1. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 1 © Graduate University, Chinese academy of Sciences. Network Design and Performance Analysis Wang Wenjie Wangwj@gucas.ac.cn

2. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 2 © Graduate University, Chinese academy of Sciences. Queueing System III

3. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 3 © Graduate University, Chinese academy of Sciences. Agenda 1.Introduction 2.Open Networks of Queues 3.Closed Networks of Queues

4. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 4 © Graduate University, Chinese academy of Sciences. 1. Introduction So far, we analyzed single queues – occupancy, delay, blocking Move toward interconnected queuing systems – performance analysis – network optimization and design – various network topologies

5. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 5 © Graduate University, Chinese academy of Sciences. Queueing Network Interconnected queues with jobs flow from one queue to another

6. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 6 © Graduate University, Chinese academy of Sciences. Classification

7. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 7 © Graduate University, Chinese academy of Sciences. Classification(2) An Open Network is one where jobs arrive from outside to one or more queues and eventually leave the network from some of the queues. If an open network has multiple job classes then it must be open for each class of jobs.

8. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 8 © Graduate University, Chinese academy of Sciences. Classification(3) An Closed Network is one where there are a constant number of jobs that continually circulate in the network with no other arrivals to the system or departures from the system. If a closed network has multiple job classes then it must be closed for each class of jobs

9. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 9 © Graduate University, Chinese academy of Sciences. Classification(4)

10. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 10 © Graduate University, Chinese academy of Sciences. Classification(5)

11. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 11 © Graduate University, Chinese academy of Sciences. 2. Open Networks of Queues

12. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 12 © Graduate University, Chinese academy of Sciences. Notation  : total mean arrival rate to the network   i : mean service rate of i th server  r sj : probability that a customer arriving from the source will be routed to queue j  r jd : probability that a customer departing from queue j will be routed to the destination  r jk : probability that a customer departing from queue j will be routed to queue k

13. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 13 © Graduate University, Chinese academy of Sciences. Application : Tandem Queues

14. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 14 © Graduate University, Chinese academy of Sciences. Application : Multi-stage Network

15. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 15 © Graduate University, Chinese academy of Sciences. Application : General Topology

16. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 16 © Graduate University, Chinese academy of Sciences. State Description of the System where n i is the # of customers in the i th system (queue + server) Our goal : find the pmf of

17. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 17 © Graduate University, Chinese academy of Sciences. Average Throughput  i is the average throughput through queue i – mean rate of entering/leaving the system Traffic Equations:

18. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 18 © Graduate University, Chinese academy of Sciences. Exercise 1 Find the avg throughput through each queue.

19. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 19 © Graduate University, Chinese academy of Sciences. Global Balance Equations (1/3) Let  ( )be the probability of being in state What is the rate of leaving state Arrival to any queue Departure from any From outside queue

20. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 20 © Graduate University, Chinese academy of Sciences. The i th Unit Vector i th position

21. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 21 © Graduate University, Chinese academy of Sciences. Global Balance Equations (2/3) What is the rate of entering state

22. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 22 © Graduate University, Chinese academy of Sciences. Global Balance Equations (3/3)

23. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 23 © Graduate University, Chinese academy of Sciences. Local Balance Equations Use these, plus global balance, plus traffic equations to get the solution

24. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 24 © Graduate University, Chinese academy of Sciences. Product Form Solution Network behaves as if all queues were statistically independent !!

25. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 25 © Graduate University, Chinese academy of Sciences. Exercise 2 In the network of queues from the previous exercise, assume Poisson arrivals at an average rate of 1,000 packets/sec and average service times at each queue of 0.2 msec. What is the probability of no packets being in the system?

26. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 26 © Graduate University, Chinese academy of Sciences. Number of customers and delay in the network

27. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 27 © Graduate University, Chinese academy of Sciences. Exercise 3 For the network in the two previous exercises, what is (a) the average number of packets in the network? (b) the average delay through the network?

28. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 28 © Graduate University, Chinese academy of Sciences. 3. Closed Networks of Queues

29. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 29 © Graduate University, Chinese academy of Sciences. Model Closed networks of queues can be used to model different types of systems – in reality, system is open Number of customers in the system at any time is a fixed value N

30. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 30 © Graduate University, Chinese academy of Sciences. Applications Finite population models (M/M/s/s/K) – Each of K users can have at most 1 call active at a time (max # of calls is K) Systems under heavy loads – Multi-stage packet switches w/ finite # of packets allowed in, and a new packet always ready to enter when one leaves Window-based flow control – Maximum # of packets in transit at any time

31. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 31 © Graduate University, Chinese academy of Sciences. Traffic Equations

32. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 32 © Graduate University, Chinese academy of Sciences. Global Balance Equations

33. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 33 © Graduate University, Chinese academy of Sciences. Product Form Solution for Closed Networks G(M,N) is a normalization constant

34. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 34 © Graduate University, Chinese academy of Sciences. Finding G(M,N)

35. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 35 © Graduate University, Chinese academy of Sciences. Example 4.1 See figure in slide “Closed Networks of Queues” Computer system allows 2 active jobs at any given time Each job requires CPU & I/O processing When job leaves CPU, there are 2 possibilities: – Job finished and instantly replaced by another (probability p) – Job requires I/O, then more CPU (prob. 1-p)

36. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 36 © Graduate University, Chinese academy of Sciences. Example 4.1 – Getting Started Write down the traffic equations: What states are possible?

37. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 37 © Graduate University, Chinese academy of Sciences. Example 4.1 – State Probabilities

38. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 38 © Graduate University, Chinese academy of Sciences. Utilization Utilization  is the proportion of time that the server is busy  i =  i ’ /  i  i ’ is the Actual arrival rate to queue i In example 4.1,  1 = ?

39. Network Design and Analysis-----Wang Wenjie Queuing Theory III: 39 © Graduate University, Chinese academy of Sciences. Example 4.2 – A Numerical Example Use the same set-up as in Example 4.1 Let  1 = 4 jobs/sec,  2 = 1 job/sec, p = 1/3 a)Calculate the state probabilities b) Calculate the utilization at each queue c) Calculate the average number of customers in each queuing system d) Calculate the average delay through each queuing system