Reliability Analysis of Multi-State Networks: Making Monte Carlo Simulation

Feasible Through Biasing Enrico ZIO1, Massimo LIBRIZZI, Giovanni SANSAVINI*

Department of Nuclear Engineering, Polytechnic of Milan, Via Ponzio 34/3, Milano 20133, Italy

Abstract. Monte Carlo simulation offers a valuable tool for capturing the complex stochastic behavior of distributed, interconnected systems. To reduce the associated computational burden, it is possible to resort to biasing techniques which improve the efficiency of the simulation. In this paper, two biasing methods are proposed for improving the efficiency of the unreliability estimate of complex multi-state network systems, in which the arcs and the nodes can stay in various states of different performance. The biasing is founded on a sample strategy tailored to encourage the multi-state system to enter failed configurations with respect to the required demand at the network target node. This is achieved by forcing the arcs to visit their lower performance states. The performance of the methods is tested on a literature case study and a sensitivity analysis is carried out with respect to the parameter controlling the intensity of the bias.

Keywords. Multi-state Networks Reliability, Biased Monte Carlo Simulation

Introduction

Several modern engineering systems are made up of components, hereafter also called nodes, interconnected in a network structure by linking arcs. Among such systems are those for water distribution, telecommunication, oil and gas supply, power generation and transmission, transport by rail and by road, etc. The growing diffusion of, and reliance on, such systems demands their rational design and operation with respect to risk and reliability.

In this paper, the problem of evaluating the network two-terminal reliability (2TR) is considered, which requires verifying the existence of the connection between a source node and a target node of the network [1-9]. Typically, the simplifying hypothesis is made that the components, the arcs and the network have a binary behavior, i.e. at any time they can only be in a fully working or fully failed state. Such an assumption is often unrealistic since in practice components and arcs operate in intermediate functioning states, by design or for degradation, and correspondingly the network can supply various performance levels according to the functioning states of its constituents [10-14].

1E-mail address: enrico.zio@polimi.it (E. Zio) * Presenting author

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Network systems for which a binary state representation is insufficient, require a multi-state modeling approach to avoid an incorrect reliability assessment and consequent erroneous decision making which may lead to unnecessary expenditures, incorrect maintenance scheduling and reduction of safety standards.

When dealing with network systems made up of multi-state elements, the two-terminal reliability concept is extended to the multi-state two-terminal reliability at demand level d (M2TRd), defined as the probability that a demand of d units can be supplied from a source to a sink through the network multi-state arcs [15-21]. Evaluation of such quantity can be based on knowledge of the network multi-state minimal cut vectors (MMCVs), defined as those system configurations for which the available capacity from source to sink is lower than the requested demand and such that configurations identical to these, but for at least one element in a higher performance state, meet the demand [1]. In other words, an MMCV is a configuration of minimum performance level of the system.

On the contrary, Monte Carlo (MC) simulation can be an effective tool for assessing in practice the M2TRd of a realistic network system. In few words, the approach amounts to using MC simulation to sample several system “histories” or “trials” which constitute realizations of the system configurations that are then compared to the system MMCVs to verify whether the required demand is satisfied or not [17-21].

Analog MC simulation methods for network unreliability evaluation have already been proposed [1-2], [4], [22-23]. However, when assessing highly reliable network systems, the large majority of the sampled configurations do not bring any contribution to the unreliability estimate, system failure being a rare event [24-26]. As a consequence, the MC estimate of system unreliability is not robust, the associated variance being large due to the poor statistics of failure realizations. To increase the robustness of the estimate one may increase the number of network simulations, yet raising up computational time remarkably. An alternative, more effective way is to resort to the so-called biasing or variance reduction techniques [27-30]. These are sampling techniques which lead to more efficient simulations in terms of reduced variance of the estimates.

In this paper, we propose a sample strategy tailored to encourage a multi-state system to enter failed configurations, with respect to the network demand, by forcing the arcs to visit their lower performance states so that a larger number of MC system realizations contributes to the unreliability estimate.

