Routing policy of stochastic-flow networks under time threshold and budget constraint

The quickest path problem is to find a path which sends a given amount of data from the source to the sink such that the transmission time is minimized. More specifically, the capacity of each arc in the network is assumed to be deterministic. However, in many real-life networks such as computer systems, telecommunication systems, etc., the capacity of each arc is stochastic due to failure, maintenance, etc. Such a network is named as stochastic-flow network. Hence, the minimum transmission time is not a fixed number. The purpose of this paper is to provide a decision procedure for a stochastic-flow network under the time and budget constraints. We try to evaluate the probability that d units of data can be sent through the network under both time threshold and budget according to the routing policy. Such a probability is named the system reliability, which is a performance index to measure the system quality. An efficient algorithm is proposed to derive the optimal routing policy with highest system reliability. The sensitive analysis can be conducted to improve the most important component which increases the system reliability most significantly.     

Optimal routing policy of a stochastic-flow network

Under the assumption that each arc’s capacity of the network is deterministic, the quickest path problem is to find a path sending a given amount of data from the source to the sink such that the transmission time is minimized. However, in many real-life networks such as computer systems, telecommunication systems, etc., the capacity of each arc is stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not fixed. We try to evaluate the probability that d units of data can be sent through the stochastic-flow network within the time constraint according to a routing policy. Such a probability is named the system reliability, which is a performance index to measure the system quality. This paper mainly finds the optimal routing policy with highest system reliability. The solution procedure is presented to calculate the system reliability with respect to a routing policy. An efficient algorithm is subsequently proposed to derive the optimal routing policy.     

Optimal Pair of Minimal Paths Under Both Time and Budget Constraints

The quickest path (QP) problem is to find a path which sends a given amount of data from the source to the sink such that the transmission time is minimized. Two attributes are involved, namely, the capacity and the lead time. The capacity of each arc is assumed to be deterministic. However, in many real-life flow networks such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We modify the QP problem to a stochastic case. The new problem is to evaluate the probability that d units of data can be sent from the source to the sink under both time T and budget B constraints. Such a probability is named the system reliability. In particular, the data can be transmitted through two disjoint minimal paths (MPs) simultaneously. A simple algorithm is proposed to generate all (d, T, B)-QPs, and the system reliability can subsequently be computed. The optimal pair of MPs with highest system reliability could further be obtained.     

Transmission reliability of k minimal paths within time threshold

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.     

A method to evaluate the routing policy with two minimal paths within time threshold

Two attributes, the capacity and the lead time, are involved in the quickest path problem which finds a path with the minimum transmission time. The capacity of each edge is assumed to be deterministic in this problem. However, in many real-life networks such as computer, telecommunication, logistics networks, etc., each edge should be multistate due to failure, maintenance, etc. Such a network is named a multistate network. Hence, the minimum transmission time through a multistate network is not fixed. We evaluate the system reliability that a specified amount of data can be sent through a pair of minimal paths simultaneously within the time threshold. A solution procedure is first proposed to calculate it. In order to boost the system reliability, the network administrator decides the routing policy in advance to indicate the first and the second priority pairs of minimal paths. The second one will be responsible for the transmission duty if the first one fails. According to the routing policy, the system reliability can be subsequently computed. The case to transmit data through more than two minimal paths can be extended easily.     

A method to evaluate routing policy through p minimal paths for stochastic case

In this paper, a stochastic-flow network is presented to model a computer network in which each arc has various possible capacities and may fail. In order to shorten the transmission time, the transmission protocol allowing the data to be sent through multiple minimal paths simultaneously is utilized for the computer network. However, the minimum transmission time to send a given amount of data is not fixed due to the property of stochastic capacity. Accordingly, the first addressed issue is to evaluate the probability that the network is able to send the data within a time constraint by adopting the transmission protocol. Such a probability is named as transmission reliability that is regarded as a performance indicator to measure the QoS for a computer network. Without knowing all minimal paths in advance, an efficient solution procedure is proposed to calculate transmission reliability. The experimental results of 35 random networks show that the proposed algorithm can be executed efficiently. Moreover, in order to increase transmission reliability, the network administrator decides the routing policy to designate the first and the second priority p minimal paths. The second addressed issue is to evaluate transmission reliability associated with the routing policy. A sort criterion is subsequently presented to find an ideal routing policy.     

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.     

