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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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.

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

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

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

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

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

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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Output from M/M/1/K

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

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

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 If Then Calculation of (Asymptotic Variance Rate of Output Process)

Yoni Nazarathy, Gideon Weiss, University of Haifa, Explicit Formula for M/M/1/K

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Counts of Transitions Book: Stochastic Process Limits,. Paper: 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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, K K – 1 Some intuition for 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/10/10 M/M/1/40 K=20 K=30 c=30 c=20

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

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

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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, Extensions

Yoni Nazarathy, Gideon Weiss, University of Haifa,

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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.

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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?

Yoni Nazarathy, Gideon Weiss, University of Haifa, Other Phenomena at

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

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, The variance function in the short range

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

Yoni Nazarathy, Gideon Weiss, University of Haifa, 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