Stochastic-flow Network Research Articles

AN EVALUATION METHOD FOR ENTERPRISE RESOURCE PLANNING SYSTEMS

An Enterprise Resource Planning (ERP) system is a complex network composed of various business processes. This paper proposes a method based on stochastic-flow network model to evaluate the performance of an ERP system depending upon the results of the ERP examination of the users involved. The nodes in the network denote the persons responsible for the business tasks during the processes. The arcs between nodes denote the process precedence relationships in the ERP system. When the process starts, the documents are initiated from the source node to its succeeding nodes. Finally, the documents are released in the destination node. Thus, the performance of an ERP system is related to the flow of the documents through the network. The failure of an ERP system is therefore described as in the condition that the flow of the system is under the acceptable level d. We propose a performance index, the probability of the maximal flow not less than d, to evaluate the performance of an ERP system. An algorithm is subsequently proposed to generate the performance index, which can be used to assess the system performance either before or after the system going live.

Two-commodity reliability evaluation of a stochastic-flow network with varying capacity weight in terms of minimal paths

A real-life manufacturing system can be modeled as a stochastic-flow network in which nodes stand for the machine stations, and arcs stand for the shipping media. In terms of minimal paths (MPs), this paper presents a stochastic-flow network model with four characteristics: (1) both nodes and arcs have multiple possible capacities, and may fail; (2) each component (arc/node) has both capacity and cost attributes; (3) two-commodity are proceeded; and (4) the capacity weight varies with arcs, nodes, and types of commodity. We study the possibility of two-commodity to be transmitted through this network simultaneously under the budget constraint. Such a possibility is named as the system reliability. The MPs play the role of media to describe the relationship among flow assignments and capacity vectors. Subsequently, a simple algorithm, in terms of MPs, is proposed to evaluate the system reliability. From the capacity management and decision making viewpoints, managers may adopt the system reliability as a performance index to measure the system capacity and finally to determine if it meets the customers’ orders or not.

An algorithm to generate all upper boundary points for [formula omitted] in terms of minimal cuts

This paper discusses a stochastic-flow network from single-commodity case to multicommodity case. We propose a performance index, namely the probability that the upper bound of the system capacity is a given vector subject to the budget constraint, to evaluate the quality level for such a network. A simple approach based on minimal cuts is presented to generate the all upper boundary points for the demand d subject to the budget B in order to evaluate the performance index.

Evaluation of the reliability of transport networks based on the stochastic flow of moving objects

In transport networks, human beings are moving objects whose moving direction is stochastic in emergency situations. Based on this idea, a new model—stochastic moving network (SMN) is proposed. It is different from binary-state networks and stochastic-flow networks. The flow of SMNs has multiple-saturated states, that correspond to different flow values in each arc. In this paper, we try to evaluate the system reliability, defined as the probability that the saturated flow of the network is not less than a given demand d. Based on this new model, we obtain the flow probability distribution of every arc by simulation. An algorithm based on the blocking cutset of the SMN is proposed to evaluate the network reliability. An example is used to show how to calculate the corresponding reliabilities for different given demands of the SMN. Simulation experiments of different size were made and the system reliability precision was calculated. The precision of simulation results also discussed.

MC-based Algorithm for a telecommunication network under node and budget constraints

It is an important issue to design some performance indexes in order to measure the performance for a telecommunication network. Network analysis is an available approach to solve the performance problem for a real-life system. We construct a two-commodity stochastic-flow network with unreliable nodes (arcs and nodes all have several possible capacities and may fail) to model the telecommunication network. In which, all types of commodity are transmitted through the same network simultaneously and compete the capacities. This paper defines the system capacity as a 2-tuple vector, and then proposes a performance index, the probability that the upper bound of the system capacity equals a demand vector subject to the budget constraint. An upper boundary point is a vector representing the capacities of arcs and nodes, and is the maximal vector exactly meeting the demand vector. A simple algorithm based on minimal cuts (or named MC-based algorithm) is then presented to generate all upper boundary points in order to evaluate the performance index. The storage and computational time complexity of this algorithm are also analyzed. The performance evaluation for the multicommodity case can be extended easily.

Reliability of a Flow Network Subject to Budget Constraints

The system capacity of a deterministic flow network is the maximum flow from the source to the destination. In a single-commodity stochastic-flow network (branches all have several possible capacities, and may fail), the system reliability, the probability that the maximum flow is larger than or equal to a given demand, is an important performance index to measure the quality level of a network. In a two-commodity stochastic-flow network, different types of commodities are transmitted through the same network simultaneously, and compete for the capacities. We concentrate on the reliability problem for such a network subject to the budget constraint. This paper defines firstly the system capacity as a pattern. We propose a performance index, the probability that the system capacity is less than or equal to a given pattern subject to the budget constraint, to evaluate the system performance. A simple algorithm based on minimal cuts is proposed to generate all maximal vectors meeting the demand and budget constraints. The performance index can then be computed in terms of all such maximal vectors