System reliability for a multistate flow network with multiple joint minimal paths under time constraint

The uncertainty of transmission time for a multistate flow network is the main issue when the quickest path problem is applied to such a network. A variant of quickest path problem focusing on the probability of complete transmission instead of finding the quickest path is studied in this paper. In transmission process, a congestion phenomenon may happen when there are joint arcs in minimal paths (MPs). That is, the some arcs are shared by some of MPs. Based on Monte Carlo Simulation, an algorithm is developed to assign the flows through MPs for evaluating the system reliability, the probability that the data can be sent within a time constraint. Experiments are performed to verify the approximate system reliability value obtained from the proposed algorithm is close to the real one. Furthermore, the experiments about CPU time of the algorithm and the characteristics of the variants of quickest path problem are also displayed.     

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.     

A label-setting algorithm for finding a quickest path

The quickest path problem is to find a path to send a given amount of data from the source to the destination with minimum transmission time. To find the quickest path, existing algorithms enumerate non-dominated paths with distinct capacity, and then determine a quickest path by comparing their transmission time. In this paper, we propose a label-setting algorithm for finding a quickest path by transforming a network to another network where an important property holds that each subpath of a quickest path is also a quickest path. The proposed algorithm avoids enumerating non-dominated paths whose transmission time is greater than the minimum transmission time. Although the computational complexity of the proposed algorithm is the same as that of existing algorithms, experimental results show that our algorithm is efficient when a network has two or more non-dominated paths.     

随机流量网络中流量分配控制的多目标优化研究

现实世界的网络比如：物流网络、通信网络、交通网络,电网等可以被抽象成一个随机流量网络。以传输成功率和整个传输所花费的成本为目标,对随机流量网络上流量的分配控制的多目标优化问题进行了研究。采用MPs的概念对问题建模,大大简化了模型的复杂程度。最后提出一个多目标遗传算法,通过实例验证,该算法较好地解决了随机流量网络上的流量分配控制问题。     

Backup reliability of stochastic imperfect-node computer networks subject to packet accuracy rate and time constraints

In a real-time computer network, arcs and nodes have multi-state capacity, lead time, and packet accuracy rate (PAR). Evaluating the reliability of a network whose nodes are imperfect is complex, because node failure results in the disablement of adjacent arcs. Such a network is named a stochastic imperfect-node computer network (SINCN). Under the strict assumption that each arc has a deterministic capacity, the quickest path problem is to find a path that sends a specific amount of data with minimum transmission time. Subject to both an assured PAR and time constraints, this paper proposes an efficient algorithm to evaluate the system reliability of an SINCN. Furthermore, a routing scheme is adopted to reinforce the system reliability. Accordingly, reliability based on the routing scheme is calculated. An application of our method on the Taiwan academic network is described to show its impact on the backup reliability for different routing schemes.     

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     

Computer network reliability optimization under double-resource assignments subject to a transmission budget

This paper discusses a double-resource assignment problem to maximize network reliability for a computer network. The resources are separated into two types: one is transmission line and another is transmission facility. In particular, each resource is multistate due to full failure, partial failure, or maintenance. Such a network assigned with multistate resources is usually modeled as a stochastic-flow network. Furthermore, each resource should have a transmission cost in reality. Hence, the network reliability is the probability that a specified demand is transmitted through the network successfully subject to a transmission budget. This paper devotes to find out the optimal double-resource assignment with maximal network reliability. An optimization algorithm combining the genetic algorithm, the minimal paths, and the Recursive Sum of Disjoint Products is developed to solve the proposed problem. The experimental results show that the proposed algorithm can be executed in a reasonable time.     

Finding the shortest path in stochastic networks

Common network parameters, such as number of nodes and arc lengths are frequently subjected to ambiguity as a result of probability law. A number of authors have discussed the calculation of the shortest path in networks with random variable arc lengths. Generally, only a subset of intermediate nodes chosen in accordance with a given probability law can be used to transition from source node to sink node. The determination of a priori path of the minimal length in an incomplete network is defined as a probabilistic shortest path problem. When arc lengths between nodes are randomly assigned variables in an incomplete network the resulting network is known as an incomplete stochastic network. In this paper, the computation of minimal length in incomplete stochastic networks, when travel times between nodes are allowed to be exponentially distributed random variables, is formulated as a linear programming problem. A practical application of the methodology is demonstrated and the results and process compared to the Kulkarni’s [V.G. Kulkarni, Shortest paths in networks with exponentially distributed arc lengths, Networks 16 (1986) 255–274] method.     

