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

3. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 3 © Graduate University, Chinese academy of Sciences. Agenda 1. Reversibility and Burke’s Theorem 2. State-dependent M/M/1 Queuing System 3. M/M/1/K QUEUE 4. M/M/∞QUEUE 5. M/M/m Queue 6. M/M/m/m System 7. Center Server CPU model 8. M/G/1 Queue 9. Priority Queuing

4. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 4 © Graduate University, Chinese academy of Sciences. 1. Reversibility and Burke’s Theorem  Introduction The input to the M/M/1 queueing system is a Poisson process, what can we say of its output? For the M/M/1, consider the inter-departure times The queueing system is not-empty The queueing system is empty

5. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 5 © Graduate University, Chinese academy of Sciences. Reversibility  For a stochastic process,reversibility means that when the direction of time is reversed, that is, if time flows backwards, the statistics of the process are the same as in the time normal case  Definition: A stochastic process, X(t), is reversible if the samples (X(t 1 ), X(t 2 ),…, X(t m )) has the same distribution as (X(  -t 1 ), X(  - t 2 ),…, X(  - t m )) for every real  ( for continuous processes) and for every t 1, t 2, t m.

6. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 6 © Graduate University, Chinese academy of Sciences. Global Balance  Transition rate from state i to j Equilibrium: or

7. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 7 © Graduate University, Chinese academy of Sciences. Reversibility vs. Satisfaction  Theorem A stationary Markov chain is reversible if and only if there is a collection of positive numbers p i, i  S, which sum to one and satisfy the detailed balance equations: p i q ij,= q j p ji, for i,j  S. These p i are naturally the equilibrium state probabilities All birth/death processes are reversible – Detailed balance equations must be satisfied

8. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 8 © Graduate University, Chinese academy of Sciences. Burke’s Theorem  The departure process from an M/M/1 queuing system, in equilibrium, is Poisson.

9. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 9 © Graduate University, Chinese academy of Sciences. Implications of Burke’s Theorem Since the arrivals in forward time form a Poisson process, the departures in backward time form a Poisson process Since the backward process is statistically the same as the forward process, the (forward) departure process is Poisson By the same type of argument, the state (packets in system) left by a (forward) departure is independent of the past departures – In backward process the state is independent of future arrivals

10. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 10 © Graduate University, Chinese academy of Sciences. NETWORKS OF QUEUES 求解两个 M/M/1 队列串联后系统的状态概率。该系统的到达过程是到达 率为 的 Poisson 过程。这两个队列的服务时间相互独立，服务时间与 到达过程相互独立。

11. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 11 © Graduate University, Chinese academy of Sciences. 2. State-dependent M/M/1 Queuing System

12. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 12 © Graduate University, Chinese academy of Sciences. Results Derive from local balance equations

13. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 13 © Graduate University, Chinese academy of Sciences. 3. M/M/1/K QUEUE(1) Finite capacity, can hold a maximum of K customers

14. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 14 © Graduate University, Chinese academy of Sciences. 3. M/M/1/K QUEUE(2) Finite capacity, can hold a maximum of K customers

15. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 15 © Graduate University, Chinese academy of Sciences. Results(1)

16. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 16 © Graduate University, Chinese academy of Sciences. Results(2)

17. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 17 © Graduate University, Chinese academy of Sciences. Exercise -1 a)Suppose that messages arrive according to a Poisson process at a rate of one message every 4 msec, and that message transmission times are exponentially distributed with mean 3 ms. The system maintains buffers for 4 messages, including the one being served. What is the blocking probability? b)What is the average # of messages in the system?

18. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 18 © Graduate University, Chinese academy of Sciences. 4. M/M/  QUEUE Every arriving customer is assigned to its own server of rate 

19. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 19 © Graduate University, Chinese academy of Sciences. Results The steady state solution is (0 <  <  ) This is a Poisson distribution, E[n]=  Due to the unlimited supply of servers,  may exceed 1

20. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 20 © Graduate University, Chinese academy of Sciences. 5. M/M/m Queue

21. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 21 © Graduate University, Chinese academy of Sciences. Markov Chain Use results from state-dependent M/M/1 systems, with:

22. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 22 © Graduate University, Chinese academy of Sciences. Results

23. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 23 © Graduate University, Chinese academy of Sciences. Erlang C Formula

24. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 24 © Graduate University, Chinese academy of Sciences. 6. M/M/m/m System No queue : blocked customers lost What does the Markov chain look like?

25. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 25 © Graduate University, Chinese academy of Sciences. Markov Chain Use results from state-dependent M/M/1 systems, with:

26. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 26 © Graduate University, Chinese academy of Sciences. Results                      m n n m m n n n m p p n p 1 0 ! 1 1 ! 1 Formula B Erlang )( 1   

27. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 27 © Graduate University, Chinese academy of Sciences. 7. Center Server CPU model Single server, finite population K State-transition diagram is:

28. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 28 © Graduate University, Chinese academy of Sciences. Result(1/2) Here

29. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 29 © Graduate University, Chinese academy of Sciences. Result(2/2) Steady-state probabilities are:

30. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 30 © Graduate University, Chinese academy of Sciences. 8. M/G/1 Queue M/M systems very tractable due to memoryless property of interarrival & service times However, exponential service times not a very good assumption – service times deterministic in ATM – there are limits on packet sizes Poisson arrival assumption somewhat better – aggregation of arrival streams

31. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 31 © Graduate University, Chinese academy of Sciences. Methods for M/G/1 In general, there are two methods 1.Residual Life Approach: This is easy to use but can only give the mean values of the desired parameters 2.Method of Imbedded Markov Chains: This is based on finding a set of a time points where the Markovian Property is retained. This is generally harder to use but will give the distribution of various parameters from which mean and higher moments may be computed

32. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 32 © Graduate University, Chinese academy of Sciences. Mean Delay in M/G/1 Let X1,X2 … be the iid sequence of service times in an M/G/1 system Suppose an arriving customer finds the server busy X1 X2 … Xj

33. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 33 © Graduate University, Chinese academy of Sciences. Derivation (1/4) ith arriving i-1, …, i-N i Server N i customers waiting for service Let Wi = waiting time in queue of ith arrival Ri = Residual service time seen by I (i.e., amount of time for current customer receiving service to be done) Ni = Number of customers found in queue by i 用户 i 的等待时间

34. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 34 © Graduate University, Chinese academy of Sciences. Derivation (2/4)

35. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 35 © Graduate University, Chinese academy of Sciences. Derivation (3/4)

36. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 36 © Graduate University, Chinese academy of Sciences. Derivation (4/4)

37. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 37 © Graduate University, Chinese academy of Sciences. Results(1/2) The Mean waiting time( is the second moment of service time distribution):

38. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 38 © Graduate University, Chinese academy of Sciences. Results(2/2)

39. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 39 © Graduate University, Chinese academy of Sciences. Exercise-2 ( a) What is the mean residual service time of a system with exponential service times with mean m? Does this make sense? (b) What is the mean residual service time of a system with constant service time m? (c) Compare the average waiting time for the M/M/1 and M/D/1 systems

40. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 40 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations(1) Suppose that at the end of each busy period, the server goes on “vacation” for some random interval of time. Thus, a new arrival to an idle system, rather than going into service immediately, waits for the end of vacation period When the queue is empty, the server takes a vacation For data networks, vacations correspond to the transmission of various kinds of control and recordkeeping packets This system is useful for polling and reservation systems (e.g., token ring)

41. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 41 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations (2)

42. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 42 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations (3)

43. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 43 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations (4)

44. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 44 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations (5)

45. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 45 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations (6)

46. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 46 © Graduate University, Chinese academy of Sciences. M/G/1 Queue with Vacations (7)

47. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 47 © Graduate University, Chinese academy of Sciences. 9. Priority Queuing When a higher priority arrival occurs at a time when a relatively lower priority customer is still in service, different choices on the strategy are: Non-Preemptive Priority Preemptive Resume Priority

48. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 48 © Graduate University, Chinese academy of Sciences. M/G/1 with Non-Preemptive Priority

49. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 49 © Graduate University, Chinese academy of Sciences. Model n priority classes of customers Type-k customers arrive according to Poisson process of rate k and have the mean service times 1/  k Separate queues for each priority, when server becomes available it selects from the highest priority non-empty queue Non-preemptive

50. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 50 © Graduate University, Chinese academy of Sciences. Utilization Server utilization for type-k customers:  k = k /  k Total utilization:  k =  1 +  2 + …+  n <1

51. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 51 © Graduate University, Chinese academy of Sciences. Waiting time for highest-priority customers R’’ :  residual time of customer (if any) found in service Nq1(t)  # of type-1 customers found in the Q

52. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 52 © Graduate University, Chinese academy of Sciences. Waiting time for type-2 customers customers

53. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 53 © Graduate University, Chinese academy of Sciences. Little’s Law in action again...

54. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 54 © Graduate University, Chinese academy of Sciences. Preliminary Results

55. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 55 © Graduate University, Chinese academy of Sciences. E[R’’] = ? The customer found in service may belong to any of the priority classes With the same arguments as used for M/G/1:

56. Network Design and Analysis-----Wang Wenjie Queueing Theory II: 56 © Graduate University, Chinese academy of Sciences. Results