Minimal cut sets of s–t networks with k-out-of-n nodes

In this paper, we extend traditional directed s–t network by letting nodes have k-out-of-n property: To generate output flows, a node must receive at least k flows from its n input links, where k is an integer assigned for the node and its value can be any number from 1 to n. To evaluate the system reliability, minimal cut sets for the extended network are defined for nodes. Under this definition, an extended network and its sink node have the same minimal cut sets. A new algorithm is designed to generate minimal cut sets for all nodes, starting with the source node and ending with the sink node. With different initializations, the algorithm can be applied for extended s–t networks with or without node failures.     

Cutset enumeration of network systems with link and node failures

Network reliability analysis has received considerable attention and is thus widely studied to predict and prevent any network failure. However, most of such works presume perfectly reliable nodes. Although a few studies have considered both link and node failures, none of these methods has utilized the minimal paths or cuts, which are considered as fundamental approaches in the network reliability evaluation. An efficient method for deducing the minimal cutsets of a system subject to both link and node failures from the minimal cutsets of the system, which assumes perfect node reliability, is presented. The proposed method does not require re-enumeration of minimal cutsets for the additional consideration of the node failures. For a simple extension of such a method, the proposed approach can be embedded in any exact or approximate algorithm to account for link failures as well as node failures. As a result, the application of this method would be more realistic and valuable in practice for the reliability evaluation of networks with unreliable nodes.     

On computing the reliability of opportunistic multihop networks with Mobile relays

Opportunistic multihop networks with mobile relays recently have drawn much attention from researchers across the globe due to their wide applications in various challenging environments. However, because of their peculiar intrinsic features like lack of continuous connectivity, network partitioning, highly dynamic behavior, and long delays, it is very arduous to model and effectively capture the temporal variations of such networks with the help of classical graph models. In this work, we utilize an evolving graph to model the dynamic network and propose a matrix‐based algorithm to generate all minimal path sets between every node pair of such network. We show that these time‐stamped‐minimal‐path sets (TS‐MPS) between each given source‐destination node pair can be used, by utilizing the well‐known Sum‐of‐Disjoint Products technique, to generate various reliability metrics of dynamic networks, ie, two‐terminal reliability of dynamic network and its related metrics, ie, two‐terminal reliabilities of the foremost, shortest, and fastest TS‐MPS, and Expected Hop Count. We also introduce and compute a new network performance metric?Expected Slot Count. We use two illustrative examples of dynamic networks, one of four nodes, and the other of five nodes, to show the salient features of our technique to generate TS‐MPS and reliability metrics.     

An Efficient Reliability Evaluation Approach for Networks with Simultaneous Multiple‐Node‐Pair Flow Requirements

This paper proposes an efficient approach for reliability evaluation of multiple‐sources and multiple‐destinations flow networks. The proposed approach evaluates multiple node pair capacity related reliability. The proposed approach is a three‐step approach; in the first step, it enumerates network minimal cut sets. These network minimal cut sets are then used to enumerate subset cuts in the second step. Third step involves the evaluation of multiple node pair capacity related reliability from the enumerated subset cuts using a multi‐variable inversion sum‐of‐disjoint set approach. The proposed approach can be used for optimal network design as it combines multiple performance requirements, that is, flow requirements between multiple node pairs and network reliability, in a single criterion. Copyright © 2016 John Wiley & Sons, Ltd.     

A generalized multistate-based path vector approach to multistate two-terminal reliability

The two-terminal reliability problem assumes that a network and its elements are either in a working or a failed state. However, many practical networks are built of elements that may operate in more than two states i.e., elements may be degraded but still functional. Multistate two-terminal reliability at demand level d (M2TR_d) can be defined as the probability that the system capacity generated by multistate components is greater than or equal to a demand of d units. This paper presents a fully multistate-based algorithm that obtains the multistate equivalent of binary path sets, namely, Multistate Minimal Path Vectors (MMPVs), for the M2TR_d problem. The algorithm mimics natural organisms in the sense that a select number of arcs inherit information from other specific arcs contained in a special set called the “primary set.” The algorithm is tested and compared with published results in the literature. Two features of the algorithm make it relevant: (i) unlike other approaches, it does not depend on an a priori knowledge of the binary path sets to obtain the MMPVs; and (ii) the use of an information sharing approach and network reduction technique significantly reduce the number of vector analyses needed to obtain all the component levels that guarantee system success. Additionally, the complexities associated with the computation of reliability are discussed. A Monte Carlo simulation approach is used to obtain an accurate estimate of actual M2TR values based on MMPVs. Examples are used to validate the algorithm and the simulation procedure.     

