Output process of the $\mathrm{M|GI|1}$ is an asymptotical renewal process

I. L. Lapatin ; A. A. Nazarov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 1, pp. 100-110

Résumé

Most of the studies on models with retrials are devoted to the research of the number of applications in the system or in the source of repeated calls using asymptotic and numerical approaches or simulation. Although one of the main characteristics that determines the quality of the communication system is the number of applications served by the system per unit of time. Information on the characteristics of the output processes is of great practical interest, since the output process of one system may be incoming to another. The results of the study of the outgoing flows of queuing networks are widely used in the modeling of computer systems, in the design of data transmission networks and in the analysis of complex multi-stage production processes. In this paper, we have considered a single server system with redial, the input of which receives a stationary Poisson process. The service time in considered system is a random value with an arbitrary distribution function $B(x)$. If the customer enters the system and finds the server busy, it instantly joins the orbit and carries out a random delay there during an exponentially distributed time. The object of study is the output process of this system. The output is characterized by the probability distribution of the number of customers that have completed service for time $t$. We have provided the study using asymptotic analysis method under low rate of retrials limit condition. We have shown in the paper that the output of retrial queue $\mathrm{M|GI|1}$ is an asymptotical renewal process. Moreover, the lengths of the intervals in output process are the sum of an exponential random value with the parameter $\lambda + \kappa$ and a random variable with the distribution function $B (x)$. The results of a numerical experiment show that the probability distributions of the number of served customers in the system are practically the same for significantly different distribution laws $B (x)$ of service time if the service times have the same first two moments.