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 Meeting of the euro working group on stochastic modeling. Koc university, Istanbul. June 23-25, 2008

2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 1 Sojourn Times Queue Sizes Server Idleness Lost Job Rates Variability of Outputs Typical Queueing Performance Measures

3. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 2 Variability of Outputs Sometimes related to down-stream queue sizes Aim for little variability over [0,T] Queueing Networks Setting Manufacturing Setting Asymptotic Variance Rate of Outputs

4. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 3 Previous work – Asymptotic Variance Rate of Outputs Baris Tan, Asymptotic variance rate of the output in production lines with finite buffers, Annals of Operations Research, 2000.

5. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 4 In this work … Analyze for simple finite queueing systems A surprising phenomenon Simple formula for Extensions Results: e.g. M/M/1/K

6. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 5 The M/M/1/K Queue  Finite Buffer NOTE: output process D(t) is non-renewal.

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

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

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

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

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

12. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 11 Output from M/M/1/K

13. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 12 Calculating Using MAPs Calculating Using MAPs

14. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 13 MAP (Markovian Arrival Process) (Neuts, Lucantoni et al.) Generator Transitions without events Transitions with events Asymptotic Variance Rate Birth-Death Process

15. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 14 Attempting to evaluate directly For, there is a nice structure to the inverse. But This doesn’t get us far…

16. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 15 Main Theorem

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

18. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 17 Explicit Formula for M/M/1/K

19. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 18 Idea of Proof

20. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 19 Counts of Transitions Book: 2001 - Stochastic Process Limits,. Paper: 1992 - Asymptotic Formulas for Markov Processes… 1) Use above Lemma: Look at M(t) instead of D(t). 2) Use Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance. 3) Use Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP. Lemma: Asymptotic Variance Rate of M(t):, Births Deaths Observe: MAP of M(t) is “Fully Counting” – all transitions result in counts of events. Proof Outline

21. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 20 More On BRAVO B alancing R educes A symptotic V ariance of O utputs

22. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 21 01 K K – 1 Some intuition for M/M/1/K …

23. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 22 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

24. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 23 BRAVO also occurs in GI/G/1/K MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions

25. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 24 The “ 2/3 property ” GI/G/1/K SCV of arrival = SCV of service

26. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 25 M/M/1+ Impatient Customers - Simulation

27. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 26 Thank You

28. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 27 Extensions

29. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 28

30. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 29 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.

31. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 30 Require: Stable Queues Push-Pull Queueing Network (Weiss, Kopzon 2002,2006) Server 2 Server 1 PUSH PULL PUSH Positive Recurrent Policies Exist!!! Asymptotic Variance Rate of the output processes?

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

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

34. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 33 Proposition: For, The y-intercept of the Linear Asymptote M/M/1/K

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

36. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 35 Lemma: Proof: Q.E.D

37. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 36 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