Mixing of Two Renewal Processes and Its Applications to Reliability Theory

The probabilistic behavior of repairable two-state systems is studied. It is assumed that the reliability of the system can be measured by the amount of time the system has been operative. The operating and repair states are a pair of renewal processes, a particular mixture of which describes the statistical behavior of the system. The object of this contribution is to extend the results of Muth who has earlier obtained the average value of the downtime and its variance, when one of the constituent renewal processes has its interval lengths distributed exponentially. This paper, by the repeated use of the method of regeneration points, obtains the mean and mean-square values of the uptime distribution. In addition the correlation of the uptime for different times has been derived and a proof of Takacs' theorem is provided. Since the criteria for the reliability of the system include the associated cost, it is worthwhile investigating the operating cost over any period of time for arbitrary distribution of the two states. In particular a demonstration of the calculation of the first two moments of the total cost is given.     

The number of working periods of a repairable Markov system duringa finite time interval

In reliability analysis, continuous parameter homogeneous irreducible finite Markov processes are used to model repairable systems with time-independent transition rates between individual states. The state space is then partitioned into the set of up states and the set of down states. The number of completed repair events during a finite time interval is an important (undiscounted) cost measure for such a system; it can be expressed in terms of the number of working periods during the same time interval. This paper derives a closed-form expression for the PMF of this latter quantity. The tool used is a recent result on the joint distribution of sojourn times in finite Markov processes. The MatLab implementation of the Markov model of a 2-unit parallel power transmission system is used to demonstrate the utility of the formula. The quantity examined is the number of completed repairs during a finite time interval. The method is viable in this case whereas the more usual randomization technique is not     

Reliability analysis and optimization applications of a two-unit standby redundant system with spare units

Consider a two-unit standby redundant system with two main units, one repair facility, and n spare units. When the main unit has failed and the other is under repair, a spare unit takes over the operation and if it fails, it is replaced by a new one until the repair of the failed unit is completed. The system fails when the last spare unit fails while one main unit is under repair and the other has failed. In this paper, we derive expressions for 1) the distribution function of the first time to system failure, 2) the probability that the total number of failed spare units during the time interval (0,t] is n and 3) the mean of the total number of failed spare units in (0,t] and its asymptotic behaviour. Introducing costs incurred for each failed main unit and each failed spare unit, the expected cost per unit of time of the system was also derived. Finally an optinmization problem is discussed in order to compare the expected cost of the system with both main units and spare units with that of spare units only, and particular cases are considered.     

Cost-benefit analysis of a one-server two-unit imperfect switch system with delayed repair

This paper deals with the cost-benefit analysis of a one-server two-unit system with imperfect switch where the server is summoned upon failure of an item (i.e. unit or switch). The amount of time the server takes to arrive is a random variable, distributed arbitrarily. The server leaves when there is no item waiting for repair. The repair times are arbitrarily distributed whereas all failure rates are constant. Initially one unit is switched on (switch is working at t = 0) and the other is kept as cold standby. Explicit expressions for the expected uptime in (0, t) of the system, busy period of the server due to repair of a unit and that of the switch are obtained to carry out the cost-benefit analysis.     

First uptime and downtime joint distribution of a duplication system with random switching time

This paper deals with a 2-unit cold standby redundant system with random switching time and imperfect switchover. In model I we study the system without preventive maintenance (PM), but in model II we study the same system with PM. For the two models we find: (1) the joint distribution of the first uptime and downtime of the system; (2) the sojourn time distribution of the system down as a marginal distribution of the joint distribution; (3) the mean downtime.     

Analysis of a two-unit system subject to repair and preventive maintenance with non-linear revenue and costs

The cost-benefit analysis of a one-server two-identical-unit cold standby system subject to repair and preventive maintenance (PM) with non-linear revenue and costs is presented in this paper. Initially, one unit is in operation and the other is cold standby. When a unit fails, it is taken up for repair or waits for repair if the server is busy. The latter results in a system breakdown. If a unit returns from service (repair or PM) and the other unit is operating, the operating unit is taken up for PM. The revenue obtained by operating a unit for an uninterrupted interval of time is some function of the length of the interval. Similarly, the cost of a repair or PM action is a function of the length of the repair or PM time, respectively, for that action. With the help of regeneration point technique, the expected net revenue over an interval (0,t] is obtained. It is shown that the results for the special case when the revenue and cost function are linear agree with previously obtained results.     

Reliability analysis of an intermittently used system

We consider a one unit system backed by a repair facility, the system itself being used intermittently with the use (need) and non-use (no need) period alternating in a Markov manner while the repair and life time of the unit are assumed to be independent random variables distributed quite generally. If a need arises in the time interval in which the unit is under repair or if the unit fails while it is in need, the need waits for a random period. If however the repair time is longer than the waiting time, a disappointment results and the need is taken care of by other means. The probabilistic analysis of the system is provided and suitable measures characteristic of the system are obtained. In particular explicit expressions are provided for the probability distribution of the time to the first disappointment as well as the expected value of the number of disappointments in an arbitrary time interval.Reliability Analysis of an Intermittently used System     

Probabilistic Analysis of a 2-Unit Cold-Standby System with a Single Repair Facility

The reliability and the availability characteristics of a 2-unit cold standby system with a single repair facility are analyzed under the assumption that the failure and the repair times are both generally distributed. System breakdown occurs when the operating unit fails while the other unit is undergoing repair. The system is characterized by the probability of being up or down. Integral equations corresponding to different initial conditions are set up by identifying suitable regenerative stochastic processes. The probability of the first passage to the down-state starting from specified initial conditions is obtained by the same method. An explicit expression for a Laplace Transform of the probability density function (pdf) of the downtime during an arbitrary time interval is obtained when the repair time is exponentially distributed. A general method is suggested for the calculation of the moments of the downtime when the repair time is arbitrarily distributed.     