A simple MC-based algorithm for evaluating reliability of stochastic-flow network with unreliable nodes

A MP/minimal cutset (MC) is a path/cut set such that if any edge is removed from this path/cut set, then the remaining set is no longer a path/cut set. An intuitive method is proposed to evaluate the reliability in terms of MCs in a stochastic-flow network subject to both edge and node failures under the condition that all of the MCs are given in advance. This is an extension of the best of known algorithms for solving the d-MC (a special MC but formatted in a system-state vector, where d is the lower bound points of the system capacity level) problem from the stochastic-flow network without unreliable nodes to with unreliable nodes by introducing some simple concepts. These concepts were first developed in the literature to implement the proposed algorithm to reduce the number of d-MC candidates. This method is more efficient than the best of known existing algorithms regardless if the network has or does not have unreliable nodes. Two examples are illustrated to show how the reliability is determined using the proposed algorithm in the network with or without unreliable nodes. The computational complexity of the proposed algorithm is analyzed and compared with the existing methods.     

Search for all MCs in networks with unreliable nodes and arcs

A simple method is proposed to search for all minimal cutsets (MCs ) for imperfect networks reliability subject to both arc and node failures under the condition that all of the MCs in the network with perfect nodes are given in advance. The proposed method does not require re-enumeration for all of the MCs for additional node failure consideration. All of the MC candidates found in the proposed algorithm are actual MCs without any need for further verification. This algorithm is more effective than the existing algorithm in which every MC candidate is not verified as a MC. No identical MCs are found using the proposed algorithm, which does not duplicate MCs and is more efficient than the existing methods. Only simple concepts are used to implement the proposed algorithm, which makes it easier to understand and implement. With considering unreliable nodes, the proposed method is also more realistic and valuable for reliability analysis in an existing network. The correctness of the proposed algorithm will be analyzed and proven. One example is used to illustrate how all MCs are generated in a network with arc and node failures solved using the proposed algorithm.     

A Monte-Carlo simulation approach for approximating multi-state two-terminal reliability

This paper describes a Monte-Carlo (MC) simulation methodology for estimating the reliability of a multi-state network. The problem under consideration involves multi-state two-terminal reliability (M2TR) computation. Previous approaches have relied on enumeration or on the computation of multi-state minimal cut vectors (MMCV) and the application of inclusion/exclusion formulae. This paper discusses issues related to the reliability calculation process based on MMCV. For large systems with even a relatively small number of component states, reliability computation can become prohibitive or inaccurate using current methods. The major focus of this paper is to present and compare a new MC simulation approach that obtains accurate approximations to the actual M2TR. The methodology uses MC to generate system state vectors. Once a vector is obtained, it is compared to the set of MMCV to determine whether the capacity of the vector satisfies the required demand. Examples are used to illustrate and validate the methodology. The estimates of the simulation approach are compared to exact and approximation procedures from solution quality and computational effort perspectives. Results obtained from the simulation approach show that for relatively large networks, the maximum absolute relative error between the simulation and the actual M2TR is less than 0.9%, yet when considering approximation formulae, this error can be as large as 18.97%. Finally, the paper discusses that the MC approach consistently yields accurate results while the accuracy of the bounding methodologies can be dependant on components that have considerable impact on the system design.     

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.     

Tracing the unreliability and recognizing the major unreliability contribution of network components

Weak part analysis of a system is a key element in a system reliability quantification process. It enables the weakest areas of a system to be recognized and assists in directing remedial measures for improving the system reliability. This paper presents a novel approach to identifying the weak parts using the unreliability tracing (UT) technique and introduces the proportional sharing principle (PSP). The model for tracing the unreliability of a complex network is presented based on reliability evaluation methods using minimal cut sets (MCSs) and the PSP. The system UT sharing factors (UTSFs) are derived to easily identify the major unreliability contributions (MUCs) in a system. The method is illustrated using three cases and the UT, UTSF and the reliability impact analysis of different components are discussed. The results show that the developed technique can be applied to complex networks for UT tracing and recognizing the MUC.     

