Reliability evaluation for acyclic consecutively-connected networks with
multistate elements
Gregory Levitin
Reliability Department, Planning, Development and Technology Division,
Bait Amir, Israel Electric Corporation Ltd., P.O. Box 10, Haifa, 31000 Israel
E-mail: levitin@iec.co.il
Abstract
In this paper, an algorithm for acyclic consecutively-connected networks (ACCN) reliability
evaluation is suggested. The ACCNs consist of a number of positions in which multistate
elements (MEs) capable of receiving and/or sending a signal are allocated. Each network has a
root position where the signal source is located, a number of leaf positions that can only receive a
signal, and a number of intermediate positions containing MEs capable of transmitting the
received signal to some other nodes. Each ME that is located in a nonleaf node can have different
states determined by a set of nodes receiving the signal directly from this ME. The probability of
each state is assumed to be known for each ME. The ACCN reliability is defined as the
probability that a signal from the root node is transmitted to each leaf node. The suggested
algorithm is based on a universal generating function technique.
Keywords: consecutively-connected network, multistate system, universal generating
function.
Abbreviations
ACCN acyclic consecutively-connected network
ME
multistate element
UGF
universal generating function
1
Nomenclature
N
total number of nodes in ACCN
M
number of leaf nodes in ACCN

set of leaf nodes in ACCN

subset of 
ci
i-th node of ACCN
ik
set of nodes receiving a signal from ME located at ci when it is in state k
Ki
number of different states of ME located at node ci
Ti
number of different states of group of MEs located at nodes c1,…,cn
pi ik
probability of a state in which the set of nodes ik receive a signal from ME located at ci
hnt
probability that a group of MEs located at nodes c1,…,cn is in state t
Vi
random binary vector representing sets of ACCN nodes receiving a signal directly from ci
Vik
value of Vi at state k (vector representing the set ik)
Wn
random binary vector representing sets of ACCN nodes receiving a signal from c1 through
MEs located at c1,…,cn
Wnt
value of Wn at state t
ui(z)
u-function corresponding to ME located at node ci (represents probabilistic distribution of
Vi)
Un(z)
u-function corresponding to group of MEs located at nodes c1,…,cn (represents
probabilistic distribution of Wn)

