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.     

Reliability of Special Redundant Systems Considering Exchange Time and Repair Time

The reliability of a 2-out-of-3 parallel redundant system having a limited number of standby spare units is derived when the exchange of the failed unit for a spare unit is not instantaneous. The reliability can be represented in the form of a failure state diagram. When the number of standby spares and the repair rate are both small, the influence of the exchange rate is small. When the number of standby spares and the repair rate are both large, the influence of the exchange rate is large. The number of standby spares and the repair rate influence the probability that system failure occurs after all spare units have failed. The exchange rate strongly influences the probability of system failure during the exchange time.     

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.     

An (m, M) machine repair problem with spares and state dependent rates: A diffusion process approach

The diffusion approximation model for (m, M) general machine repair problems with spare part support has been developed. The inter-failure time and repair times are assumed to be generally identically and independently distributed (i.i.d.). The failure rate of operating units in a short system when fewer than M units are operating and all spares are being used, is faster than a normal system. The spare units may also fail with rates different from operating units. The repairmen switch to the faster rate to reduce a backlog of down units in the case of a busy repair facility. By using reflecting boundaries, the approximate formulae for some performance measures, namely, expected number of inoperative/operative units and probability that the system is short/down have been obtained.     

s̱-expected busy period of the service facility of a 1-server n-unit system subject to arbitrary failure and different repair strategies

This paper deals with the s?-expected busy period of a n-unit system with a single repair facility. The service facility is called for whenever r (1 ≤ r ≤ n) units are in failed condition and is retained until no unit is waiting for repair. The analysis is carried out under the assumption that the time to failure of a unit is arbitrarily distributed while the service rate is constant. The case when n = 2 with the failure time distribution is 2-stage Erlangian is discussed.     

A two non-identical three-state units redundant system with common-cause failures and one standby unit

This paper presents a mathematical model for predicting a two non-identical three-state active units redundant system with common-cause failures and one standby unit. The units may fail in either of two mutually exclusive failure modes or by the occurrence of common-cause failures. System is only repaired when all the units fail (including the standby unit). The failure rates of units are constant and system repair times are arbitrarily distributed. Laplace transforms of the state probabilities are derived.     

A note on a multi-unit system with r repair facility

This paper deals with a system consisting of (n + m) identical units; n units are needed for the system to function and the remaining m units are warm standby supported by r repair facility. The online and standby units have different but constant failure rates; the repair time distribution for the standby unit is taken to be a constant and the distribution for the online unit is arbitrary. Functions describing the behaviour of all the other units when one unit is undergoing online repair are studied and an earlier result is recovered as a special case.     

Analysis of a two-unit cold standby system with three modes

This paper analyses a two-unit cold standby system under the assumption that each unit works in three different modes—normal, partial failure and total failure. Failure time distributions of units are exponential, whereas repair time distributions are arbitrary. Breakdown of the system occurs when both the units are in total failure mode. Several reliability characteristics of interest to system designers as well as operations managers have been evaluated.     

A k-out-of-N:G redundant system with cold standby units and common-cause failures

This paper presents a k-out-of N:G redundant system with M cold standby units, r repair facilities and common-cause failures. The constant failure rates of the operating and cold standby units are different. Failed system repair times are arbitrarily distributed. The system is in a failed state when (N+M?k+1) units failed or a common-cause occurred. Laplace transforms of the state probabilities, the availability of the system and the system steady-state availability are derived.     

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.     

Cost-benefit analysis of one-server two-unit imperfect switch system with degradation

This paper deals with the cost-benefit analysis of 1-server 2-unit system with imperfect switch where the items (units/switch) are not ‘as good as new’ after the repair. The successive repair times of each item are arbitrarily distributed while the successive failure rates are constants. If repair of a unit is not possible it is discarded incurring some salvage value. After a fixed number of failures the unit is scrapped without any salvage value. The switch is always repairable and is scrapped only when both the units are scrapped. The expected revenue, expected busy period due to different types of repairs of units/switch, expected salvage value and expected number of repair completions of the switch, all in [0, t], are obtained to carry out the cost-benefit analysis.     

