The paper studies the queuing model with three non-identical exponential servers S1, S2 and S3 and provides a matrix-geometric solution for an underlying quasi birth-and-death (QBD) queue of an M/M(S1),M(S2,S3)/3/(m,∞) system. Customers arrive individually according to a Poisson process and form two parallel queues, say q1 and q2. The size of q1 represents the system length (queue+server) of a finite queueing facility M/M(S1)/1/(m+1) and the size of the q2 accounts the system length (queue+server) of a two-server queue M/M(S2,S3)/2 facility. Queue management for each of q1 and q2 is through a ‘First Come First Served (FCFS)’ basis but according to the norms of an m-policy. At an arrival instant, if the size of q1 is strictly less than ‘m’, the new arrival is assigned to q1 with an unknown probability P1; otherwise it is assigned to q2 with probability (1-P1) subject to a condition that switching from q1 to q2 and vice versa is to be avoided. At every service completion epoch, the dispatching mechanism of the m-policy either assigns a customer of q1 > 0 to server S1 or a customer of q2 >0 to server S2, if available, or otherwise to server S3. The underlying QBD process representing the number of customers in the system under study is formulated as a bi-variate queue length process X=(q1= i, q2= j) defined on the two-dimensional state space ={(i, j): 0 ≤ i ≤ m, j ≥ 0}. Explicit expressions for the stationary condition, stationary distribution of X, marginal expected values of q1 and q2, and the probability P1 are obtained. The paper also constructs a formal linear programming to find an optimal value of m, corresponding to the minimum cost.

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