M/M/ 1 Retrial Queueing System with Variable Service Rate
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M/M/1 RETRIAL QUEUEING SYSTEM WITH VARIABLE SERVICE RATE M. S. Bratiichuk,1 A. A. Chechelnitsky,2 and I. Ya. Usar3,4
UDC 519.21
We consider a queueing system of the M/M/1 type with repeated claims in the case where the service rate depends on the loading of the system, i.e., on the number of claims in the queue waiting for service. We establish the existence conditions and the formulas for the ergodic distribution of the number of claims in the system with bounded and unbounded queues of repeated claims.
1. Introduction A standard retrial system of the M/M/1/1 type with repeated claims can be described as follows: The system is formed by a single server. The input flow is a Poisson flow with a rate λ. A claim arriving at the system with free server is immediately served and leaves the system after service. However, if the server is occupied, then the claim becomes repeated and enters the so-called orbit. After certain time intervals (exponentially distributed with a parameter ⌫) each claim on the orbit tries to get service independently of the other claims present on the orbit at the same time and if the server is free at the time of appeal, then this claim is served. Otherwise, it returns to the orbit and the process is repeated. Thus, each claim on the orbit generates a Poisson flow of repeated claims with a rate ⌫. The service time of each claim obeys an exponential distribution. The parameter of this distribution is determined as follows: If, at the time of service of a claim (either new or from the orbit), the system contains j claims (it is clear that j = i + 1, where i is the number of claims on the orbit), then the parameter of distribution of the service time is equal to µj . The service rate changes only at the times when new claims arrive at the server. Thus, in the analyzed system, unlike ordinary systems (see [1, 2]), the service time depends on the number of claims in the system. The mathematical models of systems with repeated calls are extensively used in practice (see [3, 4]). The necessity of investigation of models whose parameters depend on the number of orders in the system is connected with economical aspects of operation of these systems. In specific systems, it is often impossible to change the rate of arrival of new orders. At the same time, it is possible to change the intensity of operation of the server depending on the number of orders in the system. Hence, depending on the number of orders, the manager can dynamically regulate the service policy with an aim to improve its economical (cost) characteristics. Thus, in the model described in [5], the service rate changes if the queue reaches a certain level T. This is the so-called threshold strategy. Similar schemes were also considered, e.g., in [6–8]. According to the method of determination of ergodic probabilities for systems with threshold strategy (see, e.g., [8]), the equation for ergodic probabilities is first solved in the sets of states {0, 1, . . . , T − 1} and {T, T + 1, . . .} and then the solutions are “glued” on the boun
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