Central Limit Theorem for an M/M/1/1 Retrial Queue with Unreliable Server and Two-Way Communication

Abstract

AbstractIn this paper, we consider a single-server retrial queue with unreliable server and two-way communication. Inbound calls arrive according to a Poisson process. If the server is busy upon arrival, an incoming call joins the orbit and retries to occupy the server after some exponentially distributed time. Service durations of incoming calls follow the exponential distribution. In the idle time, the server makes outgoing calls. There are multiple types of outgoing calls in the system. We assume that durations of each type of outgoing calls follow a distinct exponential distribution. The server is subject to breakdowns with rates depending on the state of the server. If the breakdown occurs, the server undergoes a repair whose duration follows an exponential distribution. The aim of our research is to show that the scaled number of calls in the orbit follows a normal distribution under the condition that the retrial rate is low.KeywordsRetrial queueTwo-way communicationIncoming callsOutgoing callsAsymptotic analysisCentral limit theorem