1. 7/3/2015© 2007 Raymond P. Jefferis III1 Queuing Systems

2. 7/3/2015© 2007 Raymond P. Jefferis III2 Basic Queuing Model M/M/1 Queue

3. 7/3/2015© 2007 Raymond P. Jefferis III3 Note The queue forms because the instantaneous arrival rate exceeds the service rate. If arrivals and service completions were synchronized, there would be no queue. The average arrival rate must be less than the service rate, or the length of the queue will grow without limit.

4. 7/3/2015© 2007 Raymond P. Jefferis III4 Notation M/M/1 1= number of servers M=Service: M=exponential G=general D=deterministic (constant) M= Arrivals: Poisson (Markov) process

5. 7/3/2015© 2007 Raymond P. Jefferis III5 Input Source - Arrivals Generates arrivals according to a statistical distribution (random interarrival times) Arrivals may require random service times Average interarrival time is the mean (expected value) of the model probability density function

6. 7/3/2015© 2007 Raymond P. Jefferis III6 Expected Value (Mean) The expected value E(X i ) of a discrete random variable X i (i=1,2, …, n) is defined as,

7. 7/3/2015© 2007 Raymond P. Jefferis III7 Variance The variance of a distribution function having mean E(k) is given by,

8. 7/3/2015© 2007 Raymond P. Jefferis III8 Example Arrival Distributions Weibull distribution (component failures) (a and b are shape and scale parameters)

9. 7/3/2015© 2007 Raymond P. Jefferis III9 Example Arrival Distribution Poisson distribution (telephone call arrivals) Assumptions: –the probability of a single arrival in  t is  t, where is the average arrival rate [packets/s] –the probability of no arrivals in  t is 1-  t –arrivals are independent of previous events [Markov (memoryless) system]

10. 7/3/2015© 2007 Raymond P. Jefferis III10 Poisson Arrival Distribution The probability of k arrivals during interval T is given by, The mean and variance are,

11. 7/3/2015© 2007 Raymond P. Jefferis III11 Mean of Poisson Distribution

12. 7/3/2015© 2007 Raymond P. Jefferis III12 Poisson ( =0.5)

13. 7/3/2015© 2007 Raymond P. Jefferis III13 Poisson ( =0.8)

14. 7/3/2015© 2007 Raymond P. Jefferis III14 Poisson ( =2)

15. 7/3/2015© 2007 Raymond P. Jefferis III15 Poisson ( =5)

16. 7/3/2015© 2007 Raymond P. Jefferis III16 Observations The mean increases with , as expected. The variance increases with , as derived. If only m arrivals can be serviced in time T, there is a probability that k>m. In this case k-m arrivals must be queued. This probability increases with T.

17. 7/3/2015© 2007 Raymond P. Jefferis III17 Example A student at a terminal generates a call to the network server =2 times per hour, on average. What is the probability of two or more such calls in the next hour, assuming Poisson statistics?

18. 7/3/2015© 2007 Raymond P. Jefferis III18 Answer to example The probability of two or more calls in a one-hour interval is,

19. 7/3/2015© 2007 Raymond P. Jefferis III19 Cumulative Distribution The cumulative distribution F(x<k) is the integral of the probability density function. Thus, for the previous example:

20. 7/3/2015© 2007 Raymond P. Jefferis III20 Queue holds arrivals until serviced has maximum length parameter state is length at given time arrivals experience waiting time in queue multiple queues are possible

21. 7/3/2015© 2007 Raymond P. Jefferis III21 Service Mechanisms One or more servers (single or parallel) servers may also be queued a service discipline is imposed (FIFO, etc.) a holding time is required for service (reciprocal of the serving rate) server is busy until service is complete

22. 7/3/2015© 2007 Raymond P. Jefferis III22 Example Service Times Exponential distribution (tel. call service)

23. 7/3/2015© 2007 Raymond P. Jefferis III23 Erlang (gamma) Service Times Distribution k is integer m, k positive

24. 7/3/2015© 2007 Raymond P. Jefferis III24 Summary of System Properties interarrival times randomly distributed service times randomly distributed service discipline imposed number of servers length of queue (has maximum) Markov process if no influence of queue length on arrivals (memoryless)

25. 7/3/2015© 2007 Raymond P. Jefferis III25 Queuing Analysis

26. 7/3/2015© 2007 Raymond P. Jefferis III26 Example A certain communications channel delivers data at its capacity of C=14,400 [bits/sec]. The average packet (message) length, l is 450 bits. The packet arrival rate is thus:

27. 7/3/2015© 2007 Raymond P. Jefferis III27 Utilization Factor The ratio of arrival rate to service rate is the utilization factor.

28. 7/3/2015© 2007 Raymond P. Jefferis III28 Example If a channel transmits 35 packets/second and its service rate is 40 packets/second, the utilization factor can be calculated as:

29. 7/3/2015© 2007 Raymond P. Jefferis III29 Total Time Delay The average total time delay before transmission of an arriving packet, W, is equal to the waiting time in the queue, w q, plus the time to service the packet, w s. That is (mean values):

30. 7/3/2015© 2007 Raymond P. Jefferis III30 Delay if n Packets Are Queued If n packets are in the queue, the average time delay for an incoming packet will be: Since w s =1/ , then,

31. 7/3/2015© 2007 Raymond P. Jefferis III31 Result The expected time delay depends upon the state of the queue and the mean service rate. Note that the service rate may depend upon the average length of the message serviced. For message length l i and channel capacity C,

32. 7/3/2015© 2007 Raymond P. Jefferis III32 Number of Packets in System The expected (mean) number of packets in the system will be, where  is the average arrival rate W is the expected waiting time, including service

33. 7/3/2015© 2007 Raymond P. Jefferis III33 Packets in System vs Traffic The expected number of packets in the system is:

34. 7/3/2015© 2007 Raymond P. Jefferis III34 Graphical Representation

35. 7/3/2015© 2007 Raymond P. Jefferis III35 Number of Packets in Queue The expected (mean) number of packets in the queue will be, where  is the average arrival rate W q is the expected waiting time in the queue

36. 7/3/2015© 2007 Raymond P. Jefferis III36 Queued Packets vs Traffic The expected number of packets in the queue is:

37. 7/3/2015© 2007 Raymond P. Jefferis III37 Graphical Representation

38. 7/3/2015© 2007 Raymond P. Jefferis III38 Total Expected Waiting Time If the service time is constant at its average rate 1/ , then

39. 7/3/2015© 2007 Raymond P. Jefferis III39 Wait in Queue vs Traffic The waiting time in the queue is:

40. 7/3/2015© 2007 Raymond P. Jefferis III40 Total Waiting Time vs Traffic The total waiting time will be:

41. 7/3/2015© 2007 Raymond P. Jefferis III41 Little’s Result There are 4 variables and three independent equations, so if one is known the rest can be calculated