A simple algorithm for reliability evaluation of a stochastic-flow network with node failure

https://doi.org/10.1016/S0305-0548(00)00039-3Get rights and content

Abstract

This paper addresses a stochastic-flow network in which each arc or node has several capacities and may fail. Given the demand d, we try to evaluate the system reliability that the maximum flow of the network is not less than d. A simple algorithm is proposed firstly to generate all lower boundary points for d, and then the system reliability can be calculated in terms of such points. One computer example is shown to illustrate the solution procedure.

Scope and purpose

The maximum flow problem is a standard problem in operations research and network analysis (Ford and Fulkerson, Flows in networks. NJ: Princeton University Press, 1962). This paper discusses the maximum flow problem for a stochastic-flow network in which arcs and nodes all have several capacities and may fail. We can evaluate the probability (named system reliability here) that the maximum flow is not less than a given demand, and then treat such a reliability as a system performance index. The purpose of this paper is to provide such a performance index for many real-world systems such as computer systems, telecommunication systems, logistics systems, etc.

Introduction

In a binary-state network (no flow happens), the network or each component (arc or node) has good/bad states. Assuming each node in the network is perfectly reliable, the system reliability is the probability that source s can communicate with sink t, which can be computed in terms of minimal paths (MPs) by applying inclusion–exclusion rule [1], [2], [3] or disjoint-event method [4]. In such a network, a minimal path is a sequence of arcs from s to t, which contains no cycle. The system reliability can also be obtained without MPs, for example by applying factorial formula sequentially [5]. Aggarwal et al. [6] extended the binary-state network to node failure case. They proposed a concept that the failure of a node implies the failure of arcs incident from it. Using this concept, the original network with node failure can be modified to be a conventional network with perfect nodes. Then the system reliability can be calculated by the published methods previously. Note that the concept of Aggarwal et al. [6] is only adaptable to a binary-state network.

In a binary-state flow network, the capacity of each arc (the maximum flow passing the arc per unit time) has two levels 0 and a positive integer. Given the demand d, the system reliability is the probability that the maximum flow of the network is not less than d. Lee [7] used lexicographic ordering and labeling scheme [8] to calculate the system reliability. Aggarwal et al. [9] solved such a reliability problem in terms of MPs. Without MPs, Rueger [10] extended to the case that nodes as well as arcs all have a positive-integer capacity and may fail.

Xue [3], Lin et al. [11] and Jane et al. [12] supposed that each arc has several states/capacities but each node is perfectly reliable. Such a network is called a stochastic-flow network here. Given the demand d, the system reliability is the probability that the maximum flow of the network is not less than d. Lin et al. [11] proposed an algorithm to generate all lower boundary points for d (named d-MP in Lin et al. [11]) in terms of MPs. This paper mainly extends the stochastic-flow network to the more general case that nodes as well as arcs all have several states/capacities and may fail. A simple algorithm based on MPs is proposed firstly to find all lower boundary points for d and then calculate the system reliability in terms of such points. The proposed solution procedure can be applied to evaluate the system reliability, one performance index, for real-world systems such as computer systems, telecommunication systems, transportation systems, electric-power transmission systems and logistics systems. An illustrative example is shown in Section 5.

Section snippets

Assumptions

A minimal path is a sequence of nodes and arcs from source s to sink t, which contains no cycle. Let G=(A,N,M) be a stochastic-flow network where A={ai|1⩽i⩽n} is the set of arcs, N={ai|n+1⩽i⩽n+p} is the set of nodes, and M=(M1,M2,…,Mn+p) with Mi (an integer) being the maximum capacity of each component ai (arc or node). Such a G under study is assumed to further satisfy the following assumptions.

  • 1.

