Determining Component Reliability and Redundancy for Optimum System Reliability

The usual constrained reliability optimization problem is extended to include determining the optimal level of component reliability and the number of redundancies in each stage. With cost, weight, and volume constraints, the problem is one in which the component reliability is a variable, and the optimal trade-off between adding components and improving individual component reliability is determined. This is a mixed integer nonlinear programming problem in which the system reliability is to be maximized as a function of component reliability level and the number of components used at each stage. The model is illustrated with three general non linear constraints imposed on the system. The Hooke and Jeeves pattern search technique in combination with the heuristic approach by Aggarwal et al, is used to solve the problem. The Hooke and Jeeves pattern search technique is a sequential search routine for maximizing the system reliability, RS (R, X). The argument in the Hooke and Jeeves pattern search is the component reliability, R, which is varied according to exploratory moves and pattern moves until the maximum of RS (R, X) is obtained. The heuristic approach is applied to each value of the component reliability, R, to obtain the optimal number of redundancies, X, which maximizes RS (R, X) for the stated R.     

A Component-Testing Procedure For A Parallel System With Type II Censoring

We consider the problem of acceptance testing for a parallel (1-out-of-n:G) system of n different components with constant failure rates. The components are individually tested and the tests are terminated as soon as a preassigned number of each component fails. This paper provides a criterion for accepting or rejecting the system based on the product of the total times on test for each component. The critical level for the test statistic is chosen so as to guarantee that the specified levels of consumer and producer risks on the system reliability are not exceeded. If the testing costs depend on the number of each component tested, aminimum-cost procedure can be found from the feasible set of plans.     

Optimal Reliability Design of a System: A New Look

The reliability literature offers an abundance of methods for the optimal design of systems under some constraints. In most of the papers, the problem considered is: given reliabilities of each constituent component and their constraint-type data, optimize the system reliability. This amounts to the assignment of optimal redundancies to each stage of the system, with each component reliability specified. This is a partial optimization of the system reliability. At the design stage, a designer has many options, e.g., component reliability improvement and use of redundancy. A true optimal system design explores these alternatives explicitly. Our paper demonstrates the feasibility of arriving at an optimal system design using the latter concept. For simplicity, only a cost constraint is used, however, the approach is more general and can be extended to any number of constraints. A particular cost-reliability curve is used to illustrate the approach.     

A type-II censored, log test time based, component-testingprocedure for a parallel system

The authors consider the problem of acceptance testing for a parallel (1-out-of-n:G) system of different components with constant failure rates. The components are individually tested and the tests are terminated as soon as a preassigned number of each component fails. The authors provide a criterion for accepting or rejecting the system based on the sum of the logarithms of the total times on test for each component. The critical level for the test statistic is chosen so as to guarantee that the specified consumer and producer risks on the system reliability are not exceeded. The use of this statistic makes the computation of these critical values much simpler as compared with that of a previously used statistic based on the product of the total times on test for each component. Several approximate procedures are considered for deriving these critical values. The authors also formulate the optimization problem for deriving the minimum-cost component-testing plans when a type-II censored component-test procedure is used for a parallel system     

A system-based component test plan for a series system, withtype-II censoring

Acceptance testing is analyzed for a series system of n components, each having an unknown, different, constant failure rate. Components are tested individually, and tests are terminated when a preassigned number of failures is observed for each component. The total time-on-test for each component is noted, and a statistic is constructed using observed test times and the number of failures of the components; the statistic is based on the maximum likelihood estimation of system reliability. This statistic is then used in specifying a decision rule for accepting or rejecting the entire system. The design of the test plan is stated as an optimization problem which minimizes test costs while ensuring that specified consumer and producer risks on system reliability are not exceeded. Numerical examples are provided, and implications of the test plan are discussed     

Reliability optimization of distributed access networks with constrained total cost

In this paper, we study the system reliability optimization of distributed access networks subject to a constraint on the total cost. We first formulate the cost-constrained system reliability optimization problem as a searching process in a combinatorial tree, which enumerates all the possible solutions to the problem. Because the calculation of each possible solution for the reliability problem is extremely time-consuming, a novel algorithm, the Shrinking & Searching Algorithm (SSA), is proposed to speed up the searching process. SSA jointly considers the upper bound of the system reliability for each branch in the combinatorial tree, and the cost constraint on the possible solutions. It avoids most of the redundant calculations in the searching process by gradually shrinking the difference between lower & upper bounds of the length of a path in the corresponding combinatorial tree, which represents a feasible solution. Case study & simulation results are presented to demonstrate the performance of the SSA.     

