The Queueing Model MAP|PH|1|N with Feedback Operating in a Markovian Random Environment

A.N Dudin,L Breuer,V.I Klimenok,A.V Kazimirsky,U Krieger

doi:10.17713/ajs.v34i2.403

Abstract

Queueing systems with feedback are well suited for the description of message transmission and manufacturing processes where a repeated service is required. In the present paper we investigate a rather general single server queue with a Markovian Arrival Process (MAP), Phase-type (PH) service-time distribution, a finite buffer and feedback which operates in a random environment. A finite state Markovian random environment affects the parameters of the input and service processes and the feedback probability. The stationary distribution of the queue and of the sojourn times as well as the loss probability are calculated. Moreover, Little’s law is derived.

Highlights

The traditional assumption in queueing is the following: after the server completes the service of a customer, this customer leaves the system forever and does not affect the further operation of the system
We take into account the correlated nature of arrival streams in modern systems, e.g. message flows in telecommunication networks
The input flow of the system is determined by the following modification of the wellknown BMAP

Summary

Introduction

The traditional assumption in queueing is the following: after the server completes the service of a customer, this customer leaves the system forever and does not affect the further operation of the system. In some real systems a quality control is implemented for the service after the service completion and with some probability the customer may return to get additional service. Such a situation takes place,for instance, when a message is transmitted along a noisy wireless channel. These models with a return of a customer to get additional service (feedback queueing models) deserve special consideration. We apply so called Markovian Arrival Processes (MAP) as input of the system They are used instead of a Poisson process which is one of the most popular input streams in the literature. We deal with a finite buffer while previous studies have mainly considered feedback models with infinite buffer

Mathematical Model

Stationary Distribution of the Queue Length

Sojourn Time Distribution

Numerical Examples

Conclusion