Stat 2153 Introduction to Queiueng Theory
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The document discusses queueing theory and stochastic processes. It defines key concepts in queueing systems such as interarrival times, service times, traffic intensity, and queueing notations. The document also summarizes the steady state behavior of the M/M/1 queue where arrivals and services times follow exponential distributions. It presents the balance equations and rate equality principle to derive the geometric distribution for the steady state probabilities of the number of customers in the system.
Stat 2153 Introduction to Queiueng Theory
- 1. Stat-2153: Statistics III Section-B Chapter: Stochastic Process in Queueing and Reliability Md. Menhazul Abedin Assistant Professor Statistics Discipline Khulna University, Khulna-9208 Email: [email protected]
- 3. Definition: Queue • A queue or waiting line is formed when the units (customers, clients) needing some kind of service arrive at a service channel (counter) that offers such facility. • A queueing system can be described by the flow of units for service, forming or joining the queue, if service is not immediately available, leaving the system after being served ( or without served). • Unit: Those demanding service.
- 4. • The basic feature of a queue – The input: The manner in which units arrive and join the system. • Delay or loss system • Limited or unlimited capacity • Source of units may be finite or infinite • Enter singly or group – The service mechanism: Describe the manner in which service is rendered. • A unit may be served singly or in a batch. Basic features of Queue
- 5. – The queue discipline: Indicates the way in which the units from a queue and are served • First come fisrt serve (FCFS) or First in fisrt out (FIFO) • Last come first serve • Random ordering • A realistic service discipline processor-sharing, considered in computer science literature; envisages that if there are m jobs each receives service at the rate of 1/m. • The number of service channels: Channel may be single or more. Basic features of Queue
- 6. Interarrival time • The interval between two consecutive arrivals is called the interarrival time or interval. • Interarrival time or service time – Deterministic or chance-dependent – If chance-dependent interarrival time and service time are i.i.d.
- 7. Traffic intensity • The mean arrival rate is the mean number of arrivals per unit time. Denoted by 𝜆. And 1 𝜆 is mean of the interarrival time distribution. • The mean service rate is the mean number of units served per unit time. Denoted by μ. And 1 μ is mean service time. • The ratio 𝜌 = 𝜆 𝜇 is called the traffic intensity.
- 8. Queueing processes: Other random variables • 𝑁(𝑡): Number of units at time 𝑡 waiting in the queue including those being served. • Busy period: • 𝑊𝑛: The waiting time in the queue or the waiting time of n th arrival. • 𝑊(𝑡): The virtual waiting time i.e. the interval of time a unit would have to wait in the queue, were it to arrive at the instant t.
- 9. Queueing processes: Other random variables • 𝑁 𝑡 , 𝑡 ≥ 0 , {𝑊 𝑡 , 𝑡 ≥ 0} , {𝑊𝑛, 𝑛 ≥ 0} : are stochastic process. • 𝑁 𝑡 , 𝑡 ≥ 0 , {𝑊 𝑡 , 𝑡 ≥ 0} are continuous time. • {𝑊𝑛, 𝑛 ≥ 0} is discrete time. Study this stochastic process by Markov Chain.
- 10. Queueing processes: Notations • Universally acccepted notation is designed
- 11. Queueing processes: Notations • A and B usually take one of the following symbols: – 𝑀: For exponential (Markovian) distribution – 𝐸 𝑘: For Erlang-k distribution – 𝐺: For arbitrary (general) distribution – 𝐷: For fixed (Deterministic) interval
- 12. Queueing processes: Notations • 𝑀/𝐺/1: meant single channel queueing system having exponential interarrival time distribution and arbitrary (general) service time distribution.
- 13. Steady state distribution • 𝑁(𝑡), the number in the system at time 𝑡 and its probability distribution 𝑝 𝑛 𝑡 = Pr{𝑁 𝑡 = 𝑛|𝑁 0 = 0} 𝑝 𝑛 = 𝑙𝑖𝑚 𝑝 𝑛(t) as 𝑡 → ∞ When the limit exists it is said that the system reached equilibrium or steady state.
