(Number)
...
return with Number
end

1. Introduction


1.11 Library Functions
○○○.f 로 이루어져 있다 . 예를 들면 abs.f 는 주어진 argument
의 절대값을 return 한다 .
1.12 Text Mode Variables
텍스트를 표현하는 변수 모드이다 . real / integer 처럼 선언
한다 .
1.13 Alpha Variables
문자 하나를 변수로 선언할 때 사용되는 모드 .
1.14 Adding Performance Measurement




U.resource : 현재 이용 가능한 자원의 수
N.Q.resource : 큐에 있는 자원의 수
N.X.resource : 현재 실행되고 있는 자원의 수

2. Elementary modeling concept


Model Structure
시뮬레이션을 하는 모델은 다음의 구성요소를 가져야 한다 .
1) 새로운 객체의 도착을 표현하는 메카니즘
2) 모델된 시스템 안에서 그 객체에서 일어나는 일의 표현
3) 시뮬레이션을 종료시키는 메카니즘

Process Concept
프로세스는 모델 안에서 시뮬레이션이 수행되는 시간동안
능동적으로 행동하는 개체이다

2. Elementary modeling concept


Resource Concept


자원 (resource) 은 모델 안에서 프로세스가 요구하는 일을 행하는
수동적인 개체이다 .

Program Structure
1) Preamble : C 의 Header File 과 유사하다 .
2) Main program : 시뮬레이션이 수행되도록 하는 절차를 밟
는 부분이다 . 시스템의 컨트롤이 Timing Routine 으로 넘어가
는 동작으로 수행한다 .
3) Process routine : preamble 에서 선언된 process 의 동작을 표
현하는 routine
Timing routine


Discrete-event simulation 의 심장부로 모델 개발자에게 투명

예제 : A Simple Gas Station
Model


[ Model 개요 ]
주유펌프가 2 개인 주유소가 있다 . 이 주유소에는 고객이
random 하게 찾아온다 . 고객이 주유소에 도착하는 경우 먼저 서
비스를 기다리고 , 서비스를 받은 후 떠나게 된다 . 이러한 시스
템으로부터 이 주유소에 주유펌프가 효율적으로 작동하는지를
검사하고 주유펌프를 추가할 것인가 , 제거할 것인가를 결정하
려고 한다 .
실제로 효율성 검사를 하지 않고 주유펌프를 추가 / 제거하는 것
은 비용 문제가 있기 때문에 우리는 이 결정을 위해 시뮬레이션을
한다 .

Model


시뮬레이션에서 사용된 가정





시뮬레이션 시간은 고객 1000 명을 기준으로 한다 .
이 주유소에 도착하는 고객들의 시간 간격은 2 분에 8 분 사
이로 uniform 하게 분포되어 있다 .
고객 서비스 시간은 5 분에 15 분 사이로 uniform 하게 분포
되어 있다 .

Model


PREAMBLE
PROCESSES INCLUDE GENERATOR AND CUSTOMER
RESOURCES INCLUDE ATTENDANT
ACCUMULATE AVG.QUEUE.LENGTH AS THE AVERAGE
AND MAX.QUEUE.LENGTH AS THE MAXIMUM
OF N.Q.ATTENDANT
ACCUMULATE UTILIZATION AS THE AVERAGE
N.X.ATTENDANT
END

OF

Model
MAIN
CREATE EVERY ATTENDANT(1)
LET U.ATTENDANT(1) = 2
ACTIVATE A GENERATOR NOW
START SIMULATION
PRINT
4
LINES
WITH
MAX.QUEUE.LENGTH(1),
AND UTILIZATION(1) * 100. / 2 THUS


AVG.QUEUE.LENGTH(1),

SIMPLE GAS STATION MODEL WITH 2 ATTENDANTS
AVERAGE CUSTOMER QUEUE LENGTH IS
*.***
MAXIMUM CUSTOMER QUEUE LENGTH IS
*
THE ATTENDANTS WERE BUSY **.** PER CENT OF THE TIME.
END

