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 ORSIS Conference, Israel April 18-19, 2008 ORSIS Conference, Israel April 18-19, 2008

2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 1 The Classic Theorem on M/M/1 Outputs: Burkes Theorem (50’s): Output process of stationary version is Poisson ( ). 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 Queueing Output Process

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 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 Calculating Using MAPs Calculating Using MAPs

12. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 11 MAP (Markovian Arrival Process) (Neuts, Lucantoni et al.) Generator Transitions without events Transitions with events Asymptotic Variance Rate Birth-Death Process

13. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 12 Attempting to evaluate directly For, there is a nice structure to the inverse. But This doesn’t get us far…

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

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

16. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 15 Explicit Formula for M/M/1/K

17. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 16 Proof Outline (of part i)

18. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 17 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.

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

20. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 19 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

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 Thank You