Reliability evaluation for a manufacturing network with multiple production lines

This paper focuses on performance evaluation of a manufacturing system with multiple production lines based on the network-analysis perspective. Due to failure, partial failure, or maintenance, the capacity of each machine is stochastic (i.e., multi-state). Hence, the manufacturing system can be constructed as a stochastic-flow network, named manufacturing network herein. This paper intends to measure the probability that the manufacturing network can satisfy customers’ orders. Such a probability is referred to as the system reliability. A graphical representation is first proposed to transform a manufacturing system into a manufacturing network. Thereafter, we decompose the manufacturing network into general processing paths and reworking paths. Three algorithms are subsequently developed for different scenarios and multiple production lines to generate the minimal capacity vectors that machines should provide to satisfy demand. The system reliability can be derived in terms of such capacity vectors afterwards.     

System reliability maximization for a computer network by finding the optimal two-class allocation subject to budget

In the real world, a computer/communication system is usually modeled as a capacitated-flow network since each transmission line (resp. facility) denoted by an edge (resp. node) has multiple capacities. System reliability is thus defined to be a probability that d units of data are transmitted successfully from a source node to a sink node. From the perspective of quality management, system reliability is a critical performance indicator of the computer network. This paper focuses on maximizing system reliability for the computer network by finding the optimal two-class allocation subject to a budget, in which the two-class allocation is to allocate exactly one transmission line (resp. facility) to each edge (resp. node). In addition, allocating transmission lines and facilities to the computer network involves an allocation cost where the cost for allocating a transmission line depends on its length. For solving the addressed problem, a genetic algorithm based method is proposed, in which system reliability is evaluated in terms of minimal paths and state-space decomposition. Several experimental results demonstrate that the proposed algorithm can be executed in a reasonable time and has better computational efficiency than several popular soft computing algorithms.     

An FPTAS for Quickest Multicommodity Flows with Inflow-Dependent Transit Times

Given a network with capacities and transit times on the arcs, the quickest flow problem asks for a "flow over time" that satisfies given demands within minimal time. In the setting of flows over time, flow on arcs may vary over time and the transit time of an arc is the time it takes for flow to travel through this arc. In most real-world applications (such as, e.g., road traffic, communication networks, production systems, etc.), transit times are not fixed but depend on the current flow situation in the network. We consider the model where the transit time of an arc is given as a non-decreasing function of the rate of inflow into the arc. We prove that the quickest s-t-flow problem is NP-hard in this setting and give various approximation results, including a fully polynomial time approximation scheme (FPTAS) for the quickest multicommodity flow problem with bounded cost.     

Reliability evaluation for an intermittent production system with stochastic number of normal machines

In an intermittent production system (IPS), a number of normal machines in a workstation may present multiple levels owing to maintenance, possibility of failure, etc. It means that the number of machines in each workstation is stochastic. This paper proposes a key performance index (KPI), which reflects the probability that an IPS can complete demand d within time constraint T. Such a probability is defined as system reliability. The IPS is modeled as a stochastic network, in which each arc is regarded as a workstation with stochastic number of normal machines, and each node is represented as a buffer. The concept of minimal machine vector (MMV), which indicates the minimal capacity required at each workstation to satisfy the demand and time constraints, is presented for evaluating the system reliability. In particular, a novel algorithm based on depth-first search is proposed to derive all MMVs. This algorithm avoids searching for unnecessary child nodes, and thus increases efficiency. Two practical examples, a printed circuit board and a footwear manufacturing systems, are used to illustrate the proposed algorithm. Such a KPI can provide information to production managers to understand the probability that an order can be completed on time.     

Reliability evaluation for a manufacturing network with multiple production lines

This paper focuses on performance evaluation of a manufacturing system with multiple production lines based on the network-analysis perspective. Due to failure, partial failure, or maintenance, the capacity of each machine is stochastic (i.e., multi-state). Hence, the manufacturing system can be constructed as a stochastic-flow network, named manufacturing network herein. This paper intends to measure the probability that the manufacturing network can satisfy customers’ orders. Such a probability is referred to as the system reliability. A graphical representation is first proposed to transform a manufacturing system into a manufacturing network. Thereafter, we decompose the manufacturing network into general processing paths and reworking paths. Three algorithms are subsequently developed for different scenarios and multiple production lines to generate the minimal capacity vectors that machines should provide to satisfy demand. The system reliability can be derived in terms of such capacity vectors afterwards.