Some GI/M/1Queues With Bernoulli Vacation Interruption

In this Ph.D. thesis, we investigate some GI/M/1queues with Bernoulli vacation interruption. This thesis is organized as follows.In Chapter2, a GI/M/1queue with Bernoulli vacation interrup-tion is analyzed. During the working vacation period, if there are customers at a service completion instant, the vacation can be inter-rupted and the server is resumed to a regular busy period with prob-ability p(0≤<p≤1)(not with probability1), or continues the vacation with probability1-p. Obviously, if p=0or p=1, we will get the same results for GI/M/1queue with working vacations and without or with vacation interruption, respectively. And, we regard the vacation inter-ruption is controlled by Bernoulli. Thus, we can investigate working vacation and vacation interruption at the same time, which is differ-ent from the situation many authors considered before. Compared with a GI/M/1queue with working vacations but without vacation interruption, even if the vacation time is not end, it's still possible for our server to stop the vacation and come back to the normal busy period. On the other hand, compared with a GI/M/1queue with vacation interruption, even if there are customers at a service comple-tion instant during the working vacation period, our server can still continue the vacation with probability1-p. Using the matrix-analytic method, we get the steady-state distribution for the queue length at arrival epochs. The stochastic decomposition structure for the queue length is also derived. Using the theory of semi-Markov process, the steady-state distribution for the queue length at arbitrary epochs is obtained. Using different methods, the LST of waiting time and the LST of sojourn time are derived. Finally, taking the expected queue length as an example, we perform some numerical examples to study the effect of various parameters on the system's characteristics. In Chapter3, we consider a set-up period on the basis of the model in Chapter2. When the vacation interruption happens, the server cannot come back to the regular busy period immediately, but begins a set-up period of random length. In fact, the set-up period exists in some practical situations. For example, in a production sys-tem, improving the productivity of a machine may experience some time before processing the job. Chapter4investigates a GI/M/1queue with Bernoulli-schedule-controlled vacation and vacation inter-ruption, we consider ordinary vacation, working vacation and vacation interruption at the same time. Let parameters take proper values, many GI/M/1vacation queues will be the special cases of the model we consider. Chapter5studies a GI/M/1queue with start-up pe-riod and single working vacation, and the vacation interruption is also controlled by Bernoulli. Furthermore, we describe the correspond-ing discrete-time queue at the end of each Chapter. Using the same method, these discrete-time queues can be analyzed in a similar way.