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Flow reliability of a probabilistic capacitated-flow network in multiple node pairs case
Abstract
This article mainly generalizes the flow (or transportation) reliability problem for a directed capacitated-flow network in which the capacity of each arc a(i) has the values 0 < I < 2 < ... < M-i from s (source) to t (sink) case to a multiple node pairs case. Given the demands for all specified node pairs simultaneously in the network, a simple algorithm is proposed first to find out the family of all lower boundary points for such demands in terms of minimal paths. The flow reliability, the probability that the system allows the flow satisfying the demands simultaneously, can be calculated in terms of such lower boundary points. The overall-terminal flow reliability, one source to multiple sinks flow reliability and multiple sources to one sink flow reliability can be calculated as special cases.
... To compute the system reliability of an MM-SFN, one can find all the system state vectors by which all the given demands can be satisfied simultaneously, and then calculate the probability of such vectors according to the given probability distributions of the arcs' capacities. It is usually assumed that all the required minimal paths are at hand in advance [12][13][14][15][16], then some approaches are proposed to determine the desired system state vectors, and finally the system reliability can be computed using some techniques such as the inclusion-exclusion principle [17] or the sum of disjoint products [18]. ...
... Then, using the presented results, an improved algorithm is proposed to solve the problem. The proposed algorithm is shown to be correct and more efficient than the proposed ones in [15,16]. The remainder of our work is organized as follows. ...
... So, considering Ψ = {X | X is an SSV on which all the demands can be satisfied}, we see that R D = Pr{X | X ∈ Ψ}. Assuming Ψ = {X | X is a minimal vector in Ψ} = {X 1 , X 2 , …, X δ } and A i = {X | X ≥ X i }, one can see that Ψ = ⋃ =1 , and consequently, we can then compute the system reliability using methods such as inclusion-exclusion or sum of disjoint products [9][10][11][12][13][14][15]. Hence, to evaluate the system reliability, it is enough to find all the vectors in Ψ . ...
Reliability evaluation of a stochastic-flow network (SFN) has been extensively studied in the past decades and various algorithms have been proposed. A number of graph based algorithms are in terms of minimal paths (MPs). Most MP-based algorithms have been proposed for the case of single-source single-sink SFN and a few algorithms have considered the multi-source multi-sink case. However, there are many practical networks such as communication and telecommunication networks comprised of several sources and sinks. Here, we consider a multi-source multi-sink SFN and propose an algorithm to find all the lower boundary points in terms of the MPs. The proposed algorithm is shown to be correct. Moreover, the algorithm is shown to be more efficient than some existing ones.
... Each edge or node may have different capacity states, which may partially be invalid or valid. Such a network is called a stochastic flow network (SFN) [5,[8][9][10][11][12][13][14][15]. It is possible to gain an accurate view of this type of network and evaluate the quality of data transmission. ...
... Besides, the number of sink nodes (i.e., branches) also has a major influence on SR. In this paper, the SR of an SFN with a single source to multiple sinks will be discussed in an actual system environment [15,22,23]. We mainly address the performance of electronic transaction data transmissions from HQs of the commercial bank to multiple branches. ...
The main purpose of this paper is to assess the system reliability of electronic transaction data transmissions made by commercial banks in terms of stochastic flow network. System reliability is defined as the probability of demand satisfaction and it can be used to measure quality of service. In this paper, we study the system reliability of data transmission from the headquarters of a commercial bank to its multiple branches. The network structure of the bank and the probability of successful data transmission are obtained through the collection of real data. The system reliability, calculated using the minimal path method and the recursive sum of disjoint products algorithm, provides banking managers with a view to comprehend the current state of the entire system. Besides, the system reliability can be used not only as a measurement of quality of service, but also an improvement reference of the system by adopting sensitivity analysis.
... To quantify the impact of earthquakes on lifeline networks, various network reliability metrics, e.g., connectivity, flow capacity, and travel time, were proposed [1][2][3][4]. For example, connectivity reliability, such as two-terminal or k-terminal reliability, evaluates network accessibility in terms of the probability that at least one origin-destination (OD) pair remains connected. ...
