Abstract
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.
Highlights
Queues with server vacations have been studied extensively in the past, and have been successfully used in various applied problems
In 2002, Servi and Finn [1] introduced a new kind of semi-vacation policy called working vacation (WV) policy, in which the server works at a lower rate rather than completely terminates during a vacation period
Li and Tian [6] introduced the vacation interruption schedule in an M/M/1 queue with working vacations. Under this type of vacation policy, if there are customers waiting at the instant of a service completion in the vacation period, the server ends his/her vacation and comes back to the normal working level; otherwise, if there is no customer waiting upon completion of the service at a lower rate, he/she continues the vacation until the system is non-empty after a service or a vacation ends
Summary
Queues with server vacations have been studied extensively in the past, and have been successfully used in various applied problems. In the working vacation policy, it is generally known that server resumes normal service rate only when the system is not empty at the end of a vacation Such assumption seems much more restrictive in real world scenarios. Li and Tian [6] introduced the vacation interruption schedule in an M/M/1 queue with working vacations Under this type of vacation policy, if there are customers waiting at the instant of a service completion in the vacation period, the server ends his/her vacation and comes back to the normal working level; otherwise, if there is no customer waiting upon completion of the service at a lower rate, he/she continues the vacation until the system is non-empty after a service or a vacation ends. Under the modified vacation interruption policy, the server ends his/her vacation and resumes normal service rate as soon as at least N customers accumulate in the system upon the completion of a service in the vacation period.
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