Laplace-transforms and fast-repair approximations for multipleavailability and its generalizations

Stochastic models, describing multiple availability, are analyzed for a system with periods of operation and repair that form an alternating renewal process with exponential times to failure and repair. For the simplest case multiple availability is defined as the probability that the system is available in the interval [0, t) at each moment of demand. Instants of demand form a homogeneous Poisson process. This setting is generalized to considering a possibility of one or more points of unavailability in [0, t) as well as time redundancy. The corresponding integral equations are derived and solved (wherever possible) via the Laplace transform. A fast repair approach is also applied to each case under consideration and simple approximate relations for multiple availability are obtained. The fast repair approximation makes it possible to derive approximate solutions for problems that cannot be solved by the first approach. The accuracies of the fast repair approximations are analyzed. Generalizations to arbitrary failure and repair distributions are also discussed     

Behaviour of a two-unit standby redundant system with imperfect switching and delayed repair

A 2-unit standby redundant system with imperfect switchover is considered. A repair facility is assumed to be available only at a fixed proportion of time. Failure-time distributions of units are exponential whereas repair-time distributions, distribution of time for the repairman to become available are general. The system has been investigated in detail by the help of semi-Markov process and closed form results are obtained for mean-time-to-system-failure, steady-state availability, expected number of occurrences of a state, expected profit and second moment of time in up state. Behaviour of several parameters has also been studied and several earlier models are included as particular cases.     

The Bayesian Assessment of System Availability: Advanced Applications and Techniques

The statistical assessment of system availability within a Bayesian framework is extended here to consider four additional areas: 1) applications to more complicated availability definitions than considered in an earlier paper; 2) applications to measures of performance other than availability; 3) the use of prior distributions on the failure and repair rates more general than the gamma form; and 4) the use of a model involving nonexponential repair intervals. Under category 1) we consider applications involving: a) a demand for system performance which occurs at a random time within an initial interval; b) a situation involving demands upon a system repeated at intervals; and c) the availability statistics of a redundant configuration. Under category 2) we develop: a) the moment and distributional statistics for the accumulated repair time in a given real-time interval when the rate parameters are uncertain; and b) additional measures of performance in a repeated demand situation. Under category 3) we treat fully the case of a prior distribution composed of a linear combination of gamma distributions; this allows multimodal priors. Under category 4) we treat the case of gamma-distributed repair intervals where both the location and shape parameters are uncertain. The results obtained under all four categories can be expressed in terms of the basic measures for the Euler distribution developed in an earlier paper [1].     

An approximation analysis for the availability of a parallel redundant system with general distributions

This paper presents an approximation method for deriving the availability of a parallel redundant system with general distributions. The system discussed is composed of two identical units. A single service facility is available for the performance of preventive maintenance(PM) and repair. The failure times, repair times and PM times are assumed to be arbitrarily distributed. The presented method formulates the problem of the availability analysis of a parallel redundant system as a semi-Markov process which represents the state transitions of one specified unit in the system. This method derives the availability easily and accurately. Further, when all the distributions are exponential, the availability obtained by this method is exact.     

Interval-availability distribution of 2-state systems withexponential failures and phase-type repairs

Interval availability is a dependability measure defined as the fraction of time during which a system is in operation over a finite observation period. Usually, for computing systems, the models used to evaluate interval availability distribution are Markov models. Numerous papers using these models have been published, and only complex numerical methods have been proposed as solutions to this problem even in simple cases such as the 2-state Markov model. This paper proposes a new way to compute this distribution when the model is a 2-state semi-Markov process in which the holding times have an exponential distribution for the operational state and a phase-type distribution for the nonoperational one. The main contribution of this paper is to define a new algorithm to compute the interval availability distribution for systems having only one operational state. The computational complexity depends weakly on the number of states of the system, and sometimes it can deal also with infinite state spaces. Moreover, simple closed expressions of this distribution are shown when repair periods are of the Erlang type with eventually absorbing states     

A time-dependent availability measure for a repairable systemsubject to catastrophic failures

Attention is given to a repairable system which is subject to catastrophic failures while it is in operation or under repair. The topic of investigation is the time-dependent point availability measure of the system, which is obtained in terms of the repair and failure time distributions and in the form of convolution integrals that are readily evaluated by means of known computational algorithms. Also derived are expressions for this measure that are useful for estimating the parameters of the model according to the data available. An application of the model is considered     

Cost-benefit analysis of a system with intermittent repair and inspection under abnormal weather

This paper studies a repairable system with intermittent repair. Weather under which the system works changes randomly (in time) from normal to abnormal weather and vice-versa. By intermittent repair, we mean that the repair facility is not available instantaneously but takes random time to be available. The system operating under abnormal weather is sent for inspection randomly with Poisson process. Failure rates of the system and rates of change of weather are constant while repair times, inspection time and inter-inspection time are arbitrarily distributed. The system is analysed by using regenerative point technique to obtain various economic measures such as mean time to system failure, steady state availability, probability that the repairman is busy, expected number of visits by repairman and expected profit earned by the system.     

