Multistate Network Reliability Research Articles

Research on Reliability Prediction Methods for Dynamic Networks Based On Deep Neural Networks

In reality many networks, such as information storage networks, can be modelled as a multi-state stochastic flow network. The reliability of an information storage network is the probability that the network is able to deliver the required amount of information data to the specified server, and its an important metric to assess the performance of an information storage network. Most existing multi-state network reliability assessment algorithms are calculated using very small capacity vectors. However, calculating the reliability of such a network requires reuse of algorithms and is inefficient when the network size, demand, etc., changes rapidly, resulting in managers not making timely decisions accordingly. In order to obtain reliability metrics quickly, this paper proposes a dynamic network reliability prediction model based on deep neural networks (DNNs), which allows the corresponding reliability to be obtained quickly even when the network is changing rapidly. Finally, a local information storage network is used as an example for validation to illustrate the feasibility of the prediction model.

Optimal carrier Selection Based on Reliability Under Delivery Spoilage and Cost Constraint

In a highly competitive market environment, outsourcing of logistics service is one of the most significant strategies for enterprises to enhance competitiveness. However, how to construct an efficient and reliable logistics network by selecting suitable logistics carriers is a complex but important practical issue. In general, there are multiple carriers to provide freight service along each route of a logistics network. Moreover, the transportation capacity of each carrier has multiple states due to various factors, which can be portrayed by a capacity probability distribution. In addition, delivery spoilage and cost constraint are the key indicators affecting carrier selection. Therefore, this paper studies the logistics carrier selection problem from the perspective of multi-state network reliability under delivery spoilage and cost constraint, and apply the minimal path (MP) method to solve the problem. The practicality of the algorithm is verified by example analysis.

An Improved Method to Search All Minimal Paths in Networks

Minimal paths (MPs) play an important role in system reliability analysis, such as binary network reliability evaluation and multistate network reliability evaluation. Searching all MPs is an NP-hard problem. The direct-search-based algorithm implements the depth-first search (DFS) mechanism to find all MPs, which is a simple and efficient method. The current efficient DFS algorithm is based on backtracking and linked list by nodes. In this article, we propose several improvements to speed up the current algorithm. First, we find that the relative positional information of the network can be used to avoid visiting some unnecessary nodes. Thus, we improve the algorithm by incorporating the point in the polygon algorithm to avoid visiting nodes surrounded by other nodes in a network. Second, we further improve the current backtracking condition in which the time complexity of implementing each backtrack is reduced. In addition, we find that different search branches from a similar node are independent and can be executed in parallel. A parallel search mechanism is proposed and incorporated to speed up the searching process. Through computational experiments on several benchmark networks and real transportation networks, we demonstrate that the improved algorithm is more efficient than the existing algorithm. The proposed algorithm becomes more advantageous as the size of the network grows.

Survey of multistate network reliability and resilience evaluation methods

Survey of multistate network reliability and resilience evaluation methods

MC-based simulation approach for two-terminal multi-state network reliability evaluation without knowing d-MCs

A two-terminal multi-state network (TMSN) is a network (system) with multi-state subsystems (arcs and nodes) and two terminals (source and sink nodes). In a TMSN, system reliability is a widely applied performance indicator of demand satisfaction. To assess system reliability, previous studies relied on knowing the upper bounds (d-MCs) or lower bounds (d-MPs) for specified demand d and a given capacity probability distribution. Accordingly, a novel minimal cut (MC)-based simulation approach is proposed, which does not rely on knowing the d-MCs and capacity probability distribution. An algorithm based on Monte Carlo simulation with demand confirmation via MCs is developed to estimate system reliability. In addition, an extension with a time attribute is examined to investigate the reliability degradation with time. Additional case studies, including a real-life instance of the Taiwan Academic Network, are analyzed to validate the scalability and applicability of the proposed approaches.

