Time to Failure and Availability of Paralleled Systems with Repair

This paper discusses reliability properties of some simple paralleled or redundant systems, where repair is possible in case of failure. We are assuming here that a ``failure' may always be instantly identified, and the appropriate steps taken. In certain problems such an assumption is not warranted. The ``systems' discussed are composed of two identical ``subsystems,' e.g., computers, or radars, and the system is considered to be in a state of failure when, and only when, both subsystems are simultaneously in such a state. Such system design strategies have been proposed for various applications, but have received little analysis. Two measures of reliability are discussed: 1) the time to system failure, measured from an instant at which both subsystems are operative, and 2) the long-run availability of the system, where the latter means the average fraction of the time during which the system is able to perform its function. Analysis is based on the assumption of ``random' (Poisson-like) failure for the subsystems (for theoretical justification see Drenick [2]), and independent but otherwise arbitrarily distributed repair times. It is of some interest that several of the important operational measures deduced, depend in detail upon the form of the distribution of repair times, as it is summarized in its Laplace transform, and not simply upon certain simple averages or moments of repair time.     

Reliability Analysis of Systems Comprised of Units with Arbitrary Repair-Time Distributions

This work demonstrates the feasibility of reliability modeling of systems with repair capability using a semi-Markov process. A two-unit system with exponential failure times but general repair times is studied. Formulas for state-transition probabilities, waiting-time distribution functions, and mean time in each state are developed. These quantities are expressed in terms of the Laplace transform of repair time distribution functions. Once these quantities are known, mean time to system failure and system availability, as well as other system parameters, can be found using matrix manipulations. In addition, time-dependent results may be obtained. A numerical example varying the parameter in a repair-time law is presented. The formulas developed can be extended to larger systems with repair capability for only one unit at a time and exponential failure times.     

Stochastic behaviour of a two-unit (dissimilar) hot standby system with three modes

A hot standby system composed of two non-identical units is analysed under the assumption that each unit works in three possible modes—normal, partial failure and total failure. For each unit the failure time distribution is negative exponential and the repair time distribution is arbitrary. Breakdown of the system occurs when both the units are in total failure mode. There is only one repair service and when both the units are in the same mode, priority is given to the first unit in the matter of operation as well as repair. Several reliability characteristics of interest to system designers and operations managers have been evaluated.     

A two unit priority redundant system with three modes and correlated failures and repairs

This paper considers a two dissimilar units priority redundant system with three modes. One of the units has a priority operative mode and the other has a priority repair mode. Assuming that the joint distribution of failure and repair times is exponentially bivariate, some reliability characteristics useful to system managers have been obtained. Results for a system with two similar units are obtained as a particular case.     

On Failure-Time Distributions for Systems of Dissimilar Units

Certain reliability problems of systems of dissimilar units with repair are described. The mean time to system failure is based on the relation of mean first passage times between states of the system. The failure-time distribution is obtained from an integral equation of the renewal type. The two approaches can be also applied to a system of dissimilar units under an overload. Finally, it is shown that these results include many earlier results as special cases.     

Human error analysis of a standby redundant system with arbitrarily distributed repair times

This paper presents human error analysis of a (two units working and one on standby) system with arbitrarily distributed repair times. The supplementary vairables method is used to develop the system availability expressions. A general formula for the system steady-state availability is developed when the failed system repair times are gamma distributed. Time-dependent availability, system reliability with repair, mean time to failure and variance of time to failure formulae are developed for some particular cases. Selective plots are shown to demonstrate the impact of critical human error on system availability and reliability.     

Reliability measures for two-unit systems with a dependent structure for failure and repair times

This paper deals with reliability measures for two-unit systems with a repair facility assuming that the failure times and the repair time follow a trivariate exponential distribution of Marshall and Olkin, J. Amer. Statist. Assoc., 1967, 62, 30–44. The case where the system down-time is observed, is also discussed. The system reliability and system mean-time before failure are evaluated for standby and parallel systems. When the down-time is observed the system availability, steady-state availability and the system mean down-time are evaluated for standby, parallel and series systems.     

