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

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Abstract

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.

Introduction

In recent years, network reliability theory has been applied extensively in many real-world systems such as computer and communication systems, power transmission and distribution systems, transportation systems, etc, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Thus, the system reliability plays important roles in our modern society. It is recommended to be measured and evaluated the performances of the systems which can be modeled as stochastic networks or into fault trees first.

Traditional binary-state reliability theory is dealing only with system and elements that have two possible states—complete failure and perfect functioning [1], [2], [6], [8], [12], [13], [14], [15], [16], [17]. Modern large-scale technical systems are distinguished by their structural complexity. MNN is a special MSS which is a system where components and system itself are allowed to have a number of performance levels [18], [19], [20], [21], [22], [23], [24], [25], [26].

In MNN, every state of any node has its own performance, where performance is treated as a capability to transmit a signal to other nodes. The system has a source node that can only emit and send a signal to other nodes, a number of sink nodes that can only receive a signal, and a number of intermediate nodes (neither source nor sink nodes) that retransmit the received signal to some other non-source nodes. The signal is transmitted from a non-sink node to a number of non-source nodes along edges between these nodes, i.e. multi-states. The probability of which nodes (states) are transmitted to next is assumed to be known for each non-sink node. All of these probabilities are assumed to be statistically independent. The MNN is more practical and reasonable than traditional binary-state network which satisfies the conservation law in many real-life situations such as computer networks, cellular telephone networks [18], [19], [20], [21], [22], [23], [24], [25], [26], etc. Therefore, the MNN analysis has become a new subject in system reliability.

If no signal leaving a node can return to this node through any sequence of nodes in MNN, then this MNN is called the multi-state node acyclic network (MNAN). Up to now, in an MNN, only the MNAN was well studied. MNANs were first investigated by Malinowski and Preuss [18]. Multi-state linear consecutively connected networks were introduced by Hwang and Yao [19] and studied by Kossow and Preuss [20] and Zuo and Liang [21]. Two best-known algorithms for the MNAN reliability evaluation [22], [23] are all in terms of minimal trees (MTs). These two existing best-known algorithms are based only on the universal generating function technique (an implicit enumeration procedure) [22] or the special Branch-and-Bound algorithm (an efficient implicit enumeration procedure) [23]. Both methods are complicated and still only evaluate the MNAN reliability. The need for a method to evaluate the MNN reliability thus arises.

The only difference between the MNN and TBSN is the way the signal (or flow) is transmitted. Therefore, the main purpose of this study is to show how to evaluate the MNN reliability using the TBSN reliability algorithm. A revised heuristic SDP (sum-of-disjoint-products) algorithm based on the MC is proposed next to evaluate the exact and approximated MNN reliability with a special method for finding the probability of DPs (disjoint-products). SDP algorithm has been well studied in finding the TBSN reliability. It provides successive lower bounds on the network reliability, which converge to the exact reliability value in terms of either the MC or minimal path (MP). Abraham [14] and Locks [15] applied this idea in traditional binary networks, while Heidtman improved this method with shorter computer time and fewer DPs [16]. The SDP algorithm is a power tool for reliability evaluation and has advantages over other techniques as summarized in Ref. [17]. It is adapted and revised here as an example that the algorithms to evaluate the TBSN reliability can also be used to solve MNN reliability.

The paper is organized as follows. Section 2 describes assumptions required. Section 3 discusses the relationship between MNN and TBSN. Some important properties related to calculate the probability of the DPs obtained from the SDPs algorithm in MNN are also presented. In Section 4, a simple heuristic SDP algorithm based on MC is proposed with implementing the properties related to calculate the probability of the DPs. Section 5 illustrates how to generate all DPs based on MCs and the reliabilities between the source node and some target sets in MNN through a numerical example using the proposed algorithm. Concluding remarks are given in Section 6.

Section snippets

Assumptions

The MNN satisfies the following assumptions [22], [23]:

  • 1.

    Each node/edge is perfectly reliable and MNN is a connected network.

  • 2.

    The signal can be retransmitted without following the flow conservation law.

  • 3.

    All of these probabilities of states of each non-sink node are according to a given distribution and assumed to be statistically independent.

Preliminaries

The only difference between the MNN and TN is the way the flow is transmitted from the current node to the next nodes to satisfy the flow conservation law. Different transformation methods result in different node state expressions. For example, Table 1, Table 2, Table 3 show probability distribution samples for the node states when considering Fig. 1 is a MNN, TBSN, or multi-state TN, respectively.

Therefore, the graphs in MNN and in TN (includes binary- and multi-state) are the same without

A heuristic SDP algorithm based on MC for MNN reliability enumeration

To demonstrate that any TBSN reliability algorithm can be implemented to evaluate the MNN reliability, the SDP algorithm is proposed. The proposed SDP algorithm is implemented in a heuristic way based on MC. It cannot only find the exact MNN reliability, but also stopped anywhere in the procedure to find the approximated MNN reliability to reduce the running time. The proposed heuristic SDP algorithm based on MC to find all DPs between the source node 1 and the target set in MNN is described as

An illustrative example

Just likes the two existing known methods [22], [23], the enumeration of MNN reliability used the proposed algorithm also possesses a computational difficulty that, in the worse case, grows exponentially with the network size. Therefore, the suggested method can be applied for moderate size networks. Owing to this inherent problem, instead of presenting practically large network systems, a moderate size network discussed in Refs. [22], [23] is selected to demonstrate this methodology.

Example 1

Consider

Conclusions

MNNs are more practical and reasonable than TNs in many real-life situations such as computer networks or cellular telephone networks. Therefore, MNN analysis has become a new subject in system reliability. Basically, the known existing approach can only solve the exact reliability for an MNAN which is just a special MNN without cycle.

In this article, we showed that each MNN reliability can be solved using any TBSN reliability algorithm, but with a special coding for the state probability.

Acknowledgements

This research was supported in part by the National Science Council of Taiwan, ROC under grant no. NSC 90-2218-E-035-006.

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