- 14. General relationships in queue theory The most important relationship is (Little’s formula or result) 𝐿 = 𝜆𝑊 – 𝜆: arrival rate – 𝐿: expected number of unit in the system – 𝑊: expected waiting time in the system in steady state • If is system replaced by queue then above equation will be 𝐿 𝑄 = 𝜆𝑊𝑄
- 15. Proof of Little’s formula • Intuitive justification-1 by Foster – 𝑊: Average waiting time of a unit – 1 𝑊 :Average rate of departure per unit – 𝐿: Average number of units in the system – 𝐿 𝑊 : Average rate of departure for all the aggregate of all units – 𝜆: arrival rate • Since the system is steady state 𝐴𝑟𝑟𝑖𝑣𝑎𝑙 𝑟𝑎𝑡𝑒 = 𝐷𝑒𝑝𝑎𝑟𝑡𝑢𝑟𝑒 𝑟𝑎𝑡𝑒 𝜆 = 𝐿 𝑊 𝐿 = 𝜆𝑊
- 16. Proof of Little’s formula • Intuitive justification-2 by Foster – Study yourself (page-411)
- 17. The queueing model 𝑀/𝑀/1 (Steady state behaviour) • Input: Poisson • Service time: Exponential • Discipline: FCFS (First Come First Serve) – Such queue is known as simple queue or Poisson queue or simple Markovian queue
- 18. The queueing model 𝑀/𝑀/1 (Steady state behaviour) • The probability that an arrival occures in an infinitesimal interval of length h is 𝜆ℎ + 𝑜(ℎ) – That is distribution of interarrival time is exponential having pdf 𝑎 𝑡 = 𝜆𝑒−𝜆𝑡 . • The probability that one service is completed in an interval of infinitesimal length h is μℎ + 𝑜(ℎ) – That is distribution of service time is exponential having pdf 𝑏 𝑡 = μ𝑒−μ 𝑡 . Such model is known as 𝑀/𝑀/1 with parameters 𝜆 and μ
- 19. Balance Equations • N(t): Number in the system at instant 𝑡 (𝑡 ≥ 0). • { N(t), 𝑡 ≥ 0} is a markov process in continuous time with denumerable number af states {0, 1, 2, 3, …}. Trasition take place only two neighbouring states. – This type a type of birth and death process. • Let 𝑃𝑟{N(t)=n|N(0)=·}= 𝑝 𝑛 𝑡 , 𝑛 ≥ 0 • Thus the difference equations 𝑝′ 0 𝑡 = −𝜆𝑝0 𝑡 + μ𝑝1 𝑡 𝑝′ 𝑛 𝑡 = − 𝜆 + μ 𝑝 𝑛 𝑡 + 𝜆𝑝 𝑛−1 𝑡 + μ𝑝 𝑛+1 𝑡 , 𝑛 ≥ 1
- 20. Balance Equations • Since steady state, as t tends to infinity, 𝑝 𝑛 𝑡 tends to limit 𝑝 𝑛. • The equations of steady state probabilities Pr(𝑁 = 𝑛) = 𝑝 𝑛 can be obtained by putting 𝑝′ 𝑛 𝑡 = 0 0 = −𝜆𝑝0 + μ𝑝1 … … … (𝑖) 0 = − 𝜆 + μ 𝑝 𝑛 + 𝜆𝑝 𝑛−1 + μ𝑝 𝑛+1, 𝑛 ≥ 1 … … … (𝑖𝑖) The above equations are called balance equation.