Confidence bounds for the reliability of binary capacitated two-terminal networks

Binary capacitated two-terminal reliability at demand level d (2TR_d) is defined as the probability that network capacity, generated by binary capacitated components, between specified source and sink nodes is greater than or equal to a demand of d units. For the components that comprise these networks, reliability estimates are usually obtained from some source of testing. For these estimates and depending on the type of testing, there is an associated uncertainty that can significantly affect the overall estimation of 2TR_d. That is, an accurate estimate of 2TR_d is highly dependent on the uncertainty associated to the reliability of the network components. Current methods for the estimation of network reliability and associated uncertainty are restricted to the case where the network follows a series-parallel architecture and the components are binary and non-capacitated. For different capacitated network designs, an estimate on 2TR_d can only be approximated for specific scenarios. This paper presents a bounding approach for 2TR_d by explaining how component reliability and associated uncertainty impact estimates at the network level. The proposed method is based on a structured approach that generates a α-level confidence interval (CI) for binary capacitated two-terminal network reliability. Simulation results on different test networks show that the proposed methods can be used to develop very accurate bounds of two-terminal network reliability.     

Fast approximate methods for the reliability analysis of computer networks

The complexity of computer communication networks has taken a dramatic upswing, following significant developments in electronic technology such as medium and large scale integrated circuits and microprocessors. Although components of a computer communication network are broadly classified into software, hardware and communications, the most important problem is that of ensuring the reliable flow of information from source to destination. An important parameter in the analysis of these networks is to find the probability of obtaining a situation in which each node in the network communicates with all other remaining communication centres (nodes). This probability, termed as overall reliability, can be determined using the concept of spanning trees. As the exact reliability evaluation becomes unmanageable even for a reasonable sized system, we present an approximate technique using clustering methods. It has been shown that when component reliability ⩾ 0.9, the suggested technique gives results quite close to those obtained by exact methods with an enormous saving in computation time and memory usage. For still quicker reliability analysis while designing the topological configuration of real-time computer systems, an empirical form of the reliability index is proposed which serves as a fairly good indicator of overall reliability and can be easily incorporated in a design procedure, such as local search, to design maximally reliable computer communication network.     

A simple minimal path method for estimating the weighted multi-commodity multistate unreliable networks reliability

The weighted multicommodity multistate unreliable network (WMMUN) is a novel network composed of multistate unreliable components (arcs and nodes) capable of transmitting different types of commodities in which capacity weight varies with components. It is an extension of the multistate network. The current method for evaluating the directed WMMUN reliability has been derived from minimal cut (MC) based algorithm. The existing best-known method needed extensive comparison and verification, and failed to find the real directed WMMUN reliability. A very simple algorithm based on minimal paths (MPs) is developed for the WMMUN reliability problem. The correctness and computational complexity of the proposed algorithm will be analyzed and proven. An example is given to illustrate how the WMMUN reliability is evaluated using the proposed algorithm. The relationships among all different versions of MPs are also clarified.     

A simple algorithm for evaluating the k-out-of-n network reliability

Evaluating the network reliability is an important topic in the planning, designing, and control of systems. The minimal cut set (MC, an edge set) is one of the major and fundamental tools for evaluating network reliability. A k-out-of-n MC is a special MC in a k-out-of-n network in which some nodes must receive at least k flows from their n input edges, where k is an integer number between 1 and n. In this study, an alternative method is given first to define a MC using a node set (called MCN) in k-out-of-n networks. A very simple algorithm based on some intuitive theorems that characterize the structure of the MCN and the relationship between MC and MCN is developed to solve the k-out-of-n network reliability by finding the k-out-of-n MCs between two special nodes. The proposed algorithm is not only easier to understand and implement, but is also better than the existing algorithm. The correctness of the proposed algorithm will be analyzed and proven. Two examples are illustrated to show how all k-out-of-n MCs are generated and verified in a k-out-of-n network using the proposed algorithm. The reliability of one example is then computing using one example.     

MINE: A Method of Multi-Interaction Heterogeneous Information Network Embedding

Interactivity is the most significant feature of network data, especially in social networks. Existing network embedding methods have achieved remarkable results in learning network structure and node attributes, but do not pay attention to the multiinteraction between nodes, which limits the extraction and mining of potential deep interactions between nodes. To tackle the problem, we propose a method called MultiInteraction heterogeneous information Network Embedding (MINE). Firstly, we introduced the multi-interactions heterogeneous information network and extracted complex heterogeneous relation sequences by the multi-interaction extraction algorithm. Secondly, we use a well-designed multi-relationship network fusion model based on the attention mechanism to fuse multiple interactional relationships. Finally, applying a multitasking model makes the learned vector contain richer semantic relationships. A large number of practical experiments prove that our proposed method outperforms existing methods on multiple data sets.     