u-function simplification operator

composition operator

R
function for vector composition
probability that a signal from c1 reaches all the nodes belonging to 
2
r
probability that a signal from c1 reaches all the nodes from  and does not reach the rest of
nodes from 
1. Introduction
Acyclic consecutively-connected networks (ACCN) consist of a certain number of positions
(nodes) in which MEs capable of receiving and/or sending a signal are allocated. Each network
has a root node where the signal source is located, a number of leaf nodes that can only receive a
signal and a number of intermediate (neither root nor leaf) nodes containing MEs capable of
transmitting the received signal to some other nodes. The signal transmission is possible only
along links between the nodes. The networks are arranged in such a way that no signal leaving a
node can return to this node through any sequence of nodes (no cycles exist).
Each ME located in nonleaf node can have different states determined by a set of nodes
receiving the signal directly from this ME. The event that a ME is in a specific state is a random
event. The probability of this event is assumed to be known for each ME and for every its
possible state. All the MEs in the network are assumed to be statistically independent.
The whole network is in working condition if a signal from the root node is transmitted to
each leaf node. Otherwise, the network fails. (Note that it is not necessary for a signal to reach all
the network nodes in order to provide its propagation to the leaf ones).
The acyclic consecutively-connected network is a generalization of the tree-structured
multistate systems investigated by J. Malinowski and W. Preuss [1] and multistate linear
consecutively-connected networks introduced by Hwang & Yao [2] and studied by Kossow &
Preuss [3] and Zuo & Liang [4].
An example of the ACCN is a set of radio relay stations with a transmitter allocated at root
node and a receivers allocated at leaf nodes. Each station has retransmitters generating signals
that can reach a set of next stations. Note that the composition of this set for each station depends
3
on power and availability of retransmitter amplifiers as well as on signal propagation conditions.
The aim of the system is to provide propagation of a signal from transmitter to receivers.
In this paper, an algorithm is suggested for the evaluation of the reliability of acyclic
consecutively-connected networks. The algorithm is based on using a universal generating
function technique.
Section 2 of the paper presents a description of the acyclic consecutively-connected network
model. Section 3 describes the technique used for evaluating the network reliability. In the fourth
section, illustrative examples are presented.
2. Model description
An acyclic consecutively-connected network can be represented by acyclic directed graph
G=(C,E) with N nodes ciC (1iN), M of which are leaf ones. The nodes are numbered in such
a way that for any arc (ci,cj)E j>i and last M numbers are assigned to the leaf nodes: ={cNM+1,…,cN}
(note that such numbering is always possible in acyclic directed graph). The existence
of arc (ci,cj)E means that a signal can be transmitted directly from node i to node j. One can
define for each nonleaf node ci a set of nodes i directly following ci: cji if (ci,cj)E (see Fig
1).
Multistate elements located in each nonleaf node ci (1iN-M) can transmit a signal to the
nodes belonging to the set i. In each state k, these elements transmit a signal to some subset ik
of i (in the case of total failure, the ME cannot transmit a signal to any node: ik=; in the case
of a fully operational state: ik=i). Each ME located at ci can have Ki different states and each
Ki
state k has probability pi ik , such that
 pi ik
 1 . The states of all the MEs are independent.
k 1
A signal can be transmitted by the ME located at ci only if it reaches this node.
4
The system reliability R is defined as a probability that a signal generated at the root node
c1 reaches all the M leaf nodes cN-M+1,…,cN.
3. ACCN reliability estimation based on a universal generating function
The procedure used in this paper for network reliability evaluation is based on the universal
z-transform (also called u-function or universal generating function) technique, which was
introduced in [5] and which proved to be very effective for reliability evaluation of different
types of multi-state systems [6-11]. The u-function extends the widely known ordinary moment
generating function (OGF). The essential difference between the ordinary and universal
generating functions is that the latter allows one to evaluate probabilistic distributions of overall
performance for wide range of systems characterized by different topology, different nature of
interaction among system elements and different physical nature of elements' performance
measures. This can be done by introducing different composition operators over UGF (the only
composition operator used with OGF is the product of polynomials).
3.1. Determination of u-functions for individual MEs and their groups
The UGF (u-transform) of a discrete random variable X is defined as a polynomial
u (z) 
K
 qkzXk ,
(1)
k 1
where the variable X has K possible values and qk is the probability that X is equal to Xk.
In order to represent random sets of ACCN nodes that receive a signal, we modify the UGF
by replacing the random value X with the binary vector V={v(1)…v(N)} such that v(j)
corresponds to node cj.
5
Consider a multistate element located at position ci. In each state k (1k<Ki), the ME
provides a signal transmission from ci to a set of nodes ik. In order to represent the set ik, we
determine vector Vik as follows

1, c j   ik
vik ( j)  
.
0
,
c



j
ik

(2)
The polynomial
u i ( z) 
Ki
 pi ik z Vik
(3)
k 1
represents all the possible states of the ME located at ci by relating the probabilities of each state
k to the value of a random vector V (representing set ik) in this state.
Assume that a signal generated at c1 in state s reaches c2 (c21s which corresponds to
v1s(2)=1). If the ME located at c2 is in state g, the signal generated at c2 reaches all the nodes
belonging to 2g. Therefore, when the first ME is in state s and the second one is in state g, the set
of nodes receiving the signal is 1s2g. This set can be represented by vector V1s V2g, where
the  operator (logical OR) for two arbitrary vectors A and B is defined as follows:

a ( j)  b( j)  0 if a ( j)  b( j)  0, (1jN).
1 otherwise
(4)
If a signal generated at c1 at some state s does not reach c2 (c21s which corresponds to
v1s(2)=0), the ME located at c2 cannot transmit the signal in any of its states and, therefore, ME c2
does not affect the state of the ACCN. The set of nodes receiving the signal remains 1s
represented by the vector V1s. In the general case of arbitrary states of the two MEs, one can use
the following function  to determine the random vector W2 representing the set of nodes
receiving the signal:

