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Introduction to Queueing Theory

Introduction to Queueing Theory. Motivation. First developed to analyze statistical behavior of phone switches. Queueing Systems. model processes in which customers arrive. wait their turn for service. are serviced and then leave. Examples. supermarket checkouts stands.

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Introduction to Queueing Theory

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  1. Introduction to Queueing Theory

  2. Motivation • First developed to analyze statistical behavior of phone switches.

  3. Queueing Systems • model processes in which customers arrive. • wait their turn for service. • are serviced and then leave.

  4. Examples • supermarket checkouts stands. • world series ticket booths. • doctors waiting rooms etc..

  5. Five components of a Queueing system: • 1. Interarrival-time probability density function (pdf) • 2. service-time pdf • 3. Number of servers • 4. queueing discipline • 5. size of queue.

  6. ASSUME • an infinite number of customers (i.e. long queue does not reduce customer number).

  7. Assumption is bad in : • a time-sharing model. • with finite number of customers. • if half wait for response, input rate will be reduced.

  8. Interarrival-time pdf • record elapsed time since previous arrival. • list the histogram of inter-arrival times (i.e. 10 0.1 sec, 20 0.2 sec ...). • This is a pdf character.

  9. Service time • how long in the server? • i.e. one customer has a shopping cart full the other a box of cookies. • Need a PDF to analyze this.

  10. Number of servers • banks have multiserver queueing systems. • food stores have a collection of independent single-server queues.

  11. Queueing discipline • order of customer process-ing. • i.e. supermarkets are first-come-first served. • Hospital emergency rooms use sickest first.

  12. Finite Length Queues • Some queues have finite length: when full customers are rejected.

  13. ASSUME • infinite-buffer. • single-server system with first-come. • first-served queues.

  14. A/B/m notation • A=interarrival-time pdf • B=service-time pdf • m=number of servers.

  15. A,B are chosen from the set: • M=exponential pdf (M stands for Markov) • D= all customers have the same value (D is for deterministic) • G=general (i.e. arbitrary pdf)

  16. Analysibility • M/M/1 is known. • G/G/m is not.

  17. M/M/1 system • For M/M/1 the probability of exactly n customers arriving during an interval of length t is given by the Poisson law.

  18. Poisson’s Law n ( l t ) - l t P ( t ) = e (1) n n !

  19. Poisson’s Law in Physics • radio active decay • P[k alpha particles in t seconds] • = avg # of prtcls per second

  20. Poisson’s Law in Operations Research • planning switchboard sizes • P[k calls in t seconds] • =avg number of calls per sec

  21. Poisson’s Law in Biology • water pollution monitoring • P[k coliform bacteria in 1000 CCs] • =avg # of coliform bacteria per cc

  22. Poisson’s Law in Transportation • planning size of highway tolls • P[k autos in t minutes] • =avg# of autos per minute

  23. Poisson’s Law in Optics • in designing an optical recvr • P[k photons per sec over the surface of area A] • =avg# of photons per second per unit area

  24. Poisson’s Law in Communications • in designing a fiber optic xmit-rcvr link • P[k photoelectrons generated at the rcvr in one second] • =avg # of photoelectrons per sec.

  25. - Rate parameter l l • =event per unit interval (time distance volume...)

  26. Poisson’s Law and Binomial Law • The Poisson law results from asymptotic behavior of the binomial law in the limit as: • the prob[event]->0 • # of trials -> infinity • the number of events is small compared with the number of trials

  27. Example: let l = interarrival rate = 10 cust . per min n = the number of customers = 100 t = interval of finite length

  28. We have: P ( t ) , l = 10 10 - l t P ( t ) = e 0 D t Æ 0 - l D t P ( D t ) = l D te 1 - l t - l D t a ( t ) D t = e l D te

  29. Ú a ( t ) dt = 1 t = 0 proof : ax e ax Ú e dx = a - l t Ú l e - l t - l t l e dt = = - e - l • - l - l 0 = e + e = 1

  30. a ( t ) D t V that an interarrival interval is between t and t prob . + D t: - l t a ( t ) dt = e l dt

  31. Analysis • Depend on the condition: • we should get 100 custs in 10 minutes (max prob).

  32. In maple: P ( t ), l = 10 10

  33. To obtain numbers with a Poisson pdf, you can write a program:

  34. total=length of sequence lambda = avg arrival rate exit x() = array holding numbers with Poisson pdf local num= Poisson value r9 = uniform random number t9= while loop limit k9 = loop index, pointer to x()

  35. for k9=1 to total num=0 r9=rnd t9=exp(-lambda) while r9 > t9 num = num + 1 r9 = r9 * rnd wend x(k9)=num next k9 (returns a discrete Poisson pdf in x())

  36. Prove: • Poisson arrivals gene-rate an exponential interarrival pdf.

  37. Prove: a ( t ) D t • prob. that an interarrival interval is between t and t + • This is the prob. of 0 arrivals for time t times the prob. of 1 arrival in Dt: D t

  38. Subst into : • Poisson’s Law: (1)

  39. Now you get:

  40. We already knew: • In the limit as the exponential factor in: approaches 1.

  41. So by subset: = and (2) • Ú a ( t ) dt = 1 with t = 0

  42. Ú a ( t ) dt = 1 t = 0 ax e ax Ú e dx = a - l t l e - l t - l t Ú l e dt = = - e - l - l • - l 0 = e + e = 1

  43. We have made several assumptions: • exponential interarrival pdf. • exponential service times (i.e. long service times become less likely)

  44. The M/M/1 queue in equilibrium

  45. State of the system: • There are 4 people in the system. • 3 in the queue. • 1 in the server.

  46. Memory of M/M/1: • The amount of time the person in the server has already spent being served is independent of the probability of the remaining service time.

  47. Memoryless • M/M/1 queues are memoryless (a popular item with queueing theorists, and a feature unique to exponential pdfs). . P = equilibrium prob k that there are k in system

  48. Birth-death system • In a birth-death system once serviced a customer moves to the next state. • This is like a nondeterminis-tic finite-state machine.

  49. State-transition Diagram • The following state-transition diagram is called a Markov chain model. • Directed branches represent transitions between the states. • Exponential pdf parameters appear on the branch label.

  50. Single-server queueing system

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