Discrete-Time State Dependent Bulk Service Queue with Multiple Vacations and Changeover Times

P Vijaya Laxmi,D Seleshi

doi:10.1155/2014/356325

Abstract

This paper presents the analysis of a discrete-time renewal input multiple vacations queue with state dependent service and changeover times under (a,c,b) policy. The service times, vacation times, and changeover times are geometrically distributed. The server begins service if there are at least c units in the queue and the services are performed in batches of minimum size a and maximum size b (a≤c≤b). At service completion instant, if the queue size is less than c but not less than a secondary limit a, the server continues to serve and takes vacation if the queue size is less than a-1. The server is in changeover period whenever the queue size is a-1 at service completion instant and c-1 at vacation completion instant. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at prearrival and arbitrary epochs. Based on the queue length distributions, some performance measures of the system have been discussed. A cost model has been formulated and optimum values of the service and vacation rates have been evaluated using genetic algorithm. Numerical results showing the effect of model parameters on the key performance measures are presented.

Highlights

The interest in discrete-time queues in which time is slotted has got a spectacular growth with the arrival of the digital technologies
A fundamental motive to study discrete-time queues is that they are more appropriate than their continuous-time counterparts for analyzing computer and telecommunication systems, since nowadays these systems are more digital than analogical. In view of this they have become increasingly important due to their applications in the study of many computer and communication systems such as asynchronous transfer mode (ATM) multiplexers, broadband integrated services digital network (B-ISDN), circuit-switched time-division multiple access (TDMA) systems, slotted carrier-sense multiple access (CSMA) protocols, and traffic concentrators in which the time axis is divided into slots
To demonstrate the applicability of the theoretical investigation made in the previous sections, we present some numerical results in the form of tables and graphs

Summary

Introduction

The interest in discrete-time queues in which time is slotted (divided into fixed-length of contiguous intervals) has got a spectacular growth with the arrival of the digital technologies. A fundamental motive to study discrete-time queues is that they are more appropriate than their continuous-time counterparts for analyzing computer and telecommunication systems, since nowadays these systems are more digital (such as a machine cycle time, bits, and packets) than analogical. In view of this they have become increasingly important due to their applications in the study of many computer and communication systems such as asynchronous transfer mode (ATM) multiplexers, broadband integrated services digital network (B-ISDN), circuit-switched time-division multiple access (TDMA) systems, slotted carrier-sense multiple access (CSMA) protocols, and traffic concentrators in which the time axis is divided into slots. One can see Bruneel and Kim [1], Takagi [2], Tran-Gia et al [3], and Robertazzi [4] for extensive treatments of various types of discrete-time queues and their applications

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