V ,
v1 (2)  0,
W2  (V1, V2 )   1

V1  V2 , v1 (2)  1.
6
(5)
To represent all the possible combinations of states of the two MEs, one has to relate the
corresponding probabilities (obtained by multiplying the probabilities of corresponding states of
each ME) with the values of the random vector V in these states. For this purpose, we introduce a
composition operator  over u-functions of individual MEs which takes the following form for a
pair of MEs located at c1 and c2:
K1
U 2 (z)  (u1 (z), u 2 (z))  (  p11s z
V1s
s 1
K1 K 2
  p11s p 2 2g z
s 1g 1
( V1s , V2 g )
K2
,  p 2 2 g z
V2 g
)
g 1
T2
(6)
  h 2 t z W2 t
t 1
The resulting polynomial U2(z) represents the probabilistic distribution of the possible values
of the random vector W2 corresponding to set of nodes receiving the signal from c1 directly or
through the ME located at c2. The random vector W2 can have no more than T2=K1K2 different
values. The probability of each state t of group of MEs located at c1 and c2 is h2t.
Consider a random vector Wn representing a set of nodes receiving the signal directly from
c1 or through the MEs located at c2,…, cn. It can easily be seen that the addition of the ME located
at cn+1 changes the set of nodes receiving the signal in such a way that the random vector Wn+1,
representing this new state, takes the form:
W ,
 n
Wn 1  ( Wn , Vn 1 )  
Wn  Vn 1 ,

w n (n  1)  0,
w n (n  1)  1.
(7)
Let Un(z) be the u-function representing probabilistic distribution of Wn. Since node cn+1
cannot receive the signal from any node cm with m>n+1, the probability that the signal generated
at c1 reaches cn+1 is completely determined by Un(z). Therefore, we can obtain a recursive
expression for the u-function representing the distribution of ACCN states:
7
Tn
U n 1 (z)  ( U n (z), u n 1 (z))  (  h nt z
t 1
Tn K n 1

t 1 s 1
h nt p n 1 s z
( Wnt , Vn 1 s )

