1. Yoni Nazarathy Gideon Weiss University of Haifa Yoni Nazarathy Gideon Weiss University of Haifa The Asymptotic Variance Rate of the Departure Process of the M/M/1/K Queue The XXVI International Seminar on Stability Problems for Stochastic Models October 24, 2007, Nahariya, Israel The XXVI International Seminar on Stability Problems for Stochastic Models October 24, 2007, Nahariya, Israel

2. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 1 Poisson arrivals: Independent exponential service times: Finite buffer size: Jobs arriving to a full system are a lost. Number in system,, is represented by a finite state irreducible CTMC: The M/M/1/K Queue  Buffer Server M

3. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 2 Traffic Processes Counts of point processes: - The arrivals during - The entrances into the system during - The departures from the system during - The lost jobs during Poisson Renewal Non- Renewal Poisson Non- Renewal Renewal M/M/1/K Renewal Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987.

4. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 3 A Markov Renewal Process (Cinlar 1975). A Markovian Arrival Process (MAP) (Neuts 1980’s). Some Attributes: (Disney, Kiessler, Farrell, de Morias 70’s): Not a renewal process. Expressions for. Transition probability kernel of the Markov Renewal Process. Departures processes of M/G/. Models. What about ? D(t) – The Departure process of M/M/1/K

5. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 4 Asymptotic Variance Rate For a given system ( ), what is ?

6. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 5 Asymptotic Variance Rate For a given system ( ), what is ?

7. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 6 Asymptotic Variance Rate For a given system ( ), what is ? Similar to Poisson:

8. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 7 Asymptotic Variance Rate For a given system ( ), what is ? OUR MAIN RESULT M

9. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 8 An Explicit Formula Theorem: Corollary: Is minimal over all when.

10. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 9 What is going on? - The number of movements on the state space during Lemma: Proof: Q.E.D 01 K K-1 The State Space

11. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 10 What is going on? (continued … ) 01 K K-1 Observation: When, is minimal. As a result the “modulation” of M(t) is minimal. Rate of M(t), depends on current state of Q(t) And thus the “modulation” of D(t) is minimal. M

12. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 11 Calculations and Proof Outline

13. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 12 Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.) Note: We may similarly represent M(t), E(t), L(t) and we may also use similar methods (MMAP) to find cross-covariances. Generator Transitions without “arrivals” Transitions with “arrivals”

14. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 13 Calculation of : Option 1: Invert Numerically Option 2: For only, we have the explicit structure of the inverse… Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula for any ).

15. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 14 Proof Outline: Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula). 1)M(t) is “fully counting”: It exactly counts the number of movements in the state-space during [0,t]. 2)“Decoupling Theorem” (stated loosely): There exists a MMPP that has the same expectation and variance as a fully counting MAP. 3)Combined results of Ward Whitt (2001 book and 1992 paper) are used to find explicit formulas for the asymptotic variance rate of birth-death type MMPPs. Note: This technique can be used to find similar explicit formulas for the asymptotic variance rate of departures from M/M/c/K and other structured finite birth and death queues.

16. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 15 Open Questions:

17. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 16 Open Questions: 1)The limiting value of 2/3 also appears in the asymptotic variance rate of the losses (e.g. Whitt 2001). What is the connection? 2)Non-Exponential Queueing systems. Is minimization of the characteristic attribute of the “dip” in the asymptotic variance rate? 3)Asymptotic variance rate of departures from the null- recurrent M/M/1? 4)Variance of departure processes from more complex queueing networks (our initial motivation).

18. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007 17 Thank You