On a retrial queueing model with single/batch service and search of customers from the orbit

Abstract

A single server retrial queueing system, where customers arrive according to a batch Poisson process and are served either in single or as a batch, is considered here. An arriving batch, finding the server busy, enters an orbit. Otherwise, one customer, a few customers, or all customers from the arriving batch, depending on if the batch size exceeds a threshold value or not, enter service immediately, while the rest join the orbit. Customers from the orbit try to reach the server subsequently with the inter-retrial times, exponentially distributed. Additionally, at each service completion epoch, one of the two types of search mechanisms say, type I and type II search, to bring the orbital customers to service, is switched on—type I search when the orbit size is less than the threshold value and type II search otherwise. This means that, while the server is idle, a competition takes place among primary customers, customers who come by retrial and by one of the two types of search as the case may be. A type I search selects a single customer whereas a type II search takes a batch of customers from the orbit. In the case of primary customers and those who come by type II search, maximum size of the batch taken into service is restricted to a pre-assigned value. Both single and batch service are assumed to be arbitrarily distributed with different distributions, which are independent of each other. Steady state analysis is performed. Some important system descriptors are computed algorithmically and numerical illustrations are provided.