Wnt
Tn 1
K n 1
,

s 1
p n 1 s z
 h n 1 t
Vn 1 s
)
,
z
(8)
Wn 1 t )
t 1
where Tn+1TnKn+1.
One can obtain the u-function representing the distribution of the ACCN states when all the
MEs are considered (or, equivalently, the probabilistic distribution of random vector WN-M)
applying the Eq. (8) in sequence for n=1, n=2,…, n=N-M-1. Summing probabilities hN-M t for all
the states t in which wN-M t(j)=1 for N-M+1jN, one obtains the probability that the signal
reaches all the leaf nodes, which is equal to ACCN reliability index.
Note that the probability that the signal reaches any subset  of leaf nodes can also be easily
obtained by summing probabilities hN-M t for all the states t in which wN-M t(j)=1 for all j.
3.2. Simplification of u-functions
Observe that when u-function Un(z) is obtained, the values wn(1),…, wn(n) representing the
presence of a signal at nodes c1,…,cn are not used further for determining Um(z) for m>n. Indeed,
when determining Un+1(z), we need to know only the probabilities that the signal reaches nodes
cn+1,…,cN. It does not matter through what paths the signal reaches these nodes. For example, if
the signal reaches cn+1 through a number of different paths (represented by the same number of
different terms in Un(z)), one does not have to distinguish these paths. The only thing one has to
know is the sum of probabilities of states in which these paths exist, meaning that one can collect
the corresponding terms in Un(z) by replacing all the values wns(1),…, wns(n) in vectors Wns of
Un(z) with zeros and collecting the like terms.
If in some state t wnt(n+1)=…=wnt(N)=0, the signal cannot reach any position from cn+1 to cN
independently of states of MEs located in these positions. Therefore, this state does not contribute
8
to signal propagation to the leaf nodes and the corresponding term can be removed from the ufunction Un(z).
Taking into account the above-mentioned considerations, one can drastically simplify
polynomials Un(z) for 1nN-M using the following operator (Un(z)) which
-
zeroes wns(1),…, wns(n) in each term of Un(z) (1sTn);
-
removes all the terms in which Wns contain only zeros;
-
collects like terms in the resulting polynomial.
3.3. Algorithm for determination of ACCN reliability
Using the UGF technique described above, one can obtain the ACCN reliability for the given
set of parameters ( pi ik , ik) 1iN-M, 1kKi applying the following procedure, which is
convenient for numeric implementation:
1. Determine vectors Vik corresponding to sets ik for the MEs located at positions c1,…,cN-M
using rule (2).
2. Determine the u-functions of the MEs located at positions c1,…,cN-M using expression (3).
3. Assign U1(z)=u1(z).
4. Apply expression Un+1(z)=((Un(z)),un+1(z)) for n=1, 2, …, N-M-1 in sequence using
operator  (8) and operator  described in the previous section.
5. Simplify polynomial UN-M(z) using operator  and obtain the ACCN reliability R as the
coefficient of the term of (UN-M(z)) in which wN-M(j)=1 for all N-M+1jN.
Note that in the general case, the resulting polynomial contains 2M-1 terms. Therefore, the
suggested method can be applied for ACCNs with moderate values of M.
9
4. Illustrative examples
4.1. Analytical example
Consider ACCN with N=5 and M=2 presented in Figure 2. According to (2) and (3), the u
functions of individual MEs located at nodes c1, c2 and c3 are:
u1(z)=p1z00000+p1{2}z01000+p1{3}z00100+p1{2,3}z01100,
u2(z)=p2z00000+p2{3}z00100+p2{5}z00001+p2{3,5}z00101,
u3(z)=p3z00000+p3{4}z00010.
Following the consecutive procedure, we obtain:
U1(z)=u1(z),
(U1(z))=p1{2}z01000+p1{3}z00100+p1{2,3}z01100,
U2(z)=((U1(z)),u2(z))=p1{2}p2z01000+p1{2}p2{3}z01100+p1{2}p2{5}z01001+
p1{2}p2{3,5}z01101+p1{3}z00100+p1{2,3}p2z01100+p1{2,3}p2{3}z01100+
p1{2,3}p2{5}z01101+p1{2,3}p2{3,5}z01101,
(U2(z))=(p1{2}p2{3}+p1{3}+p1{2,3}p2+p1{2,3}p2{3})z00100+p1{2}p2{5}z00001+
(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})z00101
(note that  operator reduces the number of different terms in the polynomial from 9 to 3).
U3(z)=((U2(z)),u3(z))=(p1{2}p2{3}+p1{3}+p1{2,3}p2+p1{2,3}p2{3})p3z00100
+p1{2}p2{5}p3z00001+(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3z00101+
(p1{2}p2{3}+p1{3}+p1{2,3}p2+p1{2,3}p2{3})p3{4}z00110+p1{2}p2{5}p3{4}z00001+
(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3{4}z00111,
(U3(z))=[p1{2}p2{5}+(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3]z00001+
(p1{2}p2{3}+p1{3}+p1{2,3}p2+p1{2,3}p2{3})p3{4}z00010+
(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3{4}z00011.
10
The coefficient in the term with the vector W=00011 in (U3(z)) is the probability that the
signal reaches both c4 and c5, which is equal to ACCN reliability:
R{4,5}=(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3{4}.
One can also obtain the probabilities that the signal reaches nodes c4 and c5 by summing the
coefficients of the terms with w(4)=1 and w(5)=1, respectively.
R{4}=r{4}+r{4,5}=(p1{2}p2{3}+p1{3}+p1{2,3}p2+p1{2,3}p2{3})p3{4}+(p1{2}p2{3,5}+
p1{2,3}p2{5}+p1{2,3}p2{3,5})p3{4},
R{5}=r{5}+r{4,5}=p1{2}p2{5}+(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3+
(p1{2}p2{3,5}+p1{2,3}p2{5}+p1{2,3}p2{3,5})p3{4}.
4.2. Numerical example
Consider an ACCN with N=10 and M=3, presented in Figure 3. The list of possible states
of MEs located at positions c1,…,c7 (represented by sets ik) and corresponding probabilities
pi ik is presented in Table 1. Table 2 contains the coefficients of the resulting polynomial
(U7(z)) corresponding to different states of the leaf nodes of the ACCN. The polynomial
consists of 7 different terms corresponding to these states. The system reliability R{8,9,10} is
presented in Table 3, as well as the probabilities of signal reception by all the possible
combinations of leaf nodes.
5. Conclusions
The paper suggests a reliability evaluation method for acyclic consecutively-connected
networks which are a generalization of the tree-structured multistate networks and multistate
linear consecutively-connected systems. It can be applied for estimating reliability of
transmission networks (radio relay systems, computer networks etc.) The method is based on
using a universal generating function technique extended for representing random binary vectors.
11
The suggested method allows one to obtain probability that the signal generated at the root
node of an ACCN reaches any subset of its terminal (leaf) nodes.
A simplification procedure is developed which reduces the length of the polynomials used to
represent different states of ACCN and, therefore, drastically reduces the computational burden
of the algorithm. However, the combinatorial nature of the problem imposes limitations on the
size of ACCN which can be analyzed using the method: since the resulting polynomial can
contain up to 2M-1 terms, the method can be applied for ACCNs with moderate values of M
(number of leaf nodes).
12
References
[1] Malinowski J, Preuss W. Reliability evaluation for tree-structured systems with multistate
components, Microelectron. Reliab., 1996, vol. 36, pp. 9-17.
[2] Hwang F, Yao Y. Multistate consecutively-connected systems, IEEE Transactions on
Reliability, 1989, vol. 38, pp. 472-474.
[3] Kossow A, Preuss W. Reliability of linear consecutively-connected systems with multistate
components, IEEE Transactions on Reliability, 1995, vol. 44, pp. 518-522.
[4] Zuo M, Liang M. Reliability of multistate consecutively-connected systems, Reliability
Engineering & System Safety, 1994, vol. 44, pp. 173-176.
[5] Ushakov I. Universal generating function, Sov. J. Computing System Science, 1986, vol. 24,
No 5, pp. 118-129.
[6] Levitin G, Lisnianski A. Importance and sensitivity analysis of multi-state systems using the
universal generating function method, Reliability Engineering and System Safety, 1999, vol. 65,
pp. 271-282.
[7] G. Levitin, A. Lisnianski, Beh-Haim H, Elmakis D. Redundancy Optimization for Seriesparallel Multi-state Systems, IEEE Transactions on Reliability, 1998, vol. 47, pp. 165-172.
[8] Lisnianski A, Levitin G, Ben Haim H. Structure optimization of multi-state system with time
redundancy, Reliability Engineering & System Safety, 2000, vol. 67, pp. 103-112.
[9] Levitin G. Redundancy optimization for multi-state system with fixed resource requirements
and unreliable sources, to appear in IEEE Transactions on Reliability, 2000, vol. 49.
[10] Levitin G, Lisnianski A. Reliability optimization for weighted voting system, Reliability
Engineering & System Safety, 2001, vol. 71, pp. 131-138.
[11] Levitin G, Lisnianski A. Structure Optimization of Multi-state System with Two Failure
Modes, to appear in Reliability Engineering & System Safety.
13
Figure Captions
Figure 1: Fragment of ACCN.
Figure 2: ACCN for the analytical example.
Figure 3: ACCN for the numerical example.
14
Table 1. Probabilistic state distribution of the system MEs.
i
1
2
3
4
5
6
7
ik
pi ik
{2,3,4}
{2,3}
{3,4}
{2}
{3}

