Redundancy optimization problems for systems with dependent failure rates

The present study is concerned with some redundancy optimization models for “load-sharing” devices. The models considered here enable one to examine the extent to which the optimal allocation of redundancies is affected if a model based on the assumption of mutual independence of components is used in situations where there is actual “load-sharing”.     

First failure time of dependent parallel systems with safety periods

In this paper we consider the time to first failure of a parallel system in which the failure and repair rates of components depend on the state of the other components as well. A back-up unit with a random lifetime is employed whenever all the components of the system are down. The system fails when all the components of the system and the back-up unit are down. The first moment, the Laplace transform and the probability distribution of the time to first failure of this system are obtained. Sufficient conditions under which this distribution has the new better than used (NBU) and an exponential limit property are given. Special cases with phase type and deterministic back-up unit lifetimes are also considered. These results extend the results of Ross and Schechtman (1979).     

A k-out-of-N:G redundant system with dependent failure rates and common-cause failures

This paper presents a k-out-of-N: G redundant system with dependent failure rates, common-cause failures and r repair facilities. The failure rates of the components increase as the number of components failed increases, while the repair rates are constant. Common-cause failure is not considered in Model I. In Model II the common-cause failures are involved. Steady-state probabilities and steady-state availability are derived.     

Optimal configuration of redundant real-time systems in the face ofcorrelated failure

Real-time computers are frequently used in harsh environments, such as space or industry. Lightning strikes, streams of elementary particles, and other manifestations of a harsh operating environment can cause transient failures in processors. Since the entire system is in the same environment, an especially severe disturbance can result in a momentary, correlated, failure of all the processors. To have the system survive transient correlated failures and still execute all its critical workload on time, designers must use time redundancy. To survive permanent or transient independently-occurring failures, processor redundancy must be used, and the computer configured into redundant clusters. Given a fixed total number of processors, there is a tradeoff between processor- and time-redundancy, This paper considers the tradeoffs between configuring the system into duplexes and triplexes. There are pessimistic and optimistic reliability models for each configuration. For the range of pertinent parameters, these models are very close, indicating that these models are quite accurate. The duplex-tripler tradeoff is between the effects of permanent, independent-transient, and correlated-transient failures. Configuring the system in triplexes provides better protection against permanent and independent-transient failures, but diminishes protection against correlated-transient failures. The better configuration is given for each application     

Availability and reliability of system with dependent componentsand time-varying failure and repair rates

System reliability depends not only on the reliabilities of components in the system but also on their interactions. Generally, in a system, not only s-independent failures but also s-dependent failures among components can occur; thus there are many studies where the s-dependencies among components are taken into account in system reliability and availability analysis, but in which the failure and repair rates were assumed constant. Whereas, from a practical viewpoint, the constant failure rate assumption for components has been, and is repeatedly challenged by knowledgeable reliability practitioners. Therefore, there are other studies which handled the problem of time-varying failure rates, among which all concerned repairable systems did not involve s-dependent failures. In most cases, however, to combine s-dependent failures and time-varying failure and repair rates in system reliability and availability analysis is the most appropriate for real systems. But it is very difficult to obtain the analytic solution and, in most cases, the closed-form solution for system reliability and availability does not exist, so that numerical or simulation methods must be used. This paper studies one kind of system that endures environmental shocks, and where one or more components can fail simultaneously due to a cumulative shock-damage process. An approach for reliability and availability analysis of such kinds of repairable systems is presented, where failure and repair rates of components can be varied with time. One type of special vehicle with such mechanical systems illustrates system reliability and availability solutions     

Two-unit redundant system with inspection and adjustable rates

A two-unit redundant system is studied, in which one unit is operative and the other is a warm standby which replaces the operative failed unit instantaneously. To increase system availability, the failure rate of the operative unit and the repair rate of the failed unit adjust automatically according to the state of the standby unit. Also, after repair of the operative failed unit it is sent for inspection to decide whether the repaired unit is perfect or not. If the repaired unit is found to be imperfect, it is sent for post repair. Using a regenerative point technique in the Markov renewal process, several reliability characteristics of interest to system designers and operation managers are obtained.     

