Heavy outgoing call asymptotics for $\mathrm{MMPP|M|1}$ retrial queue with~two way communication and multiple types of outgoing calls

A. A. Nazarov ; S. V. Paul ; O. D. Lizyura

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 1, pp. 111-124

Résumé

In this paper, we consider a single server retrial queue $\mathrm{MMPP|M|1}$ with two way communication and multiple types of outgoing calls. Calls received by the system occupy the device for operating, if it is free, or are sent to orbit, where they make a random delay before the next attempt to occupy the device. The duration of the delay has an exponential distribution. The main issue of this model is an existence of various types of outgoing calls in the system. The intensity of outgoing calls is different for different types of outgoing calls. The operating time of the outgoing calls also differs depending on the type and is exponential random variable, the parameters of which in the general case do not coincide. The device generates calls from the outside only when it does not operate the calls received from the flow. We use asymptotic analysis methods under two limit conditions: high rate of outgoing calls and low rate of serving outgoing calls. The aim of the current research is to derive an asymptotic stationary probability distribution of the number of incoming calls in the system that arrived from the flow, without taking into account the outgoing call if it is operated on the device. In this paper, we obtain asymptotic characteristic function under aforementioned limit conditions. In the limiting condition of high intensity of outgoing calls, the asymptotic characteristic function of the number of incoming calls in a system with repeated calls and multiple types of outgoing calls is a characteristic function of a Gaussian random variable. The type of the asymptotic characteristic function of the number of incoming calls in the system under study in the limiting condition of long-term operation of the outgoing calls is uniquely determined.