The remainder of the paper is organized as follows: Section 1 describes the M2TRd problem in mathematical terms and introduces the Monte Carlo simulation approach to its solution. Two biasing methods are also proposed. Section 2 deals with the application of the two methods to a case study of literature. Finally, Section 0 draws some conclusions on the findings of the work performed.

1. Multi State Network Reliability Assessment by Biased Monte Carlo Simulation

1.1. Problem Description

Let G = (nn, na) represent a stochastic capacitated network required to supply a specified demand d from source node S to sink node T. Let nn be the number of nodes and na the number of arcs. The former are considered infallible, i.e. always in their

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design state, whereas each arc can occupy a number of states of different performance, according to given probability distributions. The generic state of arc i is denoted by an integer index j which takes values from j=1 to j=mi, in order of increasing performance. The corresponding occupancy probability is pi,j and the performance is wi,j, with the lower performance state j=1 having zero performance (wi,1=0), i.e. the arc is failed.

In the following, we are concerned with the assessment of the multi-state two-terminal unreliability of such network system.

1.2. Monte Carlo Simulation for Network Unreliability Estimation

Monte Carlo simulation for evaluating the unreliability of a network system requires sampling a large number M of independent network configurations, each one constituting a so-called Monte Carlo trial or history. The network configurations are sampled from the cumulative distribution functions constructed from the known discrete probability distribution functions pi,j of occupancy of state j by arc i. These probabilities are the natural ones in the analog simulation and the biased ones in the case that a variance reduction technique is adopted, as explained in the following paragraph. The vector of sampled arc performances wi,j constitutes a sampled network realization, which is compared with the MMCVs to assess whether the system is in a functioning or failed state, that is whether or not it meets the requested demand level d. When the system working state matches a MMCV, a counter Q for system failure is increased by 1 in the analog case or by the weight associated to the trial in the biasing case. At the end of the M trials, the network unreliability can be computed by dividing the quantity accumulated in Q by the number or trials performed, M.

1.3. Biasing for Network Failure

In the following, we propose a variance reduction technique to estimate the unreliability of multi-state network systems and compare their performances.

1.3.1. Method 2

The biasing technique here propounded merges a previous technique proposed by the authors [31] with the knowledge of Multi-state Minimal Cut Vectors.

For each arc, the average performance is

, ,1

im

i i jj

w w p=

= ⋅∑ i j n 1,..., ai = (1)

where pi,j is the probability of arc i being in state j. The average arc performance over all na arcs of the network is

1

1 an

iia

wn =

= ⋅∑w (2)

A performance threshold for biasing is then taken as

3

thw k= ⋅w (3)

where k > 0 is an arbitrary coefficient to be chosen by the analyst to control the number of arcs subject to biasing.

The arcs to be biased are selected based on their average performances, i.e. all arcs

i belonging to the subset { }i thi w wΓ = ≥ are biased. Then, the MMCVs are

examined to find which is the minimal performance state vi with which each arc i appears in a MMCV. On this basis, the states of arc i are divided into two subsets iB

and iB :

{ }1i iB j j v= ≤ ≤ { }i iB j j v= > (4)

Only the states are subject to biasing. ij B∈This procedure allows forcing arcs directly towards those states which contribute

to system failure, i.e. those which appear in the MMCV. In this sense, this may be considered an extension to the multi-state case of the biasing procedure for binary systems proposed in [32].

The biasing of the occupancy of state j by arc i is achieved by sampling from biased probabilities taken proportional to the natural probabilities *

,i jp ,i jp , i.e.

*, ,i j i i jp pγ= ⋅ 1,..., 1ij m= − i∀ ∈Γ (5)

where the bias factor 1iγ > , which controls the intensity of the bias, is taken the same for all favored states j < mi, without loss of generality.

