Analysis of MAP/PH/1 Queueing System with Degrading Service Rate and Phase Type Vacation

Abstract

Degradation of services arises in practice due to a variety of reasons including wear-and-tear of machinery and fatigue. In this paper, we look at MAP/PH/1-type queueing models in which degradation is introduced. There are several ways to incorporate degradation into a service system. Here, we model the degradation in the form of the service rate declining (i.e., the service rate decreases with the number of services offered) until the degradation is addressed. The service rate is reset to the original rate either after a fixed number of services is offered or when the server becomes idle. We look at two models. In the first, we assume that the degradation is instantaneously fixed, and in the second model, there is a random time that is needed to address the degradation issue. These models are analyzed in steady state using the classical matrix-analytic methods. Illustrative numerical examples are provided. Comparisons of both the models are drawn.

A Model for a Vacation Queuing Policy Considering Server’s Deterioration and Recovery

“…As an illustration of this problem, they studied a two-machine line model. Choudhary et al [25] also considered the deterioration of the server's efficiency during service, which was recovered through a vacation of a random time duration. For clarification, Table 1 summarizes previous studies that address the deterioration of the server.…”

“…The paper focuses on the time when the main server leaves service and the length of the recuperation period. While Choudhary et al [25] assumed that the working and vacation durations are exogenous variables that cannot be controlled by the system, in reality the system's manager can control these variables (e.g., by controlling the vacation duration) in many service systems. Thus, in the current work, we extend this model, taking the working and vacation durations as endogenous variables, and provide a scheme for the system's manager to optimize the vacation policy.…”