1. Yoni Nazarathy Gideon Weiss University of Haifa Yoni Nazarathy Gideon Weiss University of Haifa The Asymptotic Variance of the Output Process of Finite Capacity Queues EURANDOM QPA Seminar April 4, 2008 EURANDOM QPA Seminar April 4, 2008

2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 1 Outline Background A Queueing Phenomenon: BRAVO Main Theorem More on BRAVO Some open questions

3. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 2 Buffer size: Poisson arrivals: Independent exponential service times: Jobs arriving to a full system are a lost. Number in system,, is represented by a finite state irreducible birth-death CTMC. Assume is stationary. The M/M/1/K Queue  Finite Buffer Server “Carried load”

4. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 3 Counts of point processes: - Arrivals during - Entrances - Outputs - Lost jobs Traffic Processes Poisson Renewal Non-Renewal Poisson Non-Renewal Renewal M/M/1/K Renewal Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987.

5. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 4 Some Attributes: (Disney, Kiessler, Farrell, de Morias 70’s) Not a renewal process (but a Markov Renewal Process). Expressions for. Transition probability kernel of Markov Renewal Process. A Markovian Arrival Process (MAP) (Neuts 80’s) What about ? The Output process Asymptotic Variance Rate:

6. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 5 What values do we expect for ? Keep and fixed.

7. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 6 Work in progress by Ward Whitt What values do we expect for ? Keep and fixed.

8. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 7 Similar to Poisson: What values do we expect for ? Keep and fixed.

9. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 8 What values do we expect for ? Keep and fixed.

10. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 9 B alancing R educes A symptotic V ariance of O utputs What values do we expect for ? Keep and fixed.

11. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 10

12. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 11

13. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 12 Explicit Formula for M/M/1/K

14. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 13 Calculating Using MAPs Calculating Using MAPs

15. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 14 MAP (Markovian Arrival Process) (Neuts, Lucantoni et al.) Generator Transitions without events Transitions with events Asymptotic Variance Rate Birth-Death Process

16. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 15 Attempting to evaluate directly For, there is a nice structure to the inverse. But This doesn’t get us far…

17. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 16 Main Theorem

18. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 17 Main Theorem Part (i) Part (ii) Scope: Finite, irreducible, stationary, birth-death CTMC that represents a queue. and If Then Calculation of (Asymptotic Variance Rate of Output Process)

19. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 18 Proof Outline (of part i)

20. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 19 Define The Transition Counting Process Lemma: Proof: Q.E.D - Counts the number of transitions in [0,t] Asymptotic Variance Rate of M(t):, BirthsDeaths MAP of M(t) is “Fully Counting” – all transitions result in counts of events.

21. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 20 Proof Outline Whitt: Book: 2001 - Stochastic Process Limits,. Paper: 1992 - Asymptotic Formulas for Markov Processes… 1) Lemma: Look at M(t) instead of D(t). 2) Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance. 3) Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP.

22. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 21 Fully Counting MAP and associated MMPP MMPP (Markov Modulated Poisson Process) Example: rate 4 Poisson Process rate 2 rate 3 rate 4 rate 2 rate 4 rate 3 rate 2 rate 3 rate 4 rate 2 Proposition Transitions without events Transitions with events Fully Counting MAP

23. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 22 More On BRAVO B alancing R educes A symptotic V ariance of O utputs

24. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 23 01 K K – 1 Some intuition for M/M/1/K …

25. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 24 Intuition for M/M/1/K doesn ’ t carry over to M/M/c/K But BRAVO does M/M/40/40 M/M/10/10 M/M/1/40 K=20 K=30 c=30 c=20

26. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 25 BRAVO also occurs in GI/G/1/K MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions

27. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 26 The “ 2/3 property ” GI/G/1/K SCV of arrival = SCV of service

28. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 27 Other Phenomena at

29. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 28 Asymptotic Correlation Between Outputs and Overflows For Large K M/M/1/K

30. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 29 Proposition: For, The y-intercept of the Linear Asymptote M/M/1/K

31. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 30 The variance function in the short range

32. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 31 Why looked at asymptotic variance rate?

33. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 32 Require: Stable Queues Push-Pull Queueing Network (Weiss, Kopzon 2002,2006) Server 2 Server 1 PUSH PULL PUSH Positive Recurrent Policies Exist!!! Low variance of the output processes? PROBABLY NOT WITH THESE POLICIES!!!

34. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 33 Queue Size Realizations BURSTY OUTPUTS

35. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 34 Can we calculate ? Diffusion Approximations of the Outputs. Is the right measure of burstines? Which policies are “good” in terms of burstiness? Work in progress Server 2Server 1 PUSH PULL PUSH

36. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 35 Heavy Traffic Scaling, Whitt. Prove the 2/3 Property for GI/G/1/K. BRAVO - What is going on? M/M/1 with. Formulas for asymptotic variance of outputs from other systems. Other Questions

37. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 36 In Progress by Ward Whitt Question: What about the null recurrent M/M/1( ) ? Some Guessing 1970, Iglehart and Whitt Standard independent Brownian motions. 2008, (1 week in progress by Whitt) Uniform Integrability Simulation Results

38. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 37 M/M/1+ Impatient Customers - Simulation

39. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 38 Thank You