On a batch arrival queue with second optional service, random breakdowns, delay time for repairs to start and restricted availability of arrivals during breakdown periods

Abstract

We study a batch arrival single service channel queuing system where the server (service channel) provides two stages of general service to customers, the first essential service followed by the second optional service.  It is assumed that the service channel is subject to breakdowns and on the occurrence of a breakdown, the service channel waits for the repairs to start and this waiting time (termed as the set-up time or delay time for repairs) is assumed general.  Further, the repair times are also assumed general. We employ the supplementary variable technique using four supplementary variables, one each for the elapsed service time of the first essential service, the elapsed service time of the second optional service, the elapsed delay time and the elapsed repair time. In addition, we add an important assumption that during breakdown periods, the arriving batches are admitted into the system based on a policy of restricted admissibility. We derive queue size distribution for this system at a random epoch under the steady state conditions. Further, we derive some important performance measures of this system. This extends many models studied earlier by several authors. Finally, a few interesting particular cases are discussed.