A discrete-time Geo<SUP align=right>X</SUP>/G/1 retrial queueing system with starting failures and optional service

Abstract

This paper is concerned with the analysis of a single-server batch arrival retrial queue with starting failures in discrete time in which all the arriving customers need essential service (ES) necessarily whereas only some may require the optional service (OS) after ES. The customers are assumed to arrive at the system according to a geometric process during the consecutive slots. The service times and retrial times are generally distributed. The generating functions of the number of customers in the retrial group and in the system are obtained along with the marginal distributions of the number of customers in the retrial group when the server is idle, busy (in providing ES or OS) or broken down. The explicit formulae for the stationary distribution and other performance measures at equilibrium are obtained. Some special cases are deduced by setting appropriate parameters. The stochastic decomposition property has also been discussed. The numerical examples are facilitated to explore the effect of different parameters on various performance measures.

Mathematical Analysis of Queue with Phase Service: An Overview

“…Retrial queue, bulk arrival, starting failure, stochastic decomposition, and -stage service Jain and Agrawal [16,17] Probability generating function Retrial queue, feedback, and phase service Liu and Gao [165] Geo/Geo/1 Second optional service Lin et al [166] Matrix analytic approach, quasi-Newton method MAP/PH/1 Discrete time, multiple working vacations, and two-phase service Goswami and Selvaraju [167] M a t r i x a n a l y t i c m e t h o d GI/D-MSP/1 Discrete time, multiple vacations, Samanta and Zhang [168] Matrix geometric method…”

“…To illustrate, this phenomenon we can cite a clinical physical examination procedure as a doctor's clinic wherein the patient goes through a series of physical checkup such as ear, nose and throat examination, blood test, electrocardiogram, eye examination which all are considered as essential phases of service to reach the final decision by any medical doctor to declare a candidate to be fit. Recently, Jain and Agrawal [16,17] examined a batch arrival queueing system with modified Bernoulli vacation under -policy wherein the customer needs -stage of service in succession that is, the first stage service (FSS) is followed by the second stage service (SSS), the second stage service followed by third stage service (TSS) and so on up to -stages of service.…”