Performance and Optimization Analysis of a Queue with Delayed Uninterrupted Multiple Vacation and N-Policy

This paper considers an M/G/1 queue with delayed uninterrupted multiple vacation and N-policy, in which (i) the server remains dormant from vacation to a non-empty system with no more than N customers, and (ii) when the system becomes exhausted, the server waits for a random time instead of immediately going on vacation (applicable to many scenarios and common to human behavior). We first study the transient queue length distribution from an arbitrary initial state and obtain its Laplace transform expressions. Then, the recursive formulas of the stationary queue length distribution are developed. Meanwhile, some crucial performance measures are presented. Finally, the cost optimization problems are discussed with and without the average waiting time constraint. Four numerical examples are illustrated to determine the optimal control policy for minimizing cost under the conditions that the random variables obey different phase-type (PH) distributions.