Semi-Markov Model of a Single-Server Queue with Losses and Maintenance of an Unreliable Server
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SEMI-MARKOV MODEL OF A SINGLE-SERVER QUEUE WITH LOSSES AND MAINTENANCE OF AN UNRELIABLE SERVER A. I. Peschanskya and A. I. Kovalenko b
UDC 519.872
Abstract. A semi-Markov model of the single-server GI / G / 1/ 0 queue is constructed with allowance for the maintenance of an unreliable server. Stationary reliability and economic characteristics of the queue are found, and two-criteria optimization of maintenance periodicity is carried out. Keywords: single-server queue with an unreliable server, server maintenance, stationary distribution of embedded Markov chain, stationary characteristic, two-criteria optimization. INTRODUCTION The possibility of taking into account the probability of failure and recovery of servers was first mentioned in [1]. Investigations of queues (queuing systems) in this direction are conducted, for example, in [2–6]. One of possible methods for improving stationary characteristics of queues with unreliable servers is their maintenance service (MS). In this article, a model is constructed that describes the operation of a single-server queue with losses, an unreliable server, and its MS when a prescribed time between failures is reached. Under the assumption that the form of random quantities describing the queue is general, stationary reliability and economic indices of the queue and also its optimal maintenance periodicity are found with the help of the apparatus of the theory of semi-Markov processes with a discrete-continuous phase space of states [7, 8]. PROBLEM STATEMENT In a single-server queue with losses, a recurrent incoming flow of demands arrives that is generated by a random quantity b with an arbitrary distribution function G ( t ) = P{ b £ t }. The duration of serving a demand is a random quantity a with a distribution function F ( t ) = P{a £ t }. In serving a demand, when the server reaches the accumulated operating time represented by a random quantity g with a distribution function F( t ) = P{g £ t }, the server failed and its emergency recovery service (ERS) immediately begins. The counting of the time between failures of the server begins with the moment of the beginning of serving the first demand after completing its ERS. The duration of an ERS is a random quantity s a with a distribution function Ya ( t ) = P {s a £ t }. Moreover, it is assumed that a server MS is performed immediately after reaching the accumulated operating time by the server; this time is equal to a deterministic quantity t. After the performed MS, this time is zeroized and the counting of the accumulated operating time for the following MS begins with the moment of serving the first demand arriving after the completion of the previous MS. The duration of performing an MS is a random quantity s p with a distribution function Y p ( t ) = P{ s p £ t } . The demand served at the moment of a server failure or at the beginning of a server TO is lost as well as all the demands arrived in the system during repair or prophylactic works. It is supposed that random quantities a, b, g , s a , and a
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