A Stochastic Service System with N-Policy, Bernoulli Interruption Vacations and Patient Server

This paper studies a stochastic service system with N-policy, Bernoulli interruption vacations, and a patient server. When the number of waiting customers reaches the threshold N during a vacation, the server interrupts the vacation with probability p(0≤p≤1), otherwise continuing the vacation until its completion. If the system is empty when a vacation ends, the server waits for a patience period before starting a new vacation. Service station failures occur during service, and interrupted service resumes after repair. We derive the Laplace transform expressions for the transient queue length probabilities and recursive formulas for the stationary queue length distribution. In addition, cost optimization models for the threshold N and the vacation length T are developed, both without and with an average waiting time constraint. Using the Particle Swarm Optimization algorithm, numerical examples under phase-type distributions illustrate how the probability p affects the optimal control policy.