QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL

Qual. Reliab. Engng. Int. 2001; 17: 373–378 (DOI: 10.1002/qre.418)

RELIABILITY EVALUATION FOR LINEARCONSECUTIVELY-CONNECTED SYSTEMS WITH MULTISTATE

ELEMENTS AND RETRANSMISSION DELAYS

GREGORY LEVITIN∗Reliability Department, Planning, Development and Technology Division, Bait Amir, Israel Electric Corporation Ltd.,

PO Box 10, Haifa, 31000 Israel

SUMMARYA linear consecutively-connected system consists of N linear ordered positions at which statistically-independentmultistate elements with different characteristics are allocated. Each element can provide propagation of a signalit receives from the previous positions to the next few positions. Each element can have different states determinedby a number of adjacent nodes receiving the signal directly from this element. The signal retransmission processis associated with delays. The system fails if the signal generated at the first position (source) cannot reachthe N th position (receiver) within a specified time. An algorithm based on the universal generating functionmethod is presented in this paper for determination of reliability of the linear consecutively-connected systemwith delays. Copyright 2001 John Wiley & Sons, Ltd.

KEY WORDS: consecutively-connected system; multistate system; retransmission delay; universal generatingfunction

NOTATION

R reliability of LCCS (probability thatthe time of signal propagation fromtransmitter to receiver does not exceeda specified value)

E expected delay in LCCSN number of positions in LCCSei ith ME of LCCSτi retransmission delay of ith MECi ith position of LCCSK maximal number of different states of

MEsSi random state of ith MEpik Pr{Si = k}Pi probabilistic distribution of state of ith

ME: Pi = {pi1, . . . , piK }V vector representing minimal times of

signal arrival to LCCS nodesui(z) u-function for ith MEU1,...,i (z) u-function for the group of MEs

e1, . . . , ei�, ϕ operators over u-functionsθ(x) min{x,N}∗Correspondence to: G. Levitin, Reliability Department, Planning,Development and Technology Division, Bait Amir, Israel ElectricCorporation Ltd., PO Box 10, Haifa, 31000 Israel.Email: levitin@iec.co.il

1. INTRODUCTION

Linear consecutively-connected systems (LCCS) con-sist of a certain number of linearly-ordered positions(nodes) in which multistate elements (MEs) capableof receiving and/or sending a signal are allocated. Thesignal source is located in the first position C1. Theintermediate positions contain MEs transmitting thereceived signal to a few next positions. The elementlocated in the last position CN can only receive asignal.

Each ME located in each node can have differentstates determined by the number of adjacent nodesreceiving the signal directly from it. The event thata ME is in a specific state is random event. Theprobability of this event is assumed to be known foreach ME and for every one of its possible states. Allthe MEs in the network are assumed to be statisticallyindependent.

An example of the LCCS is a set of radio relaystations (Figure 1) with a transmitter allocated atC1 and a receiver allocated at CN . Each station Ci(1 < i < N) has MEs (retransmitters) generatingsignals that reach the next Si stations. Note that Si isa random value dependent on power and availabilityof retransmitter amplifiers as well as on the signalpropagation conditions. The aim of the system is to

Received 7 February 2001Copyright 2001 John Wiley & Sons, Ltd. Revised 28 June 2001

374 G. LEVITIN

provide propagation of a signal from transmitter toreceiver.

The LCCS with MEs was first introduced byHwang and Yao [1] as a generalization of the lin-ear consecutive-k-out-of-n:F system and the linearconsecutively-connected system with two-state ele-ments, studied by Shanthikumar [2,3]. Algorithms forLCCS reliability evaluation were developed by Hwangand Yao [1], Kossow and Preuss [4] and Zuo andLiang [5].

In all the mentioned works, the issue of signalretransmission delay was not addressed, although indigital telecommunication systems the retransmissionprocess is usually associated with a certain delay.Since the delay is equal to the time needed for thedigital retransmitter to processes the signal, it can beexactly evaluated and treated as a constant for anygiven type of signal. When it is so, the total time Tof the signal propagation from transmitter to receivercan vary depending only on the combination of thestates of MEs (retransmitters). The whole system isconsidered to be in working condition if T is notgreater than a certain specified level T ∗. Otherwise,the network fails.