{4,6,8}
{4,6}
{4,8}
{6,8}
{4}
{6}
{8}

{4,5}
{4}
{5}

{6,7,10}
{6,7}
{6,10}
{7,10}
{6}
{7}
{10}

{6,7}
{6}
{7}

{8,9}
{8}
{9}

{9,10}
{9}
{10}

0.75
0.1
0.08
0.02
0.01
0.04
0.65
0.08
0.05
0.08
0.05
0.02
0.05
0.02
0.85
0.06
0.04
0.05
0.62
0.08
0.06
0.02
0.05
0.05
0.07
0.05
0.83
0.04
0.07
0.06
0.8
0.06
0.1
0.04
0.6
0.35
0.02
0.03
15
Table 2. Probabilities of states of ACCN leaf nodes
r{8,9,10}
r{8,9}
r{8,10}
r{9,10}
r{8}
r{9}
r{10}
0.8185
0.0905
0.0068
0.0312
0.0045
0.0037
0.0024
Table 3. Probabilities of signal arrival to ACCN leaf nodes
R{8,9,10}
R{8,9}
R{8,10}
R{9,10}
R{8}
R{9}
R{10}
0.8185
0.9090
0.8253
0.8497
0.9202
0.9437
0.8588
16
Fig. 1
Ci
 ik
i
17
Fig. 2
1
2
3
4
5
18
Fig. 3
1
2
3
4
6
8
5
7
9
10
19