Simulation approaches for multi-state network reliability estimation: Practical applications

In this study, simulation approaches for the reliability estimation of a multi-state network (MSN) are developed; the approaches obviate the reliance on known capacity probability distributions. Two procedures are proposed to randomly generate the system (capacity) state, which is further compared with the lower bounds of demand, for estimating system reliability. In particular, the simulation approaches consider the time-varying reliability function for components in the MSN. Two examples, i.e., with and without considering time-varying reliability functions, are investigated to determine the effectiveness (accuracy, convergence, and variance) and efficiency of the proposed procedures. Additionally, a practical case of the National Science Foundation Network is investigated to validate the scalability and applicability of the proposed procedures. Experimental results show that both proposed procedures are effective and efficient for estimating the system reliability of an MSN.

Pareto-based grouping meta-heuristic algorithm for humanitarian relief logistics with multistate network reliability

This article considers a biobjective location-routing problem to deliver relief resources to the victims affected by a disaster under uncertainty in demand, transportation infrastructure, and travel time. Since transportation networks are exposed to a considerable level of uncertainty, choosing the reliable path for relief goods to be transmitted to the affected areas ensures the arrival of these supplies. For the first time, route reliability is calculated based on the multistate theory, and the universal generating function technique is used for network reliability assessment. The problem is formulated as a multiperiod robust biobjective mixed-integer programming model. Two objective functions are considered: (a) decreasing the sum of arrival times of relief vehicles at the demand nodes for delivering aids to the affected areas, and (b) increasing the minimum route reliability for all the serving vehicles. A novel multiobjective grouping algorithm is proposed to obtain the Pareto-optimal solutions of the problem. Then, its performance is compared with two other multiobjective grouping algorithms. To evaluate the solution method, the algorithms are implemented on various test problems and compared statistically. A case study is presented to illustrate the potential applicability of our model. Additionally, to determine the effect of the changes in the main parameters of the problem on the value of objective functions, the sensitivity analyses are performed and the managerial insights are given.

An improved bounding algorithm for approximating multistate network reliability based on state-space decomposition method

The evaluation of two-terminal multistate network reliability precisely is a NP-hard problem. Approximating the reliability bounds with state space decomposition is an effective method to tradeoff the computational effort and the acceptable reliability approximate value when the scale of the network is relatively large. This decomposition process cannot avoid the shortage of a large amount of computational effort consumed by sets of unspecified states with little contributions to reliability bounds. Thus, preset critical values are used to filter out sets of unspecified states with less probability. However, managers cannot predict the preset critical value that satisfies their demand. Therefore, we first proposed a serial bounding algorithm based on the breadth-first mechanism, wherein sets of unspecified states on the same generation are all decomposed before moving to the next generation. The mechanism of combining serial computing with parallel computing is further developed to fully utilize computer capability. In addition, the preset critical values are not required before running the algorithm. The results of efficiency comparison demonstrate that the proposed algorithm can significantly improve the efficiency for approximating multistate network reliability. Stability investigations show that the computational efficiency of the proposed algorithm is more stable and effective under various component state distributions.

A Minimal Path-Based Method for Computing Multistate Network Reliability

Most of modern technological networks that can perform their tasks with various distinctive levels of efficiency are multistate networks, and reliability is a fundamental attribute for their safe operation and optimal improvement. For a multistate network, the two-terminal reliability at demand level d, defined as the probability that the network capacity is greater than or equal to a demand of d units, can be calculated in terms of multistate minimal paths, called d-minimal paths (d-MPs) for short. This paper presents an efficient algorithm to find all d-MPs for the multistate two-terminal reliability problem. To advance the solution efficiency of d-MPs, an improved model is developed by redefining capacity constraints of network components and minimal paths (MPs). Furthermore, an effective technique is proposed to remove duplicate d-MPs that are generated multiple times during solution. A simple example is provided to demonstrate the proposed algorithm step by step. In addition, through computational experiments conducted on benchmark networks, it is found that the proposed algorithm is more efficient.