In this paper, an algorithm is suggested for theevaluation of the reliability of linear consecutively-connected systems consisting of MEs with fixedretransmission delays. The algorithm is based on usinga universal generating-function technique.

Section 2 of the paper presents a description ofthe LCCS model and measures of its performance.Section 3 describes the technique used for evaluatingthe network reliability. In the fourth section, anillustrative example is presented.

2. MODEL DESCRIPTION

The LCCS consists of N consequently orderedpositions Ci , i ∈ [1, N]. At each position Ci , MEei is allocated. This element provides connectionsbetween the position to which it is allocated andfurther positions.

Each element ei hasK states, where state Si of ei isa discrete random value with distribution

Pr{Si = k} = pik,

K−1∑k=0

pik = 1 (1)

All the states Si are statistically independent. Si = k

for element ei allocated at position Ci implies thatconnection exists between Ci and each of Ci+1, Ci+2,. . . , Cθ(i+k), where θ(x) = min{x,N}. Si = 0 impliesthe total failure state of ei (no signal leaves Ci ).

Note that although different MEs can have differentnumbers of states, one can define the same numberof states for all the MEs without loss of generality.Indeed, if ME ei has Ki states and ME em has Kmstates (Ki ≤ Km), one can consider both MEs ashaving K = max{Ki,Km} states while assigningpik = 0 for Ki ≤ k < K .

Each ME ei receiving the signal can retransmit it tofurther positions after a fixed delay τi (it is assumedthat the signal source e1 has no delay and, therefore,τ1 = 0). The signal generated at ei reaches all the Sinext positions immediately. Since the states of eachME are random values, the total time T of the signalpropagation from C1 to CN is also a random value.The signal propagation time tj can be determined forevery combination of ME states and the probabilitythat T = tj can be determined is

Pr{T = tj } = qj ,

J∑j=1

qj = 1 (2)

where J is the total number of LCCS statescharacterized by different values of T (note thatdifferent combinations of individual MEs’ states canresult in the same T ). One can see that somecombinations of MEs’ states lead to the absence ofconnection between C1 and CN . In this case tj = ∞for this specific state.

The system reliability R(T ∗) is defined as aprobability that the signal generated in C1 can bedelivered to CN at time not greater than T ∗. Havingthe distribution (2) one can obtain this reliability as

R(T ∗) =∑tj≤T ∗

qj (3)

Another important measure of LCCS performanceis the expected time of signal delivery E, which alsocan be obtained from (2) as

E =∑tj<∞

tj qj

/ ∑tj<∞

qj (4)

In the next section we present a techniquefor obtaining the distribution (2) based on statedistributions of the individual MEs and their delaytimes.

3. LCCS RELIABILITY ESTIMATION BASEDON A UNIVERSAL GENERATING

FUNCTION

The procedure used in this paper for LCCSreliability evaluation is based on the universalz-transform (also called u-function or universal

Copyright 2001 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2001; 17: 373–378

RELIABILITY OF LINEAR CONSECUTIVELY CONNECTED SYSTEMS 375

generating function) technique, which was introducedin [6] and which proved to be very effective forreliability evaluation of different types of multi-statesystems [7–11]. The u-function extends the widely-known ordinary moment generating function (OGF).The essential difference between the ordinary anduniversal generating functions is that the latter allowsone to evaluate probabilistic distributions of overallperformance for a wide range of systems characterizedby different topology, a different nature of interactionamong system elements and a different physical natureof the performance measures of elements. This can bedone by introducing different composition operatorsover UGF (the only composition operator used withOGF is the product of polynomials).

3.1. Determination of u-functions for individualMEs and their groups

The UGF (u-transform) of a discrete randomvariable X is defined as a polynomial

u(z) =K∑k=1

qkzXk (5)

where the variable X has K possible values and qk isthe probability that X is equal to Xk .

In order to represent the signal arrival timedistribution in our LCCS, we modify the UGF byreplacing the random value X with the random vectorV = {v(1) . . . v(N)} such that v(j) corresponds totime of signal arrival to the node Cj .