By normalization, the probability of the nominal state mi is

1*,

11

i

i

m

i m i i jj

,p pγ−

=

= − ⋅∑ (6)

Hence, the increase forced onto the natural probabilities ,i jp of the favored states

of subset iB leads to a reduction in the probability of the unfavored nominal state mi in

iB by an amount (see Figure 1):

( ) ,1i

i i ij B

p pγ+

∈

= − ⋅∑ j n 1,..., ai = (7)

4

This surplus is taken away from the probabilities ,i jp of the unfavored states

ij B∈ , in proportion to their values ,i jp (Figure 1):

,*, ,

, 'i

i ji j i j i

i jj B

pp p

pp+

′∈

= − ⋅∑

1,..., ai n= 1,...,i ij v m= +

, '

,, '

1i

i

i ij B

i ji j

j B

pp

p

γ′∈

′∈

⎡ ⎤− ⋅⎢ ⎥

= ⋅ ⎢⎢ ⎥⎣ ⎦

j

⎥

∑∑

(8)

Figure 1. Biased probabilities (method 2)

The value of bias factor iγ is determined such as to reduce the average

performance of the biased arcs down to the threshold : *iw thw

* *, , , ,

ii

i j i j i j i j thj B j B

w p w p w∈ ∈

⋅ + ⋅ =∑ ∑ (9)

From Eq. (5) and Eq. (8) and after some mathematical manipulation

5

, ,

, , , ,

1

i

ii

th i j i jj B

ii j i j i j i j

j B j B

w w p

w p w pα

γβ∈

∈ ∈

− ⋅ ⋅=

⋅ − ⋅ ⋅

∑∑ ∑

(10)

where

,i

i jj B

pα∈

= ∑ (11)

,

, ''

i

i

i jj B

i jj B

p

pβ ∈

∈

=∑∑

(12)

Due to the normalization of the state probabilities, there is a lower limit in the choice of the factor k in Eq. (3): for all favored states iB , the total biased probability

cannot exceed unity, otherwise for unfavored states in iB the biased probabilities would loose physical significance. This constraint reads

*, 1

i

i jj B

p∈

<∑ (13) 1,..., ai = n

Using Eq. (5) and Eq. (11), an upper limit for iγ is found

11iγ α

<−

(14)

Inserting Eq. (10) into this inequality

, ,

, , , ,

11

1i

ii

th i j i jj B

i j i j i j i jj B j B

w w p

w p w pα

β α∈

∈ ∈

− ⋅ ⋅<

⋅ − ⋅ ⋅ −

∑∑ ∑

(15)

According to Eq. (3), after some mathematical manipulation, and noticing that the denominator of the first term is negative, a condition for a minimum value of k results

6

, ,

,

i

i

i j i jj B

i jj B

w pk

w p∈

∈

⋅>

⋅

∑

∑ (16)

For a probabilistically correct MC biasing procedure, a value of k above this lower limit has to be chosen.

2. Application of the Biasing Methods

The biasing methods previously proposed have been applied to a literature case study reported in [1]. All calculations have been performed by the Fortran code NUMA (Network Unreliability Monte carlo Analysis), developed at the Laboratorio di Analisi di Segnale ed Analisi di Rischio (LASAR, http://lasar.cesnef.polimi.it). The network, named ARPA, is depicted in Figure . The number of performance states which each arc can visit is mi=4 for i=1,2 and 8,9, mi=2 for i=3,5 and 7 and mi=3 for i=4 and 6.

Figure 2. ARPA network [1]

At the beginning of each trial, all the arcs are in their nominal state of maximum performance. The state occupancy natural probabilities of each arc are reported in [1]. The MMCVs for the ARPA network with respect to a required demand d=10 in arbitrary units are also reported in [1].

With respect to the original case study, the reliability of the ARPA network has been here increased by reducing by 4 orders of magnitude the occupancy probabilities of non-nominal states ( ) for each arc i. By so doing, the probability that a demand of 10 units be supplied from source to target is very high or, from a dual perspective, system failure is a very rare event. Thus, tackling the evaluation of the network unreliability by Monte Carlo simulation would result in a great majority of the trials giving no contribution to the system unreliability estimate, so that a biasing technique is in order.

ij m≠

7

2.1. Application of Biasing Method 1

Several MC simulations have been performed with the biasing method 1 illustrated in Section Error! Reference source not found. and with values of k ranging from 0.527 to 1. As explained in Section 1.3.1, the choice of the value of k sets the performance threshold wth for arc biasing: each arc i whose average performance wi is above threshold, i.e. , is forced to a new average performance level by increasing the occupancy probabilities of its non nominal states j<m

iw w> th th

th

*iw w=

i. Hence, a biased simulation with k > 1, resulting in , would be meaningless because it would

increase the occupancy probability

*iw w>

, ii mp of the arc nominal state mi, which is exactly the opposite goal of the biasing. As for the lower limit of 0.527, it derives from the normalization to 1 of the state probabilities, as explained in Section 1.3.1.