Efficient Enumeration of d‐Minimal Paths in Reliability Evaluation of Multistate Networks

A number of real‐world complex networks can be modeled as multistate networks for performance analysis. A multistate network consists of multistate components and possesses multiple different performance levels. For such a network, reliability is concerned with the probability of the network capacity level greater than or equal to a predetermined demand level d. One major method for multistate network reliability evaluation is using d‐minimal paths. This paper proposes an efficient algorithm to find d‐minimal paths. First, a new concept of qualified state vector is defined so as to fix a relatively smaller search space of d‐minimal paths, and a sufficient and necessary condition for a qualified state vector to be d‐minimal path is established. Then, the max‐flow algorithm and the enumeration algorithm are integrated to search for d‐minimal paths in the determined search space that is recursively divided into subspaces such that the searching efficiency can be increased as much as possible. Both analytical and numerical results show that the proposed algorithm is more efficient in finding all d‐minimal paths. In addition, a case study related to power transmission network is performed to demonstrate the implication of network reliability.

Single minimal path based backup path for multi-state network

Backup path is an important mechanism to sustain the reliability of a multi-state network. As a popular backup path method, the double minimal path based backup path algorithm can improve the multi-state network’s reliability when the main paths fail. However, this algorithm cannot work efficiently when a single main minimal path fails. To improve the reliability in the first main minimal path failure case, we propose a single minimal path based backup path algorithm. In the single minimal path based backup path algorithm, two disjoint minimal paths are used as the main routing pair to transmit the data, and one single minimal path, which is disjoint with the main minimal paths, acts as the backup path. In the second main minimal path failure case, we propose a double–single minimal path based backup path algorithm to improve the multi-state network reliability. To develop the single minimal path based backup path and the double–single minimal path based backup path algorithms, this article first formulates the multi-state network reliability analysis problem. Then, a solution procedure is proposed to calculate the multi-state network reliability. Furthermore, numerical examples are given to validate the effectiveness of the algorithms. Finally, some comparisons are made between the single minimal path based backup path/double–single minimal path based backup path and the double minimal path based backup path/double minimal path based backup path algorithms. The comparison results indicate that the single minimal path based backup path and the double–single minimal path based backup path algorithms lead to considerable improvement in terms of the multi-state network reliability in the first and the second main minimal path failure cases, respectively, which are verified by both the mathematical analysis and numerical experiments.

Capacitated stochastic coloured Petri net-based approach for computing two-terminal reliability of multi-state network

Classical network reliability problems assume both networks and components have only binary states, fully working or fully failed states. But many actual networks are multi-state, such as communication networks and transportation networks. The nodes and arcs in the networks may be in intermediate states which are not fully working either fully failed. A simulation approach for computing the two-terminal reliability of a multi-state network is described. Two-terminal reliability is defined as the probability that d units of demand can be supplied from the source to sink nodes under the time threshold T. The capacities of arcs may be in a stochastic state following any discrete or continuous distribution. The transmission time of each arc is also not a fixed number but stochastic according to its current capacity and demand. To solve this problem, a capacitated stochastic coloured Petri net is proposed for modelling the system behaviour. Places and transitions respectively stand for the nodes and arcs of a network. Capacitated transition and self-modified token colour with route information are defined to describe the multi-state network. By the simulation, the two-terminal reliability and node importance can be estimated and the optimal route whose reliability is highest can also be given. Finally, two examples of different kinds of multi-state networks are given.

Optimal resource assignment to maximize multistate network reliability for a computer network

Computer network is a major tool to transmit data in our modern society. How to evaluate and enhance network reliability is thus an important issue for organizations, especially to maximize network reliability. A computer network is a multistate network in which each edge has several possible capacities with a probability distribution and may fail. The multistate network reliability is the probability that the maximal flow is no less than a given demand. From the standpoint of quality management, a further problem is to reassign the existing resources for maximizing multistate network reliability without changing the network topology. Hence, this paper focuses on the resource assignment problem to propose an efficient approach based on the simple genetic algorithm. In which, a resource assignment is represented as a chromosome and the corresponding multistate network reliability is the fitness value of the chromosome. The experimental results show that the proposed algorithm can derive the optimal resource assignment in a reasonable time.