Confidence limits for steady state availability of systems

A 100% confidence interval for the steady state availability of a system is derived when (i) both the operating time and the repair time distributions are lognormal and (ii) the operating time distribution is inverse Gaussian (IG) and the repair time distribution is lognormal.     

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.     

A note on the confidence interval for the availability ratio

If one is interested in determining the worth of equipment to perform a given task, often the primary concerns are reliability, maintainability, and availability. Availability consider both reliability and maintainability since it is a measure of the ratio of the operating time of the system to the operating time plus the downtime or time to repair. One method commonly used to establish a confidence interval for availability is to establish a confidence interval for the meantime between failure of the equipment and then determine a confidence interval for the meantime to repair the equipment. These two confidence intervals may then be utilized to obtain an “at least” type confidence interval for availability. This approach is not entirely satisfactory since points outside of both of the above mentioned intervals can give points inside the confidence interval for availability. This paper proposes a method for establishing an exact confidence interval for availability under the assumption that the time between failures and the time to repair are independent gamma and lognormal random variables respectively. The method requires computations of the distribution of , where Ut X²(q) and . In order to make the results usuable, tables of the cumulative distribution of W are included.     

Confidence limits for steady state availability of systems with lognormal operating time and inverse Gaussian repair time

A 100 p% confidence interval for the steady state availability of a system is derived, when the operating time distribution is lognormal and the repair time distribution is Inverse Gaussian (IG). It is assumed that one of the parameters of lognormal distribution and also the ratio of parameters of IG distribution are known.     

A complex two-unit system with random breakdown of repair facility

This paper deals with a two-unit warm standby system with a single repair facility. The lifetime of the functioning (online) unit has a general distribution, while the standby unit has a phase-type distribution. The repair facility is subjected to random breakdown and is restored after a random time. The statistical characteristics, such as reliability, availability and interval reliability, are provided for the model.     

Excess repair time for a repairable system

In this paper we deal with a repairable system and the topic of investigation is the total of all excess repair times—an excess repair being that repair the duration of which exceeds a given constant τ. The mean of this total excess repair time to the mean of the total repair time in (0, t), provides a useful measure of overtime repair. We also derive expressions for the distribution function of the time to the instant of the beginning of the first excess repair.     

Confidence limits for steady-state availability of systems with a mixture of exponential and gamma operating time and lognormal repair time

A 100% confidence interval for the steady-state availability of a system is derived, when the operating time distribution is a mixture of exponential and gamma and the repair time distribution is lognormal. It is assumed that one of the parameters of the operating time distribution and also one of the parameters of the repair time distribution are known.     

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].     

First uptime and disappointment time joint distribution of an intermittently used system

This paper derives the explicit expression for joint distribution of first uptime and disappointment time of an intermittently used 2-unit cold-standby redundant system. In particular we deduce the first disappointment time distributions as marginal distributions of the joint ones. At the end some special cases are derived.     

Approximation Formulas for Reliability with Repair

A system consists of n-identical parallel subsystems, each having an exponential distribution of times to failure and an exponential distribution of times to repair. The system reliability with repair is the probability of no more than q out of n subsystems being simultaneously in a failed state during time t. Under conditions frequently met in practice, system reliability with repair R(t) can be approximated by: R(t) ? exp [?t/Tm] where Tm is the mean time for the system to pass for the first time from zero to (q + 1) simultaneous subsystem failures. Exact and approximate methods of calculating Tm are developed. A detailed error analysis is presented showing the limitations of using Tm to calculate system reliability with repair.     

Optimal replacement policy for induced draft fans

This paper derives the optimal block replacement policies for four different operating configurations of induced draft fans. Under the usual assumption of higher cost of repair or replacement on failure compared to preventive replacement, the optimal preventive replacement interval is found by minimising the total relevant cost per unit time. Specifically, this paper finds optimal preventive maintenance strategies for the following two situations.

1. (i)|Both the time to failure and time to carry out minimal repair or replacement are exponentially distributed.

2. (ii)|The time to failure follows the Weibull distribution and there is no possibility of on-line repair or replacement.

For both situations closed form expressions are derived whose solutions give optimum preventive maintenance intervals.     

Profit evaluation of a two unit cold standby redundant system with two operating modes

This paper considers a two unit cold standby system subject to a single repair facility with exponential failure time and arbitrary repair time distribution. Each unit has three modes—normal (N), partial (P) and total failure (F). By using the regenerative point technique the system has been analysed to determine mean time to system failure and profit earned by the system. A numerical example is used to highlight the important results.