Two Queueing Systems With Working Vacations

In2002, Servi and Finn first introduced working vacation policy, which plays an important role in the performance analysis of gateway router in optical networks. Queueing system with working vacations has become an emerging research hot spot in recent years. In this paper, we investigate two queueing systems with working vacations. The main contents are organized as follows.Chapter1is preface and preliminary knowledge. The historical background, the subject, the method, the recent development of four queueing systems and the main results of this paper are introduced.In chapter2, we consider the case where customers’impatience is due to the slow service rate when the server is in a working vacation period. We assume the system has different arrival rates when the server is in a regular busy period or a working vacation period. We analyze an M/M/1queue with impatient customers and multiple working vacations or single working vacation, respectively. For the M/M/c queueing system with impatient customers, we adopt the multiple synchronous working vacation policy. Using balance equation and probability generating function, we can get the steady-state distribution for the queue length. Finally, we consider the sojourn time of a customer who completes his service. Let parameters take proper values, many queueing systems with impatient customers will be special cases of the model we consider.In Chapter3, we analyze a GI/M/1queue with negative customers and Bernoulli working vacation interruption. The arrival of a negative customer can remove a positive customer at the head (if present), and we also regard this positive customer completes his service. During the working vacation period, if there are positive customers at a service completion instant, the vacation can be interrupted and the server comes back to a regular busy period with probability p(0≤p≤1)(not with probability1), or continues the working vacation with probability1-p. And, we regard the working vacation interruption is controlled by Bernoulli. Thus, we can study working vacation and working vacation interruption at the same time. Using the matrix-analytic method, we get the steady-state distribution for the queue length at positive customers’ arrival epochs. Finally, we obtain the LST of sojourn time.