A Time-dependent Tandem BMAP with Balking and Batch Service with Possible Breakdown and Delayed Service

Abstract

In this paper, a time-dependent queueing system containing two stages in tandem has been considered. Carriers carry jobs by bulks of various sizes and arrive at the first stage according to a Poisson distribution. There is a possibility of balking and, hence, jobs attend with some probability at an infinite size buffer, located at the entrance of the first stage. As the jobs attend, they will be placed randomly in the buffer with some type of identification for later to be served based on that order, that is, first-come – first-served rule. For jobs to be served by batches, they will be grouped with a minimum and a maximum limit and will be moved to be served by a single server. However, before being served, jobs must go through a procedure that causes service be performed with delay. There is also a possibility of a server breakdown that would require to be repaired, which will affect arrivals, that cause another possible delay in service. As a batch exits the first stage, some may leave the system at that point with some probability. The rest of the batch attend an infinite buffer at the second stage with the complement of the probability of leaving. The attending batch es will be numbered and they will move to service as they are. As it can be anticipated for a time-dependent case, the system is a complicated one. Nonetheless, the time-dependent probability generating function for the number of jobs at each stage and the system as a whole, as well as the first and the second moments are found. The probability generation function and convolution of exponential functions and generating functions have been used to obtain moments for each stage as well as the system. Several special cases have been
illustrated to show the validity of the results.