Multi-server M/M/c queue with N-policy and d-Type working vacation

Abstract

Introducing the N-policy and half exhaustive d-type working vacation into a multi-server M/M/c queue. If d or more than d servers are idle, we let d idled servers start a multiple synchronous working vacation, the servers on working vacation keep on serving new customers in a low rate, and the rest c-d servers work as usual. When a vacation is finished, once the customers in the system is greater or equal to N, terminating the vacation of these d servers, otherwise starting another working vacation. When a service is completed in a vacation period, what kind of service has been accepted by the customer who has left, depends on the customers in the system whether is more than c-d, if the number of customer is more than c-d, a customer who has left maybe served by normal rate or low rate service, if the number of customer isn’t more than c-d, a customer who has left served by normal rate service. Using a quasi-birth and death process and matrix geometric method, we obtain the stationary distributions. Finally, the result is applied to hierarchical cellular system, and we make the numerical analysis.