The results are reported in Error! Reference source not found. and in Figure 1 and Figure 2. Table 1. Results of the ARPA network of Figure obtained with the biasing method 1. The unreliability estimate is reported with one standard deviation. The figure of merit is the reciprocal of the product of the simulation time multiplied by the variance of the unreliability estimate.

k Γ (Biased arcs) Unreliability Minimum weight Maximum weight Figure of merit

1 1 9 6 8 2 (12.029 ± 2.521)·10-5 1.35·10-20 7.59·100 3.12·108

0.8 1 9 6 8 2 (7.562 ± 0.0612)·10-5 5.53·10-22 3.6·10-3 4.86·1011

0.766 1 9 6 8 2 4 (7.446 ± 0.0626)·10-5 3.38·10-22 4.94·10-3 4.56·1011

0.7 1 9 6 8 2 4 (8.582 ± 0.125)·10-5 3.21·10-26 5.41·10-2 1.09·1011

0.6 1 9 6 8 2 4 (8.614 ± 0.398)·10-5 5.56·10-27 2.70·10-1 3.72·1010

0.55 1 9 6 8 2 4 7 5 3 (8.931 ± 1.486)·10-5 1.53·10-37 2.53·100 2.81·109

0.527 1 9 6 8 2 4 7 5 3 (1.304 ± 0.170)·10-5 1.62·10-38 1.33·10-1 2.15·1011

Analog simulation (8.502 ± 0.030)·10-5 1 1 2.00·109

Figure 1 shows the system unreliability estimate and the associated uncertainty, measured by one standard deviation, for different values of the biasing parameter k. The proposed method gives good results when the value of k lies within 0.55 and 0.7, with the best performance at 0.7 in terms of reduced standard deviation. Conversely, a bias in the estimate of the unreliability is introduced when k lies outside of this range, in spite of higher values of the figure of merit (Figure 2). This result can be explained by analyzing the weight distribution of the sampled system configurations, which is strictly correlated with the intensity of the biasing as regulated by k.

8

Figure 1. Unreliability vs. k for the ARPA network of Figure and the biasing method 1. The solid line gives the result of the analog simulation

9

Figure 2. Figure of merit for the simulations of the ARPA network of Figure 3 with the biasing method 1. The solid line gives the result of the analog simulation

Consider, for instance, the case of k = 1. From Figure 1 and Error! Reference source not found. it can be seen that a bias is introduced in the estimate of the unreliability and that the standard deviation is large. In Figure 3, the corresponding weight distribution is reported in a bi-logarithmic graph where the x and y axes represent the weight values collected for the unreliability estimate and the number of corresponding failed system configurations sampled, respectively.

Figure 3. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the system failed configurations sampled, for the case of k=1. The arrow points to a weight larger than unity.

As reported in Error! Reference source not found., only 5 arcs undergo biasing in the simulation with k=1. This results in the weight distribution being subdivided in five bins of weight value less than unity. Each bin corresponds to a given number of biased arcs, either 1, 2, 3, 4 or 5, sampled to go below the nominal performance state in a given network failed configuration. The failed configurations in which only 1 biased arc is sampled to lower its state below the initial nominal one, lay in the weight range

[10-5 - 100]. Since only one out of the possible 5 arcs is biased, a number of

different weight values are found in this bin. Indeed, as explained earlier, the system failed configurations sampled by biasing contribute to the unreliability estimate with a

weight equal to the product of the weight factors

55

1⎛ ⎞

=⎜ ⎟⎝ ⎠

,*,

i j

i j

pp

of their constitutive arc states.

