Abstract

This paper considers a discrete-time Geo/G/1 queue, in which the server operates a single vacation at end of each consecutive service period. After all the messages are served in the queue exhaustively, the server immediately leaves for a vacation. Upon returning from the vacation, the server inspects the queue length. If there are some messages waiting in the queue, the server either resumes serving the waiting messages (with probability p) or remains idle in the system (with probability 1-p) until the next message arrives; and if no message presents in the queue, the server stays dormancy in the system until at least one message arrives. Using the generating functions technique, the system state evolution is analyzed. The probability generating functions of the system size distributions in various states are obtained. Some system characteristics of interest are 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.

Key words: Cost, busy period, discrete time queue, markov chain, randomized vacation.