Retrial queues with variable service rate
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RETRIAL QUEUES WITH VARIABLE SERVICE RATE E. A. Lebedeva† and V. D. Ponomarov a‡
UDC 519.217
Abstract. This paper deals with a Markov model for a retrial queue in which service rate depends on the number of calls in orbit. The investigation method is based on an approximation of the initial system by a system with a finite state space for which explicit formulas of stationary probabilities are found. The accuracy of such an approximation is also discussed. Keywords: stochastic system, retrial queue, stationary mode.
A retrial queuing system is a model of a server with a special configuration of queue. It is assumed that calls that have not gained access to servers upon arrival are in orbit (see [1]). All calls in orbit are equal, disordered, and independently interrogate the servers in random time (orbit cycle). If there are idle servers, then a call is served by some of them and leaves the system. Otherwise, it remains in orbit and repeats attempts to be served. A call that is for the first time refused to be served and is sent to orbit may not know how many calls are currently in orbit. For example, this really occurs in telecommunication networks when the subscriber presses the redial button. The classical retrial queuing system can be considered as a stochastic queuing network with two nodes. The main node that receives arriving calls includes servers and serves arrivals if there are idle servers. If all servers are busy, the main node sends calls to the auxiliary node, which collects calls that are in orbit and interrogates servers. Once servers have become available, the call is sent to the main node for service. To make a mathematical model more realistic and to account for the configuration and control algorithm, it is sometimes necessary to introduce more than two nodes into a stochastic network. Thus, the service process in a retrial queuing system becomes multidimensional and, consequently, more difficult to analyze than a system without retrials (see [2]). However, we managed to overcome these difficulties by setting up an informal theory of retrial queuing systems. The state of the art in this theory can be judged from the monograph [3] and the bibliography therein. The development of the theory is stimulated by applications in many fields: telephone and mobile phone networks, call centers and local computer networks, air traffic control. Examples of using retrial queuing models in these fields can be found in [4, 5]. One of the important applications of retrial queuing models is in formulating and solving problems of finding the optimal parameters of the system. During the solution of these problems, it is necessary to study models with the local characteristics of the servicing process dependent on the current phase state of the system. In these conditions, the method of generating functions, which proved well for classical models (see, e.g., [6]), is no longer applicable. We will develop an approximate approach to systems with variable service rate. A truncated, finite model will be studied first
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