In this respect, the non-biased arcs contribute a weight factor equal to unity, whereas the biased arcs give a weight factor greater or lower than 1 according to whether a nominal or a non-nominal state is actually sampled in the system configuration. When only 1 biased arc is sampled to a lower, non nominal state, the total network

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configuration weight associated to this trial is made up of 4 factors equal to unity, for the non-biased arcs, 4 factors greater than unity, for the arcs biased but sampled to the nominal maximum performance state, and 1 factor lower than unity, for the only arc sampled to move to a state below nominal performance.

In a similar way, when 2 biased arcs are sampled in a state below the nominal,

different weight values are found, ranging in [105

102⎛ ⎞

=⎜ ⎟⎝ ⎠

-10 – 10-5]. In this case, the

total network configuration weight has contributions lower than 1 from 2 weight factors, thus resulting in smaller overall system configuration weights. A similar argument holds for bins with 3 and 4 biased arcs sampled to move in a state below the nominal.

The single lower weight in bin 5 derives from failed configurations in which all 5 biased arcs are sampled to move below nominal performance. Only one weight value is possible for these network configurations because the weight factors associated to

transitions of arcs to non-nominal states j are the same and equal to 1

iγ, independently

of the arrival state j (Eq.Error! Reference source not found.). Finally, the network may be in a failed configuration due to transitions to degraded

performances of non-biased arcs (i.e. sampled from the natural probabilities), even if the biased arcs are in their nominal, best performing states (e.g. MMCV 5 and MMCV 7 in Error! Reference source not found.). Though such configurations are very unlikely, the occupancy probabilities of non nominal states in biased arcs being very high, if sampled, they give rise to no weight factor lower than 1 and a total weight well above 1 associated to the sampled network configuration. The contribution to the unreliability estimate of such few unfavoured configurations with relatively large weights results in a significant degradation of the estimate statistics. This is a common ghost of biasing techniques where, together with configurations with small weights favoured by the biasing there are also unfavoured configurations with large weights. On the other hand, when these few configurations are not sampled during the simulation, the unreliability estimate may be biased, because computed without spanning the entire spectrum of possible system configurations. This conflicting problem is commonly tackled by resorting to a greater number of Monte Carlo trials, with the increased computational time somewhat undoing the benefits of the biasing, or more effectively by introducing splitting techniques [32]. We shall not investigate further this issue.

Continuing the analysis of the proposed biasing method 1, we have seen in Error! Reference source not found. and Figure 1 that it gives the best results for k=0.7. In Figure 4, the associated weight distribution is drawn.

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Figure 4. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.7

As before, the distribution turns out to be organized in bins according to the number of biased arcs (in this case 6) which reside in non-nominal states in the sampled network failed configurations. Hence, in the range [10-5 - 100] there are

possible weight values corresponding to the 6 failed network configurations

for which only 1 among the 6 biased arcs occupies a non-nominal state. Similarly,

when 3 of the 6 biased arcs are in a non-nominal state,

66

1⎛ ⎞

=⎜ ⎟⎝ ⎠

620

3⎛ ⎞

=⎜ ⎟⎝ ⎠

possible weight

values are found in [10-15 – 10-10]. The remarkable difference from the case with k=1 resides in the fact that here no failed system configuration is sampled with the 6 biased arcs in their nominal states so that it is impossible to find total weights greater than 1,

because any one of the 6 weight factors ,*,

i j

i j

pp

is small enough to reduce the total

weight below unity. Then, system unreliability is estimated by averaging all relatively small weights distributed over the entire set of failed configurations, thus leading to an unbiased and robust statistics and a figure of merit of 2 orders of magnitude larger than the one of the analog simulation.

For the case of k=0.8, a situation similar to the one for k=1 is obtained. The graph in Figure 5 shows a weight distribution shifted towards lower values than the ones for k=1 (Figure 3), coherently with the increased magnitude of the bias. The biased arcs are still 5 and the number of different weights in each weight range still reflects the number of biased arcs occupying a non-nominal state.