Analysis of 7 Models for the 2-Dissimilar-Unit, Warm Standby, Redundant System

The paper is in 2 parts. In all models the failure rates are constant, but repair rates need not be constant. The method of supplementary variables is used for solving the models. Part I considers the effect of priorities on reliability and availability for 4 basic models; 1) priority in both repair and operation; 2) priority in repair; 3) priority in operation; 4) no priority. Models 1 and 2 treat 2 repair disciplines: a) preemptive-repeat, b) preemptive-resume. We obtain 1) Laplace transforms of availability and reliability and 2) explicit expressions for steady state availability and for mean time to system failure. The effect of priority assignment to maximize steady state availability is discussed. Part II considers the effect of having different repair rates, depending on whether the failure was from standby or from operation. We obtain 1) Laplace transforms of availability and reliability and 2) explicit expressions for mean time to system failure.     

Reliability of imperfect switching of cold stanby systems with multiple non-critical and critical errors

Reliability and availability analysis of having k active, N cold standby units with repair facilities and multiple non-critical and critical errors while the switching mechanism subjected to failure is presented. Failed (active and/or by any one of the multiple non-critical errors) units will be repaired at a constant repair rate. The system is in a failed state when any one of the multiple critical errors has occurred, (N + 1) units have failed or there is a failure of switching mechanism. A failed system will be repaired with repair times arbitrarily distributed. The expressions for reliability and steady-state availability are given.     

Analysis of a repairable redundant system with delayed replacement

The paper deals with a redundant system with two types of spare units—a warm standby unit for instantaneous replacement at the time of failure of the active unit and a cold standby (stock) unit which can be replaced after a random amount of time. Failure time distributions of operative and standby units are exponential whereas all repair times follow arbitrary distributions. The system has been studied in detail by applying the results from the theory of semi-Markov process and mean-time-to-system-failure, steady-state availability, expected number of visits to a state, second moment of time in an up-state and expected profit of the system have been obtained.     

Reliability of multi-state systems with two failure-modes

Systems with two failure modes (STFM) consist of devices, which can fail in either of two modes. For example, switching systems can not only fail to close when commanded to close but can also fail to open when commanded to open. This paper considers systems consisting of different elements characterized by nominal performance level in each mode. Such systems are multi-state because they have multiple performance levels in both modes, depending on the combination of elements available at the moment. The system availability is defined as the probability of satisfaction of given constraints imposed on system performance in both modes. The paper suggests reliability measures for multi-state systems with 2 failure modes, and presents a procedure for evaluating these measures. The procedure is based on the use of a universal moment generating function (UMGF). It allows one to estimate availability and s-expected performance of complex systems with series-parallel and bridge topology. Basic UMGF technique operators are developed for two types of systems, based on transmitting-capacity and on operation-time.     

A note on the statistical characteristics of a two-unit system

Expressions for the Laplace transforms of reliability and availability functions are obtained for a two-unit system, with different repair times for the units which have failed from online and standby states, and a dead time value for the repair facility by the use of regeneration point technique. The system consists of two-units with one repair facility. The repair facility is not available for a random time immediately after each repair completion. From the Laplace transforms of reliability and availability functions the steady state availability, reliability and mean time to system failure can be obtained.     

Cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair

This paper deals with the cost analysis of a two dissimilar-unit cold standby redundant system subject to inspection and two types of repair where each unit of the system has two modes, normal and failed. It is assumed that the failure, repair, replacement and inspection times are stochastically independent random variables each having an arbitrary distribution. The cold standby unit replaces the failed operative unit after a random amount of time. An inspection is required to decide whether it needs type I (minor repair) or type 2 (major repair). In this system the repairman is not always available with the system, but is called whenever the operative unit fails. The system is analysed by the semi-Markov process technique. Some reliability measures of interest to system designers as well as operations managers have been obtained. Pointwise availability, steady-state availability, busy period by a server and the expected cost per unit time of the system are obtained. Certain important results have been derived as particular cases.     

