Queue Length and Server Content Distribution in an Infinite-Buffer Batch-Service Queue with Batch-Size-Dependent Service

U C Gupta,S Pradhan

doi:10.1155/2015/102824

U C Gupta, S Pradhan

Open Access

https://doi.org/10.1155/2015/102824

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Abstract

We analyze an infinite-buffer batch-size-dependent batch-service queue with Poisson arrival and arbitrarily distributed service time. Using supplementary variable technique, we derive a bivariate probability generating function from which the joint distribution of queue and server content at departure epoch of a batch is extracted and presented in terms of roots of the characteristic equation. We also obtain the joint distribution of queue and server content at arbitrary epoch. Finally, the utility of analytical results is demonstrated by the inclusion of some numerical examples which also includes the investigation of multiple zeros.

Highlights

Banerjee and Gupta [1] analyzed a finite-buffer batch-service queue with batch-size-dependent service and obtained joint distribution of queue and server content at departure and arbitrary epochs
This paper considers an infinite-buffer single server queue with Poisson arrivals and general service time distribution where customers are served in batches according to general bulk service, (a, b) rule, and service times of the batches depend on the number of customers within the batch under service process
Our main objective in this paper is to develop a tractable yet implementable procedure to obtain the joint distribution of the queue and the server content at departure and arbitrary epochs

Summary

Introduction

Banerjee and Gupta [1] analyzed a finite-buffer batch-service queue with batch-size-dependent service and obtained joint distribution of queue and server content (i.e., number in the queue as well as with the server) at departure and arbitrary epochs. Repeated roots occur for Erlang (Em), m ≥ 2; service time distribution and number of roots depend on the threshold value “a” and the maximum capacity “b” and multiplicity of each root depends on “m” Recent studies on this model have been carried out by Bar-Lev et al [2], who derived the pgf of only queue content at departure epoch but did not use it due to the complexity involved in the inversion process.

Joint Distribution of Queue Content and Number with the Departing Batch

Joint Distribution of Queue and Server Content at Arbitrary Epoch

Numerical Illustration

Conclusion