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Figure 5. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.8

No failed system configuration is sampled with all 5 biased arcs residing in their nominal, maximum performance states, this being a very unfavoured configuration. Yet, precisely for this reason, the estimate of the system unreliability turns out to be biased. On the other hand, as shown in Figure 6, which refers to a simulation with k=0.8 with initial random seed different than before, the estimate of the system unreliability would be biased even if one realization of such an unfavoured configuration were sampled, and with large variance since it brings a large weight in the estimate.

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Figure 6. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.8. The arrow points to a weight larger than unity.

When k is set to 0.766, unsatisfying results are found which look very similar to those for k=0.8, due to the similar magnitude of biasing (see Error! Reference source not found.). In this case there are 6 biased arcs, but the weight distribution in Figure 7 does not seem to show the usual bin structure which in this case seems split in a bottom and top part. The reason is once again understood by looking at the weights. The value k=0.766 is really the boundary between the biasing of 5 and 6 arcs. The added 6th arc is number 4 and it is biased very little, i.e. , since its mean performance is very close to the threshold performance w

*4, 4,jp p∼ j 4w

th. For this “almost analog” arc, the biased occupancy probabilities are almost equal to the natural ones *

4, jp 4, jp , and the

associated weight factors 4,*4,

j

j

pp

are very close to unity. A visual comparison between

analog and biased occupancy probabilities for arcs 1 and 4 is presented in Figure 9, where the y-axis is in logarithmic scale. It is clear that arc 4 is “almost analog”, that is, even if it is biased, the occupancy probabilities are very similar to the natural ones. In Figure 10, the weight factors for the same two arcs are plotted in logarithmic scale. It can be seen that weight factors for arc 4 are very near to unity. That is why in Figure 7 some configurations, unlikely to be sampled, are found in the bottom part of the graphs in Figure 7 and Figure 8. Indeed, all unlikely configurations are characterized by arc 4 occupying a non-nominal state, since the occupancy probabilities associated to these states are four orders of magnitude smaller than the probabilities of any other state of other biased arcs, as pictorially shown in Figure 9 with a comparison between occupancy probabilities for arc 1 and 4. Arc 4 is then much more likely to be sampled to its nominal state, the corresponding probability being “almost analog” and consequently very high.

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Figure 7. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.766

Even if there are no failed configurations with all biased arcs in nominal state and non-biased arcs in states under the nominal, best-performing one, as for k=0.7, yet a failed configuration with a relatively large weight is present. This is MMCV 5 in Error! Reference source not found., characterized by arc 4 being in a non-nominal state. For the argument above, this failed configuration is very unlikely to be sampled, being arc 4 in a non-nominal state. Figure 8 presents the weight distribution for case k=0.766 starting from a different initialization of the random number chain which happens to lead to the realization of such an unlikely large-weight network configuration.

15

Figure 8. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.766. The arrow points to a weight larger than unity.

As explained previously, the weight of such configuration is the product of 5 weight factors greater than 1, for biased arcs in nominal state, and one weight factor slightly less than 1, for arc 4 which is sampled to an unlikely state below the nominal one. Yet, in this case the latter contribution is not small enough to reduce below unity the total system configuration weight (Figure 10).

Figure 9: Natural and biased occupancy probabilities for arcs 1 and 4 in the case of k=0.766

16

Figure 10: Weight factors for arcs 1 and 4 in the case of k=0.766. The solid line gives the result of the analog simulation.

2.2. Application of Biasing Method 2

Several MC simulations have been performed with the biasing method illustrated in Section 1.3.1, with values of k ranging from 0.287 to 1. As explained in Section 1.3.1, the choice of the value of k sets the performance threshold wth for arc biasing: each arc i whose average performance wi is above threshold, i.e. , is forced to a new

average performance level by increasing the occupancy probabilities of its

non nominal states j<m

iw w> th

th

h

*iw w=

i. Hence, a biased simulation with k > 1, resulting in ,

would be meaningless because it would increase the occupancy probability

*i tw w>

, ii mp of the arc nominal state mi, which is exactly the opposite goal of the biasing. As for the lower limit of 0.527, it derives from the normalization to 1 of the state probabilities, as explained in Section 1.3.1.

