Level Dependent Quasi-Birth and Death Processes of M M M B B / , / ( ) 1 2 1 2 2 / ( , )queues with Stalling

This Paper analyses a two server M M M B B / , / ( ) 1 2 1 2 2 / ( , ) queueing system with stalling where customers are served by one fast server and one slow server. The servers are allowed to work in parallel. It has provided a finite buffer B 1 of size `K   ' to stall customers in queue-1 which is meant to feed the fast sever only. There is one more buffer B 2 of infinite capacity, called the waiting space to accept further arrivals in queue-2 when the buffer B 1 is full. The primary task of B 2 is to feed customers to queue-1 and to the slow server as and when it is warranted. An arriving customer who finds the queue-1 is full and queue-2 is empty joins the slow server. If the queue-2 is non-empty at a time epoch when the slow server finishes a service, he accepts a customer from the head of the queue-2. Both arrival and service rates of customers are assumed to be state dependent parameters for the first (K+2) states starting from state `0'. Formulating the queue length (queue +service) process of the whole system as a QBD processes, steady state results to state probabilities and mean queue length have been obtained using matrix-analytical methods.