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.     

Analyses of Two Different 1-Server Systems

In this paper, the availability and the reliability of two 1-server systems with redundancy have been obtained. System 1 consists of n subsystems in series; each subsystem consists of two redundant i.i.d. components in `parallel' (cold standby) and one server. The times to failure of the components are exponentially distributed; their repair time distributions are arbitrary and different. System 2 consists of n dissimilar units and one server. The times to failure of the units are arbitrarily distributed; the repair rates are constant but all different. Explicit expressions for the Laplace transform of the mean down-time of the system in (0, t) and for the mean time to system failure have been obtained. A few particular cases are discussed.     

Failure Time for a Redundant Repairable System of Two Dissimilar Elements

The distribution of time to failure for a system consisting of two dissimilar elements or subsystems operating redundantly and susceptible to repair is discussed. It is assumed that the times to failure for the two system elements are independent random variables from possibly different exponential distributions, and that the repair times peculiar to each element are independently distributed in an arbitrary fashion. For this basic model a derivation is given of the Laplace-Stieltjes transform of the distribution function of time to system failure, i.e, the time until both elements are simultaneously down for repair, measured from an instant at which both are operating. An explicit formula is given for the mean or expected time to system failure, a natural approximation to the latter is exhibited, and numerical comparisons indicate the quality of this approximation for various repair time distributions. In a second model the possibility of system failures due to overloading the remaining element after a single element failure is explicitly recognized. The assumptions made for the basic model are augmented by a stochastic process describing the random occurrence of overloads. Numerical examples are given. Finally, it is shown how the above models may be easily modified to account for delays in initiating repairs resulting from only occasional system surveillance, and to account for random catastrophic failures.     

Analysis of a two-dissimilar unit cold standby redundant system subject to inspection and random change in units

This paper deals with the stochastic behaviour of a two-dissimilar unit cold standby redundant system with random change in units. In this system each unit works in two different modes—normal and total failure. It is assumed that the failure, repair, post repair, interchange of units and inspection times are stochastically independent random variables, each having an arbitrary distribution. 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. Explicit expressions for the Laplace-Stieltjes transforms of the distribution function of the first passage time, mean time to system failure are obtained. Certain important results have been derived as particular cases.     

A Method for Predicting System Downtime

A system whose components, upon failure, are repaired or replaced is considered. Only two system states, the ``operating' state and the ``failed' state, are distinguished. The system is by defined a reliability network and by the failure rate and repair rate of each component. The time to failure and the time to repair of the components are assumed to be exponentially distributed. A criterion of system worth is the random variable ``downtime,' denoted by D(t), which is defined as the time the system is down during the time interval (0, t). The following questions are answered: 1) What is the distribution function of D(t)? 2) What are the mean and the variance of D(t)? 3) What is the asymptotic behavior of D(t) for large values of t? 4) How can one make approximate probability statements about D(t)? It is shown that the beta distribution is a suitable approximation for the conditional distribution of D(t)/t, given that at least one failure has occurred, and that for t greater than 20 mean failure times the distribution of D(t) is practically normal.     

Analysis of a two unit standby system with partial failure and two types of repairs

A two dissimiliar unit standby system is analysed. The priority unit can either be in normal or partial operative mode. When the unit fails from the partial mode, it undergoes minor repair and the unit becomes operative with different failure rate. If this unit fails again, it goes to major repair after which it works as good as new. The standby unit while in use is either operative or failed. This non priority unit fails without passing through the partial failure mode and undergoes only one type of repair with different repair time distribution. Failure and repair time distributions are negative exponential and general respectively. Regenerative technique in MRP is applied to obtain several reliability characteristics of interest to system designers.     

Two models for two-dissimilar-unit standby redundant system with three types of repair facilities and perfect or imperfect switch

This paper deals with two models for two-dissimilar-unit cold standby redundant systems under the assumption that each unit works in three different modes—normal, partial failure and total failure. In model I, the switch is perfect but it is imperfect in model II. The failure and repair times are assumed to have different arbitrary distributions.Explicit expressions for the mean time to system failure and the availability analysis are obtained in each model. A computer program is shown for comparison between the two models.     

