The probabilistic model of sharing system with collisions, H-persistence and rejections data processing

A queuing system with repeated calls, one server, collisions (conflicts) of calls, /-persistence and rejections is considered. The input flow of calls is the Poisson process with an intensity o. A call that found the device free occupies it, and service begins, which ends successfully if no other requests were received during it. If the server is busy, then a conflict (collision) arises between the call that has come for service and the ones being serviced, and in the general case, both calls instantly go to the orbit and repeat the attempt to successfully serve after a random time. In this article, in the event of a collision, one of the calls, for example, which was in service (on the device), goes into orbit with probability 1, the other goes into orbit with probability H, and with probability (1-H) refuses service and leaves the system. We have two types of arriving calls: primary and repeated. Primary calls are calls received from the outside into the system; repeated calls are calls in the orbit after the collision and making a repeated attempt to occupy the device for servicing. When building models with repeated calls, it is usually assumed that the arrival of primary calls obeys Poisson's law. Nevertheless, this does not mean that the incoming flow of calls in the analyzed system is Poisson, we are talking only about the flow of primary calls. The total incoming arrival flow consists of a Poisson flow of primary calls and a flow of repeated calls. The service time of a primary or repeated claim does not depend on its type and has an exponential distribution with the parameter li. and one unit of line resource is used to service requests. The random delay that a call carries out in orbit is exponentially distributed with the parameter g. The problem is to find the probabilities distribution of the calls number in the orbit. We write the Kolmogorov differential equations system for the stationary regime and then construct a recurrent algorithm to solve it. The numerical results obtained with recurrent algorithm made it possible to conclude that with a decrease in the random delay, which is carried out by a call in orbit under a conflict, the distribution of the number of calls in the orbit has the form of a Gaussian. The technical characteristics of the system, which are of practical importance for its design, are found.

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