Asymptotic analysis of an open queueing networks with fixed number of claims of two classes and random waiting time in queues
Queueing networks are used in the mathematical modeling of various economic, computers and other technical systems related to the servicing of client requests. The total number of messages being processed by these systems changes over time, that does not allow to consider closed queueing networks with fixed number of claims. In the introduction it is described the object of research - an open network with a limited number of claims of two classes, the waiting time of which in the queues of network systems is limited by random variables with exponential distribution. The network maintenance parameters are constant, the claims route is determined by an arbitrary stochastic transition probability matrix. The service time of claims in each system line is distributed according to the exponential law, claims are selected for maintenance in accordance with the FIFO discipline. The total number of claims is limited. The network state is described by a Markov random process. The aim of the study is to asymptotically analyze this process with a large number of claims and find the average relative number of claims in the network systems at any time. In the main part it is shown that the probability density of the process under study satisfies the Kolmogorov-Fokker-Planck equation with a certain accuracy. A system of ordinary linear differential equations of the first order with discontinuous right-hand sides is obtained for the average relative number of claims in each of the service systems at the time of interest. It is given an example of calculating the average relative number of applications in the network systems, which is a model of the processing of customer applications in the insurance company.