Reliability analysis of a M1/M/G1, G21 Queueing system with a repairable service station

A queueing situation often encountered in practice is that in which the service station may fail and can be repaired. This paper considers a priority queueing system with a repairable service station, where two types of customers arrive in batches according to two homogeneous independent Poisson processes, their service time distributions are general and only one customer can be served at a time. Assume that the service station has a constant failure rate and arbitrary repair time distribution. The focus of this paper is on the reliability. Using the supplementary variable method, we obtain the Laplace transform of the pointwise availability, the pointwise failure frequency and the reliability of the service station. Furthermore, we obtain the mean steady-state availability, steady-state failure frequency and mean time to first failure of the service station.     

Stochastic analysis of systems with primary and secondary failures

This paper presents the stochastic analysis of repairable systems involving primary as well as secondary failures. To this end, two models are considered. The first model represents a system with two identical warm standbys. The failure rates of units and the system are constant and independent while the repair times are arbitrarily distributed. The second system modeled consists of three repairable regions. The system operates normally if all three regions are operating, otherwise it operates at a derated level unless all three regions fail. The failure rates and repair times of the regions are constant and independent. The first model is analyzed using the supplemental variable technique while the second model is analyzed using the regenerative point technique in the Markov renewal process. Various expressions including system availability, system reliability and mean time to system failure are obtained.     

On a two dissimilar unit cold standby system with switchover time and proper initialization of connect switching

This paper investigates the stochastic behaviour of a two dissimilar unit cold standby system with connect switching. The connect switch keeps this system in good connection with other systems. The standby unit takes random switchover time to assume the operative state when the operative unit fails. The failure times of the units and connect switch and the repair times of the units are assumed to have different arbitrary distributions. The mean waiting times in the states of the system and expression for the steady state availability of the system are obtained. The results obtained by Kumar and Lal and Laprie [1, 2] are derived from the present results as special cases.     

Stochastic analysis of a 2-unit standby system with two failure modes and slow switch

This paper considers the cost-benefit analysis of a one-server two unit system subject to two different failure modes and slow switch. The failure rates of the units are constant. The repair times and the switchover time are assumed to be arbitrarily distributed. The server repairs the units and puts the standby unit into operation. Detailed analysis of the system is done by using regenerating point technique and results are obtained for mean time to system failure, steady state availability, busy period of a repair man, expected number of visits by the repair man and expected profit earned by the system.     

Reliability of Some Redundant Systems with Repair

A mathematical model is established for the reliability of modularly redundant systems with repair. The model allows different hazard rates for active units and for standby units. The hazard rates are assumed to be constant. The cases of constant repair rate and constant repair time for a two unit system are evaluated using the reliability and mean time between failure. The approach is then extended to systems with more than two units. A system parameter, relating to certain types of sensing, switching, and/or recovery has a very significant impact on system reliability for modularly redundant systems with repair.     

Analysis of a two-unit warm standby system subject to degradation

This paper deals with the reliability analysis and the mean time to system recovery of a single server, two-unit (priority and ordinary) warm standby subject to degradation. Initially the priority unit is operative and the ordinary unit is kept as a warm standby. The priority unit passes through three different operative stages (excellent, good and satisfactory) before it fails. The priority unit enters into the total failure mode only from the satisfactory stage, and after repair it enters into the normal mode with any of the ‘excellent’, ‘good’ and ‘satisfactory’ stages with different probabilities. The failure, repair and degradation time distributions are assumed to be general and arbitrary. The system is observed at suitable regenerative epochs in order to carry out the expected first passage time analysis. Moreover, three special cases have been considered. The results of Gupta [Int. J. Systems Sci.22 (11) 2329–2338 (1991)] are derived from the present results as a special case. A computer program for calculating the mean time to system failure and the mean time to system recovery is made.