Abstract: In this paper, the classical Geo/Geo/1 queueing system is investigated. First of all, a new GI/M/1 type Markov Chain model for the queue is proposed. Moreover, by using the matrix analytic method, the joint stationary distribution for the Markov chain is given, the results enable us not only obtain an explicit expression for the stationary queue length distribution of the queueing system, but also give the probability of the exact number of vacations that the sever has taken. Such accurate descriptions for the status of the server are new results for the queueing model. Finally, numerical examples are demonstrated to illustrate the effectiveness of the results.

Key words: Geo/Geo/1 queue, working vacation, GI/M/1 type Markov process, matrix geometric solution, difference equation