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 P n (t)  ( t) n n! e  t (1)

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  =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  interarrival rate  10 cust. per min n  the number of customers 100 t  interval of finite length 

28. We have: P 10 (t),  10 P 0 (t)  e  t  t  0 P 1 (  t)  te  t a(t)  t  e  t  te  t

29. a(t)dt t  0    1 proof: e ax dx   e a e  t dt   e  t   e  t  e    e  0  1 ax

30. a(t)dt  e  t dt a(t)  t prob. V that an interarrival interval is between t and t +  t:

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

32. In maple: P 10 (t),  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: – 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: a(t)  t  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) with a(t)dt t  0    1

42. e ax dx   e ax a e  t dt   e  t   e  t  e   e  0  1 a(t)dt t  0    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 k  equilibrium prob. 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

51. Symbles: P 0  mean number of transitions/ sec from state 0 to 1  P 1  mean number of transitions/ sec from state 1 to 0

52. States  State 0 = system empty  State 1 = cust. in server  State 2 = cust in server, 1 cust in queue etc...

53. Probalility of Given State  Prob. of a given state is invariant wrt time if system is in equilibrium.  The prob. of k cust’s in system is constant.

54. Similar to AC  This is like AC current entering a node  is called detailed balancing  the number leaving a node must equal the number entering

55. Derivation 3 3a 4 4a

56. by 3a = 4 since 5

57. then: 6 where = traffic intensity < 1

58. since all prob. sum to one 6a Note: the sum of a geometric series is 7

59.  Suppose that it is right, cross multiply and simplify:  k k  0    1 1  So Q.E.D.

60. subst 7 into 6a P 0  k k  0    1 6a 7aand =prob server is empty 7b

61. subst into 6 yields: 8

62. Mean value:  let N=mean number of cust’s in the system  To compute the average (mean) value use: 8a

63. Subst (8) into (8a) 8 8a 8b we obtain

64. differentiate (7) wrt k 7 we get 8c

65. multiply both sides of (8c) by  8d 9

66. Relationship of, N  as r approaches 1, N grows quickly.

67. T and  T=mean interval between cust. arrival and departure, including service.

68. Little’s result:  In 1961 D.C. Little gave us Little’s result: 10

69. For example:  A public bird bath has a mean arrival rate of 3 birds/min in Poisson distribution.  Bath-time is exponentially distributed, the mean bath time being 10 sec/bird.

70. Compute how long a bird waits in the Queue (on average): = mean arrival rate = mean service rate

71. Result:  So the mean service-time is 10 seconds/bird =(1/ service rate) sec for wait + service

72. Mean Queueing Time  The mean queueing time is the waiting time in the system minus the time being served, 20-10=10 seconds.

73. M/G/1 Queueing System  Tannenbaum says that the mean number of customers in the system for an M/G/1 queueing system is: 11 This is known as the Pollaczek-Khinchine equation.

74. What is C b of the service time.

75. Note:  M/G/1 means that it is valid for any service-time distribution.  For identical service time means, the large standard deviation will give a longer service time.

76. Introduction to Queueing Theory