Multistate stochastic network interdiction via reliability modelling and evolutionary optimization

This paper extends the current stochastic network interdiction problem (SNIP) approaches to allow for the inclusion of multistate behaviour of link flow when a given link is targeted for interdiction. Under this modelling scenario, described as the multistate SNIP, a new optimization model is described for maximizing the reliability associated with an interdiction strategy constrained to a prespecified s— t flow and a prespecified cost. An evolutionary algorithm, known as the probabilistic solution discovery algorithm, is used to obtain quasi-optimal solutions for the multistate SNIP problem. This algorithm comprises a three-step process that implements multistate network reliability computation techniques along with an evolutionary optimization routine to find interdiction strategies. The solutions to different test cases show that the optimization routine is able to identify high-quality solutions in a significantly reduced search space.

An efficient method for reliability evaluation of multistate networks given all minimal path vectors

The multistate networks under consideration consist of a source node, a sink node, and some independent failure-prone components in between the nodes. The components can work at different levels of capacity. For such a network, we are interested in evaluating the probability that the flow from the source node to the sink node is equal to or greater than a demanded flow of d units. A general method for reliability evaluation of such multistate networks is using minimal path (cut) vectors. A minimal path vector to system state d is called a d-MP. Approaches for generating all d-MPs have been reported. Given that all d-MPs have been found, the issue becomes how to evaluate the probability of the union of the events that the component state vector is greater than or equal to at least one of the d-MPs. There is a need for a more efficient method of determining the probability of this union of events. In this paper, we report an efficient recursive algorithm for this union probability evaluation based on the Sum of Disjoint Products (SDP) principle, and name it the Recursive Sum of Disjoint Products (RSDP) algorithm. The basic idea is that, based on the SDP principle and a specially defined “maximum” operator, “⊕”, the probability of a union with L vectors can be calculated via calculating the probabilities of several unions with L−1 vectors or less. The correctness of RSDP is illustrated. The efficiency of this algorithm is investigated by comparing it with an existing algorithm that is generally accepted to be efficient. It is found that RSDP is more efficient than the existing algorithm when the number of components of a system is not too small. RSDP provides us with an efficient, systematic and simple approach for evaluating multistate network reliability given all d-MPs.

A new approach to evaluate reliability of multistate networks under the cost constraint

A multistate network is a system composed of multistate components. The network reliability under the cost constraint for level ( d, c) can be computed in terms of ( d, c)-MP (MP stands for minimal path) which is a vector such that d units of flow can be transmitted between two specified nodes with the total cost not greater than c. In this study, a new algorithm was developed to evaluate the reliability of multistate networks under cost constraint in terms of the entire ( d, c)-MPs. The proposed method is more efficient than the best-known existing algorithm. One example is illustrated to show how all ( d, c)-MPs are generated by the proposed algorithm. The reliability of this example is then computed. The computational complexity of the proposed algorithm is also analyzed.

Multistate network reliability evaluation under the maintenance cost constraint

Many real-world systems (such as electric power, transportation, etc.) are multistate systems composed of multistate components. Such systems may be regarded as flow networks whose arcs have independent, discrete, limited and multivalued random capacities. Their reliability can be computed in terms of minimal cut (MC) vectors to level ( d, c) (named ( d, c)-MCs here), using the probability that d units of flow can be transmitted from the source node to the sink node such that the total maintenance cost of each d-MCs is less than or equal to c. In this study, all MCs are assumed to be known in advance and we developed an intuitive algorithm based on some simple concepts that were found in this study to find the entire ( d, c)-MCs before calculating the reliability value of a network. One example is illustrated to show how all ( d, c)-MCs are generated by the proposed algorithm. Then the reliability of this example is computed. The computational complexity of the proposed algorithm is also analyzed.