Reliability optimization for weighted voting system

The voting system studied consists of n units, each of which provides a binary decision (0 or 1), or abstains from voting. The system output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise the system output is 0.The existing method for evaluating the reliability of a weighted voting system can not be applied to real systems without imposing some restrictions because of its combinatorial complexity. In this paper a method is suggested which allows the reliability of weighted voting system to be exactly evaluated without imposing constraints on unit weights or threshold value. The approach is based on using a universal generating function technique. Using the suggested method an optimization procedure is developed for system reliability maximization by choosing proper unit weights and threshold values. A genetic algorithm is used as the optimization tool. Examples are presented.     

Asymmetric weighted voting systems

The weighted voting system (WVS) studied consists of n units, each of which provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise, system output is 0. The system input is either 0 or 1. Every unit is characterized by the probability of making decisions 0 and 1 and by probability of abstaining for each input. The system fails if the system output is not equal to its input.In this paper, an asymmetric WVS is suggested in which each voting unit has two weights. The first weight is applied when the unit's decision is 0 and the second weight is applied when the unit's decision is 1. The asymmetry of unit weights allows the WVS designer to take advantage of the knowledge of statistical asymmetry of voting units (asymmetric probabilities of making correct decisions with respect to the input). The paper presents an algorithm for asymmetric WVS reliability evaluation. This algorithm is based on using a universal generating function technique.For a system consisting of voting units with given reliability characteristics, one can maximize the entire system reliability by choosing proper unit weights and threshold value. An algorithm is suggested which finds the optimal unit weights and the threshold. A genetic algorithm is used as the optimization tool. An example is presented in which the superiority of asymmetric WVS over regular symmetric one is demonstrated.     

Optimal unit grouping in weighted voting systems

The voting system studied consists of n units that each provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise system output is 0.For a system consisting of voting units with given reliability characteristics one can maximize the entire system reliability by choosing proper unit weights and threshold values τ. In this paper it is shown that additional system reliability improvement can be achieved by grouping units in voting subsystems and tallying the weighted votes of these subsystems (groups) to make a final decision. An algorithm is suggested which finds the optimal element grouping as well as unit weights within each group, weights of the groups and corresponding system thresholds. The approach is based on using a universal generating function technique for evaluating system reliability. A genetic algorithm is used as the optimization tool. Examples are presented.     

Maximizing survivability of vulnerable weighted voting system

The weighted voting system (WVS) studied consists of N units that each provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise system output is 0.For a system consisting of voting units (VUs) with given reliability characteristics one can maximize the entire system reliability by choosing proper unit weights and threshold values τ.When a system operates in battle conditions or is affected by a corrosive medium or other hostile environment, its survivability (the ability to tolerate intentional attacks or accidental failures or errors) is becoming especially important. One of the ways to enhance voting system survivability is to separate its VUs.We formulate the problem of maximizing survivability of WVS by proper choice of unit weights and system threshold value and by unit separation.An algorithm for solving the problem is based on using a universal generating function technique for evaluating system survivability. A genetic algorithm is used as the optimization tool. Examples are presented.     

Modeling the reliability of threshold weighted voting systems

In many applications, ranging from target detection to safety monitoring systems, we are interested in determining whether or not to accept a hypothesis based on the information available. In this paper we model the reliability of threshold weighted voting systems (WVS) with multi-failure-modes, where a general recursive reliability function of the WVS is presented. We also develop approximation formulas for calculating the reliability of WVS based on a large number of units. We also develop reliability functions of time-dependent threshold weighted voting systems, where each unit is a function of time. Finally, the optimal stopping time that minimizes the total cost of the systems subject to a reliability constraint is discussed.     

Holistic reliability analysis of weighted voting systems from a multi-state perspective

The computation of the reliability of weighted voting systems is an important problem in reliability theory due to its potential application in security, target identification, safety and monitoring areas. Voting systems are used in a wide variety of applications where an acceptance or rejection decision has to be made about a binary proposition presented to the system. For these systems, it is of interest to obtain the probability so that based on the vote of decision-making units, the system aggregates these votes into the right decision when presented with such a proposition. This paper presents a holistic work on weighted voting system reliability by presenting modeling, computation, estimation and optimization techniques. The modeling part takes advantage of the structure of weighted voting systems to present a model of its reliability as a multi-state system. Next, based on the multi-state view of the system, an exact computational approach based on multi-state minimal cut and path vectors is introduced. The paper then acknowledges the computational complexity of the problem and provides a Monte Carlo simulation approach that estimates system reliability accurately and in an efficient computational time. Finally, an optimization heuristic that generates quasi-optimal solutions is presented that is able to solve the problem of maximizing the reliability of a weighted voting system based on a specified number of decision-making units with known reliability characteristics.     

