An MX/GI/1/N queue with close-down and vacation times

Andreas Frey,Yoshitaka Takahashi

doi:10.1155/s1048953399000064

Andreas Frey, Yoshitaka Takahashi

Open Access

https://doi.org/10.1155/s1048953399000064

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Abstract

An MX/GI/1/N finite capacity queue with close-down time, vacation time and exhaustive service discipline is considered under the partial batch acceptance strategy as well as under the whole batch acceptance strategy. Applying the supplementary variable technique the queue length distribution at an arbitrary instant and at a departure epoch is obtained under both strategies, where no assumption on the batch size distribution is made. The loss probabilities and the Laplace-Stieltjes transforms of the waiting time distribution of the first customer and of an arbitrary customer of a batch are also given. Numerical examples give some insight into the behavior of the system.

Highlights

There has been much interest in batch arriving queueing systems during the last three and a half decades, both from theoretical and practical points of view
As for the batch-Poisson arrival MX/GI/1/N finite capacity queue, the embedded Markov chain (EMC) technique cannot be straightforwardly applied under the whole batch acceptance strategy, WBAS, while the supplementary variable (SV) technique can be applied with an artificial condition by Baba [1]
We will apply the above-derived results to the following setting: the batch size is deterministic and equals 5, i.e., g5 1; the number of waiting places, including the customer that may be in service, equals 11, i.e., N- 11; the service time, close-down time and vacation time distribution is either deterministics (Det), Erlang of order 2 (Erl), exponential (Exp)or hyperexponential of order 2 (Hyp); by Det(x), Erl(x), Exp(1/x), Hyp(x), we denote the corresponding distribution with mean x

Summary

Introduction

There has been much interest in batch arriving queueing systems during the last three and a half decades, both from theoretical and practical points of view. Those systems are frequently encountered in the real world as can be seen in Chaudhry and Printed in the U.S.A. Many techniques have been developed or extended to deal with the additional analytical complexities that result from the introduction of batch arrivals. Both the embedded Markov chain (EMC) technique and the supplementary variable (SV) technique can be applied to the Poisson (non-batch) arrival M/GI/1/N finite capacity queue. The main purpose of the paper is to present an SV-technique based analysis for a batch-Poisson arrival MX/GI/1/N finite capacity queue with closedown time and server vacation, asserting that Baba’s [1] assumption can be omitted

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