| Summary: | 博士 === 國立交通大學 === 工業工程與管理系所 === 98 === In this dissertation, we analyze the <N, p>-policy and the <T, p>-policy M/G/1 queues with second optional service, server breakdowns and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service which follows a general distribution. By <N, p>-policy we mean that the server is turned on when N customers have accumulated, but the server is turned off with probability p as the system becomes empty. The so-called <T, p>-policy queue is characterized by the fact that if at least one customer is present in the queue after T time units have elapsed since the end of the busy period, the server can be switched on with probability p or leaves for a vacation of the same length T with probability (1-p). The main difference between the two randomized policies is that for an <N, p>-policy a decision maker selects actions randomly at completion epoch when there are no customers in the system, whereas for a <T, p>-policy a decision maker selects actions randomly at the beginning epoch of the service when at least one customer appears.
For those two queueing systems, we develop various system performances by the convex combination property and the renewal reward theorem. The expected cost functions are established to determine the joint optimal threshold values of (N, p) and (T, p), respectively. Then we obtain the explicit closed-form of the joint optimal solutions for those two policies. Because of sensitivity investigation on the queueing system with critical input parameters may provide some answers for the system analyst. A sensitivity analysis is provided to discuss how the system performances can be affected by the input parameters (or cost parameters) in the investigated queueing service model. For illustration purpose, numerical results are also presented.
Furthermore, it is rather difficult to derive the steady-sate probability explicitly for those two queueing systems. The maximum entropy approach has been widely applied in queueing theory to analyze different queueing models. Here, we use this approach to approximate the steady-state probability distributions of the queue length for those two queues. Then the approximate expected waiting time in the queue can be obtained by the maximum entropy solutions. Subsequently, we perform comparative analysis between the approximate results with established exact results for various service time, repair time and startup time distributions with specific parameter values. Our numerical investigations showed that the relative error percentages of the approximate method are quite small. Based on the numerical results, one can demonstrate that the maximum entropy approach is accurate enough and provides a helpful method for analyzing more complicated queueing models.
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