Quasi-birth and death processes of two-server queues with stalling

This paper investigates an optimal K-policy for a two-server Markovian queueing system M/(M1,M2)/2/(B1,B2), with one fast server S1 and one slow server S2, using the matrix analytic method. Two buffers B1 and B2 are organized to form waiting lines of customers in which, buffer B1 is of finite size K(<infinity) and buffer B2 is of infinite capacity. Buffer B1 stalls customers who arrive when the system size (queue + service) is less than (K+1) and dispatches a customer to the fast server S1 only after S1 completes its previous service. This K-policy is of threshold type which deals with controlling of informed customers and hence the customers have better choice of choosing the fast server routing through the buffer B1. The (K+2)-nd customer who arrives when the number of customers present in the system is exactly (K+1) has the Hobson's choice of getting service from the slow server S2. Buffer B2 accommodates other customers who arrive when the number of customers present in the system is (K+2) or more and feeds them one after another to either buffer B1 or the sever S2 whichever event can first accept the customer at the head-of-the-line in B2. Queue length processes of interest are (1) q1=limt ->infinity X1(t) and (2) q2=limt ->infinity X2(t), where X1(t) represents the number of customers who are in the buffers B1 and B2 and also in the service with server S1 at time 't' and X2(t) represents the number of customers available with server S2 only. The bi-variate random sequence X(t)=(X1(t),X2(t)) of the system size (queue + service) forms a quasi-birth and death process (QBD). Steady state characteristics, and some of the performance measures such as the expected queue length, the probability that each server is busy etc are obtained. Numerical illustrations are provided based on the average cost function to explore the methodology of finding the best K-policy which minimizes the mean sojourn time of customers.