Stationary Characteristics of an Unreliable Single-Server Queueing System with Losses and Preventive Maintenance
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NTROL IN TECHNICAL SYSTEMS
Stationary Characteristics of an Unreliable Single-Server Queueing System with Losses and Preventive Maintenance A. I. Peschansky Sevastopol State University, Sevastopol, Russia e-mail: peschansky [email protected] Received September 30, 2019 Revised January 22, 2020 Accepted January 30, 2020
Abstract—This paper considers an unreliable restorable single-server queueing system with losses in which server failures may occur during operation. The random variables describing the system have general distributions. For increasing the efficiency of this system, it is proposed to carry out preventive maintenance of the server as soon as the accumulated operating time exceeds an upper permissible threshold. A semi-Markov model of the system’s evolution over time is constructed. The explicit-form expressions for the final probabilities and mean sojourn times of the system in different physical states are derived using the stationary distribution of the embedded Markov chain. The frequency of preventive maintenance of the server is optimized via maximizing the average specific profit and minimizing the average specific costs of the system. Keywords: single-server queueing system with losses, preventive maintenance, stationary distribution of embedded Markov chain, final probabilities of states, stationary characteristics of efficiency DOI: 10.1134/S0005117920070061
1. INTRODUCTION The interest in queueing systems with possible server failures is connected with the practical importance of an adequate description for the operation of modern technical and information systems. A survey of unreliable queueing systems can be found, e.g., in [1, 2]. Discuss in brief some results of research in this field. In [3], a multi-server queueing system with servers of finite reliability and a particular form of the distribution functions of random variables describing the system processes was studied. In [4], the system M/G/1/1 with two types of requests and a rather general form of the distribution functions of the restoration time and the time between failures was described. In [5], an optimal group policy for restoring failed servers was designed. In [6], the service discipline in a heterogeneous two-server queueing system with different reliability characteristics and service rate of requests was investigated. It should be emphasized that the most complete results were obtained under the condition that the random factors describing unreliable queueing systems have exponential distributions. In [7, 8], models of unreliable single-server queueing systems in the case of general-form distributions were constructed. The publications cited above do not claim to be a complete list of studies on the characterization of queueing systems with possible occurrence of server failures. Queueing systems with unreliable servers are often the subject of modern research, since the development of technology leads to new applications-relevant mathematical problems. Along with 1243
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determining the characteristics of queueing sys
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