Economic application in a finite capacity multi-channel queue with second optional channel

We consider a finite capacity M/M/R queue with second optional channel. The interarrival times of arriving customers follow an exponential distribution. The service times of the first essential channel and the second optional channel are assumed to follow an exponential distribution. As soon as the first essential service of a customer is completed, a customer may leave the system with probability (1 − θ) or may opt for the second optional service with probability θ (0 ⩽ θ ⩽ 1). Using the matrix-geometric method, we obtain the steady-state probability distributions and various system performance measures. A cost model is established to determine the optimal solutions at the minimum cost. Finally, numerical results are provided to illustrate how the direct search method and the tabu search can be applied to obtain the optimal solutions. Sensitivity analysis is also investigated.