Various simulation-based and analytical methods have been developed to evaluate the seismic fragilities of individual structures. However, a community's seismic safety and resilience are substantially affected by network reliability, determined not only by component fragilities but also by network topology and commodity/information flows. However, seismic reliability analyses of networks often encounter significant challenges due to complex network topologies, interdependencies among ground motions, and low failure probabilities. This paper proposes to overcome these challenges by a variance-reduction method for network fragility analysis using subset simulation. The binary network limit-state function in the subset simulation is reformulated into more informative piecewise continuous functions. The proposed limit-state functions quantify the proximity of each sample to a potential network failure domain, thereby enabling the construction of specialized intermediate failure events, which can be utilized in subset simulation and other sequential Monte Carlo approaches. Moreover, by discovering an implicit connection between intermediate failure events and seismic intensity, we propose a technique to obtain the entire network fragility curve with a single execution of specialized subset simulation. Numerical examples demonstrate that the proposed method can effectively evaluate system-level fragility for large-scale networks.
... Recently, Lin and Yuan (2003) addressed the flow reliability problem, in terms of the probability that flow satisfies the demands simultaneously for multiple node pairs of a directed capacitated-flow network, through calculating a family of lower boundary points for servicing such demands in terms of minimal paths. Also, Yeh (2004) evaluated the multistate network reliability as the probability that flow satisfies the demands such that the total budget (maintenance cost) constraints are satisfied. ...
This paper investigates the continuous version of the stochastic Network Design Problem (NDP) with reliability requirements. The problem is considered as a two-stage Stackelberg game with complete information and is formulated as a stochastic bi-level programming problem, which is extended to include reliability as well as physical and budget constraints. The estimation procedure combines the use of Monte Carlo simulation for modelling the stochastic nature of the system variables with a Genetic Algorithm (GA), for treating the complexity of this new formulation. The computational experience obtained from a test road network application demonstrates the ability of the proposed methodology to address the need for incorporating reliability requirements and stochasticity into the various system components in the design process. The results can provide useful insight into the evaluation of alternative reliable network capacity improvement plans under the effect of uncertainty on the demand, supply and route choice process of travellers.
Computer and communication applications such as big data analytics services, social networks, enterprise business and mobile applications often require to simultaneously pass data of different sizes between multiple source and multiple destinations in the network. In practice, the nodes in the communication networks are often connected through multiple intermediate links of different types with varying failure probabilities. Failure of intermediate links can adversely affect data transmission between source and destination nodes; thus, they can impact the quality of service. Therefore, reliability evaluation of such networks is of subtle importance in today’s service dependent world. This article presents an efficient method for reliability evaluation of stochastic flow networks that can pass various demands simultaneously from multiple source nodes to multiple destination nodes. The proposed method has three steps: First step obtains the combined minimal cut sets between the given set of source and destination nodes. Second step generates the set of simultaneous upper boundary flows for the varying demands using the combined minimal cut sets. Third step calculates the network unreliability by applying the Sum of Disjoint Product method on the upper boundary flows. The reliability of the network, i.e., the probability that the network can simultaneously pass the set of demands, is calculated as 1-Unreliability. The MATLAB simulation of the proposed method on bench mark networks show that the proposed approach takes less computational time than the existing method.
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.
This paper proposes a cut set based approach to evaluate reliability of undirected capacitated networks for given simultaneous capacity requirements for specified multiple node pairs of the network. The capacity related reliability measures presented in literature have focused primarily on single node pair capacity requirements. However, networks need to support simultaneous demands for network resources for various node pairs. Therefore, a new reliability measure, as multi node pair capacity related reliability (MNPCRR), and an algorithm to evaluate this measure are proposed in this paper. The proposed algorithm takes link capacity and reliability values; minimal cut set and required capacities for each of the specified node pair as inputs to compute MNPCRR. Evaluation of this measure is expected to help network designers in assessing and optimizing network performance with ease and effectiveness. It will further help by providing sensitivity of the MNPCRR index to various link reliability parameters and capacities.
In this paper, we study the transportation network flow problem with stochastic flow distribution. According to the stochastic properties of transportation network flows, we first discuss the functional relations between the arcs and the nodes for modeling the transportation network systems, and originally construct an incidence matrix of capacity in the frequency domain. Then a new frequency-domain spanning graph model to analyze the network stochastic maximum flow and the stochastic minimum flow problem is presented. A corresponding algorithm is designed to resolve the model. In our solution, through the mutual transformations of probability functions between time-domain and frequency-domain in our model, the dynamic process of origin-destination flows is analyzed qualitatively; hence the minimum flow and maximum flow are derived quantitatively. An advantage of our approach is that it can handle uniformly both the continuous probability distribution case and the discrete probability distribution case. Numerical experiment results illustrate the feasibility and effectiveness of our approach.
