Analysis of G-network with unreliable service systems

Abstract

In this paper, the object of research is Markov’s network with positive and negative customers and unreliable service lines with single-line queuing systems (QS). The discipline of service of customers in the systems – FIFO (“first come first served”) and the service time of customers in each line of the QS network are distributed according to the exponential law with their parameters for each QS. The service lines in each QS are defeated by accidental breakdowns, and the time of correct operation of the service line in each SMO has an exponential distribution, with different parameters for each QS. After the breakdown, the line immediately begins to recover, and the recovery time also has an exponential distribution, the parameters of which are different for each QS. The aim of the study is to find the non-stationary probabilities of network states. To find them, a modified method of successive approximations combined with the method of series is proposed. This method allows one to remove the condition of high load. The properties of successive approximations are proved. On the basis of the obtained data, using a computer, a model example illustrating the finding of the time-dependent probabilities of network states is calculated. The results of this work can be applied to the modeling of various information systems and networks.