A study of service reliability and availability for distributed systems

Distributed systems are usually designed and developed to provide certain important services such as in computing and communication systems. In this paper, a general model is presented for a centralized heterogeneous distributed system, which is widely used in distributed system design. Based on this model, the distributed service reliability which is defined as the probability of successfully providing the service in a distributed environment, an important performance measure for this type of systems, is investigated. An application example is used to illustrate the procedure. Furthermore, with the help of the model, various issues such as the release time to achieve a service reliability requirement, and the sensitivity of model parameters are studied. This type of analysis is important in the application of this type of models.     

Analyzing the behavior and reliability of voting systems comprising tri-state units using enumerated simulation

Voting is a common technique used in combining results from peer experts, for multiple purposes, and in a variety of domains. In distributed decision making systems, voting mechanisms are used to obtain a decision by incorporating the opinion of multiple units. Voting systems have many applications in fault tolerant systems, mutual exclusion in distributed systems, and replicated databases. We are specifically interested in voting systems as used in decision-making applications.In this paper, we describe a synthetic experimental procedure to study the behavior of a variety of voting system configurations using a simulator to: analyze the state of each expert, apply a voting mechanism, and analyze the voting results. We introduce an enumerated-simulation approach and compare it to existing mathematical approaches. The paper studies the following behaviors of a voting system: (1) the reliability of the voting system, R; (2) the probability of reaching a consensus, P_c; (3) certainty index, T; and (4) the confidence index, C. The configuration parameters controlling the analysis are: (1) the number of participating experts, N, (2) the possible output states of an expert, and (3) the probability distribution of each expert states. We illustrate the application of this approach to a voting system that consists of N units, each has three states: correct (success), wrong (failed), and abstain (did not produce an output). The final output of the decision-making (voting) system is correct if a consensus is reached on a correct unit output, abstain if all units abstain from voting, and wrong otherwise. We will show that using the proposed approach, we can easily conduct studies to unleash several behaviors of a decision-making system with tri-state experts.     

Extended great deluge algorithm for the imperfect preventive maintenance optimization of multi-state systems

This paper deals with preventive maintenance optimization problem for multi-state systems (MSS). This problem was initially addressed and solved by Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203]. It consists on finding an optimal sequence of maintenance actions which minimizes maintenance cost while providing the desired system reliability level. This paper proposes an approach which improves the results obtained by genetic algorithm (GENITOR) in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203]. The considered MSS have a range of performance levels and their reliability is defined to be the ability to meet a given demand. This reliability is evaluated by using the universal generating function technique. An optimization method based on the extended great deluge algorithm is proposed. This method has the advantage over other methods to be simple and requires less effort for its implementation. The developed algorithm is compared to than in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203] by using a reference example and two newly generated examples. This comparison shows that the extended great deluge gives the best solutions (i.e. those with minimal costs) for 8 instances among 10.     

A universal generating function approach for the analysis of multi-state systems with dependent elements

The paper extends the universal generating function technique used for the analysis of multi-state systems to the case when the performance distributions of some elements depend on states of another element or group of elements.     

Optimal separation of elements in vulnerable multi-state systems

In this paper we consider vulnerable systems, which can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a system performance rate, which is the quantitative measure of a system’s ability to perform its task. Both the impact of external factors (attack) and internal causes (failures) affect system survivability, which is determined as probability of meeting a given demand.One of the ways to enhance system survivability is to separate elements with the same functionality (parallel elements). Since system elements can have different performance rates and different availability, the way in which they are separated strongly affects system survivability. In this paper we formulate the problem of how to separate the elements of series-parallel system in order to achieve a maximal possible level of system survivability by the limited cost.An algorithm based on the universal moment generating function method is suggested for determination of the vulnerable series-parallel multi-state system survivability. A genetic algorithm is used as optimization tool in order to solve the structure optimization problem.     

Survivability of series–parallel systems with multilevel protection

In this paper, we consider vulnerable systems which can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a performance rate, which is the quantitative measure of a system's ability to perform its task. Both the impact of external factors (attack) and internal causes (failures) affect system survivability, which is determined as the probability of meeting a given demand.In order to increase the system's survivability a multilevel protection can be applied to its subsystems. In such systems, the protected subsystems are destroyed by external impacts only if all of the levels of their protection are destroyed.The paper describes an algorithm for evaluating the survivability of series–parallel systems with arbitrary configuration of multilevel protection. The algorithm is based on a composition of Boolean and the Universal Generating Function techniques. The adaptation of the algorithm for numerical implementation is suggested.Illustrative examples are presented.     

Reliability evaluation of multi-state weighted -out-of- systems

The k-out-of-n systems have been extensively studied in recent years. A binary weighted k-out-of-n model has also been reported in the literature. In this paper, we first compare two approaches for reliability evaluation of binary weighted k-out-of-n systems. We then provide two models of multi-state weighted k-out-of-n system models. Recursive algorithms are presented for reliability evaluation of these new models. The universal generating function approach is also used for reliability evaluation of multi-state weighted k-out-of-n systems.     

