Reliability and Criticality Analysis of Communication Networks by Stochastic Computation

Transcription

1 > EPLACE HIS LINE WIH YOU PAPE IDENIFICAION NUMBE (DOUBLE-CLICK HEE O EDI) < 1 eliability and Criticality Analysis of Communication Networks by Stochastic Comutation Peican Zhu, Jie Han, Yangming Guo and Fabrizio Lombardi - Abstract eliability is an imortant feature in the design and maintenance of a large-scale network. In this aer, the reliability of information transmission between a transmitter and a receiver (i.e. a two-terminal network) is considered as a generalized connectivity framework of terminal nodes. As network comlexity increases, existing aroaches to reliability analysis are encountering significant challenges. In this aer, stochastic comutational models are resented to efficiently analyze the reliability and criticality of a two-terminal network. Non-Bernoulli sequences with fixed numbers of 1 s and 0 s are utilized to encode the signal robabilities and to imrove the comutational efficiency and accuracy. Both unidirectional and bidirectional links are considered for the robabilistic information transition rocess by imerfect links. Imerfect nodes are also modeled by the stochastic model of an imerfect unidirectional link. Non-exonential failure distributions and correlated signals in a two-terminal network are readily handled by the stochastic aroach. he reliability of a system with external deterministic failures on a link is comared to that of the system rior to the occurrence of the failures. he difference in reliability is referred to as the criticality of the link. An analysis is ursued for the critical links based on the value of criticality. he roosed aroach can be used to analyze and imrove network reliability when utilizing a limited redundancy for rotecting the links. N I. INODUCION etworks exist in many large-scale systems such as those for communication, ower delivery, water distribution and gas suly. For these systems, large economic losses and security risks are likely to be incurred if the underlying network fails. Hence, the robust oerations of these systems are critical, and reliability is a very imortant requirement. A network often connects its two end terminals, also referred to as the transmitter and the receiver; hence, this tye of network is commonly referred to as a two-terminal P. Zhu was with the Deartment of Electrical and Comuter Engineering, University of Alberta, Edmonton, AB 6G 1H9, Canada. He is currently with the School of Comuter Science and echnology, Northwestern Polytechnical University (NWPU), Xi an, Shaanxi, China. J. Han is with the Deartment of Electrical and Comuter Engineering, University of Alberta, Edmonton, AB 6G 1H9, Canada ( jhan8@ualberta.ca). Y. Guo is with the School of Comuter Science and echnology, Northwestern Polytechnical University (NWPU), Xi an, Shaanxi, China. F. Lombardi is with the Deartment of Electrical and Comuter Engineering, Northeastern University, Boston 02115, MA. lombardi@ece.neu.edu. network. Such a network is usually modeled by a robabilistic grah, i.e., G(V, E), consisting of a node set V and a link set E [1]. he set V consists of a transmitter ( in Figure 1), a receiver ( in Figure 1) and the intermediate nodes (I 1 I m, IN k and J 1 J n in Figure 1). he set E consists of the links between the nodes of the network (either unidirectional or bidirectional). Such a network is likely to be affected by various catastrohic failures due to externally induced events (e.g., earthquakes and hurricanes). It fails when the anticiated delivery service cannot be comleted through a fault-free ath connecting the transmitter and the receiver. herefore, it is imortant to redict its erformance under these failures, for examle, to evaluate whether a gas suly network can reliably function following a hurricane. he reliability of a two-terminal network is defined as the robability of a successful communication (or the anticiated information transmission) between the transmitter and the receiver of a network. Hence, reliability evaluation usually deals with the connectivity robability between the transmitter and the receiver [1]. edundancy is often utilized to guarantee the network to be oerational in the resence of failures. However, it requires a detailed analysis to determine the critical links [2]. Methods found in the technical literature can be classified into two major categories: Monte Carlo (MC) simulation and robabilistic analysis. In MC simulation, a large samle size is usually required for achieving a suitable accuracy and obtaining a stable robability due to the slow convergence [3]. hus, a long simulation time is usually incurred. For robabilistic analyses such as the ath- [4] and cut-based algorithms [5], an accurate result can be derived by decomosing a network into disjoint aths; however, a large comutational comlexity results for ath-based algorithms. he reliability evaluation rocess is cumbersome, because the number of aths increases exonentially with the number of network edges or nodes. [6] has resented an imroved recursive algorithm to calculate the reliability; however, only aroximate results (within an error bound) can be obtained. [7] has rovided an exact algorithm to find the reliability of two-terminal networks; however, the rocess for deriving the exressions is rather comlex. Binary decision diagram (BDD) based aroaches have also been roosed [8] and the reliability assessment of multile-valued networks has been

2 > EPLACE HIS LINE WIH YOU PAPE IDENIFICAION NUMBE (DOUBLE-CLICK HEE O EDI) < 2 Service rovider Intermediate ransmission I 1 ransmitter I m IN k J 1 Customer J n eceiver Figure 1. A general structure for a two-terminal network. I i, i {1, 2,, m}, is a node connected transmitter. IN k is a node connected to the intermediate nodes. J i, i {1, 2,, n}, is a node connected receiver. ursued in [9]. he methods using BDDs have been shown to be very efficient for network analysis; however, the construction of a BDD is not always simle. Additionally, a reduced order BDD (OBDD) (as the most efficient form of BDD) requires variable ordering, thus further increasing the comlexity of BDD construction rocess. Finally, the failure distributions of imerfect links/nodes are not limited to fixed values, but they vary with the mission time. he mission time is defined as a time eriod and the status of the system is investigated within this time eriod. he analysis of exonentially or non-exonentially distributed failure events is even more comlex. Hence, a more comrehensive and efficient aroach is required for the reliability analysis of a network. ecently, stochastic comutation has been roosed for reliability analysis of logic circuits [10] and dynamic fault trees (DFs) [11][12]. he use of non-bernoulli sequences leads to an efficient evaluation [10]. Signal correlations are reserved in stochastic sequences, thus they are correctly accounted in a network. Non-exonential distributions can also be efficiently modeled [11][12]. In this aer, a stochastic comutational aroach is roosed for analyzing the reliability of a two-terminal network, i.e., a network in which information flows from the transmitter to the receiver through intermediate nodes and links. his can also be generalized to other networks with minor modifications [13]. Stochastic models are roosed for imerfect unidirectional and bidirectional links between two nodes, as well as for the imerfect nodes considered in [1]. Non-Bernoulli sequences are used to encode the failure robabilities, and the evaluation accuracy is found to be high at a reasonable sequence length. A stochastic analysis is further erformed for modeling information transmission for reliability and criticality assessment. In the sequel, we first resent a review of stochastic comutation. hen, we resent the stochastic models for the imerfect links and nodes resectively. A criticality analysis and an evaluation rocedure are then rovided; they can be used to imrove the erformance of a network by utilizing a limited redundancy. II. EVIEW OF SOCHASIC COMPUAION Stochastic comutation has been roosed in the ast for reliable circuit design [14]. In a stochastic sequence, a roortional number of bits is set to a secific value; for examle, a fixed number of 1 s is assigned in a non-bernoulli sequence to encode a robability. Comutations can be efficiently imlemented by logic gates; the stochastic sequence at the outut encodes the (outut) robability. herefore, Boolean logic oerations are transformed into robabilistic comutation in the real domain. In stochastic comutation, the number of 1 s in the outut sequence is not deterministic, but robabilistic due to stochastic fluctuations [10] [11][12]. However, the use of non-bernoulli sequences (as initial inuts) can greatly reduce the stochastic fluctuation [10]. A longer sequence length is usually required for achieving a higher accuracy. Stochastic logic circuits can handle correlated signals (usually caused by reconvergence of fanout signals), because signal deendencies are maintained and roagated [11][12]. his is useful when analyzing common comonents in a network. III. SOCHASIC MODELS FO IMPEFEC LINKS A link connecting two intermediate nodes is bidirectional, while the link connecting the transmitter or the receiver to an intermediate node is unidirectional. here are two ossible states for an imerfect link: oerational (fault-free) and disconnected (faulty). Usually, the state of an imerfect link is

