Analysis of state dependent M[X]/G(a, b)/1 queue with multiple vacation second optional service and optional re-service

Abstract

The objective of this paper is to analyse an M[X]/G(a, b)/1 queueing model with second optional service, multiple vacation, state dependent arrival and optional re-service. After completing the first essential service, a batch of customers either requests for re-service or leaves the system without re-service. After the completion of first essential service (with or without re-service), the batch of customers either requests for second optional service or leaves the system. At the completion moment of second optional service, the batch of customers either requests for re-service or leaves the system after the second service. Whenever the queue size is less than 'a', the server commences vacation. At the instant of vacation completion, if at least 'a' customers wait for service, the server starts a busy period. Otherwise, the server resumes another vacation. Using supplementary variable technique, the steady-state probability generating function (PGF) of the queue size is obtained.