A new dynamic programming method for reliability & redundancy allocation in a parallel-series system

Reliability & redundancy allocation is one of the most frequently encountered problems in system design. This problem is subject to constraints related to the design, such as required structural, physical, and technical characteristics; and the components available in the market. This last constraint implies that system components, and their reliability, must belong to a finite set. For a parallel-series system, we show that the problem can be modeled as an integer linear program, and solved by a decomposition approach. The problem is decomposed into as many sub-problems as subsystems, one sub-problem for each subsystem. The sub-problem for a given subsystem consists of determining the number of components of each type in order to reach a given reliability target with a minimum cost. The global problem consists of determining the reliability target of subsystems. We show that the sub-problems are equivalent to one-dimensional knapsack problems which can be solved in pseudopolynomial time with a dynamic programming approach. We show that the global problem can also be solved by a dynamic programming technique. We also show that the obtained method YCC converges toward an optimal solution.     

Estimation of system reliability for uncooled optical transmittersusing system reliability function

This paper investigates the system reliability for 155 Mb/s optical transmitters by deriving a system reliability function from reliability data of each component for transmitters, laser diode, photodiode, optical assembly, and driver IC. The reliability data for each component reliability function have been obtained from accelerated aging test. The reliability parameters such as failure rate, mean time-to-failure (MTTF), standard deviation are obtained from a probability plotting method. From the system reliability function, the MTTF of the optical transmitter at 65°C was estimated to be 47000 h with 95% confidence. In this estimation, we introduced modified lifetime of laser diodes and reliability function of optical assembly     

Reliability optimization of systems by a surrogate-constraintsalgorithm

A method for solving the problem of optimizing both, redundancy (number of redundant components) and component reliability in each stage of a system under multiple constraints is presented. A mixed-integer nonlinear programming formulation and the surrogate dual method are used. The solution of the surrogate dual problem is not always feasible in the original problem, that is, a `surrogate gap' exists. Two countermeasures to surrogate gaps are considered: (1) modifying the original problem to tighten the constraints, with the modification being continued until the solution of the surrogate dual problem of the modified problem becomes feasible in the original problem, and (2) decreasing component reliabilities in the vertical direction to the tangential plane of the objective function. The method applies to reliability optimization problems for general systems, enabling complex systems such as communication networks to be treated. Some computational results are shown and compared with other approaches; they show the efficiency of the method     

System optimization with component reliability estimation uncertainty: a multi-criteria approach

Summary & Conclusions-This paper addresses system reliability optimization when component reliability estimates are treated as random variables with estimation uncertainty. System reliability optimization algorithms generally assume that component reliability values are known exactly, i.e., they are deterministic. In practice, that is rarely the case. For risk-averse system design, the estimation uncertainty, propagated from the component estimates, may result in unacceptable estimation uncertainty at the system-level. The system design problem is thus formulated with multiple objectives: (1) to maximize the system reliability estimate, and (2) to minimize its associated variance. This formulation of the reliability optimization is new, and the resulting solutions offer a unique perspective on system design. Once formulated in this manner, standard multiple objective concepts, including Pareto optimality, were used to determine solutions. Pareto optimality is an attractive alternative for this type of problem. It provides decision-makers the flexibility to choose the best-compromise solution. Pareto optimal solutions were found by solving a series of weighted objective problems with incrementally varied weights. Several sample systems are solved to demonstrate the approach presented in this paper. The first example is a hypothetical series-parallel system, and the second example is the fault tolerant distributed system architecture for a voice recognition system. The results indicate that significantly different designs are obtained when the formulation incorporates estimation uncertainty. If decision-makers are risk averse, and wish to consider estimation uncertainty, previously available methodologies are likely to be inadequate.     