Consider a ME ei . In each state Si (1 ≤ Si < K),the ME provides a signal transmission from Ci to aset of nodes {Ci+1, . . . , Ci+Sk } with delay τi . In orderto represent the times taken to retransmit the signalentered the node Ci to the rest of LCCS nodes, wedetermine vector ViSi as follows

viSi (j) ={τi, i < j ≤ θ(i + Si),∗, j ≤ i, j > θ(i + Si) (6)

where ∗ stands for ∞ (∗ can be represented by anynumber much greater than T ∗). Note that viSi (j) isthe time taken by the ME ej to deliver the signal to thenode Cj when ei is in state Si .

The polynomial

ui(z) =K−1∑k=0

pikzVik (7)

represents all the possible states of the ME located atCi by relating the probabilities of each state k to the

value of a random vector V (representing signal arrivaltimes) in this state.

Assume that a signal generated at Cm in state Smreaches Cn (which corresponds to vmSm(n) = τm).If the ME located at Cn is in state Sn, the signalgenerated at Cn reaches all the nodes belonging to theset {Cn+1, . . . , Cθ(n+Sn)} when the time τm + τn haspassed since the signal arrived Cm.

Note that if there are different ways of signalpropagation to certain position, characterized bydifferent times, the minimal time determines the signaldelay. For example when em is in state 3 and em+1 isin state 2 the same signal can reach position Cm+2 bythe time τm (since it reached Cm) directly from em orby time τm+τm+1 being retransmitted by em+1. In thiscase the signal is received by em+2 after time τm sinceit has reached Cm.

Therefore, in order to determine the delaydistribution for a signal retransmitted by two MEsallocated at two adjacent positions Cm and Cm+1 andbeing in states Sm and Sm+1 respectively one canuse the following function over vectors VmSm andVm+1 Sm+1 ,

ω(VmSm, Vm+1 Sm+1)

where for each individual term form+ 1 < j ≤ N

ω(vmSm(j), vm+1 Sm+1(j))

= min{vmSm(j), vmSm(m+ 1)+ vm+1 Sm+1(j)}(8)

To represent all the possible combinations of statesof the two MEs, one has to relate the correspondingprobabilities (obtained by multiplying the probabili-ties of corresponding states of each ME) with the val-ues of the random vector ω(VmSm, Vm+1 Sm+1) in thesestates. For this purpose, we introduce a compositionoperator � over u-functions of individual MEs whichtakes the following form for a pair of MEs located atCm and Cm+1:

Um,m+1(z) = �(um(z), um+1(z))

= �

(K−1∑k=0

pmkzVmSk ,

K−1∑j=0

pm+1j zVm+1 Sj

)

=K−1∑k=0

K−1∑j=0

pmkpm+1j zω(VmSk ,Vm+1 Sj )

(9)

The resulting polynomial Um,m+1(z) represents theprobabilistic distribution of the delay times for a set ofnodes receiving the signal from em directly or throughem+1. The random vector ω(VmSm, Vm+1 Sm+1) canhave no more than K2 different values.

Copyright 2001 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2001; 17: 373–378

376 G. LEVITIN

One can obtain the u-function for the LCCS con-taining all the MEs (or, equivalently, the probabilis-tic distribution of delay times in the entire LCCS)defining U1(z) = u1(z) and consecutively applyingequation

U1,...,m+1(z) = �(U1,...,m(z), um+1(z)) (10)

for m = 1,m = 2, . . . ,m = N − 2. The resultingpolynomial takes the form

U1,...,N−1(z) =J∑j=1

qj zVj (11)

This polynomial relates the probabilities of allthe possible LCCS states qj with the signaldelay times vj (N) (the last element of the vectorVj ) corresponding to these states. Since U1,...,N−1determines delay distribution (2) the LCCS reliabilityas well as the expected delay can be determined inaccordance with (3) and (4) as

R(T ∗) =∑

vj (N)≤T ∗qj (12)

and

E =∑

vj (N)<∗vj (N)qj

/ ∑vj (N)<∗

qj (13)

3.2. Simplification of u-functions

Observe that when u-function U1,...,m(z) is ob-tained, the values v(1), . . . , v(m) representing thedelays of a signal arrival to nodes C1, . . . , Cm arenot used further for determining U1,...,m+1(z) (and allthe U1,...,n(z) for n > m) according to (8). Indeed,when determining U1,...,m+1(z), we need to knowonly the probabilities that the signal reaches nodesCm+1, . . . , CN and the corresponding delays. It doesnot matter through what paths the signal reaches thesenodes. For example, if in different LCCS states thesignal reaches Cm+1 through a number of differentcombinations of paths (represented by the same num-ber of different terms in U1,...,m(z)) resulting in thesame delay, one does not have to distinguish thesecombinations. The only thing one has to know is thesum of probabilities of states in which combinations ofpaths with the given minimal delay exist, meaning thatone can collect the corresponding terms in U1,...,m(z)

by replacing all the values v(1), . . . , v(m) in vectorsV of the polynomial with ∗ symbols and collecting thelike terms.

