The Asymptotic Variance of the Output Process of

The Asymptotic Variance of the Output Process of Finite Capacity Queues Yoni Nazarathy Gideon Weiss University of Haifa ORSIS Conference, Israel April 18 -19, 2008

Queueing Output Process A Single Server Queue: Buffer State: 0 1 2 3 Server 4 5 6 … M/M/1 Queue: • Poisson Arrivals: • Exponential Service times: • State Process is a birth-death CTMC The Classic Theorem on M/M/1 Outputs: Burkes Theorem (50’s): Output process of stationary version is Poisson ( ). Output Process: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 1

The M/M/1/K Queue m Finite Buffer Server “Carried load” • 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. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 2

Traffic Processes Counts of point processes: • • - Entrances • - Outputs • M/M/1/K - Arrivals during - Lost jobs Poisson Renewal Non-Renewal Poisson Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 3

The Output process • 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 ? Asymptotic Variance Rate: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 4

What values do we expect for Keep ? and fixed. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 5

What values do we expect for Keep ? and fixed. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 6

What values do we expect for Keep ? and fixed. Similar to Poisson: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 7

What values do we expect for Keep ? and fixed. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 8

What values do we expect for Keep ? and fixed. Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 9

Calculating Using MAPs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 10

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

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

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

Main Theorem (Asymptotic Variance Rate of Output Process) Scope: Finite, irreducible, stationary, birth-death CTMC that represents a queue. Part (i) Part (ii) Calculation of If and Then Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 14

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

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

Define The Transition Counting Process - Counts the number of transitions in [0, t] Births Deaths Asymptotic Variance Rate of M(t): , MAP of M(t) is “Fully Counting” – all transitions result in counts of events. Lemma: Proof: Q. E. D Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 17

Proof Outline 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. Whitt: Book: 2001 - Stochastic Process Limits, . Paper: 1992 - Asymptotic Formulas for Markov Processes… Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 18

Fully Counting MAP and associated MMPP Example: Transitions without events Transitions with events Fully Counting MAP MMPP (Markov Modulated Poisson Process) Proposition rate 2 rate 4 Poisson Process rate 4 rate 3 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 rate 2 rate 4 rate 3 19

More On BRAVO Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 20

Some intuition for M/M/1/K 0 1 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 … K– 1 K 21

Intuition for M/M/1/K doesn’t carry over to M/M/c/K But BRAVO does M/M/1/40 c=20 c=30 K=20 M/M/10/10 M/M/40/40 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 22

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, 2008 23

The “ 2/3 property” • GI/G/1/K • SCV of arrival = SCV of service • Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 24

Thank You Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008 25