Some decomposition results for a class of vacation queues

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Abstract

We analyze the MAP/PH/1 vacation system at arbitrary times using the matrix-analytic method, and obtain decomposition results for the R and G matrices. The decomposition results reduce the amount of computational effort needed to obtain these matrices. The results for the G matrix are extended to the BMAP/PH/1 system. We also show that in the case of the Geo/PH/1 and M/PH/1 systems with PH vacations both the G and R matrices can be obtained explicitly.

Introduction

Polling systems which occur frequently in the medium access control aspects in communication systems are usually approximated by vacation queueing models. A performance measure (delay) associated with several Medium Access Control (MAC) protocols is strongly dependent on the medium sharing algorithm. Exhaustive polling system, the simplest, is often used as a good approximation to studying performance measures. Hence whatever we can do to reduce the workload involved in computing the key matrices associated with this system and computation of the system performance is always a welcome. In this paper we present new decomposition results that make the analysis of the MAP/PH/1 vacation much easier to implement when it is studied in arbitrary times using the matrix-analytic method. For both the discrete time and continuous time MAP/PH/1 systems with PH vacations the model can be set up as quasi-birth-and-death (QBD) process. Sometimes the length and type of vacation models considered lead to huge sizes of the associated matrices R and G of order n(m+r)×n(m+r), where n,m and r are dimensions related to the matrices that represent the arrival, service and vacation processes. Computing these matrices is usually the heaviest computation load associated with the model. With computational focus in mind we present decomposition results associated with these matrices. We show that most of the work in computing these matrices is simply that of computing the equivalent matrices for the MAP/PH/1 system (without vacation), and that the remaining work only involves solving linear equations. In so doing we reduce the size of the main matrix that requires major computation to order nm×nm. By being able to decompose the matrices and then capitalizing on the features of this decomposition we can reduce the associated computational efforts. We show that the results for the matrix G can be extended to the BMAP/PH/1 vacation system. Finally, we also show that these matrices can be obtained explicitly for the Geo/PH/1 with discrete PH type vacations and also the M/PH/1 with PH vacations. This is an extension of the well known results by Ramaswami and Latouche  [7] for a QBD with rank one matrix at the sub-diagonal.

Section snippets

The model

First we consider the discrete time MAP/PH/1 system with phase type vacation. Let the sub-stochastic matrices Dk,k=0,1, of order n represent the discrete-time arrival process where (Dk)ij,1in,1jn, represents transition probability from phase i to phase j with k arrivals. The matrix D=D0+D1 is stochastic and irreducible. The service is discrete phase type with representation (β,S) of order m and the vacation duration is also discrete phase type with representation (α,T) of order r. Let t=1T1

Matrix G

It is known from  [6] that the matrix G which is the minimal non-negative matrix that satisfies the following equation G=A2+A1G+A0G2. It is the matrix that captures the probability of going down one level. Several queueing performance measures such as busy period are based on G. We show that G has a special structure that can be explored. Based on the structure of A2 we see that the G matrix is of the form G=[Gss0Gvs0],where Gss is an nm×nm matrix and Gvs is an nr×nm matrix.

Hence we have

Proposition 3.1

Gss=A2s

Continuous-time MAP/PH/1 queue with vacations

Now we consider the continuous time MAP/PH/1 system with phase type vacation. Let the matrix sequence Dk,k=0,1, of order n represent the arrival process where (Dk)ij,1in,1jn, represents transition rate from phase i to phase j with k arrivals. The matrix D=D0+D1 is a generator and irreducible. The service is continuous phase type with representation (β,S) of order m and the vacation duration is also continuous phase type with representation (α,T) of order r. Let t=T1 and s=S1, where 1 is a

Conclusions

The results presented in this paper are going to help us reduce the bulk of the computations regarding the R and G matrices from dealing with an n(m+r)×n(m+r) matrix to mainly dealing with an nm×nm matrix. Our focus in this paper is the exhaustive vacation type system. We plan to extend the idea to non-exhaustive type of vacation systems but would require some modifications first. Finally, the results developed can be easily extended to the case of discrete time GI/G/1 system in which both the

Acknowledgments

This research is partially supported by grants from the Natural Sciences and Engineering Research Council (NSERC) of Canada. The author acknowledges the anonymous referee and the Area Editor who provided some useful suggestions that helped to improve the presentation of the paper. The author also thanks Chamara Devanarayanan and Kamal Darchini for going through the final draft of this paper to check for typos and statements that need further clarifications.

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