3 > EPLACE HIS LINE WIH YOU PAPE IDENIFICAION NUMBE (DOUBLE-CLICK HEE O EDI) < 3 Data 1 Data 2 (a) P P 0. 6 P 1 P (b) P P P P P P 0.8 S / S Li Li Data 1 S / S I i / I i (c) Ii / Data 2 S Ii / S S L ij Data 1 S I i (d) Data 2 S I j Figure 2. (a) Probabilistic transition between Data 1 and Data 2; let the robability of imerfect link be. (b) he logic gates used for stochastic comutation. (c) A stochastic model for the imerfect unidirectional link from Data 1 to Data 2. he transition can occur from the transmitter to an intermediate node I i or from an intermediate node I i to the receiver. (d) Stochastic model for an imerfect bidirectional link between two intermediate nodes I i and I j. denoted by a binary variable a, a {0,1}, where 1 or 0 indicates that the link is either faulty (disconnected) or oerational (fault-free). he information is assed on by an oerational link, i.e., when a = 0. In the following analysis, it is also assumed that all imerfect links and nodes fail with known robabilities; the failure robabilities are either fixed or time-deendent (i.e. they vary with the mission time, either exonentially or non-exonentially distributed). Inut arameters for evaluating the reliability include network configuration, mission time, and failure arameters or failure robability for each link/node. In a network, the anticiated information, e.g. the telecommunication data in a wireless network or gas for a gas suly network, is delivered from the transmitter to the receiver. For simlicity, denote the information from the transmitter by 1, which indicates that the transmitter is totally reliable. hen the initial stochastic sequence for the signal robability of the transmitter, i.e., S, is a sequence of 1s, i.e., the ith bit of S is given by S,i = 1, i {1, 2,, L} where L is the length of the stochastic sequence. he reliability is determined by analyzing the stochastic sequence received by the receiver in the network with imerfect links and nodes. A. A stochastic model for unidirectional links A unidirectional link usually exists between either the transmitter and an intermediate node I i, or an intermediate node I i and the receiver (as in the unidirectional transition from Data 1 to Data 2 in Figure 2(a)). For simlicity, denote the unidirectional link between and I i by L i and the unidirectional link between I i and the receiver by L i. As an imerfect link, L i fails with a robability (that can be either fixed or having a value deendent on the mission time), then, the two nodes and I i is connected with a robability of 1. he stochastic model for the unidirectional link is shown in Figure 2(c). Let S Li denote the stochastic sequence generated for the failure robability of the imerfect link L i ; then S Li,j denotes the jth bit of the stochastic sequence S Li. Furthermore, let S Ii,j denote the jth bit of the stochastic sequence for the reliability of an intermediate node I i, i.e. S Ii. If S Li,j = 1, the link between the transmitter and the destination node is disconnected. hus, S Ii,j = 0, i.e., the intermediate node I i receives no information from the transmitter, because the link L i is disconnected for the jth trial (or bit). Otherwise, S Ii,j = S,j, i.e., the information can be delivered to node I i through the fault-free link. As an imerfect link L i fails with robability, then the robability of the information reliably received by node I i from the transmitter is given by 1. he stochastic model of Figure 2(c) accurately imlements the information transmission from transmitter through an imerfect link. A similar analysis can be alied to a unidirectional link between the intermediate node I i and the receiver. If the reliability of the imerfect link is rovided (instead of the failure robability), then the NO gate in the stochastic model can be removed, because the reliability is reresented as = 1. his also alies to the bidirectional stochastic model, as resented next. B. A stochastic model for bidirectional links A bidirectional link usually exists between two intermediate nodes (unless secific assumtions are made), e.g. I i and I j, as shown in Figure 2(a) for a bidirectional transition. he bidirectional link between I i and I j is reresented by L ij. he stochastic model for the imerfect link is illustrated in Figure 2(d). S Lij denotes the stochastic sequence generated for the failure robability of the imerfect link L ij. For the kth trial, the state of the imerfect link is denoted by S Lij,k. If S Lij,k = 1, then the link L ij is disconnected; hence, the states of the two intermediate nodes I i and I j do not affect each other. Otherwise,