Modeling and maximizing burn-in effectiveness

System burn-in can get rid of many residual defects left from component and subsystem burn-in since incompatibility exists not only among components but also among different subsystems and at the system level. Even if system, subsystem, and component burn-in are performed, the system reliability often does not achieve the requirement. In this case, redundancy is a good way to increase system reliability when improving component reliability is expensive. This paper proposes a nonlinear model to: estimate the optimal burn-in times for all levels, and determine the optimal amount of redundancy for each subsystem. For illustration, a bridge system configuration is considered; however, the model can be easily applied to other system configurations. Since there are few studies on system, subsystem, and component incompatibility, reasonable values are assigned for the compatibility factors at each level     

Optimal apportionment of reliability and redundancy in seriessystems under multiple objectives

A multiobjective reliability apportionment problem for a series system with time-dependent reliability is presented. The resulting mathematical programming formulation determines the optimal level of component reliability and the number of redundant components at each stage. The problem is a multiobjective, nonlinear, mixed-integer mathematical programming problem, subject to several design constraints. Sequential unconstrained minimization techniques in conjunction with heuristic algorithms are used to find an optimum solution. A generalization of the problem in view of inherent vagueness in the objective and the constraint functions results in an ill-structured reliability apportionment problem. This multiobjective fuzzy optimization problem is solved using nonlinear programming. The computational procedure is illustrated through a numerical example. The fuzzy optimization techniques can be useful during initial stages of the conceptual design of engineering systems where the design goals and design constraints have not been clearly identified or stated, and for decision making problems in ill-structured situations     

Redundancy allocation to maximize a lower percentile of the systemtime-to-failure distribution

An algorithm is presented which solves the redundancy-allocation problem when the objective is to maximize a lower percentile of the system time-to-failure distribution. The algorithm uses a genetic algorithm to search the prospective solution-space and a bisection search as a function evaluator. Previously, the problem has most often been formulated to maximize system reliability. For many engineering-design problems, this new formulation is more appropriate because there is often no clearly defined mission time on which to base component and system reliability. Additionally, most system designers and users are risk-averse, and maximization of a lower percentile of the system time-to-failure distribution is a more conservative (less risky) strategy compared to maximization of the mean or median time-to-failure. Results from over 60 examples clearly indicate that the preferred system design is sensitive to the user's perceived risk. We infer from these results that engineering-design decisions need to consider risk explicitly, and use of mean time-to-failure as a singular measure of product integrity is insufficient. Similarly, the use of system reliability as the principal performance measure is unwise unless mission time is clearly defined     

一类多状态连续n中取k(G)系统的可靠性计算

为了计算多状态连续厅中取后（G）系统的可靠性,引入4个定理,将满足引理的多状态系统转换为二元状态系统。分别推导了多状态线形连续k／n（G）系统和环形连续k／n（G）系统的可靠性计算公式。证明了固定k值增加一个新部件,若部件可靠性独立同分布,线形和环形系统可靠性均增加;若部件可靠性独立但不同分布,环形系统存在一个极值,新增加部件可靠性大于这个极值时得到的新系统可靠性增加,反之系统可靠性下降。     

一种提高数据传输可靠性的成分载波选择算法

载波聚合是LTE-Advanced的关键技术之一,它能有效地解决LTE-A系统带宽扩展问题。合理的选择成分载波将使得数据业务在传输时具有高的可靠性。本文提出了一种提高数据传输可靠性的成分载波选择算法,综合考虑成分载波信道质量和负载均衡两方面,对成分载波进行选择。通过仿真表明,该算法可以有效地选择成分载波进行聚合,降低数据传输的误比特率和误块率,减少平均传输次数,从而提高数据传输的可靠性,降低系统开销。     

