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.     

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.     

A priority multi-component system with spares

A system with n similar components in series with constant failure rates and m spares in warm standby supported by a single repair facility is studied. The repair time distributions of on-line and standby failures are taken to be different and arbitrary. Two models, one with pre-emptive repeat priority and another with non-pre-emptive priority are discussed. Equations for the reliability and availability functions are obtained for both cases and earlier results are recovered as special cases of the non-pre-emptive model.     

Common-cause failure analysis of a redundant system with non-repairable units

This paper presents a newly developed generalized expression for mean time to failure, MTTF, of a non-repairable identical unit parallel system with warm standby and common-cause failures. Also presented is a modified version of this formula that takes into account the repairability of the warm standby and/or switching mechanism. Generalized expressions for system reliability and variance of time to failure are presented along with some tables and plots of system reliability and MTTF.     

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 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.     

Reliability and availability analysis of warm standby systems in the presence of chance with multiple critical errors

This paper presents a reliability and availability analysis of k active, N warm standby units in the presence of chance with M multiple critical errors. The system is in a failed state when (N + 1) units have failed (active and/or warm standby units have failed) or one of the multiple critical errors has occurred. Failed units are not repaired but a failed system will be repaired with repair times arbitrarily distributed. The expressions for reliability, availability and steady-state availability are derived.     

General measures of a two-unit system

In this paper, we consider a two-unit standby system with single repair facility. The bivariate random variables (X₁, Y₁); i = 1, 2, …, n are defined as the busy and idle periods respectively of the repair facility. General measures like joint point availability and joint interval reliability are obtained using the regeneration point technique. Numerical example illustrates the results.     

The reliability function and the availability of a duplication redundant system with a single service facility for preventive maintenance and repair

This paper concerns a two-unit system with a cold standby and a single service facility for the performance of preventive maintenance and repair. Explicit expressions for the Laplace transforms of the availability of the system, the reliability, the mean down time during (0, t) and for the mean time to system failure have been obtained under the assumption that the failure times, the inspection times, the repair times and the preventive maintenance times of the two units are governed by distinct arbitrary general distributions. The results obtained by Srinivasan and Gopalan and by Gopalan and d'Souza are derived from the present results as special cases.     

A note on the cost analysis of an n-unit system with spares

This paper deals with the cost analysis of a non-repairable standby system consisting of (n + m) identical units; n-units are needed for the system to function, while the remaining m units are warm standbys. The online and standby units have different constant failure rates. There is no facility for repair. Functions expressing the probability that in (0, t) there are i on-line failures and j standby failures and thereby the reliability of the system. MTTF and the expected profit are obtained. Finally, a numerical example with graphs is also given to highlight important results like the reliability of the system, MTTF and the expected profit.     

Stochastic analysis of standby system with duplex units

This paper investigates a mathematical model of a system composed of two units, one operative and the other cold standby. Each unit of the system is made of two non-identical parallel components and each component is made of n-elements. Henceforth we call each unit of the system a duplex unit. Failure and repair time distributions of each element of a component are negative exponential and vary from element to element, whereas the repair time distribution of a unit is arbitrary. Upon the failure of the operative unit the standby unit does not operate instantaneously. This type of situation may be found in many electronic networks. Several reliability characteristics of interest have been obtained.     

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.     

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.     

Two 1-Server n-Unit Systems with Preventive Maintenance and Repair

This paper deals with the availability and reliability analysis of two different 1-server n-unit systems with preventive maintenance and repair. Initially, one unit operates and the remaining n - 1 units are kept as cold standbys. In the first system the time to failure and the time to preventive maintenance of a unit are arbitrarily distributed. In the second system, each unit consists of 2 components connected in series. When a unit fails, the failed component is taken up for repair while the other waits for preventive maintenance. 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 are obtained. Steady-state availability of the system is also discussed. A few special cases have been studied.     

A two-unit cold standby system with allowed down-time and correlated failures and repairs

This paper considers the stochastic behaviour of a two-unit cold standby system with allowed down-time and a single repair facility to repair the failed unit. It is assumed that the repair and failure times of a unit are associated with each other in some way or other and their joint distribution is bivariate exponential.     

A computer algorithm for the analysis of maintained standby redundant systems

A computer algorithm is developed to analyse standby redundant systems with repair maintenance. The method is applicable to a generalized class of standby systems. Failure-time distributions of units need to be exponential whereas repair-times can follow general distributions. A large number of input parameters which may be of interest to system designers have been incorporated, e.g. repair efficiency, transfer switch failure, connect switch failure, preoccupied repair facility etc. The analysis procedure consists in defining a system model by writing different states and transitions between them. Once this is done, the underlying process is visualized as a semi-Markov process and various results from this theory are applied to develop a computer algorithm in FORTRAN IV for obtaining a fairly large number of system output parameters viz., mean-time-to-system failure, steady-state availability, expected number of visits to a state, expected profit rate of the system etc. Two examples are included to illustrate the usage of the procedure developed.     

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.     

Optimal redundancy allocation for high availability routers

How to optimally allocate redundant routers for high availability (HA) networks is a crucial task. In this paper, a 5‐tuple availability function A (N, M, λ, µ, δ) is proposed to determine the minimum required number of standby routers to meet the desired availability (ρ) of an HA router, where N and M are the numbers of active routers and standby routers, respectively, and λ, µ, and δ are a single router's failure rate, repair rate, and failure detection and recovery rate, respectively. We have derived the availability function, and analytical results show that the failure detection and recovery rate (δ) is a key parameter for reducing the minimum required number of standby routers of an HA router. Thus, we also propose a High Availability Management (HAM) middleware, which was designed based on an open architecture specification, called OpenAIS, to achieve the goal of reducing takeover delay (1/δ) by stateful backup. We have implemented an HA Open Shortest Path First (HA‐OSPF) router, which consists of two active routers and one standby router, to illustrate the proposed HA router. Experimental results show that the takeover delays of the proposed HA‐OSPF router were reduced by 6, 37.3, and 98.6% compared with those of the industry standard approaches, the Cisco‐ASR 1000 series router, the Juniper MX series router, and the Virtual Router Redundancy Protocol (VRRP) router, respectively. In addition, in contract to the industry routers, the proposed HA router, which was designed based on an open architecture specification, is more cost‐effective, and its redundancy model can be more flexibly adjusted. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis of consecutive k-out-of-n: F systems with single repair facility

In this paper, a consecutive k-out-of-n:F system with non-identical units and with a single repair facility is analyzed in connection with the reliability and availability of the system. The failure rates of the units are constant and the repair time is arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. A particular case with k = 2 and n = 3 is analyzed numerically to study the effect of various parameters on the availability and reliability of the system.     

Analysis of reliability bounds for two-unit parallel and standby reparable systems

This paper investigates the least upper bounds (LUB) and greatest lower bounds (GLB) for the reliability of two-unit parallel and standby reparable systems. The bounds are in terms of the failure and repair rates of a unit, when the distributions of failure and repair times are assumed to be negative exponential.