Introduction to Queueing Theory

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