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

Alka Choudhary,Srinivas R Chakravarthy,Dinesh C Sharma

doi:10.3390/math9192387

Alka Choudhary, Srinivas R Chakravarthy + Show 1 more

Open Access

https://doi.org/10.3390/math9192387

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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.

Highlights

We see that for the processes NeA and ErS, Model 1 has less Ls and less Pbusy than the classical Markovian arrival process (MAP)/PH/1 model. This is because the server becomes frequently idle in Model 1 and the service rate is instantaneously restored to the original rate, which results in having a lower Ls
We studied the degradation that commonly occurs in many industries, notably in service sectors, in the context of MAP/PH/1-type queues
In Model 1, the degrading service rate is reset to its initial rate instantaneously after a fixed number of services or when the server becomes idle

Summary

Introduction

It is common to see services get degraded for a variety of reasons including wear-andtear, fatigue, and other inherent issues. We incorporate the degradation of services in the context of a single server, Markovian arrival process (MAP), and phase type (PH −). In the context of a multiple server queueing system, Chakravarthy [8,9] studied the synchronous vacations and phase type distributed vacation periods. He discussed an optimization problem and did the steady-state analysis of the model. Sreenivasan et al [14] studied queueing models with MAP arrivals, phase type services and with working vacations, N-policy, and vacation interruptions.

Generator of the Model

Result

Steady-State Probability Vector

Performance Measures

Numerical Examples

Conclusions