The quickest path problem involving two attributes, the capacity and the lead time, is to find a single path with minimum transmission time. The capacity of each arc is assumed to be deterministic in this problem. However, in many practical networks such as computer networks, telecommunication networks, and logistics networks, each arc is multistate due to failure, maintenance, etc. Such a network is named a multistate flow network. Hence, both the transmission time to deliver data through a minimal path and the minimum transmission time through a multistate flow network are not fixed. In order to reduce the transmission time, the data can be transmitted through k minimal paths simultaneously. The purpose of this paper is to evaluate the probability that d units of data can be transmitted through k minimal paths within time threshold T. Such a probability is called the transmission reliability. A simple algorithm is proposed to generate all lower boundary points for (d,T), the minimal system states satisfying the demand within time threshold. The transmission reliability can be subsequently computed in terms of such points. Another algorithm is further proposed to find the optimal combination of k minimal paths with highest transmission reliability.
Here is a multi-criteria approach to identify and evaluate network-flow monitoring strategies in a stochastic environment. We propose an analytical approach for assessing flow disturbance, based on limited sampling of arc-flow information in multi commodity, or multiple origin-destination (O–D), networks. A disturbance is defined here as abnormal network traffic beyond routine fluctuations. The network is characterized by arcs that are available with certain probabilities, suggesting a busy packet-switching/circuit-switching network, or simply network with failing components. Given the huge state space of such stochastic networks, our first objective is to bound the expected flow for computational efficiency. This is accomplished by extending available single-commodity-network results to the multi commodity case. The second objective is to determine the best placement of flow monitors to obtain the most accurate estimates of traffic flow, as represented by O–D pair traffic volumes in order to detect a disturbance. We followed a multi-criteria approach in defining and evaluating all possible monitor-placement strategies satisfying monitor availability. To assess a traffic pattern, O–D volumes are estimated using the l
p
-metric for p = 1, 2 and ∞, representing a decision-maker's risk-preference structure. The final objective is to define a disturbance metric providing confident assessments on the occurrence of a flow disturbance. In this regard, we use linear-regression to generate confidence intervals around the expected flow for each O–D pair. This is supplemented by the Fisher Method of Randomization, a non-parametric test, for small networks. Unlike the Fisher-test results, it appears the linear-regression “disturbance metric” is very sensitive to detecting “disturbance.” Preliminary results suggest that among the three l
p
metrics in O–D estimation, it appears that the best is the l
2-metric, which provides the appropriate sensitivity and accuracy. Counter to intuition, it is also observed that the “disturbance” estimation error generally decreases as the number of sampled arcs decreases.
The reliability literature has recently introduced several multistate models. This study discusses reliability bounds in the most general of these models. It presents inclusion-exclusion bounds and compares them with disjoint subset bounds. The latter bounds are based on a generalization of Abraham's recursive disjoint products.
In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance.
An important problem in reliability theory is to determine the reliability of a complex system given the reliabilities of its components. In real life, the system and its components can be found in a range of states varying from perfect functioning through various levels of performance degradation to complete failure. This paper presents some models and their applications, in terms of reliability analyses, to situations where the system can have a whole range of states and all its components can also have a wide range of multiple states. Properties of the system structure function and computation and approximation of system state probabilities and system reliability measures are given.
Many real-world systems such as electric power transmission and distribution systems, transportation systems, and manufacturing systems can be regarded as flow networks whose arcs have independent, finite, and multivalued random capacities. Such a flow network is indeed a multistate system with multistate components and so its reliability for the system demand d, i.e., the probability that the maximal flow is no less than d, can be computed in terms of minimal path vectors to level d (named d-MPs here). The main objective of this paper was to present a simple algorithm to generate all d-MPs of such a system for each system capacity level d in terms of minimal pathsets. Analysis of our algorithm and comparison to Xue's algorithm shows that our method has the following advantages: (1) the family of d-MP candidates that it generates is smaller in size and so d-MPs can be generated more efficiently, (2) it is expressed more intuitively and so easier to understand, and (3) whenever applied in a series-parallel case, both algorithms are essentially the same, but in a non series-parallel case, Xue's algorithm needs the extra work to transform the system into a series-parallel in advance. Two examples are illustrated to show how all d-MPs are generated by our algorithm and then the reliability of one example is computed.
