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.     

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.     

Effect of intermittent repair in a two unit redundant system with standby failure

A two-unit standby redundant system with exponential failure time distribution of the operative unit is considered. It is assumed that failure of the standby unit is detected only at the time of use and is available with a known probability. Repair facility is also available at the time of need with a known probability. Repair and preoccupation times are general. Stochastic behaviour of the system has been studied by Semi-Markov process and parameters of interest eg. MTSF, steady state availability, expected profit have been obtained. Expected profit of this model is compared with that of an earlier model and a condition on extra incurred cost is obtained.     

Cost analysis of a two unit cold standby system under different weather conditions

This paper deals with the cost analysis of a two unit cold standby system under two different weather conditions—normal and abnormal. Failure rates of units and rates of change of weather conditions are constant, whereas repair rates are general. The system is analysed in detail using regenerative point technique and results are obtained for mean time to system failure, steady state availability, busy period analysis, expected number of visits by repairman and expected profit earned by the system.     

Two models for two-dissimilar-unit cold standby redundant system with partial failure and two types of repairs

This paper deals with the availability function and the mean time to the first failure for two models of a cold standby redundant system with two different types of repair. Each model consists of two dissimilar units. In the first model, the operative unit has two modes of operation, normal mode and partial failure mode. The standby unit has one mode of operation, normal mode. In the second model, each unit has three modes of operation, normal mode, failure mode and total failure mode. Both models are analyzed by the semi-Markov process technique, assuming that the failure time and repair time distributions are general and arbitrary. Some reliability measures of interest to system designers as well as operations managers have been obtained. Moreover, we give computer programs for calculating the MTSF for each model (see Appendix).     

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.     

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 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.     

Stochastic behaviour of a two-unit redundant system with switchover time

There is a two-unit standby redundant system. Standby is kept in cold state. Whenever one unit is operating and the other is in the waiting standby state, switching is initiated on the latter after a random time. If the operative unit fails before the time to initiate switching action on the waiting standby unit, the system fails and the failed unit immediately undergoes repair type 2 along with the switching. However, if the operative unit fails when the other unit is as standby (after switchover), the failed unit undergoes repair type 1 and the unit as standby takes over the operation. All the distributions are arbitrary except failure-time, which is exponential.The system is defined with the help of states which generate a pseudo semi-Markov process. Abundant use of results from the theory of SMP has been made to obtain a large number of parameters which measure reliability characteristics of the system viz. MTSF, steady-state availability, expected number of visits to a state, conditional transition probabilities, first passage time distributions, expected profit rate, etc. Numerical examples are included to illustrate the results.     

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.     

Some characteristics of a two-dissimilar-unit cold standby redundant system with three modes

This paper deals with the stochastic behaviour of a two-dissimilar-unit cold standby redundant system in which each unit works in three different modes—normal, partial failure and total failure. It is assumed that the failure and repair 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-Stiektjes transforms of the distribution function of the first passage time, mean time to system failure, pointwise availability and steady state availability of the system are obtained. Certain important results have been derived as particular cases.     

Reliability and availability analysis of a system with warm standby and common cause failures

This paper presents reliability and availability analyses of a two unit parallel system with warm standby and common-cause failures. The standby and switching mechanisms are subject to failure. The failed system repair times are assumed to be arbitrarily distributed. Expressions for Laplace transforms of system state probabilities, steady state system availability, system reliability, and mean time to failure are developed.     

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

This paper deals with the cost-benefit analysis of a one-server two-unit system with imperfect switch where the unit, if available, is preferred to the standby. The repair time of an item (unit/standby/switch is arbitrarily distributed while failure rate of an item is constant. Initially, the unit is switched on (switch is working at t = 0). The repair of the switch is given preference after the current repair of the unit or standby is over. Explicit expressions for the expected up-time due to the unit, expected up-time due to the standby; expected busy period of the server due to the unit, due to the standby and that due to the switch are obtained to carry out the cost-benefit analysis.     

Higher moments in cost-benefit analysis

This paper deals with the cost-benefit analysis of one-server two-unit hot standby system with imperfect switch where the repair densities depend upon the type of items (unit/standby/switch) waiting for repair. Initially one unit is put to operation (switch is working at t = 0) and the other unit is kept as a hot standby. The revenue as well as the cost of repair are arbitrary functions of time. Explicit expressions are obtained for expected net gain in [0,t] from which other characteristics like expected busy period due to repair of unit/standby/switch, expected up-time, expected number of repairs completed of unit/standby/switch etc. can be obtained.     

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.     

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.     

Cost-benefit analysis of a single server three-unit redundant system with inspection, delayed replacement and two types of repair

This 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. The type of the failure of operative or warm standby unit is detected by inspection only. The service facility plays the triple role of replacement, inspection and repair of a unit. Failure time distributions of operative and warm standby units are negative exponential whereas the distributions of replacement time, inspection time and repair times are arbitrary. The system has been studied by using regenerative points.     

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.     

Probabilistic analysis of a two-unit system with a warm standby subject to preventive maintenance and a single service facility

The object of this paper is to carry out the availability and the reliability analysis of a two-unit system with a warm standby having a single service facility for the performance of preventive maintenance and repair. The failure times, the repair times, the inspection times and the preventive maintenance times of the main unit and of the standby one are assumed to be arbitrarily distributed. The system is characterized by the probability of its being in the up or the down state. Explicit expressions for the mean down time of the system and for the mean time to system failure are obtained. Some previous results are derived from this work as special cases.