MX/G/1 retrial G-queue with multistage and multi-optional services, feedback, randomized J vacation and orbital search

Abstract

A single server batch arrival retrial G-queue with orbital search is considered. Positive customers arrive in batches and the negative customers arrive in single. After repair, the server starts servicing the new customer. The server provides services in two phases. The first phase is essential to all the arriving customers, whereas the second phase is optional. There are L stages of sequential services in the second phase. In each stage, there are multi optional heterogeneous services. After the completion of essential service, the customer may depart the system with probability d0, feedback to the orbit with probability δ0 or choose any one of the optional service in 1st stage of phase2 with probability pk1 (k1 = 1,2,. …,j1). In general, after the completion of mth (m=1,2, ,L-1) stage in second phase, the customer may depart the system with probability dm, feedback to the orbit with probability δm or choose kthm+1(km+1 = 1,2, … ,jm+1) optional service in (m+1)th stage of phase 2. After the completion of Lth stage, the customer may depart the system with probability dM or feedback to the orbit with probability δL. Whenever the system becomes empty, the server goes for vacation and takes at most J vacations repeatedly until at least one customer is recorded in the orbit. At the end of Jth vacation, even if the orbit is empty, the server remains idle in the system. During the idle period in the non-empty system, the server may search for customers in the orbit or remain idle. The arrival of negative customer makes the server down and pushes out the customer being in service. The probability generating functions and the number of customers in the orbit and in the system are obtained by supplementary variable technique. Expected system size, expected orbit size, availability of the server and failure frequency are derived. Numerical results are also presented.