Model
PROCESS GENERATOR
FOR I = 1 TO 1000,
DO
ACTIVATE A CUSTOMER NOW
WAIT UNIFORM.F(2.0,8.0,1) MINUTES
LOOP
END
PROCESS CUSTOMER
REQUEST 1 ATTENDANT(1)
WORK UNIFORM.F(5.0,15.0,2) MINUTES
RELINGQUISH 1 ATTENDANT(1)
END


3. Modeling Individual Objects


3.1. Attribute Concept





프로세스나 자원 (resource) 은 속성이 주어질 수 있다 .
Resources
Every Pump has a Grade
Create Every Pump (3)
1

2

3
U.Pump
N.X.Pump
N.Q.Pump
Grade



3.2 Variables





변수는 전역 또는 지역변수 (default) 로 될 수 있다 . 전역
변수는 Preamble 에 정의된다 .
모든 변수는 mode 를 가지고 있다 .(integer, real, alpha, text)
Background mode 는 real 이며 다음의 문장에 의해 변경된
다.




NORMALLY, MODE IS mode

변수의 길이는 전형적으로 80 자 이내이며 문자 , 숫자 ,
마침표의 조합이다 .
올바른 예 ) ABC, NO.OF.CUSTOMERS, 5.12.38, ABC...
틀린 예 ) 567, 2+2, 5.12



3.3 Program Control Structures
IF Statement
IF STATUS = BUSY
ADD 1 TO BACK.LOG
ALWAYS

LOOPING
FOR EACH resource
is equivalent to
FOR resource = 1 TO N.resource
FOR EACH resource CALLED name
is equivalent to
FOR name = 1 TO N.RESOURCE
FOR EACH PUMP,
WITH GRADE(PUMP) = DESIRED.GRADE
AND RESERVE(PUMP) >= 10.0,
FIND THE FIRST CASE



3.4 The Representation of Time








시뮬레이션 시계 (clock) 는 시스템에서 정의한 Real 변수
TIME.V 에 의해 표현되며 초기에 0 의 값을 가진다 .
시간의 기본 값 단위는 일 (day) 이다 .
HOURS.V = 24
MINUTES.V = 60
시스템 설계자는 이 기본 값을 원하는 단위로 변경할 수
있다 . 컴퓨터 시스템을 생각해 보면 , DAYS 를 SECONDS
로
HOURS 를
MILLISECONDS, MINUTES 를
MICROSECONDS 로 바꿀 수 있다 .

PREAMBLE
DEFINE .seconds TO MEAN days
DEFINE .milliseconds TO MEAN hours
DEFINE .microseconds TO MEAN minutes
END
MAIN
LET HOURS.V = 1000
LET MINUTES.V = 1000
END


예제 : A Bank with a Separate Queue for Each
Teller

일반적인 은행의 경우에 , 고객은 은행에 도착해서 바로 이용
가능한 은행원에게 서비스를 받고 은행을 떠나게 된다 . 그러
나 만약 모든 은행원들이 이용가능하지 않다면 고객은 가장
짧은 줄에 줄을 서게 될 것이다 .
이러한 은행을 시뮬레이션 해 보자 . 성능 측정의 요소는 큐
( 대기열 ) 의 평균 , 최대 길이 , 은행원 각각의 이용률 , 그리
고 전체 고객의 평균 대기 시간이다 .
이 시뮬레이션에서 사용되는 파라미터는 모델 설계자가 직접
입력한다 .
은행원의 수 (Teller), 고객 도착 시간 (λ : 지수 분포를 따라
도착한다 ), 은행의 영업 시간

Teller

PREAMBLE
PROCESSES INCLUDE GENERATOR AND CUSTOMER
RESOURCES INCLUDE TELLER
DEFINE MEAN.INTERARRIVAL.TIME, MEAN.SERVICE.TIME,
DAY.LENGTH AND WAITING.TIME AS REAL VARIABLES
ACCUMULATE UTILIZATION AS THE AVERAGE OF N.X.TELLER
ACCUMULATE AVG.QUEUE.LENGTH AS THE AVERAGE,
MAX.QUEUE.LENGTH AS THE MAXIMUM OF N.Q.TELLER
TALLY MEAN.WAITING.TIME AS THE MEAN OF WAITING.TIME
END