Reliability and maintainability of a multicomponent series-parallel system under several repair disciplines

This paper considers a system consisting of two subsystems connected in series with a single repair facility. One subsystem is K-out-of-N:G system consisting of N identical units, while the other consists of M different units connected in series. The life-times of the active units depend on each other in having simultaneous failure of all the operating units and repair times are distributed quite generally. Initially all the units are operating. The system breaks down if more than (N-K+1) units in the parallel group are simultaneously in the failed states or if any failure occures in the series group. The availability and reliability function of the system under several repair disciplines are obtained simultaneously. We use a suitable transformation to deduce the reliability from the availability function. Explict expressions for the steady-state availability of the system and the mean time to system failure under several repair disciplines are obtained. Finally some properties of the steady-state availability for each repair discipline are given.     

Analysis of a two-unit hot standby system with three modes

This paper discusses the stochastic behaviour of a two-unit hot standby redundant system having exponential failure and a single repair facility with general repair time distribution. Each unit of the system has three possible states—normal, partial failure and total failure. Breakdown of the system occurs when both the units are in total failure mode. Several reliability characteristics of interest to system designers as well as operations managers have been evaluated.     

Availability, MTTF and cost analysis of a three-state parallel redundant multi-component system under critical human failures

This paper deals with the availability, MTTF and the cost analysis of a single server complex system consisting of two classes A and B with three possible states, viz. good, degraded, failed and suffers two types of failure, viz. unit failure and failure due to critical human errors.Sub-system A has two dissimilar units arranged in parallel whereas sub-system B has three non-identical units arranged in series. Failure and repair times have exponential and general distributions respectively. There is only one repair facility when the system is in failed state due to failure of sub-system B. Several parameters of interest are obtained using the supplementary variable method. A numerical example has been appended. Five graphs have also been given in the end.     

Cost analysis of a two-unit standby system with delayed replacement and better utilization of units

This paper develops the model for a system, having two identical units—one operative and the other cold standby. Each unit of the system has three modes—normal, partial failure and total failure. The replacement time of a failed unit by a standby unit is not negligible but is a random variable. System fails when both the units fail totally. Failure time distributions of units are exponential, whereas repair time distributions are arbitrary. Several reliability characteristics of interest to system designers and operations managers have been evaluated using the theory of regeneration point technique.     

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 analysis of a 3-state 2-unit reparable system

This paper presents the cost analysis of a 2-unit system with 3 states: good, degraded and failed. The units suffer from two types of failure: partial and catastrophic. The partial failure brings a unit to degraded state, whereas the catastrophic failure breaks down a unit completely. There is one repair facility, which is availed only when the system is either degraded or failed. The failure and repair times for the system follow exponential and general distributions respectively. Laplace transforms of various probability states have been obtained along with steady-state behaviour of the system. Inversions have also been computed so as to obtain time dependent probabilities, which determine expected profit as well as availability of the system at any time.     

Stochastic behaviour of a two-unit cold standby system with three modes and allowed down time

This paper discusses the stochastic behaviour of a two-unit cold standby redundant system under two very general sets of conditions: (i) each unit of the system having three different modes of working—normal, partial failure and total failure; (ii) breakdown of the system occurring when with both the units in total failure mode, the system is not regarded as failed (the system fails only when the breakdown does not terminate within the allowed down time). Failure-time distributions of units are exponential, whereas repair time distributions are arbitrary. Several reliability characteristics of interest to system designers as well as operations managers have been evaluated.     

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 and maintainability of a multicomponent series-parallel system with simultaneous failure under preemptive repeat repair discipline

This paper considers a system consisting of two subsystems connected in series with a single repair facility. One sybsystem is K-out-of-N:G system consisting of N identical units, while the other consists of M different units connected in series. The life-times of the active units depend on each other in having simultaneous failure of all the operating units and repair times are distributed quite generally. The system breaks down if more than (N−K+1) units in parallel group are simultaneously in the failed states or if any failure occurs in the series group or if simultaneous failure occurs. The availability and reliability function of the system are obtained simultaneously. Explict expressions for the steady state availability of the system and the mean time to the first system failure are obtained.