An M^(?₁,?₂)/G/1(Repairable) Queueing System With Uninterrupted Single Vacation Under The Control Of D-Policy Based On Server's Workload

In this thesis,we propose to establish an M(?1,?2)/G/1(repairable)queueing model with uninterrupted single vacation and variable arrival rate under the control of D-policy based on server's workload,and analyze the transient and equilibrium properties of the queueing model.It is divided into two chapters as follows:1)In the first chapter of this dissertation,we propose to establish an M(?1,?2)/G/1 queue-ing model with uninterrupted single vacation and variable arrival rate under the control of D-policy based on server's workload,in which the customers's arrival rate is variable,and when the server is transferred on vacation,the server starts service immediately if the total server's workload required by all the waiting customers in the system is not less than a given server's workload threshold D(?0).First of all,Employing the renewal process theory,the total probability decomposition technique and the Laplace transform(L transform)tool,we discuss the transient solution of queue length distribution under the any initial state,and obtain the expressions of the Laplace transform of the transient solution of the queue-length distribution with respect to time Furthermore,using the L' Hospital's rule and some algebraic operations,some important queueing indicators such as the recursive formulas of the steady queue-length distribution,the probability generating function of the steady-state queue length distribution and the expression of the expected queue length,are presented.Secondly,in the steady state,we discuss the changes of the probability p0 that the system is idle and the expected additional queue size E[Ld]with different values of some system's parameters.Applying the steady-state queue-length distribution {pj,j=0,1,2,…},we investigate the optimal design of the system capacity and illustrate the important role of the steady-state queue-length distribution in the optimal design of system capacity.Finally,based on a given cost structure model,the explicit expression of the long-run expected cost per unit time of the system is derived by the renewal reward theory.To demonstrate the model's application,we further consider a practical situa-tion related to a certain manufacturer processing products,numerical examples are provided to discuss the optimal control policy D*for economizing the system cost as well as the combined control policy(D*,T*)when the vacation time is fixed duration T.2)In the second chapter of this dissertation,based on the decline in the performance of the service equipments in the system,we introduce“the service station can be broken down and repairable,and the repair equipment can be broken down and replaceable”into the M(?1,?2)/G/1 queueing system studied in the first chapter,and propose to establish a variable arrival rate M(?1,?2)/G/1 repairable queueing model with uninterrupted single vacation and a replaceable repair facility under the control of D-policy based on server's workload.Firstly,following the research route in Chapter 1 above,both the transient queue-length distribution and the the steady queue-length distribution are studied.Some important queueing indicators are presented.Secondly,a series of reliability indexes of the service station and repair facility such as the unavailability and the failure frequency are discussed.The changes of the steady state unavailability and the steady state failure frequency of the service station and repair facility with different values of some system's parameters are studied by numerical examples.At the end of this chapter,based on a given cost structure model,and applying the renewal reward theory,we numerically investigate the optimal control policy for minimizing the long-run expected cost per unit time of the system under considering the case of“the service station can be broken down and repairable,and the repair equipment can be broken down and replaceable”.