If in some term of the polynomialU1,...,m(z) v(m+1) = · · · = v(N) = ∗, the signal cannot reach any

position from Cm+1 to CN independently of the statesof MEs located in these positions. Therefore, this statedoes not contribute to signal propagation to the lastnode and the corresponding term can be removed fromthe u-function U1,...,m(z).

Taking into account the above-mentioned con-siderations, one can drastically simplify polynomialsU1,...,m(z) for 1 ≤ m ≤ N − 1 using the followingoperator ϕ(U1,...,m(z)) which

• assigns ∗ symbols to v(1), . . . , v(m) in each termof U1,...,m(z),

• removes all the terms in which vectors V containonly ∗ symbols, and

• collects like terms in the resulting polynomial.

3.3. Algorithm for the determination of LCCSreliability

Using the UGF technique described above, onecan obtain the LCCS reliability for the given set ofparameters (τi, pik) 1 ≤ i ≤ N − 1, 0 ≤ k < K

applying the following procedure, which is convenientfor numeric implementation.

(1) Determine vectors Vik corresponding to states0 ≤ k < K for the MEs e1, . . . , eN−1 using rule(6).

(2) Determine the u-functions of the MEse1, . . . , eN−1 using expression (7).

(3) Assign U1(z) = u1(z).(4) Apply expression

U1,...,m+1(z) = �(ϕ(U1,...,m(z)), um+1(z))

for m = 1,m = 2, . . . ,m = N − 2

in sequence using operator � (9) and operator ϕdescribed in the previous section.

(5) Simplify polynomialU1,...,N−1(z) using operatorϕ and obtain the LCCN reliability R andexpected delay E using equations (12) and (13)respectively.

Consider for example a LCCS with N = 4 inwhich each ME can provide connection with the restof further allocated MEs with the given probabilities:Pr{Si0} = pi0,Pr{Si1} = pi1 for 1 ≤ i ≤ 3,Pr{Si2} = pi2 for 1 ≤ i ≤ 2 and Pr{S13} = p13. Thedelays of the MEs are τ1 = 0, τ2 and τ3 respectively.Maximal allowable delay is T ∗ = τ2.

According to (6) and (7), the u functions ofindividual MEs e1, e2 and e3 are

u1(z) = p10z∗∗∗∗+p11z

∗0∗∗ + p12z∗00∗ + p13z

∗000,

u2(z) = p20z∗∗∗∗ + p21z

∗∗τ2∗ + p22z∗∗τ2τ2,

u3(z) = p30z∗∗∗∗ + p31z

∗∗∗τ3

Copyright 2001 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2001; 17: 373–378

RELIABILITY OF LINEAR CONSECUTIVELY CONNECTED SYSTEMS 377

Table 1. Parameters of system units

State probability distribution pikNo. ofME i τi Si = 5 Si = 4 Si = 3 Si = 2 Si = 1 Si = 0

1 0.00 0.00 0.75 0.10 0.08 0.02 0.052 0.19 0.65 0.08 0.05 0.08 0.05 0.093 0.30 0.00 0.00 0.85 0.06 0.04 0.054 0.45 0.62 0.15 0.06 0.07 0.05 0.055 0.42 0.00 0.00 0.00 0.83 0.11 0.066 0.24 0.00 0.00 0.80 0.06 0.10 0.047 0.37 0.00 0.00 0.60 0.35 0.02 0.038 0.14 0.00 0.00 0.00 0.70 0.27 0.039 0.21 0.00 0.00 0.00 0.00 0.80 0.20

Following the consecutive procedure, we obtain

U1(z) = u1(z),

ϕ(U1(z)) = p11z∗0∗∗ + p12z

∗00∗ + p13z∗000,

U1,2(z) = �(ϕ(U1(z)), u2(z))

= p11p20z∗0∗∗ + p11p21z

∗0τ2∗

+ p11p22z∗0τ2τ2 + p12p20z

∗00∗

+ p12p21z∗00∗+p12p22z

∗00τ2 +p13z∗000,

ϕ(U1,2(z)) = p11p21z∗∗τ2∗ + p11p22z

∗∗τ2τ2

+ (p12p20 + p12p21)z∗∗0∗

+ p12p22z∗∗0τ2 + p13z

∗∗00.