4 > EPLACE HIS LINE WIH YOU PAPE IDENIFICAION NUMBE (DOUBLE-CLICK HEE O EDI) < 4 S Li Stochastic model for unidirectional link in Figure 2(c) ransmitter eceiver ransmitter I j Data j J i Data i I j Data j S I j Stochastic model for bidirectional link in Figure 2(d) Combinatorial relationshi for Data i Stochastic model for bidirectional link in Figure 2(d) J i Data i S Ji S S L ji I k I k (a) Figure 3. (a) Links connected to the node J i. (b) A stochastic model for determining the state of J i. I j,, I k indicate the intermediate nodes connected to J i, with links denoted as L ji,, L ki. (b) Data k Data k S Ik S Lki S Ii,k = S Ij,k. If either I i or I j has received the information (indicated by a 1 ), then the other node also receives the information through the fault-free link L ij. Given the stochastic sequence S Lij (for the failure robability of the imerfect link L ij ), the stochastic model in Figure 2(d) accurately evaluates the reliability of the bidirectional link. For the bidirectional link, there is no order in the transmission of the anticiated information, and this is also the case in the model of Figure 2(d). Hence, the system is considered as stable, unless the stochastic sequences for each node at two adjacent time stes are different. C. A stochastic model for nodes with multile links Multile links can be connected to a node (Figure 3(a)). If all incoming links to a node J i (either an intermediate node or receiver) are disconnected, then J i cannot receive any information. However, if any of the links is connected, i.e., at least for a link connecting I j and J i, say, L ji, S Lji,j = 0, then J i receives the information once any of the nodes connected to J i has already received it ( I j can be the transmitter or an intermediate node, i.e., I j {S, I j,, I k }) for jth trial. he state of the intermediate node J i is determined by the states of other intermediate nodes connected to J i. If any of the nodes connected to J i has already received information (e.g. S Ii,j = 1, I i {S, I j,, I k }) and the link L ji is fault free, i.e., S Lji,j = 0, then information can be received by J i, i.e. S Ji,j = 1. he relationshi in Figure 3(a) can be modeled by an n-inut O gate where n denotes the number of nodes connected to J i. Moreover, if J i is an intermediate node connected to the receiver () (Figure 3(a)), then has no effect on the state of J i ; hence, the connection from J i to the receiver can be neglected for the urose of determining the state of node J i. IV. SOCHASIC MODELS FO IMPEFEC NODES In addition to links, nodes can also be imerfect; an analysis of an imerfect node I d is ursued in this work. here are m incoming links, n outgoing links and h bidirectional links connected to other nodes. A bidirectional link is effectively considered as two unidirectional links. he imerfect node I d is further assumed to be divided into two virtual erfect nodes, I d and I d. he imerfection of a node is therefore imlemented by the imerfect unidirectional link, L d d, between the two virtual nodes. Let S I d and S I d be the stochastic sequences encoding the robabilities of a successful information arrival at I d and I d, resectively. he sequence encoding the failure robability of the imerfect link L d d is denoted by S I dd. If I d is the robability of the node to be oerational, then S Id is the stochastic sequence for the failure robability of the imerfect virtual unidirectional link between the two nodes. If the ith bit in S Id is 1, i.e., S Id,i = 1, the link between I d and I d is disconnected, then the information is lost after the imerfect node; otherwise, S Id,i = S Id,i. Hence, the robabilistic information assing through an imerfect node can be also imlemented by the stochastic model for the unidirectional link in Figure 2(c). he stochastic sequence S Id consists of 1s and 0s. If S I d,i = 1, the node successfully receives the information for the ith trial; otherwise, no information is received. If S Idd,i = 1, then the imerfect unidirectional link is faulty and the information cannot be assed to the virtual node; hence, the state of node I d is 0, i.e., S Id,i = 0. If S Id,i = 0, irresective whether the link fails or not, there is no information received, i.e. S Id,i = 0. Furthermore, if the imerfect links in/out of the imerfect node I d are considered, the stochastic models for imerfect links resented reviously can be alied.