Modeling repairable systems with failure rates that depend on ageand maintenance

A general repairable system model that includes age and preventive maintenance (PM) dependent failure rates is proposed. This model introduces the concept of system availability as a random variable, leading to availability measures for individual realizations, rather than for the average over many trials. Thus this model generalizes the classical definition, and serves as a performance index for system design. A design scheme suggested to maximize the probability of achieving a specified stochastic cycle availability with respect to the duration of the operating interval between PMs. Numerical computations of the results for Weibull-like distributions illustrate the design criteria. Asymptotic behavior of the performance with no PM has been studied. Extension to other distributions is straightforward     

A reliability test-plan for series systems with components havingstochastic failure rates

This paper proposes a reliability test plan for a series system, by considering the parameter λ_j of the exponential distribution to be a random variable having uniform distribution over [0, &thetas;_j], j = 1, 2,..., n. Explicit expressions are obtained for the optimal values of the t_j, when the number of components in the system is 2. The general solution, albeit implicit, has also been obtained when the number of components in a given system is ⩾3. Mathematical programming is used to find the optimal solution and to illustrate it with numerical results     

Reliability calculation of redundant systems with non-identical units

The present paper develops mathematical models for evaluating the exact reliability and mean time to failure of k-out-of-m: G systems with different unit failure probabilities. The ith unit is assumed to be characterized by a general hazard rate h_i(t) = λ_it^b.The models are based on the concepts of tie sets; they are fairly simple and can be used for any values of m and k. The algorithm developed for these models is suitable for adoption for a computer code.     

Mean time to first failure of repairable systems with one coldspare

A general (nonMarkov) 1-out-of-2:G system with statistically-identical components, repair, and cold standby is reviewed. Coverage is considered, viz, failures of the switching mechanism for activating the spare. The explicit derivation of the mean time-to-first-failure and its in-depth discussion appear to be new. The state transition graph and the Petri net both show the way to general (n-1)-out-of-n:G systems of this category. However, for n>2, results are given only for statistically-identical components which are all as good as new, when a repaired component is put to use. This limits applicability of results to electronic systems     

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.     

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.     

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.     

Stochastic analysis of a redundant system with two types of failure

This paper deals with the analysis of a system model consisting of two units, in which one is operative and the other is on cold standby. The failure of an operative unit may be caused by a machine as well as by random shocks which occur after a certain interval of time.Using the regenerative point technique in Markov renewal processes, several effective measures of reliability are obtained.     

Reliability of a 3-unit cold redundant system with Weibull failure

Redundancy technique is applied to increase the reliability of a system where maintenance or repair is either not possible or is too costly. Reliability of a cold redundant system is always higher than a hot one. Moreover the increase in reliability due to adding of a unit in sequence can always be exactly determined.Literature is full of derivation of reliability expressions of a cold redundant system whose units obey exponential failure density. Due to extreme difficulty in evaluation of integrals involving Weibull density function, few attempts on derivation of reliability of redundant systems have been made. We have here derived reliability of a simple case of a 3-unit cold standby system whose units obey Weibull failure density.     

Time-varying failure rates in the availability and reliabilityanalysis of repairable systems

This paper combines time varying failure rates and Markov chain analysis to obtain a hybrid reliability and availability analysis. However, combining these techniques can, depending on the size of the system, result in solutions of the Markov chain differential matrix equations that are intractable. This paper identifies solutions that are tractable, These form the analytical baseline for the reliability and availability analysis of systems with time varying failure rates. Tractable solutions were found for the 1-component 2-state and the 2-component 4-state configurations. Time varying failure rates were characterized by a general polynomial expression. Constant, linear, and Weibull failure rate functions are special cases of this polynomial. The general polynomial failure rate provides flexibility in modeling the time varying failure rates that occur in practice     

Reliability prediction for repairable redundant systems

It is assumed that the mean time to failure and mean repair time are known for each of the subsystems of a system. The subsystems conform to the usual exponential failure (and repair) laws and their behaviors are mutually independent. The system includes redundant subsystems in active standby status. Whenever, after a system failure, repair of a failed subsystem re-establishes an adequate configuration, the system as a whole is returned to active status while repair of other failed subsystems (if any) continues. Under this set of assumptions, equations are developed which permit prediction of mean time to failure and mean down time for the system. The development differs somewhat from the use of birth-and-death equations which has been customary for similar problems in the past.     

Stochastic analysis of a two-unit redundant system with two types of failure

The present paper investigates a stochostic model of a two-unit warm standby system with a single repair facility. Before repair, the failed unit is sent for fault detection to decide whether it failed due to machine defect or critical human error. The probability of having machine defect and C.H.E. has been fixed. Using the regenerative point technique in the Markov renewal process various measures of system effectiveness are obtained.     

On a two-unit cold standby redundant system with random switching time

This paper, which contains two sections, deals with a two-unit cold standby redundant system with random switching time and imperfect switchover. In Section I, we use the survival functions to find the mean TFSF and the L.S.T. of the distribution to the first system failure. Finally, we find the limiting distribution to the first system failure. In Section II, we find the mean TFSF considering two types of preventive maintenance (PM), type (a) and type (b). Finally we prove that the mean TFSF with type (b) PM is greater than or equal to the mean TFSF with type (a) PM.