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.     

Optimal multilevel protection in series–parallel systems

This paper considers vulnerable systems that 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 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.In such systems, different protections play different roles in providing for the system's survivability. Subject to budget limitations a question arises which protections should be applied to obtain the desired survivability. An algorithm for solving the protection cost minimization problem subject to survivability constraint is presented in the paper. The algorithm is based on a universal generating function technique used for system survivability evaluation and on a genetic algorithm used as an optimization engine.Illustrative example is 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.     

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.     

Block diagram method for analyzing multi-state systems with uncovered failures

The paper suggests a modification of the generalized reliability block diagram (RBD) method for evaluating reliability and performance indices of multi-state systems with uncovered failures. Such systems (or their subsystems) can fail to perform their task in the case of undetected failure of any one of their elements. Examples of this effect can be found in computing systems, electrical power distribution networks, pipe lines carrying dangerous materials etc. The suggested method based on a universal generating function technique allows performance distribution of complex multi-state series-parallel system with uncovered failures to be obtained using a straightforward recursive procedure. Illustrative examples are presented.     

Consecutive sliding window systems

This paper proposes a new model that generalizes the linear multi-state sliding window system to the case of m consecutive overlapping windows. In this model the system consists of n linearly ordered multi-state elements. Each element can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. The system fails if in each of at least m consecutive overlapping groups of r consecutive elements (windows) the sum of the performance rates of elements belonging to the group is lower than a minimum allowable level. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of evaluating system reliability and elements' reliability importance indices are presented.     

Optimal load distribution in series-parallel systems

This paper presents an algorithm for determining an optimal loading of elements in series-parallel systems. The optimal loading is aimed at achieving the greatest possible expected system performance subject to repair resource constraint. The model takes into account the dependence of elements’ failure rates on their load. The optimization algorithm uses a universal generating function technique for evaluating the expected system performance, and a genetic algorithm for determining the optimal load distribution. An illustrative example of load distribution optimization is presented.     

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.     

Multi-state systems with multi-fault coverage

The paper introduces a new model of fault level coverage for multi-state systems in which the effectiveness of recovery mechanisms depends on the coexistence of multiple faults in related elements. Examples of this effect can be found in computing systems, electrical power distribution networks, pipelines carrying dangerous materials, etc. For evaluating reliability and performance indices of multi-state systems with imperfect multi-fault coverage, a modification of the generalized reliability block diagram (RBD) method is suggested. This method, based on a universal generating function technique, allows performance distribution of complex multi-state series–parallel system with multi-fault coverage to be obtained using a straightforward recursive procedure. Illustrative examples are presented.     

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.     

Generalised importance measures for multi-state elements based on performance level restrictions

In this paper we consider some commonly used importance measures in a generalised version proposed by some of the authors for application to multi-state systems constituted by multi-state elements. Physically, these measures characterize the importance for a multi-state element of achieving a given level of performance and their definitions entail evaluating the system availability and/or performance when the functioning of the element of interest is restricted in performance.With reference to a predefined threshold of element performance, two different types of restrictions are considered. The first one limits the elements' reachable states to those corresponding to performances either larger or not larger than the threshold level. The second one allows the element to visit all its states but limits its performance to values larger or not larger than the performance threshold.An approach based on the universal generating function technique is proposed for the evaluation of the introduced importance measures. A numerical application is provided in order to highlight the informative content of the introduced measures.     

Minmax defense strategy for complex multi-state systems

This paper presents a general optimization methodology that merges game theory and multi-state system survivability theory. The defender has multiple alternatives of defense strategy that presumes separation and protection of system elements. The attacker also has multiple alternatives of its attack strategy based on a combination of different possible attack actions against different groups of system elements. The defender minimizes, and the attacker maximizes, the expected damage caused by the attack (taking into account the unreliability of system elements and the multi-state nature of complex series-parallel systems). The problem is defined as a two-period minmax non-cooperative game between the defender who moves first and the attacker who moves second. An exhaustive minmax optimization algorithm is presented based on a double-loop genetic algorithm for determining the solution. A universal generating function technique is applied for evaluating the losses caused by system performance reduction. Illustrative examples with solutions are presented.     

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.     

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.     

Optimal reserve management for restructured power generating systems

This paper presents a technique to determine the optimal reserve structure (reserve providers and the corresponding reserve capacity) for a restructured power generating system (GS). The reserve of a GS can be provided by its own generating units and can also be purchased from other GSs through the reserve agreements. The objective of reserve management for a GS is to minimize its total reserve cost while satisfying the reliability requirement. The reserve management is a complex optimization problem, which requires a large amount of calculations. In order to simplify the evaluation, a complex generating system (CGS) consisting of different GSs and the corresponding transmitting network is represented by its multi-state reliability equivalents. The universal generating functions (UGFs) of these equivalents are developed and the special operators for these UGFs are defined to evaluate the reliability of a particular GS, which has reserve agreements with other GSs in the CGS. The genetic algorithm (GA) has been used to solve the optimization problem. An improved power system-IEEE reliability test system is used to illustrate the technique.     

Optimization of series–parallel multi-state systems under maintenance policies

In the redundancy optimization problem, the design goal is achieved by discrete choices made from components available in the market. In this paper, the problem is to find, under reliability constraints, the minimal cost configuration of a multi-state series–parallel system, which is subject to a specified maintenance policy. The number of maintenance teams is less than the number of repairable components, and a maintenance policy specifies the priorities between the system components. To take into account the dependencies resulting from the sharing of maintenance teams, the universal generating function approach is coupled with a Markov model. The resulting optimization approach has the advantage of being mainly analytical.     

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.     

Extended block diagram method for a multi-state system reliability assessment

The presented method extends the classical reliability block diagram method to a repairable multi-state system. It is very suitable for engineering applications since the procedure is well formalized and based on the natural decomposition of the entire multi-state system (the system is represented as a collection of its elements). Until now, the classical block diagram method did not provide the reliability assessment for the repairable multi-state system. The straightforward stochastic process methods are very difficult for engineering application in such cases due to the “dimension damnation”—huge number of system states. The suggested method is based on the combined random processes and the universal generating function technique and drastically reduces the number of states in the multi-state model.     

Redundancy analysis for repairable multi-state system by using combined stochastic processes methods and universal generating function technique

This paper discusses a type of redundancy that is typical in a multi-state system. It considers two interconnected multi-state systems where one multi-state system can satisfy its own stochastic demand and also can provide abundant resource (performance) to another system in order to improve the assisted system reliability. Traditional methods are usually not effective enough for reliability analysis for such multi-state systems because of the “dimensional curse” problem. This paper presents a new method for reliability evaluation for the repairable multi-state system considering such kind of redundancy. The proposed method is based on the combination of the universal generating function technique and random processes methods. The numerical example is presented to illustrate the proposed method.