5 > EPLACE HIS LINE WIH YOU PAPE IDENIFICAION NUMBE (DOUBLE-CLICK HEE O EDI) < 5 V. CIICALIY ANALYSIS A two-terminal network can be evaluated by using the stochastic models roosed reviously for imerfect links and nodes. Catastrohic failures can occur due to externally induced events. edundancy is often introduced to imrove the reliability and guarantee the network to remain oerational. When only a limited amount of redundancy can be rovided, it is imortant to determine the most critical links. In this manuscrit, the reliability of an imerfect network without external catastrohic (i.e., deterministic) failures is considered as a reference. Criticality of a link is defined as the reliability difference of an imerfect two-terminal network with a set of failure arameters, when that link works with a redefined failure robability and fails deterministically (e.g. due to a catastrohic failure). If multile network comonents fail at the same time (so there is a common cause failure), this scenario can be readily taken into account by the roosed stochastic models. A stochastic model can be constructed using the stochastic models resented in revious sections to analyze the criticality of a link, e.g., L i, in the network. S Li encodes the failure robability of the imerfect link L i not affected by a catastrohic failure, while S Li encodes the failure robability of L i when it is affected by a catastrohic error. Due to the deterministic nature of a catastrohic failure, S Li is a sequence of 1s. For the stochastic two-terminal network at time t, the stochastic sequences obtained by the receiver under these two failure scenarios of L i are given by S and S resectively. An XO gate is utilized to measure the difference in these two sequences, this is denoted by S. he criticality of L i is then encoded in S. his rocess can be alied to other links to find the most critical link to ensure the functionality of the two-terminal network. he rocess of evaluating the reliability of a two-terminal network (with an analysis of criticality) is shown as follows: (1) Determine failure robabilities of the imerfect links and nodes at the secified mission time oint based on the rovided failure arameters and distributions; (2) Generate the non-bernoulli sequences for the different failure robabilities of the links and nodes; (3) For unidirectional or bidirectional links, the roosed stochastic models resented reviously are alied to build the stochastic model; (4) If more than one link are connected to a node, the stochastic O gate model is alied; (5) If an imerfect node is considered, then the stochastic model for the imerfect node is alied; (6) Generate a stochastic sequence (consisting of 1s) for the transmitter and roagate the sequence through the network (made of stochastic logic gates); (7) Calculate the robability of the information obtained by the receiver; then the reliability of the network is determined; (8) By varying the failure robability of a secific link or node for the mission time, the criticality of each link or node in the network reliability is found by analyzing the stochastic sequence S (t) obtained. VI. ANALYSIS AND APPLICAIONS he stochastic analysis is erformed for an imerfect bridge network. he network [15] consists of four nodes (2 intermediate nodes, A and B ; a transmitter, a receiver ) (Figure 4). Initially, the nodes are assumed to be erfect, while each link fails with a fixed robability i = 0.1. he failures of different links are assumed to be mutually indeendent; then, the reliability of the links is given as r i = 1 i = 0.9, i {1, 2,, 5}. As in [15], the algebraic exression for the reliability of this network is given by S = r 5 (r 1 + r 2 r 1 r 2 )(r 3 + r 4 r 3 r 4 ) + (1 r 5 )(r 1 r 3 + r 2 r 4 r 1 r 2 r 3 r 4 ), where S denotes the system reliability; r i denotes the link reliability, i {1, 2,, 5}. he exact reliability is calculated as While the accurate analytical exression can be derived here, the required comutational comlexity increases exonentially with the number of connections among nodes and the scale of a network. It becomes difficult, if not imossible, to erform an accurate analysis for large networks. he stochastic structure of the bridge network in Figure 4 can be constructed with the alication of the stochastic models for imerfect links resented in Figure 2(c) and (d). Here, the stochastic sequences for the failure robabilities of links L A, L B, L AB, L A and L B are denoted as S LA, S LB, S LAB, S LA and S LB resectively; while S and S denote the information sequences sent by the transmitter and received by the receiver resectively. hen, a stochastic analysis can be erformed by roagating the stochastic sequences through the constructed stochastic model. he mean and the variance of the network reliability found using a stochastic analysis with 30 simulations are given in able 1 for different sequence lengths. he mean and variance of the average run time are also rovided to show the efficiency of the roosed analysis. r 1 r 2 r 5 Figure 4. A bridge network [15]. 1 2 A 5 B As shown in able 1, the accuracy of the stochastic aroach is affected by the sequence length. A more accurate result can be obtained by increasing the sequence length. However, a longer run time is also incurred, because longer stochastic sequences must be generated to reresent the signal robabilities. Hence, the stochastic sequence length is determined by a tradeoff between accuracy and efficiency. Nevertheless, with a reasonable sequence length, the stochastic aroach rovides a relatively accurate estimate of the reliability of a two-terminal network. he recision of the stochastic analysis is affected by: (1) failure robabilities of imerfect nodes/links and the inherent random fluctuations, i.e., ermutations of 1s, in the sequences; (2) the numbers of imerfect links/nodes and the network toology. As in [10], quantization error is incurred when converting a robability into a r 3 3 r 4 4

