Cost Analysis of a Discrete-time Queue

 Abstract—This paper considers a Geo/G/1 queue, in which the server operates a single vacation at end of each consecutive service period. After all the customers in the system are served exhaustively, the server immediately leaves for a vacation. Upon returning from the vacation, the server inspects the queue length. If there are customers waiting in the queue, the server either resumes serving the waiting customers (with probability p) or remains idle in the system (with probability 1-p) until the next customer arrives; and if no customer presents in the queue, the server stays dormancy in the system until at least one customer arrives. Using the generating function technique, the system state evolution is analyzed. The probability generating functions of the system size distributions in various states are obtained. The waiting time distribution is also derived. With the vacation of fixed length time (say T), the long run average cost function per unit time is analytically developed to determine the joint optimal values of T and p at a minimum cost.