Availability evaluation of flow networks with varyingthroughput-demand and deferred repair

The author presents some simple formulas for availability of flow networks with several levels of performance and variable throughput demands. The system comprises highly available independent components. Two main types of component failure are considered: (1) those where the corresponding repair can be deferred to a more convenient time, and (2) all other failures. This more convenient time is set to a fixed date in the present model. This model is relevant to gas/oil production and transportation systems. Such availability concepts as throughput, demand, and onstream are discussed; the throughput availability is usually the most important. The computation of these measures can be time consuming; therefore, efficient algorithms and formulas are vital. The calculations are fast, so that sensitivity of various factors can be studied easily. The calculations show which components are contributing the most to unavailability     

Conditional Availability of Intermittently-Used Systems in Non-Markov Environment

This paper discusses the steady-state conditional availability of intermittently-used systems during the periods of demand. All the distributions governing system and demand status are arbitrary. The history dependence of the system and demand behavior is tackled by introducing history and cumulative-history functions. The system is assumed to fail only in use. Two processing disciplines regarding an interrupted demand (due to a system failure) are treated: fail-resume and failrepeat. The steady-state conditional availability under the fail-resume discipline is MTBF/(MTBF + MTTR), but not under the fail-repeat discipline. Therefore, care must be exercised not to misuse the formula.     

Cost-benefit analysis of a one-server two-unit cold standby system with repair and preventive maintenance

This paper deals with the cost-benefit analysis of a one-server two-identical-unit cold standby system with repair and preventive maintenance (PM). The PM is of the type where the operating unit is taken up for PM whenever the other unit is available for operation. Initially, one unit is placed in operation and the other unit is kept as a cold standby. When the operating unit fails while the other unit is under service (repair or PM), the system breaks down. The busy period of the server in a time interval (O, t] is divided into time spent for repair and time spent for PM. By identifying regenerative epochs, suitable expressions for the expected values of these times are obtained. The pointwise availability is also derived. With the assumptions that a revenue is earned linearly with up-time, and repair and PM costs are incurred linearly with repair and PM times, respectively, the net expected revenue for a period (O, t] is derived. A particular case where the time to failure of the operating unit is 2-Erlang and the times for repair and PM are exponential has been analysed.     

Reliability and availability analysis of a k-out-of-N:G redundant system with repair in the presence of chance of multiple critical errors

The paper presents a reliability and availability analysis of a k-out-of-N:G redundant system with repair facilities in the presence of chance of multiple critical errors. The system is in a failed state when N−k+1 units have failed or any one of the multiple critical errors has occurred. Failed units and failed system will be repaired with constant repair rate to state with N−k+1 failed units. Laplace transforms of the state probabilities, the reliability and the availability of the system are derived. The system steady-state availability is also given.     

A complex two-unit parallel system

Reliability and availability functions are obtained for a complex two-unit parallel system by the use of regeneration point technique. The system consists of two dissimilar units. There is only one repair facility. The repair facility is not available for a random time immediately after each repair completion.     

Some applications of semi-regenerative processes to two-unit warm standby system

We deal with a two-unit warm standby system with a single repair facility; a failure of an operating unit can be detected immediately but a failure of a warm standby unit can not be observed until the system is inspected. According to whether the operation of the system is stopped or not during each inspection, we consider two models. We describe the stochastic behavior of the system as a semi-regenerative process, and the pointwise availability and the steady state availability of the system are derived applying the limit theorem of semi-regenerative processes. Further, we shall discuss the optimum interinspection time maximizing the steady state availability for Model 1.     

Output and delay process analysis for slotted CDMA wirelesscommunication networks with integrated voice/data transmission

This paper presents the output and delay process analysis of integrated voice/data slotted code division multiple access (CDMA) network systems with random access protocol for packet radio communications. The system model consists of a finite number of users, and each user can be a source of both voice traffic and data traffic. The allocation of codes to voice calls is given priority over that to data packets, while an admission control, which restricts the maximum number of codes available to voice sources, is considered for voice traffic so as not to monopolize the resource. Such codes allocated exclusively to voice calls are called voice codes. In addition, the system monitoring can distinguish between silent and talkspurt periods of voice sources, so that users with data packets can use the voice codes for transmission if the voice sources are silent. A discrete-time Markov process is used to model the system operation, and an exact analysis is presented to derive the moment generating functions of the probability distributions for packet departures of both voice and data traffic and for the data packet delay. For some cases with different numbers of voice codes, numerical results display the correlation coefficient of the voice and data packet departures and the coefficient of variation of the data packet delay as well as average performance measures, such as the throughput, the average delay of data packets, and the average blocking probability of voice calls