Analysis of an M ^(λ1,λ2) / M /1 / WV Queue with Controlled Vacation Interruption and Variable Arrival Rate

This paper studies a Markovian queue with multiple working vacations and controlled vacation interruption. If there are at least N customers waiting upon completion of a service at a lower rate, the vacation is interrupted and the server returns to the system to resume the normal working level. Otherwise, the server continues the vacation until the system is non-empty after a vacation ends or there are at least N customers after a service ends. Moreover, the variable arrival rate of the customers is taken into account. Under such assumptions, by using the quasi-birth-and-death process, the matrix- geometric method and the difference equation theory, the steady-state queue length distribution along with various performance measures are derived. Additionally, under a certain cost structure, the optimal threshold N* that minimizes the long-run expected cost function per unit time is numerically determined.