Research On M/G/1 (Repairable) Queueing System With Min(N,D,V)-Policy And Uninterrupted Single Vacation

This dissertation studies the transient and equilibrium properties of the M/G/1(repairable)queueing system with Min(N,D,V)-policy and uninterrupted vacation.It is divided into two chapters as follows.1)In the first chapter of this dissertation,we introduce the vacation rule of Uninterrupted single vacation "into the M/G/1 queueing system with the two-dimensional control strategy Min(N,D),and establish an M/G/1 queueing model with Min(N,D,V)-policy and unin-terrupted single vacation,in which when the server comes back to the system from vacation,the server starts service immediately if the number of customers in the system is less than a given positive integer threshold N or the total server's workload required by all the waiting cus-tomers in the system is not less than a given server's workload threshold D,whichever occurs first.Firstly,Employing the total probability decomposition formula in Probability Theory,L transform and probability generating function,the distribution of the queue length at any time t,i.e.the transient solution of the queue length is discussed.The expressions of the Laplace transform of the transient solution about time t are obtained.Secondly,the steady probability distribution of the system queue-length is investigated.And then using the L'Hospital's rule and through some algebraic operations,the analytical expression of the probability generat-ing function of the steady queue-length distribution and the explicit expression of the expected queue size are presented.Moreover,the stochastic decomposition structure of the steady queue size,and the analytical expressions of the additional queue-length distribution caused by the Min(N,D,V)-policy and the mechanism of single vacation without interruption,are shown.At the same time,some special cases,such as N??,or D??,or P {V=0}=1,are discussed.In the end,based on a given cost structure,we study the cost problem of the long-run expected cost per unit by renewal reward theorem.With the help of Matlab software,the optimal value(N*,D*)of two-dimensional decision variable which makes the cost objective function minimum is determined in the form of a numerical example.2)In practice,due to the wear and tear of the service station itself,it is inevitable that the service station will fail.Therefore,in the second chapter of this dissertation,we introduce the "service station can be broken down and repairable" into the M/G/1 queueing system studied in the first chapter,and propose to establish an M/G/1 repairable queueing model with Min(N,D,V)-policy and uninterrupted single vacation.Firstly,we regard the customer's generalized service time in this system as the customer's service time in the system studied in the first chapter above,and discuss both the transient queue-length distribution and the steady queue-length distribution.Some corresponding queueing indicators are presented.Secondly,some reliability indexes,such as the transient unavailability and the steady unavailability of the service station,and the expected number of the service station failures during(0,t],are discussed in detail.The explicit expressions of the steady unavailability and the steady failure frequency of the service station are obtained.