- 21. Rate Equality Princilpe • Rate of flow out =rate of flow into • For tate 0, 𝜆𝑝0 = μ𝑝1 Same as equation (i) • Again state other than 0, i.e. (𝑘 ≠ 0) – Rate of flow out of state k equals 𝜆 + μ 𝑝 𝑘 – Rate of flow into of state k equals 𝜆𝑝 𝑘−1 + μ𝑝 𝑘+1 Thus 𝜆 + μ 𝑝 𝑘 = 𝜆𝑝 𝑘−1 + μ𝑝 𝑘+1 Same as qeuation (ii)
- 23. Steady State Solution • Recall equation (𝑖) and (𝑖𝑖). Dividing by μ and putting 𝜌 = 𝜆 μ • Which produce 𝑝1 − 𝜌𝑝0 = 0 … … … 𝑖𝑖𝑖 𝑝 𝑛+1 − 1 + 𝜌 𝑝 𝑛 + 𝜌𝑝 𝑛−1 = 0, 𝑛 ≥ 1 … … … (𝑖𝑣) • The solution is 𝑝 𝑛 = 𝑝0 𝜌 𝑛 , 𝑛 ≥ 0 … … … (𝑣) (will be discussed next slide)
- 24. Steady State Solution • 𝑝 𝑛 is probability distribution 𝑛=0 ∞ 𝑝 𝑛 = 1 Thus 𝑝0 𝑛=0 ∞ 𝜌 𝑛 = 1 ⇒ 𝑝0 1 1 − 𝜌 = 1 ⇒ 𝑝0 = 1 − 𝜌 Form equation (𝑣) 𝑝 𝑛 = (1 − 𝜌)𝜌 𝑛, 𝑛 ≥ 0 The distribution {𝑝 𝑛} of random varaiable N, the number in the system in steady state, is geometric. 𝑛=0 ∞ 𝜌 𝑛 is a geometric series
- 25. Supplementary Materials • The probability function Pr(𝑁 = 𝑛) = 𝑝 𝑛, n=0, 1, 2, 3,… Of random variable N satisfies the the difference equatons. 𝑝1 − 𝑎𝑝0 = 0, 0 < 𝑎 < 1 … … … 𝐴 𝑝 𝑛+1 − 1 + 𝑎 𝑝 𝑛 + 𝑎𝑝 𝑛−1 = 0, 𝑛 ≥ 1 … … … (𝐵) And 𝑛=0 ∞ 𝑝 𝑛 = 1 Thus solution is 𝑝 𝑛 = 𝑎 𝑛 𝑝0 ⇒ 𝑝 𝑛 = (1 − 𝑎)𝑎 𝑛 , 𝑛 ≥ 0
- 26. Supplementary Materials • Proof: – Method of characteristic equation – Method of generating functions – Method of induction • Let, the equation 𝑝 𝑛+1 − 1 + 𝑎 𝑝 𝑛 + 𝑎𝑝 𝑛−1 = 0, 𝑛 ≥ 1 𝑝 𝑛+1 − 𝑎𝑝 𝑛 = 𝑝 𝑛 − 𝑎𝑝 𝑛−1, 𝑛 ≥ 1 Putting (n-1) for n , we get 𝑝 𝑛 − 𝑎𝑝 𝑛−1 = 𝑝 𝑛−1 − 𝑎𝑝 𝑛−2
- 27. Supplementary Materials • So on 𝑝 𝑛+1 − 𝑎𝑝 𝑛 = 𝑝 𝑛 − 𝑎𝑝 𝑛−1 = 𝑝 𝑛−1 − 𝑎𝑝 𝑛−2 = 𝑝 𝑛−2 − 𝑎𝑝 𝑛−3 … … … = 𝑝1 − 𝑎𝑝0, 𝑛 ≥ 0 Thus 𝑝1 − 𝑎𝑝0 = 0 ⇒ 𝑝1 = 𝑎𝑝0
- 28. Supplementary Materials • Hence 𝑝 𝑛 = 𝑎𝑝 𝑛−1 = 𝑎 𝑎𝑝 𝑛−2 = 𝑎2 (𝑎𝑝 𝑛−3) = 𝑎3 (𝑎𝑝 𝑛−4) = 𝑎4 (𝑎𝑝 𝑛−5) … … … = 𝑎 𝑛 𝑝0
- 29. Supplementary Materials Since 𝑛=0 ∞ 𝑝 𝑛 = 1 Thus 𝑝0 𝑛=0 ∞ 𝑎 𝑛 = 1 ⇒ 𝑝0 1 1 − 𝑎 = 1 ⇒ 𝑝0 = 1 − 𝑎 Hence 𝑝 𝑛 = (1 − 𝜌)𝜌 𝑛 , 𝑛 ≥ 0 • This is the required solution
- 30. Supplementary Materials • Learn about difference equation.