Reliability Evaluation for an Information Network With Node Failure Under Cost Constraint

The quality of service is an important index to measure the performance of an information system. This paper constructs a stochastic-flow network to model the information system. In this network, each node and arc having a designated capacity will have different lower levels due to various partial and complete failures. The studied problem is to evaluate the possibility that a given amount of multicommodity can be sent through an information network under the cost constraint. Such a possibility, which is named the mission reliability, is an appropriate performance index to measure the quality level. The terminology "flow" represents the quantity of data transmitted via such a network, and "demand" represents the required data from clients. Based on the properties of minimal paths, a simple algorithm is first proposed to generate all lower boundary points for the demand; then, the mission reliability can be calculated in terms of such points. The lower boundary point for the demand is a minimal vector, which represents the capacity of each component (arc or node), such that the demand can be fulfilled. Extending the stochastic-flow network to the node failure case, another algorithm is proposed to calculate the mission reliability

On a multicommodity stochastic-flow network with unreliable nodes subject to budget constraint

From the quality management and decision making view point, reliability and unreliability are important indices to measure the quality level for a stochastic-flow network. In a multicommodity stochastic-flow network with unreliable nodes, the branches and nodes all have several possible capacities and may fail. Different types of the commodity, which are transmitted through the same network simultaneously, compete the capacities of branches and nodes. In this paper we first define the system capacity as a vector for a multicommodity stochastic-flow network with unreliable nodes. Then we design a performance index which is the probability that the upper bound of the system capacity is a given pattern subject to the budget constraint. It can be applied to evaluate the quality level for such a network. A simple approach based on minimal cuts is thus presented to evaluate the performance index.

Measure the quality level for a supplier-demand system by a multicommodity stochastic-flow network

From the point of view of quality management, it is important to meet the customer's demand. The probability that the system can satisfy the customer's demand is an important performance index, and can be used to measure the quality level of the system. In this paper, we use a multicommodity stochastic-flow network to describe the relationship between the supplier and the customer. Each node as well as each arc has several possible capacities and may fail. The network allows multiple types of commodities to be transmitted from the source to the sink. Given the demand for each commodity at the sink, evaluation of the probability that the system meets the demands is performed. Such a probability, named the system reliability, is a performance index of quality level. At first, a simple algorithm is proposed to generate all lower boundary points for the demand, and the system reliability can be calculated in terms of such points. The computational complexity of the proposed algorithm is polynomial time in number of arcs, nodes and minimal paths.

06/02847 Evaluate the performance of a stochastic-flow network with cost attribute in terms of minimal cutsLin, Y.-K. Reliability Engineering & System Safety, 2006, 91, (5), 539–545.

06/02847 Evaluate the performance of a stochastic-flow network with cost attribute in terms of minimal cutsLin, Y.-K. Reliability Engineering & System Safety, 2006, 91, (5), 539–545.

System performance of a two-commodity stochastic-flow network with cost attributes in terms of minimal cuts

In the single-commodity case, the system capacity of a stochastic-flow network is the maximum flow from the source to the sink. We concentrate on a two-commodity stochastic-flow network (each arc has several possible capacities and may fail) in which each arc has both capacity and cost attributes. Two types of commodities are transmitted through the same network simultaneously and compete for the capacities. The system capacity is defined as a pattern, and a performance index, the probability that the upper bound of the system capacity equals a given pattern under the budget constraint, is proposed to evaluate the system performance. Such a performance index can be computed in terms of the maximal capacity vectors which satisfy both the demand and budget constraints. A simple approach based on minimal cuts is proposed to generate all such maximal capacity vectors.

SYSTEM RELIABILITY OF A STOCHASTIC-FLOW NETWORK WITH NODES HAVING CAPACITY AND COST ATTRIBUTES

A stochastic-flow network in which nodes as well as arcs all have both capacity and cost attributes is studied. We evaluate the possibility that the maximum flow of the network is not less than a demand under the budget constraint. Such a possibility is in general called the system reliability. A minimal path, an order sequence of arcs & nodes from the source to the sink without cycles, is used to assign the flow to each component (arc or node). Based on minimal paths, an efficient algorithm is proposed to generate all (d, C)-MPs that are the minimal capacity vectors meeting the demand d and the budget C. The system reliability can then be calculated in terms of all (d, C)-MPs.

A simple algorithm to generate all ( d,B)-MCs of a multicommodity stochastic-flow network

Both minimal paths and minimal cuts are important media to evaluate the performance indexes, the system reliability or unreliability, for a single-commodity stochastic-flow network. This paper concentrates on a multicommodity stochastic-flow network in which each arc has both the capacity and cost attributes. Different from the single-commodity case, the system capacity is a pattern for multicommodity case. Since the traditional performance indexes are not suitable for multicommodity case, we propose a new performance index, the probability that the system capacity is less than or equal to a given pattern under the budget constraint. A simple algorithm based on minimal cuts is presented to generate all ( d, B)-MCs that are the maximal capacity vectors meeting the demand d and budget B. The proposed performance index can be evaluated in terms of ( d, B)-MCs.