U1,2,3(z) = �(ϕ(U1,2(z)), u3(z))

= p11p21p30z∗∗τ2∗ + p11p22z

∗∗τ2τ2

+ (p11p20 + p12p21)p30z∗∗0∗

+ p12p22p30z∗∗0τ2

+ p11p21 + p31z∗∗τ2τ2+τ3

+ (p12p20 + p12p21)p31z∗∗0τ3

+ p12p22p31z∗∗0min{τ2,τ3} + p13z

∗∗00,

ϕ(U1,2,3(z)) = p13z∗∗00 + αz∗∗∗τ2 + βz∗∗∗τ3

+ p11p21p31z∗∗∗τ2+τ3

where

α = (p11 + p12)p22,

β = p12(p20 + p21)p31 if τ3 > τ2,

α = (p11 + p12p30)p22,

β = p12p31 if τ3 > τ2

(note that ϕ operator reduces the number of differentterms in the polynomialU1,2,3 from 8 to 4). Now using

Figure 1. Example of LCCS (each ME i is in its maximal possiblestate Si )

(12) one obtains

R(τ2) = α + p13

= (p11 + p12)p22 + p13 if τ3 > τ2,

R(τ2) = α + β + p13

= (p11 + p12p30)p22

+ p12p31 + p13 if τ3 ≤ τ2,

and using (13) one obtains

E = (ατ2 + βτ3 + p11p21p31(τ2 + τ3))/

(p13 + α + β + p11p21p31)

4. ILLUSTRATIVE NUMERICAL EXAMPLE

Consider a LCCS presented in Figure 1. The LCCSconsists of ten nodes with nine MEs allocated atnodes C1 . . . C9. The ME delays and probabilisticdistributions of the ME states are presented in Table 1.The LCCS reliability obtained for T ∗ = 0.8 using thesuggested algorithm is R(0.8) = 0.908, the expecteddelay is E = 0.6. In Figure 2 one can see the systemreliability as a function of allowable delay T ∗. Thisfunction can be easily obtained using equation (12)over the polynomial (11) for different values of T ∗.

The method of LCCS reliability and expected delayevaluation provides a simple and effective way ofestimating the influence of delays of the individualMEs on the entire LCCS performance. This givesuseful information about the importance of effortsto reduce the units’ delays. Such an analysis is a

Copyright 2001 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2001; 17: 373–378

378 G. LEVITIN

Figure 2. Reliability of LCCS as a function of allowed delay

Figure 3. LCCS reliability and expected delay as functions of delaysof individual MEs

key point in tracing bottlenecks in systems and inidentifying the most important units. It is a useful toolto help the analyst find weaknesses in design and tosuggest modifications for a system upgrade.

In Figure 3 the examples of LCCS reliability andexpected delay obtained as functions of delays of MEse2, e4 and e9 are presented. One can see that reductionof τ4 is the most beneficial from both reliability andexpected delay points of view.

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Author’s biography:

Gregory Levitin received BS and MS degrees in ElectricalEngineering from Kharkov Politechnical Institute (Ukraine)in 1982, a BS in Mathematics from Kharkov State Universityin 1986 and a PhD in Industrial Automation from MoscowResearch Institute of Metalworking Machines in 1989. From1982 to 1990 he worked as a software engineer and aresearch associate in the field of industrial automation. From1991 to 1993 he worked at the Technion-Israel Instituteof Technology as a postdoctoral fellow at the faculty ofIndustrial Engineering and Management. Dr. Levitin ispresently an engineer-expert at the Reliability Department ofthe I.E.C. and adjunct lecturer at the Technion. His currentinterests are in operations research and artificial intelligenceapplications in reliability and power engineering. He is asenior member of IEEE.

Copyright 2001 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2001; 17: 373–378