    The capacity of each component ai is an integer-valued random variable which takes values 0<1<2<⋯<Mi

Stochastic-flow network model

Suppose mp1,mp2,…,mpm are totally the MPs from s to t. The stochastic-flow network is described in terms of two vectors: capacity vector X=(x1,x2,…,xn+p) and flow vector F=(f1,f2,…,fm) where xi denotes the (current) capacity of component ai and fj denotes the (current) flow on mpj. Let Lj denote the maximum capacity of mpj, i.e., Ljmin{Mi|ai∈mpj}. Then such a vector F is feasible if and only iffj⩽Ljforeachj=1,2,…,m,j=1m{fj|ai∈mpj}⩽Miforeachi=1,2,…,n+p.Constraint (1) says that the flow on each

Algorithm to generate all lower boundary points for d

Suppose all MPs have been pre-computed. All lower boundary points for d can be derived from the following steps.

.

Step 1.Find all feasible solutions F=(f1,f2,…,fm) of the constraints , by applying implicit enumeration.j=1m{fj|ai∈mpj}⩽Miforeachi=1,2,…,n+p,j=1mfj=d.
Step 2.Transform each F into X=(x1,x2,…,xn+p) viaxi=j=1m{fj|ai∈mpj}foreachi=1,2,…,n+p.
Step 3.Suppose the result of step 2 is: X1,X2,…,Xq. Remove those non-minimal ones in {X1,X2,…,Xq} to obtain all lower boundary points for d as

Numerical example

Example 1

Fig. 1 shows a simple computer network in which each arc represents a transmission line and each node represents a switch. We number the arcs in order of a1 to a6 and the nodes in order of a7 to a10. The capacity distribution of each component is given in Table 1. The supervisor would like to know the system reliability that the computer system can transmit at least 5 messages from s to t simultaneously (i.e., R5). There are 4 MPs: mp1={a7,a1,a8,a2,a10},mp2={a7,a1,a8,a3,a9,a6,a10},mp3={a7,a5,a9

Conclusion

Based on minimal paths, this paper proposes a simple algorithm to generate all lower boundary points for a given demand for a stochastic-flow network in which each arc and node have several capacities and may fail. The system reliability that the maximum flow of the network is not less than a given demand is calculated in terms of such points. In the case without node failure, the proposed algorithm reduces the number of constraints when comparing to the best existing method [11] since this

Acknowledgements

I would like to thank the anonymous referees for helpful and constructive comments. This work was supported in part by the National Science Council, Taiwan, Republic of China, under Grant No. NSC 89-2213-E-238-002.

Yi-Kuei Lin is currently an Associate Professor at the Department of Information Management, Van Nung Institute of Technology, Taiwan, Republic of China. He received a Bachelor degree in Applied Mathematics Department from National Chiao Tung University, Taiwan. He obtained his Master degree and Ph.D. degree in the Department of Industrial Engineering and Engineering Management at National Tsing Hua University, Taiwan, Republic of China. His research interest includes stochastic network

References (13)

  • J. Yuan et al.

    A factoring method to calculate reliability for systems of dependence components

    Reliability Engineering and System Safety

    (1988)
  • J.C. Hudson et al.

    Reliability bounds for multistate systems with multistate components

    Operations Research

    (1985)
  • Y.K. Lin et al.

    A new algorithm to generate d-minimal paths in a multistate flow network with noninteger arc capacities

    International Journal of Reliability, Quality, and Safety Engineering

    (1998)
  • J. Xue

    On multistate system analysis

    IEEE Transactions on Reliability

    (1985)
  • R. Yarlagadda et al.

    Fast algorithm for computing the reliability of communication network

    International Journal of Electronics

    (1991)
  • K.K. Aggarwal et al.

    A simple method for reliability evaluation of a communication system

    IEEE Transactions on Communications

    (1975)
There are more references available in the full text version of this article.

Cited by (0)

Yi-Kuei Lin is currently an Associate Professor at the Department of Information Management, Van Nung Institute of Technology, Taiwan, Republic of China. He received a Bachelor degree in Applied Mathematics Department from National Chiao Tung University, Taiwan. He obtained his Master degree and Ph.D. degree in the Department of Industrial Engineering and Engineering Management at National Tsing Hua University, Taiwan, Republic of China. His research interest includes stochastic network reliability, telecommunication management, multimedia communications and operations research. He has published several papers in refereed journals including Computers and Mathematics with Applications, Computers and Operations Research, International Journal of Reliability, Quality and Safety Engineering and Journal of Chinese Institute of Industrial Engineer.

View full text