Performance index of a stochastic-flow network with node failure under the budget constraint

This paper deals with a stochastic-flow network in which each node and arc has a designated capacity, which will have different lower levels due to various partial and complete failures. We try to evaluate the probability that the maximum flow of the network is equal to the demand d without exceeding the budget B. Such a probability is treated as a performance index to measure the quality level of the network. A simple algorithm in terms of minimal cuts is first proposed to generate all upper boundary points for (d, B), and then the probability that the maximal flow is less than or equal to d can be calculated in terms of such points. The upper boundary point for (d, B) is a maximal vector, which represents the capacity of each component (arc or node), such that the maximum flow of the network under the budget B is d. The performance index can be calculated by repeating the proposed algorithm to obtain all upper boundary points for (d−1, B). A computer simulation example is shown to illustrate the solution procedure.

Improving efficiency of solving d-MC problem in stochastic-flow network

Solving d-MC problem is often a tedious process. Three ways are suggested to improve the efficiency of solving d-MC problem. The first way is to make the best use of some special properties of network. A property of Network with Joint Parallel Part, which is more common than series–parallel network, is illustrated. The second way is to reduce the number of d-MC candidates and then to reduce the cost of testing. Two theorems on how to find the d-MCs with only one element unsaturated and on how to set the Lower Capacity Limits (LCLs) of elements to some values higher than zero are proved. These two theorems will be helpful to reduce d-MC candidates without any loss of real d-MC. The third way is to efficiently remove the duplicated d-MCs. A theorem, elucidating which Minimal Cuts (MCs) the duplicated d-MCs will be generated from, is proved. Finally, an algorithm is proposed by adding a pre-numerating step to the algorithm presented in Yeh [A new approach to the d-MC problem. Reliab Eng Syst Safety 2002;77(2):201–6], and two examples are employed to illustrate the proposed algorithm, especially the pre-numerating step.

Evaluate the performance of a stochastic-flow network with cost attribute in terms of minimal cuts

This paper proposes a performance index to measure the quality level of a stochastic-flow network in which each node has a designated capacity, which will have different lower levels due to various partial and complete failures. The performance index is the probability that the maximum flow of the network equals the demand d without exceeding the budget b. A simple algorithm in terms of minimal cuts is first proposed to generate all upper boundary points for ( d, b), and then the probability that the maximum flow is less than or equal to d can be calculated in terms of such points. The upper boundary point for ( d, b) is a maximal vector representing the capacity of each arc such that the maximum flow of the network under the budget b is d. The performance index can be calculated by repeating the proposed algorithm to obtain all upper boundary point for ( d−1, b). A benchmark example is shown to illustrate the solution procedure.

Perturbation Analysis and Optimization of Stochastic Flow Networks

We consider a stochastic fluid model of a network consisting of several single-class nodes in tandem and perform perturbation analysis for the node queue contents and associated event times with respect to a threshold parameter at the first node. We then derive infinitesimal perturbation analysis (IPA) derivative estimators for loss and buffer occupancy performance metrics with respect to this parameter and show that these estimators are unbiased. We also show that the estimators depend only on data directly observable from a sample path of the actual underlying discrete event system, without any knowledge of the stochastic characteristics of the random processes involved. This renders them computable in online environments and easily implementable for network management and optimization. This is illustrated by combining the IPA estimators with standard gradient based stochastic optimization methods and providing simulation examples.

Reliability of a Stochastic-Flow Network With Unreliable Branches & Nodes, Under Budget Constraints

System reliability evaluation for flow networks is an important issue in quality management. This paper concentrates on a stochastic-flow network in which nodes as well as branches have several possible capacities, and can fail. The possibility is evaluated that a given amount of messages can be transmitted through the stochastic-flow network under the budget constraint. Such a possibility, system reliability, is a performance index for a stochastic-flow network. A minimal path, an order sequence of nodes & branches from the source to the sink without cycles, is used to assign the flow to each component (branch or node). A lower boundary point for (d, C) is a minimal capacity vector, which enables the system to transmit d messages under the budget C. Based on minimal paths, an efficient algorithm is proposed to generate all lower boundary points for (d, C). The system reliability can then be calculated in terms of all lower boundary points for (d, C) by applying the inclusion-exclusion rule. Simulation shows that the implicit enumeration (step 1) of the proposed algorithm can be executed efficiently.

An algorithm to evaluate the system reliability for multicommodity case under cost constraint

System reliability evaluation for flow networks is an important issue in our modern society. This paper studies the system reliability that a given amount of multicommodity can be transmitted through a stochastic-flow network, in which each are has several capacities, under the cost constraint. A simple algorithm is proposed to generate all ( d 1, d 2,…, d p ; C)-MPs where d i is the demand of commodity i. The system reliability can then be calculated in terms of ( d 1, d 2,…, d p ; C)-MPs.

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.