1. Yoni Nazarathy Gideon Weiss University of Haifa Yoni Nazarathy Gideon Weiss University of Haifa On the Variance of Queueing Output Processes Haifa Statistics Seminar February 20, 2007 Haifa Statistics Seminar February 20, 2007 With Illustrations and Animations for “Non-Queueists” (Statisticians)

2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 1 Outline Background A Queueing Phenomenon: BRAVO Main Theorem More on BRAVO Current, parallel and future work

3. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 2 Some Background on Queues

4. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 3 A Bit On Queueing and Queueing Output Processes A Single Server Queue: Buffer Server 01 2345 6 … State:

5. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 4 The Classic Theorem on M/M/1 Outputs: Burkes Theorem (50’s): Output process of stationary version is Poisson ( ). A Bit On Queueing and Queueing Output Processes A Single Server Queue: Buffer Server 01 2345 6 … State: Output Process: Poisson Arrivals: M/M/1 Queue: Exponential Service times: State Process is a birth-death CTMC

6. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 5 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: The M/M/1/K Queue  Finite Buffer Server M “Carried load”

7. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 6 Traffic Processes Counts of point processes: - The arrivals during - The entrances into the system during - The outputs from the system during - The lost jobs during (overflows) Poisson Renewal Non- Renewal Poisson Non- Renewal Renewal M/M/1/K Renewal Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987.

8. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 7 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 1980’s). What about ? D(t) – The Output process: Asymptotic Variance Rate:

9. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 8 Asymptotic Variance Rate of Outputs: What values do we expect for ?

10. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 9 Asymptotic Variance Rate of Outputs: What values do we expect for ? Work in progress by Ward Whitt

11. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 10 Similar to Poisson: Asymptotic Variance Rate of Outputs: What values do we expect for ?

12. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 11 Asymptotic Variance Rate of Outputs: What values do we expect for ?

13. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 12 M B alancing R educes A symptotic V ariance of O utputs Asymptotic Variance Rate of Outputs: What values do we expect for ?

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

15. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 14

16. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 15 Asymptotic Variance of M/M/1/K:

17. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 16 Calculating Using MAPs Calculating Using MAPs

18. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 17 Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.) Generator Transitions without events Transitions with events Asymptotic Variance Rate Birth-Death Process

19. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 18 Attempting to evaluate directly … For, there is a nice structure to the inverse… But This doesn’t get us far…

20. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 19 Main Theorem Paper submitted to Queueing Systems Journal, Jan, 2008: The Asymptotic Variance Rate of the Output Process of Finite Capacity Birth-Death Queues.

21. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 20 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)

22. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 21 Proof Outline

23. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 22 Use the Transition Counting Process Lemma: Proof: Q.E.D - Counts the number of transitions in the state space in [0,t] Asymptotic Variance Rate of M(t): BirthsDeaths

24. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 23 Idea of Proof of part (i): Whitt: Book: Stochastic Process Limits, 2001. 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 an associated MMPP with same variance. 2) Results of Ward Whitt: An explicit expression for the asymptotic variance rate of MMPP with birth-death structure. Proof of part (ii), is technical.

25. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 24 Proposition (relating Fully Counting MAPs to MMPPs) Fully Counting MAP MMPP (Markov Modulated Poisson Process) Example: rate 4 Poisson Process rate 1 Poisson Process rate 3 Poisson Process rate 4 Poisson Process rate 1 Poisson Process rate 4 Poisson Process rate 3 Poisson Process rate 1 Poisson Process rate 3 Poisson Process rate 4 Poisson Process rate 1 Poisson Process The Proposition:

26. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 25 More On BRAVO B alancing R educes A symptotic V ariance of O utputs

27. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 26 01 K K-1 Some intuition for M/M/1/K:

28. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 27 Intuition for M/M/1/K doesn ’ t carry over to M/M/c/K … But BRAVO does … M/M/40/40 M/M/K/K K=30 K=20 K=10 M/M/c/40 c=1 c=20 c=30

29. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 28 BRAVO also occurs in GI/G/1/K … MAP is used to evaluate Var Rate for PH/PH/1/40 queue with Erlang and Hyper-Exp

30. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 29 The “ 2/3 property ” seems to hold for GI/G/1/K!!! and increase K for different CVs

31. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 30 Other Phenomena at

32. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 31 Asymptotic Correlation Between Outputs and Overflows M For Large K: M/M/1/K

33. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 32 Proposition: If, then: The y-intercept of the Linear Asymptote of Var(D(t)) M/M/1/K

34. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 33 The variance function in the short range

35. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 34 The “ kick-in ” time for the BRAVO effect Departures from M/M/1/K with Yet another singularity

36. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 35 How we got here … and where are we going?

37. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 36 A Novel Queueing Network: Push-Pull System (Weiss, Kopzon 2002,2006) Server 2Server 1 PUSH PULL PUSH “Inherently Stable” “Inherently Unstable” For Both Cases, Positive Recurrent Policies Exist Require: Low variance of the output processes? PROBABLY NOT WITH THESE POLICIES!!!

38. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 37 Some Queue Size Realizations: BURSTY OUTPUTS

39. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 38 Work in progress with regards to the Push-Pull system: Server 2Server 1 PUSH PULL PUSH Can we calculate ? Is asymptotic variance rate really the right measure of burstines? Which policies are “good” in terms of burstiness?

40. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 39 Future work (or current work by colleagues): View BRAVO through a Heavy Traffic Perspective, using heavy traffic limits and scaling.

41. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 40 “ Fresh ” in Progress work by Ward Whitt: Question: What about the null recurrent M/M/1( ) ? Some Guessing: Iglehart and Whitt 1970: Standard independent Brownian motions. 2008 (1 week in progress by Whitt): Uniform Integrability Simulation Results To be continued…

42. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 41 Thank You