Yoni Nazarathy Gideon Weiss University of Haifa Yoni Nazarathy Gideon Weiss University of Haifa On the Asymptotic Variance Rate of the Output Process of Finite Capacity Queues Queueing Analysis, Control and Games December 20,2007 Technion, Israel Queueing Analysis, Control and Games December 20,2007 Technion, Israel

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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 birth-death CTMC: The M/M/1/K Queue  Buffer Server M

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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:

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

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

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

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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Some Results

Yoni Nazarathy, Gideon Weiss, University of Haifa, Results for M/M/1/K: Other M/M/1/K results: Asymptotic correlation between outputs and overflows. Formula for y-intercept of linear asymptote when.

Yoni Nazarathy, Gideon Weiss, University of Haifa, Calculating Using MAPs Calculating Using MAPs

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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Main Theorem

Yoni Nazarathy, Gideon Weiss, University of Haifa, Main Theorem: Part (i): Part (ii): Scope: Finite, irreducible, stationary, birth-death CTMC that represents a queue: and or If: Then: Calculation of : (Asymptotic Variance Rate of Output Process)

Yoni Nazarathy, Gideon Weiss, University of Haifa, Proof Outline

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Idea of Proof of part (i): Whitt: Book: Stochastic Process Limits, Paper: 1992 –Asymptotic Formulas for Markov Processes… 1) Look at M(t) instead of D(t). 2) The MAP of M(t) has an associated MMPP with same variance. 2) Results of Ward Whitt allow to obtain explicit expression for the asymptotic variance rate of MMPP with birth-death structure. Proof of part (ii), is technical.

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, K K-1 Trying to understand what is going on …. M/M/1/K:

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Thank You