Analysis Of Complex Stochastic Systerms Based On The Phase Type Process

Motivated by some practical problems arising from the real world situa-tion, some complex stochastic models in operations research are studied in this thesis. The research contents of our thesis relate to several different fields of the stochastic operations research, including reliability models, queueing theory and inventory theory. During the current research process, we try to combine some contents of the above three areas, and develop some new stochastic models with strong application background in their blend. Specifically, the following four new models are considered in our thesis:(1) A PH deteriorating repairable system with phase type compo-nent procurement lead-time. In this Chapter, combining PH distribution with the geometric process, and through constructing the infinitesimal generator matrix of a high dimensional Markov process, the stationary probability distri-bution vectors of the system state and their numerical solutions are obtained. Meanwhile, according to the above research results, several important reliabil-ity indices of the system are also reported. Furthermore, an ordering policy and a replacement policy based on the number of failures of the component are considered in detail. Finally, employing the standard results in renewal reward process, the explicit expression for the long-run average cost rate of the system is derived, and a numerical example for determining the optimal (N—1,N) policy is also given.(2) An M/PH(M/PH)/1/K repairable queue with new service ma-chine procurement lead time. The maintenance policy (N—1, N) based on the number of failures of the service machine is introduced into the stochastic service system. Assuming that a failed service machine after repair will not be "as good as new", and the new spare service machine for replacement is only available by an order. More specifically, we suppose that the procurement lead time for delivering the new spare service machine follows a PH distribution. Under such assumptions, we apply the matrix-analytic method to develop the queue-length distribution of the system, and then we obtain some important performance measures of this queueing system. Finally, using a key lemma, the explicit expression of the long-run average cost rate for the service machine is derived. In addition, the direct search method is also implemented to determine the optimal value of N for minimizing the average cost rate.(3)Vacation inventory system with non-persistent retrial demand. We consider a continuous review (s, S) inventory system with multiple server vacations and non-persistent retrial demands. We assume that the primary de-mands from outside the system constitute a Markovian arrival process (MAP), and the lead time for inventory replenishment follows a PH distribution. To save on operating costs, when the inventory level reaches zero, the server leaves for a vacation of a phase type distributed duration. At the end of a vacation, if the inventory is still empty, the server immediately takes another vacation. The primary demands that occur during the stock out period or during the server vacation period may leave the system forever or enter a retrial orbit with infinite capacity and repeat their demands after some random time. Similarly, if the server is on vacation or the inventory replenishment had not taken place at the time of arrival of a retrial demand, with probability1—q, it leaves the system by giving up item, and with probability q, it goes back to the orbit again. Under these assumptions the underlying level-dependent quasi birth and death (LDQBD) process is analyzed. Furthermore, we give a simple method to determine the truncation level of the LDQBD process, and based on a matrix continued fraction approach, we also develop an efficient algorithm to com-pute the joint stationary distribution of the number of demands in the orbit and the inventory level. Employing the joint stationary distribution, various performance measures along with some interesting numerical experiments have been discussed. Finally, through direct search method, we numerically find the optimal values of.s and S under a given cost structure.(4)Waiting time analysis for an M/M/c queue with cutting in line behavior and a simple approximation method for the mean value based on the IPH distribution. An M/M/c queueing system with cutting in line behavior is considered. The arriving customers are dispersed into common customers and queue jumpers. A common customer joins the queue at the end, and for reducing the waiting time in the queue, a queue jumper tries to cut in the queue and occupy a position as close to the head of the queue as possible. The cutting in line behavior can be described in two ways, namely, the percentage of customers interjecting and the tolerance probability of interjection by individual customers who are already waiting in the queue. The formulae for calculation three kinds of mean waiting time are obtained. Furthermore, in order to improve the computational efficiency, based on the infinite phase type distribution, a simple approximation method for the mean value of the waiting time is given. Finally, numerical results show that the approximation method considered here is quite accurate and very effective.