On improved confidence bounds for system reliability

In this paper, new bounding strategies are presented to improve confidence interval estimation for system reliability based on component level reliability, and associated uncertainty data. Research efforts have been focused on two interdependent areas: 1) development & improvement of analytical approaches for quantifying the uncertainty associated with the system reliability estimate when data regarding component reliability is available; and 2) based on these analytical approaches, generating statistical inference methods that can be used to make accurate estimations about the reliability of a system. The analytical approach presented relies on a recursive rationale that can be applied to obtain the variance associated with the system reliability estimate, provided the system can be decomposed into a series-parallel configuration. The bounding procedure is independent of parametric assumptions regarding component time to failure, and can be applied whenever component reliability data are available. To assess the validity of the proposed procedure, three test cases have been analyzed. For each case, Monte-Carlo simulation has been used to generate component failure data, based on nominal component reliability values. Based on these simulated data, lower bounds have been constructed, and then compared against nominal system reliability to generate an expected confidence level. The results obtained exhibit a significant improvement in the accuracy of the confidence intervals for the system reliability when compared with existing approximation methods. The procedure described is effective, relatively simple, and widely applicable.     

Reliability evaluation of a first order standby system with module redundancy

By defining a module to be a coherent subsystem of independently operating components each with a constant failure rate, this article derives expressions for the reliability of a standby redundant system consisting of an operating module together with a cold or warm standby module. The closed form reliability expressions are dependent upon the minimal path sets of each module as well as the component failure rates. Expressions are also derived for the mean time to system failure as well as the variance of the system time to failure distribution.     

An improved algorithm for symbolic reliability analysis

The authors describe an efficient Boolean algebraic algorithm to compute the probability of a union of nondisjoint sets as applied to symbolic reliability analysis. Coherent networks and fault-trees with statistically-independent components characterized by their minimal pathsets or cutsets are used as examples for generating the nondisjoint sets. The algorithm uses the concept of multiple variable inversion originally proposed by A. Grnarov et al. (1979). The authors illustrate improvements in the use of the multiple variable inversion technique for this problem using two examples. The algorithm is extended to compute the reliability importance of a given component (sensitivity of system reliability to the component reliability). A computer program implementing the modified algorithm is used to solve and obtain measured time complexities for a large set of network and fault tree models     

Variance of system-reliability estimates with arbitrarily repeatedcomponents

A model is developed to determine the variance of system reliability estimates and to estimate confidence intervals for series-parallel systems with arbitrarily repeated components. For these systems, different copies of the same component-type are used several or many times within the system, but only a single reliability estimate is available for each distinct component-type. The single estimate is used everywhere the component appears in the system design, and component estimation-error is then magnified at the system-level. The "system-reliability estimate" variance and confidence intervals are derived when the number of component failures follow the binomial distribution with an unknown, yet estimable, probability of failure. The "system-reliability estimate" variance and confidence intervals are obtained by expressing system reliability as a linear sum of products of higher order moments for component unreliability. The generating function is used to determine the moments of the component-unreliability estimates. This model is preferable for many system reliability estimation problems because it does not require independent component and subsystem reliability estimates; it is demonstrated with an example     

Optimization Procedure for the Analysis of Coherent Structures

The determination of the reliability level at which to manufacture the components of a coherent structure so that the system reliability h(p) is at a certain level and the overall system cost is minimized is considered. The cost of utilizing component ci at reliability level pi, Ci(pi), is assumed to be a convex increasing function of pi with a continuous first derivative and Ci'(qi)>0 where qi is the lower bound on the reliability level for component ci. Since for most coherent structures the constraint set defines a nonconvex set, any mathematical programming procedure blindly applied to the program converges to a local optimum rather than a global optimum. However, in certain cases, the global optimum can be found for the series and parallel (SP) type of systems. The key to the solution is to optimize each module separately and then to substitute a component for each module where the cost function for the component is the value of the objective function for the module. As long as the cost function for each module maintains the convexity property with In R or In(1 - R) as the argument (R being the reliability of the module), the optimization procedure can continue and a global optimum found.