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**1 - 7**of**7**### (Senior Member, IIE) Associate Professor of Industrial Engineering

"... A common criticism leveled at prior research on the design of component test plans for obtaining inferences on system reliability is that it has assumed the interfaces between components to be perfect. Practitioners are often uncomfortable with testing only the components because system failures are ..."

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A common criticism leveled at prior research on the design of component test plans for obtaining inferences on system reliability is that it has assumed the interfaces between components to be perfect. Practitioners are often uncomfortable with testing only the components because system failures are often caused by imperfect interfaces. This paper addresses this criticism by explicitly considering the reliability of interfaces and also allowing for system testing in addition. A series system is considered and optimum test plans are derived by solving a two stage mathematical program. The results indicate that depending on costs and interface reliabilities, optimal plans may require testing only the system, only the components, or both. 1 1. Introduction An important step in the development of any system is a program of testing to ensure that the system will perform as desired. Such tests can often be quite expensive and thus one commonly looks for minimum cost test plans. The const...

### (Senior Member, IIE) Associate Professor of Industrial Engineering

"... Since mathematical models based on component reliabilities are frequently used for prediction of system reliability, it stands to reason that cost-effective inferences on the reliability of a system could be made on the basis of tests of its constituent components. Prior research in the area of syst ..."

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Since mathematical models based on component reliabilities are frequently used for prediction of system reliability, it stands to reason that cost-effective inferences on the reliability of a system could be made on the basis of tests of its constituent components. Prior research in the area of system-based component testing has for the most part addressed the development of plans that test only the components. From a practitioner's point of view, this is an issue of concern since system failures are often caused by imperfect interfaces and other causes that are not directly attributable to component failures. The exclusion of system tests may thus be unappealing. This paper addresses the development of test plans that explicitly consider the possibility of interface failures. The paper analyzes a series system to determine when testing should be performed on the system alone, on the components only, and on both, depending on test costs and interface reliabilities. Optimum test plans a...

### Modular Test Plans for Certification of Software Reliability

"... This paper considers the problem of certifying the reliability of a software system that can be decomposed into a finite number of modules. It uses a Markovian model for the transfer of control between modules in order to develop the system reliability expression in terms of the module reliabilities ..."

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This paper considers the problem of certifying the reliability of a software system that can be decomposed into a finite number of modules. It uses a Markovian model for the transfer of control between modules in order to develop the system reliability expression in terms of the module reliabilities. A test procedure is considered in which only the individual modules are tested and the system is certified if, and only if, no failures are observed. The minimum number of tests required of each module is determined such that the probability of certifying a system whose reliability falls below a specified value R 0 is less than a specified small fraction b. This sample size determination problem is formulated as a two-stage mathematical program and an algorithm is developed for solving this problem. Two examples from the literature are considered to demonstrate the procedure. Keywords: Software reliability; Modular Tests; Sample Size Determination; Mathematical Programming 1 1. Introduc...

### Minimum Cost Test Plans for a Series System with Imperfect Interfaces

"... In order to draw cost-effective inferences on the reliability of a system one approach is to design test plans that only test the components of the system. However, such an approach is inappropriate when the component failures are not independent or when interfaces between components cause system fa ..."

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In order to draw cost-effective inferences on the reliability of a system one approach is to design test plans that only test the components of the system. However, such an approach is inappropriate when the component failures are not independent or when interfaces between components cause system failures. In such instances one must plan on testing the system as well as its components and a natural question of interest is the relative extent to which the system and the components should be tested. This paper analyzes such a situation for a series system and determines the optimal plan for allocating the total testing effort between component and system tests. It also illustrates the use of mathematical programming techniques for obtaining test plans that guarantee conventional statistical properties. Running Title: Optimum Test Plans Key Words: Reliability, Mathematical Programming, Series System, Test Plans AMS Subject Classification: 62 Statistics, 90 Economics, operations research...

### System-Based Component Test Plans For Reliability Inferences

"... This paper addresses the design of system-based component test plans for demonstrating reliability of a series system. There are two primary contributions of this work. First, unlike most of the prior work in this area which has relied on the component failure times being exponentially distributed, ..."

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This paper addresses the design of system-based component test plans for demonstrating reliability of a series system. There are two primary contributions of this work. First, unlike most of the prior work in this area which has relied on the component failure times being exponentially distributed, this paper examines another common failure time distribution, namely the Weibull distribution and develops a test plan that exploits the relationship between the Weibull and the exponential distributions. Second, it introduces the notion of imperfect interfaces between components within the system which is another issue that all prior work has ignored, and results in plans that also call for system level tests. 1. Introduction Dedicated programs of testing are an integral part of the process by which new systems are designed and developed. The purpose of such programs is to ensure that the system will satisfy performance criteria, prior to their actual deployment in the field. While there are many different formats that such test plans could follow, the two basic options are (a) system testing, and (b) component testing. With the first option, the entire system is assembled and then tested in order to draw inferences on its reliability, while in the latter case the components that make up the system are tested, and then based on the results of these component tests one draws inferences about the system reliability. Several variations of these two basic options are possible. For instance, one may conduct tests at the level of subassemblies (or subsystems) that are composed of individual components, and that are themselves part of the entire system. Alternatively, one could adopt some combination of system, subsystem and component testing. The choice of an appropriate test plan...

### Words: System Reliability, Component Testing, Series System, Type-II Censoring, Producer and Consumer Risk, Optimization.

"... This paper considers the design of a minimum cost, system based, component test plan for evaluating the reliability of a series system of n different components, each of which has a constant, unknown failure rate. The basic problem was first considered by Gal [4], and later generalized and extended ..."

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This paper considers the design of a minimum cost, system based, component test plan for evaluating the reliability of a series system of n different components, each of which has a constant, unknown failure rate. The basic problem was first considered by Gal [4], and later generalized and extended by Mazumdar [5]. Different aspects of the test design problem for a series system have been addressed in [1, 6, 7]; parallel systems have also been addressed in [8, 9]. Earlier work on test procedures for series systems has employed Type I censoring where components are tested with replacement of failed components until the end of a preassigned length of time, with the number of failures observed being a random variable. One of the drawbacks of this approach is that it often leads to optimum test times that might be very long. In this paper, we consider a test procedure that uses Type-II censoring, where a predetermined number of components of each type is tested to failure with replacement after each failure, and the total observed time of testing for each component is a random variable. The design of the test plan is stated as an optimization problem to minimize test costs with constraints to ensure that specified levels of producer and consumer risk are not exceeded. This formulation is used frequently in common test plans, e.g., MIL-HDBK-781D [10]. 3 After stating the assumptions and the notation employed in this paper, we motivate the logic used in formulating the optimization problem by first considering the general problem with a normal approximation. We then describe a simpler formulation that dispenses with the approximation and show how this may be readily solved to arrive at the optimum test parameters. An illustrative numerical example is provided and implicatio...

### Reliability of Module Based Software System

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"... This paper consider the problem of determining the reliability of a software system which can be decomposed in a number of modules. We have derived the expression of the reliability of a system using the Markovian model for the transfer of control between modules in order. We have given the expressi ..."

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This paper consider the problem of determining the reliability of a software system which can be decomposed in a number of modules. We have derived the expression of the reliability of a system using the Markovian model for the transfer of control between modules in order. We have given the expression of reliability by considering both benign and catastrophic failure. The expression of reliability presented in this work is applicable for some control software which are designed to detect its own internal errors. 1