Optimizing survivability of vulnerable series–parallel multi-state systems

In this paper we consider vulnerable systems which can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a system performance rate, which is the quantitative measure of a system's ability to perform its task. Both the impact of external factors (attack) and internal causes (failures) affect system survivability which is determined as probability of meeting a given demand.We formulate the problem of finding structure of series–parallel multi-state system (including choice of system elements, their separation and protection) in order to achieve a desired level of system survivability by the minimal cost.An algorithm based on the universal generating function method is suggested for determination of the vulnerable series–parallel multi-state system survivability. A genetic algorithm is used as optimization tool in order to solve the structure optimization problem.     

Structure optimization of multi-state system with time redundancy

In this article, a multi-state system with time redundancy where each system element has its own operation time is considered. In addition, the system total task must be performed during the restricted time. The reliability optimization problem is treated as finding the minimal cost system structure subject to the reliability constraint. A method for reliability optimization for systems with time redundancy is proposed. This method is based on the universal generating function technique for the reliability index computation and on genetic algorithm for the optimization. It provides a solution for the optimization problem for the complex series–parallel system and for the system with bridge topology. Two types of systems will illustrate the approach: systems with ordinary hot reserve and systems with work sharing between elements connected in parallel. Numerical examples are also given.     

Structure optimization of multi-state system with two failure modes

When systems with two failure modes (STFM) are considered, introducing redundant elements may either increase or decrease system reliability. Therefore the problem of system structure optimization arises. In this paper we consider systems consisting of elements characterized by different reliability and nominal performance rates. Such systems are multi-state because they can have different levels of output performance depending on the combination of elements available at the moment. The algorithm that determines the structure of multi-state STFM, which maximizes system reliability and/or expected performance is presented. In this algorithm, system elements are chosen from a list of available equipment. Reliability is defined as the probability of satisfaction of given constraints imposed on system performance in both modes.The procedure developed to solve this problem is based on the use of a universal moment generating function (UMGF) for the fast evaluation of multi-state system reliability and a genetic algorithm for optimization. Basic UMGF technique operators are developed for two different types of systems, based, respectively, on transmitting capacity and on processing time. Examples of the optimization of series–parallel structures of both types are presented.     

Service reliability and performance in grid system with star topology

The paper considers grid computing systems in which the resource management systems (RMS) can divide service tasks into subtasks and send the subtasks to different resources for parallel execution. In order to provide desired level of service reliability the RMS can assign the same subtasks to several independent resources for parallel execution.The service reliability and performance indices are introduced and a fast numerical algorithm for their evaluation for arbitrary subtask distribution in grid with star architecture is presented. This algorithm is based on the universal generating function technique.Illustrative examples are presented.     

Optimal service task partition and distribution in grid system with star topology

The paper considers grid computing systems in which the resource management systems (RMSs) can divide service tasks into execution blocks (EBs) and send these blocks to different resources. In order to provide a desired level of service reliability the RMS can assign the same blocks to several independent resources for parallel (redundant) execution.By the optimal service task partition into the EBs and their distribution among resources, one can achieve the greatest possible service reliability and/or expected performance. The paper suggests an algorithm for solving this optimization problem. The algorithm is based on the universal generating function technique and on the evolutionary optimization approach.Illustrative examples are presented.     

Optimizing survivability of multi-state systems with multi-level protection by multi-processor genetic algorithm

In this paper we consider vulnerable systems which can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a performance rate, which is the quantitative measure of a system's ability to perform its task. Both the impact of external factors (stress) and internal causes (failures) affect system survivability, which is determined as probability of meeting a given demand.In order to increase the survivability of the system, a multi-level protection is applied to its subsystems. This means that a subsystem and its inner level of protection are in their turn protected by the protection of an outer level. This double-protected subsystem has its outer protection and so forth. In such systems, the protected subsystems can be destroyed only if all of the levels of their protection are destroyed. Each level of protection can be destroyed only if all of the outer levels of protection are destroyed.We formulate the problem of finding the structure of series–parallel multi-state system (including choice of system elements, choice of structure of multi-level protection and choice of protection methods) in order to achieve a desired level of system survivability by the minimal cost. An algorithm based on the universal generating function method is used for determination of the system survivability. A multi-processor version of genetic algorithm is used as optimization tool in order to solve the structure optimization problem. An application example is presented to illustrate the procedure presented in this paper.     

Importance and sensitivity analysis of multi-state systems using the universal generating function method

A method for the evaluation of element reliability importance in a multi-state system is proposed. The method is based on the universal generating function technique. It provides an effective importance analysis tool for complex series–parallel multi-state systems with a different physical nature of performance and takes into account a required performance (demand). The method is also extended for the sensitivity analysis of important multi-state system output performance measures: mean system performance and mean unsupplied demand during operating period. Numerical examples are given.