Optimal series–parallel topology of multi-state system with two failure modes

In this paper, an algorithm for optimal allocation of multi-state elements (MEs) in acyclic transmission networks (ATNs) is suggested. The ATNs consist of a number of positions (nodes) in which MEs capable of receiving and sending a signal are allocated. Each network has a root position where the signal source is located, a number of leaf positions that can only receive a signal, and a number of intermediate positions containing MEs capable of transmitting the received signal to some other nodes. Each ME that is located in a nonleaf node can have different states determined by a set of nodes receiving the signal directly from this ME. The probability of each state is assumed to be known for each ME. The ATN reliability is defined as the probability that a signal from the root node is transmitted to each leaf node.The optimal distribution of MEs with different characteristics among ATN positions provides the greatest possible ATN reliability. The suggested algorithm is based on using a universal generating function technique for network reliability evaluation. A genetic algorithm is used as the optimization tool. Illustrative examples are presented.     

Optimal allocation of multi-state retransmitters in acyclic transmission networks

In this paper, an algorithm for optimal allocation of multi-state elements (MEs) in acyclic transmission networks (ATNs) is suggested. The ATNs consist of a number of positions (nodes) in which MEs capable of receiving and sending a signal are allocated. Each network has a root position where the signal source is located, a number of leaf positions that can only receive a signal, and a number of intermediate positions containing MEs capable of transmitting the received signal to some other nodes. Each ME that is located in a nonleaf node can have different states determined by a set of nodes receiving the signal directly from this ME. The probability of each state is assumed to be known for each ME. The ATN reliability is defined as the probability that a signal from the root node is transmitted to each leaf node.The optimal distribution of MEs with different characteristics among ATN positions provides the greatest possible ATN reliability. The suggested algorithm is based on using a universal generating function technique for network reliability evaluation. A genetic algorithm is used as the optimization tool. Illustrative examples are presented.     

A new approach to the d-MC problem

Many real-world systems (such as cellular telephones, transportation, etc.) are multistate-node acyclic network (MNAN) composed of multistate-nodes. Such network has a source node (position) where the signal source is located, a number of sink nodes that only receive the signal, and a number of intermediate nodes that retransmit the received signal to some other nodes. The non-sink node has different states determined by a set of nodes receiving the signal directly from it. The reliability of MNAN can be computed in terms of minimal trees (MTs). Based on the Branch-and-Bound algorithm, we developed an intuitive algorithm that is simpler than the best-known existing method. The computational complexity of the proposed algorithm is also analyzed. One example is illustrated to show how all MTs are generated by the proposed algorithm. The reliability of this example is then computed.     

An evaluation of the multi-state node networks reliability using the traditional binary-state networks reliability algorithm

A system where the components and system itself are allowed to have a number of performance levels is called the Multi-state system (MSS). A multi-state node network (MNN) is a generalization of the MSS without satisfying the flow conservation law. Evaluating the MNN reliability arises at the design and exploitation stage of many types of technical systems. Up to now, the known existing methods can only evaluate a special MNN reliability called the multi-state node acyclic network (MNAN) in which no cyclic is allowed. However, no method exists for evaluating the general MNN reliability. The main purpose of this article is to show first that each MNN reliability can be solved using any the traditional binary-state networks (TBSN) reliability algorithm with a special code for the state probability. A simple heuristic SDP algorithm based on minimal cuts (MC) for estimating the MNN reliability is presented as an example to show how the TBSN reliability algorithm is revised to solve the MNN reliability problem. To the author's knowledge, this study is the first to discuss the relationships between MNN and TBSN and also the first to present methods to solve the exact and approximated MNN reliability. One example is illustrated to show how the exact MNN reliability is obtained using the proposed algorithm.     

Reliability-oriented multi-resource allocation in a stochastic-flow network

A stochastic-flow network consists of a set of nodes, including source nodes which supply various resources and sink nodes at which resource demands take place, and a collection of arcs whose capacities have multiple operational states. The network reliability of such a stochastic-flow network is the probability that resources can be successfully transmitted from source nodes through multi-capacitated arcs to sink nodes. Although the evaluation schemes of network reliability in stochastic-flow networks have been extensively studied in the literature, how to allocate various resources at source nodes in a reliable means remains unanswered. In this study, a resource allocation problem in a stochastic-flow network is formulated that aims to determine the optimal resource allocation policy at source nodes subject to given resource demands at sink nodes such that the network reliability of the stochastic-flow network is maximized, and an algorithm for computing the optimal resource allocation is proposed that incorporates the principle of minimal path vectors. A numerical example is given to illustrate the proposed algorithm.