6 > EPLACE HIS LINE WIH YOU PAPE IDENIFICAION NUMBE (DOUBLE-CLICK HEE O EDI) < 6 stochastic sequence; however, quantization error can be avoided if using an aroriate sequence length to encode the robabilities. For instance, for a robability of 0.001, a sequence length of 1000 bits is sufficient to avoid quantization error. Furthermore, due to the existence of a limited resolution, a resolution error is incurred in the rocess of stochastic comutation; hence, a minimal sequence length should be determined by the rovided signal robabilities. For instance, in order to encode a robability of 0.01, at least a sequence length of 100 bits should be used. Any robability with a recision higher than that of 0.01 cannot be reresented by a sequence length of 100 bits. Let ε denote the resolution requirement and Ng indicate the number of gates in a network, a stochastic sequence with a length L N g /ε is usually sufficient [10]. Although the accuracy of an analysis is affected by the network toology, the number of nodes, the number of imerfect links and so on, for a network with thousands of gates, a sequence length of 1000k bits is usually able to meet the recision requirement of hus in ractice, a shorter sequence length is often utilized to roduce a result with accetable accuracy. able 1 Mean and variance for the bridge network s reliability obtained by the stochastic aroach. Stochastic comutation Sequence length eliability Average run time (s) (L) Mean Variance Mean Variance 1k k k o assess criticality, link j is considered with a failure robability that varies with the mission time t. Assume that the failure of link j is exonentially distributed; its robability density function (df) and cumulative density function (cdf) are f(t) = λ e λt t andf(t) = f(t) dt = 1 e λt resectively, 0 where λ is the (constant) failure rate of link j for an exonential distribution. Let t = 100 hours, λ = and L = 100,000 bits. he failure robabilities of the other links are ket constant at 0.2; the reliability for the bridge network from to is shown in Figure 5. able 2 Different failure scenarios analyzed for the bridge network. i denotes the failure robability of link i, i {1, 2, 3, 4, 5}. case 1 i = 0.2, case 2 the failure of link 1 is exonentially distributed; i = 0.2, i 1 case 3 the failure of link 2 is exonentially distributed; i = 0.2, i 2 case 4 the failure of link 3 is exonentially distributed; i = 0.2, i 3 case 5 the failure of link 4 is exonentially distributed; i = 0.2, i 4 case 6 the failure of link 5 is exonentially distributed; i = 0.2, i 5 Figure 5. eliability by varying the failure robability of different links by the roosed stochastic aroach (sequence length L = 100,000 bits). he failure scenarios are illustrated in able 2. If the failure robability of link j, j {1, 2, 3, 4, 5}, is changed, then the reliability varies (Figure 5). Links i, i {1, 2, 3, 4} have the same criticality on the reliability of the bridge network. Link 5 is the least critical because the reliability change is the smallest, as also found by the comarison of the reliabilities for case 1 and case 6. As shown in the simulation results, the inaccuracy decreases with an increase of sequence length. In general, the use of the tye of stochastic sequences remains a key issue to be investigated, because a different tye of sequences such as low-discreancy sequences may result in a shorter length for achieving a secific accuracy, thus significantly imroving the efficiency of stochastic comutation. VII. CONCLUSION eliability of a network is of great imortance. he raid increase in network comlexity causes substantial difficulty in evaluating its reliability when communication is required between a transmitter and a receiver through intermediate nodes and links in a two-terminal network. In this aer, a stochastic comutational aroach is roosed to efficiently investigate the reliability of such general two-terminal networks. Both imerfect links and nodes have been considered by arranging the failure robabilities with values either fixed or varying with the mission time. he imerfect links, either unidirectional or bidirectional, have been modeled stochastically for the rocess of information transmission by utilizing a combination of logic gates. Non-Bernoulli sequences have been used to imrove the comutational efficiency and accuracy of the stochastic aroach. he imerfect nodes can be effectively modeled by the roosed stochastic model for unidirectional imerfect links. Hence, the stochastic aroach can be used to evaluate a general two-terminal network under any failure distribution. It has been shown that the reliability of a two-terminal network decreases with an external failure. Moreover, the reliability of the network varies if the external failure affects different links. Hence, the criticality of different links is analyzed using the roosed stochastic aroach for imroving the reliability of a two-terminal network by alying redundancy for those critical links. In the future, transmission of multile signals through networks with multile transmitters and receivers will be investigated using a stochastic analysis. EFEENCES [1] K.K. Aggarwal, J.S. Guta and K.B. Misra, A simle method for reliability evaluation of a communication system, IEEE rans. Communications, vol. 23, no. 5, 1975, [2] J. Andrews and. Moss, eliability and isk Assessment, Essex, U.K.: Longman Scientific and echnical, [3] J. Li, Lifeline Earthquake Engineering-basic Methods and Alications, Beijing: Science Press, [4] L. Fratta and U. G. Montanari, A Boolean Algebra Method for Comuting the erminal eliability in Communication Network, IEEE rans. Circuit heory, vol. 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