On impatience in Markovian M/M/1/N/DWV queue with vacation interruption

In this paper, we establish a cost optimization analysis for an M/M/1/N queuing system with differentiated working vacations, Bernoulli schedule vacation interruption, balking and reneging. Once the system is empty, the server waits a random amount of time before he goes on working vacation during which service is provided with a lower rate. At the instant of the service achievement in the vacation period, if there are customers present in the system, the vacation is interrupted and the server returns to the regular busy period with probability beta' or continues the working vacation with probability 1 - beta'. Whenever the working vacation is ended, the server comes back to the normal busy period. If the system is empty, the server can take another working vacation of shorter duration. In addition, it is supposed that during both busy and working vacation periods, arriving customers may become impatient with individual timers exponentially distributed. Explicit expressions for the steady-state system size probabilities are derived using recursive technique. Further, interesting performance measures are explicitly obtained. Then, we construct a cost model in order to determine the optimal values of service rates, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Finally, numerical illustrations are added to validate the theoretical results.