Teller

MAIN
READ N.TELLER, MEAN.INTERARRIVAL.TIME, MEAN.SERVICE.TIME,
AND DAY.LENGTH
CREATE EVERY TELLER
FOR EACH TELLER,
LETU.TELLER(TELLER) = 1
PRINT 8 LINES WITH N.TELLER, MAEN.INTERARRIVAL.TIME,
MEAN.SERVICE.TIME AND DAY.LENGTH THUS
SIMULATION OF A BANK WITH * TELLERS
(EACH WITH A SEPARATE QUEUE)
CUSTOMERS ARRIVE ACCORDING TO AN EXPONENTIAL DISTRIBUTION
OF INTER ARRIVAL TIMES WITH A MEAN OF *.** MINUTES.
SERVICE TIME IS ALSO EXPONENTIALLY DISTRIBUTED
WITH A MEAN OF *.** MINUTES.
THE BANK DOORS ARE CLOSED AFTER *.** HOURS.
(BUT ALL CUSTOMERS INSIDE ARE SERVED.)

Teller

ACTIVATE A GENERATE NOW
START SIMULATION
PRINT 6 LINES WITH TIME.V * HOURS.V,
AND MEAN.WATING.TIME * HOURS.V * MINUTES.V THUS
THE LAST CUSTOMER LEFT THE BANK AT *.** HOURS.
THE AVERAGE CUSTOMER DELAY WAS *.** MINUTES.
TELLER

UTILIZATION
AVERAGE

QUEUE LENGTH
MAXIMUM

FOR EACH TELLER,
PRINT 1 LINE WITH TELLER, UTILIZATION(TELLER),
AVG.QUEUE.LENGTH(TELLER), MAX.QUEUE.LENGTH(TELLER) THUS
*
*.**
*.**
*
END

Teller

PROCESS GENERATOR
DEFINE ARRIVAL.TIME AS A REAL VARIABLE
LET TIME.TO.CLOSE = DAY.LENGTH / HOURS.V
UNTIL TIME.V >= TIME.TO.CLOSE,
DO
ACTIVATE A CUSTOMER NOW
WAIT EXPONENTIAL.F(MEAN.INTERARRIVAL.TIME,1) MINUTES
LOOP
END

Teller

PROCESS CUSTOMER
DEFINE ARRIVAL.TIME AS A REAL VARIABLE
DEFINE MY.CHOICE AS A INTEGER VARIABLE
LET ARRIVAL.TIME = TIME.V
FOR EACH TRELLER, WITH N.X.TELLER(TELLER) = 0,
FIND THE FIRST CASE
IF FOUND,
LET MY.CHOICE = TELLER
ELSE
FOR EACH TELLER,
COMPUTE MY.CHOICE AS THE MINIMUM(TELLER)
OF N.Q.TELLER(TELLER)
ALWAYS
REQUEST 1 TELLER(MY.CHOICE)
LET WAITING.TIME = TIME.V - ARRIVAL.TIME
WORK EXPONENTIAL.F(MEAN.SERVICE.TIME,2) MINUTES
RELINQUISH 1 TELLER(MY.CHOICE)
END

Teller

[ 예제의 OUTPUT ]
SIMULATION OF A BANK WITH 2 TELLERS
(EACH WITH A SEPARATE QUEUE)
CUSTOMERS ARRIVE ACCORDING TO AN EXPONENTIAL DISTRIBUTION
OF INTER ARRIVAL TIMES WITH A MEAN OF 5.00 MINUTES.
SERVICE TIME IS ALSO EXPONENTIALLY DISTRIBUTED
WITH A MEAN OF 10.00 MINUTES.
THE BANK DOORS ARE CLOSED AFTER 8.00 HOURS.
(BUT ALL CUSTOMERS INSIDE ARE SERVED.)
THE LAST CUSTOMER LEFT THE BANK AT *.** HOURS.
THE AVERAGE CUSTOMER DELAY WAS *.** MINUTES.
TELLER
1
2

UTILIZATION
.97
.91

QUEUE LENGTH
AVERAGE
MAXIMUM
1.73
6
2.06
7

Simulation

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