1 / 18

Yoni Nazarathy Gideon Weiss University of Haifa

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

alan-joseph
Download Presentation

Yoni Nazarathy Gideon Weiss University of Haifa

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. The M/M/1/K Queue m Server Buffer • 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: M Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  3. Traffic Processes M/M/1/K • 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 Renewal Non-Renewal Renewal Non-Renewal Renewal Poisson Poisson Poisson Poisson Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  4. D(t) – The Departure process of M/M/1/K • 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 ? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

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

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

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

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

  10. K-1 K The State Space 0 1 What is going on? - The number of movements on the state space during Lemma: Proof: Q.E.D Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  11. What is going on? (continued…) K-1 K 0 1 Rate of M(t), depends on current state of Q(t) Observation: When , is minimal. As a result the “modulation” of M(t) is minimal. And thus the “modulation” of D(t) is minimal. M Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  12. Calculationsand Proof Outline Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

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

  15. Proof Outline: Option 3: Find an associated Markov Modulated Poisson Process (MMPP) to the MAP of (Proof of the explicit formula). • M(t) is “fully counting”: It exactly counts the number of movements in the state-space during [0,t]. • “Decoupling Theorem” (stated loosely): There exists a MMPP that has the same expectation and variance as a fully counting MAP. • 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. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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

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

  18. ThankYou Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

More Related