Tylyseveral Kinds Of Repairable Systems With Vacation And Their Reliability Analysis

Repairable systems with vacation have been received increasing attention during nearly two decades due to their wide applications in various fields, such as the replacement of service station and optimal control of server in repairable queue systems, the management of labourer and security maintenance of production part in manufacturing enterprises, the acquisition of best benefit in machine repair systems, etc. This dissertation is devoted to studying some repairable systems with vacation and their reliability analysis, and concerns the following two parts:single-unit system with an unreliable repair facility and single vacation and muitiple-unit systems with an unreliable repair facility (or repairman) and multiple adaptive vacations. The thesis is organized as follows.Chapter1is preface. The definitions, classifications and recent developments of repairable system and repairable system with vacation are introduced, and the main results and innovative points are also given.Chapter2treats a single-unit repairable system with a replaceable repair facility and a repairman who takes single vacation, where we extend the exponential failure distribution for the repair facility in most of the existing literatures to more general Erlangian failure distribution, and the other distributions are general continuous. By the supplementary variable method and the theory of differential equation, the Laplace transforms for state probabilities are successfully obtained. On this basis, main reliability quantities of the system and the repair facility, such as the mean renewal interval time, availability, failure frequency of the system and the unavailability and replacement frequency for the repair facility etc. are devied. Special cases show that the results presented by this paper are general than those in existing reliability literatures. Finally, we analyze the effects of facility replacement and single vacation on system performances, optimal vacation rate and optimal profit.Chapter3relates to a repairable n-unit series system with a replaceable repair facility and multiple adaptive vacations. After repair completion, the repairman takes a random maximum number of vacations.i.e.the maximum vacation number the repairman may take is a random variable with a general probability distribution. Under the vacation policy, the possible states of repairman include idle state, busy state and vacation state. However, the planned vacation process may be interrupted. If any failure occurs in the system the repairman will start a new busy period at the end of a vacation. Tian Naishuo named this kind of vacation policy for "multiple adaptive vacation policy", which is a generalization of multiple vacations, single vacation, and variant vacations, etc. With the help of renewal process theory and total probability decomposition method, some important reliability indices of the system are obtained, such as the system availability, mean failure number during (0, t], mean renewal time, and the steady-state probability that the system is waiting for repair etc. Also, under multiple adaptive vacation policy, the repairman's idle period, actual vacation number, vacation period and generalized busy period etc. are discussed. By the results obtained, we prove that the n-unit repairable series system with a replaceable repair facilty and delay repair is a special vacation system. Finally, we compare the reliability indices among three repairable series systems and analyze the affluences of facility replacement and vacation on system performances.In Chapter4we will continue to discuss the n-unit series system with a replaceable repair facility and multiple adaptive vacations presented by Chapter3. The main purpose of this chapter is to present a reliability analysis of repair facility in this system by using renewal process theory and total probability decomposition. For this purpose, we first define generalized repair time of failed unit and idle perioed and generalized busy period of repair facility, and obtain mean idle perioed, mean generalized busy period and the probability that the facility is during its generalized busy period. Next, we get the unavailability and mean replacement number in (0,t]. It is showed that the unavailability at time t and mean replacement number in (0, t] are two convolutions between the counterpart in classical single-unit system and the probability that the facility is during its generalized busy period at time t, respectively. Furthermore, in steady state, the unavailability and replacement frequency are two products between the counterpart in classical single-unit system and the probability that the facility is during its generalized busy period, respectively. Finally, two asymptotic formulas for calculating the expected failure number and expected failure time during (0, t], conveniently, are also derived.Chapter5considers a Poisson shock model for two-unit cold standby repairable systems with repairman breakdowns and general startup times under multiple adaptive vacation policy. The main purpose of this chapter is to present a reiliability performance analysis of this model. For this purpose, we first give the definitions of states and state probabilities and construct a vector Markov process by using supplementary variable method. A set of differential equations with boundary conditions and regular condition was established to describe the steady-state behavior of the system. Next, by means of the theory of differential equation and the Laplace transform, we successfully deduce the closed form expressions of state probabilities. Based on these expressions, we present the stationary reliability indices of the system as well as some performance indices of the repairman. We give two steady-state decomposition formulas for the unavailability and recovery frequency of the repairman. Additionally, we investigate mean time to first failure of system and present an analysis of several special models. Finally, some numerical examples show the influences of shock, break, startup and vacation on some crucial performance characteristics of the system.In chapter6, we study a basic model for two-unit repairable systems with repairman breakdowns and general startup times under multiple adaptive vacation policy. The basic model includes four special cases:two-unit cold standby system, two-unit warm standby system, two-unit parallel system and (n-l)/n/(G) voting system. Under the assumption that the failure distributions are exponential and the other distributions are general, the basic model with vacation is analyzed using the supplementary variable, the theory of differential equation and the Laplace transform. We obtain steady-state reliability measures of system, such as availability, failure frequency, mean renewal time and the probability that the failed system is waiting for repair. We establish a vector Markov process with finite absorbing states to obtain mean vacation period. On this basis, we derive idle period, sartup period, generalized busy period, unavailability and recovery frequency of the repairman etc. Also, we give some special modes and analyze the effects of system parameters on main reliability indices. Finally, as an application of the model, the reliability measures and profits for three special models are numerically compared.