Frank and Frisch have considered the problem of determining the maximum flow probability distribution in networks where each branch has capacity that is a continuous random variable. In this paper, we consider the branch capacity distributions to be discrete and investigate some theoretical properties of the problem under this assumption.
This paper presents two efficient algorithms for reliability evaluation of monotone multistate systems with s-independent multistate components. The algorithms are based on the Doulliez & Jamoulle decomposition method. Algorithm 1 requires the minimal paths to be known; Algorithm 2 requires the minimal cuts to be known (the state of the system need not be specified for each vector of component states). Computer programs for implementing the algorithms are given. Computational-times are presented, and compared with the ``Inclusion-Exclusion Method'' and the ``State Enumeration Method''. The results demonstrate clearly the superiority of the algorithms to the two other methods.
Discrete function theory, which extends switching function theory and multiple-valued logic function theory, is introduced into multistate system analysis. Some theoretical conclusions and algorithms which play key roles in multistate system analysis are presented. The concepts of s-coherence and duality in binary-state system analysis are generalized. The set of minimal upper (maximum lower) vectors for level j, which play the role of min path (cut) set, is introduced to represent the states of a monotonic multistate system. Two approaches to computing state probability of multistate systems are given, one is based on inclusion-exclusion, the other is based on enumeration. Binary-state fault-tree is extended to multistate fault-tree. A computer code (MSTA1) has been programmed and is used to evaluate a multistate fault-tree. Multistate fault-tree and the computer code have been applied to paper-making industry; the results are consistent with the field data.
This paper presents a simple method for deriving a symbolic reliability expression of some practical systems such as a communication system having fixed channel capacities of its links, a power distribution system having limited power ratings of its power lines, a transport system which might not allow traffic more than a particular value, or a chemical system in which oil or gas flow through pipes is permissible only up to some safe limits. A system is good if and only if it is possible to transmit successfully the required capacity from source node to the sink node. This paper defines a group as a set of branches such that success of these branches ensures system success, as defined above. All such groups are obtained from a knowledge of the minimal paths of the system graph. The method is computerized and implemented on DEC-20 computer. Two examples are considered and their solutions presented to illustrate the technique.
Boolean algebra has been used to find the probability of communication between a pair of nodes in a network by starting with a Boolean product corresponding to simple paths between the pair of nodes and making them disjoint (mutually exclusive). A theorem is given, the use of which enables the disjoint products to be found much faster than by existing methods. An algorithm and results of its implementation on a computer are given. Comparisons with existing methods show the usefulness of the algorithm for large networks.
Discrete function theory, which extends switching function theory and multiple-valued logic function theory, is introduced into multistate system analysis. Some theoretical conclusions and algorithms which play key roles in multistate system analysis are presented. The concepts of s-coherence and duality in binary-state system analysis are generalized. The set of minimal upper (maximum lower) vectors of level j, which play the role of min path (cut) set, is introduced to represent the states of a monotonic multistate system. Two approaches to computing state probability of multistate systems are given, one based on inclusion-exclusion, the other on enumeration. The binary-state fault tree is extended to become a multistate fault tree. A computer code (MSTA1) has been programmed and is used to evaluate the multistate fault tree. This multistate fault tree and the computer code have been applied to the papermaking industry; the results are consistent with the field data.
Many systems can be regarded as flow networks whose arcs have
discrete and multi-valued random capacities. The probability of the
maximum flow at each various level and the reliability of such a flow
network can be calculated in terms of K-lattices which are generated
from each subset of the family of all MCs (minimal cutsets). However the
size of such a family 2<sup>m</sup>-1 ( m =number of MCs) grows
exponentially with m . Such a flow network can be considered as
a multistate system with multistate components so that its reliability
can be evaluated in terms of upper boundary points of each level d
(named d -MCs here). This work presents an algorithm to
generate all d -MCs from each MC for each system capacity level
d . The new algorithm is analyzed and compared with the
algorithm given by J. Xue (1985). Examples show how all d -MCs
are generated; the reliability of one example is computed