For each simulation, the network unreliability, with respect to the delivery of a demand d of 10 units at the target node, and the associated standard deviation have been computed. All the biased simulations are made up of 5·105 trials; an analog simulation has also been performed with 109 trials. A commonly accepted figure of merit has been used to compare the performances of the biased simulations. This is defined as the reciprocal of the product of the simulation time multiplied by the variance of the unreliability estimate: the higher the value of the figure of merit, the more efficient the simulation.

As for the non-nominal states j which are not favoured, these are: j=3 for arc i=1, j=2 and j=3 for arc i=2, j=2 for arc i=4, j=2 for arc i=6, j=2 and j=3 for arc i=8 and, j=3 for arc i=9.

17

The results are reported in Table 1 and in Figure 11 and Figure 12. All the biased simulations are made up of 5·105 trials. Table 1. Results of the ARPA network of Figure 2. The unreliability estimate is reported with one standard deviation. The figure of merit is the reciprocal of the product of the simulation time multiplied by the variance of the unreliability estimate.

k Γ (Biased arcs) Unreliability Minimum weight Maximum weight Figure of merit

1 1 9 6 8 2 (10.152 ± 1.922)·10-5 8.12·10-22 4.39·100 5.04·108

0.8 1 9 6 8 2 (10.726 ± 3.130)·10-5 3.34·10-23 1.58·101 1.86·108

0.766 1 9 6 8 2 4 (11.026 ± 3.431)·10-5 1.95·10-23 1.74·101 1.43·108

0.7 1 9 6 8 2 4 (8.491 ± 0.038)·10-5 1.41·10-27 4.08·10-3 1.22·1012

0.6 1 9 6 8 2 4 (8.513 ± 0.055)·10-5 2.43·10-28 4.23·10-3 5.80·1011

0.5 1 9 6 8 2 4 7 5 3 (8.386 ± 0.105)·10-5 1.38·10-40 1.35·10-2 1.58·1011

0.4 1 9 6 8 2 4 7 5 3 (8.633 ± 0.301)·10-5 4.21·10-42 1.04·10-1 1.91·1010

0.3 1 9 6 8 2 4 7 5 3 (5.521 ± 1.773)·10-5 4.93·10-43 3.21·100 5.41·108

Analog simulation (8.502 ± 0.030)·10-5 1 1 2.00·109

Figure 11. Unreliability vs. k for the ARPA network of Figure 2. The solid line gives the result of the analog simulation

18

Figure 12. Figure of merit for the simulations of the ARPA network of Figure 2. The solid line gives the result of the analog simulation

Figure 11 shows the system unreliability estimate and the associated uncertainty measured by one standard deviation, for different values of the biasing parameter k. The proposed biasing method gives good results when the value of k lies within 0.4 and 1, with the best performances between [0.4 - 0.7] in terms of reduced standard deviation. Conversely, a bias in the estimate of the unreliability is introduced when k lies outside of this range. Overall, the values of the figure of merit are larger than the ones obtained with the method proposed in [31] (see Table 1 and Figure 12). Also for the present method, the value of k=0.7 gives the best results in terms of precision of unreliability estimate and reduced associated standard deviation (see Table 1). Good results are obtained also for low values of k up to k=0.4. For k<0.4 the bias of the probabilities becomes too strong and correspondingly the weights become greater than unity even for system configurations in which one of the 9 biased arcs is sampled in a favored state with weight factor well below unity. As a result, the unreliability estimate is biased and the associated standard deviation is large.

For values of k larger than 0.766 the unreliability estimate turns out to be biased due to the presence of system configurations characterized by a total weight greater than unity (see for example Figure 13 for k=0.8).

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Figure 13. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.8. The arrow points to a weight larger than unity.

The value k=0.766 is still a boundary between the biasing of 5 and 6 arcs and as in the previous case of method 1, the presence of the “almost analog” arc 4 causes a total system weight larger than 1 for configurations with arc 4 failed (e.g. Figure 14).

Figure 14. Distribution, in bi-logarithmic scales, of the weights collected in correspondence of the sampled system failed configurations, for the case k=0.766. The arrow points to the weight larger than unity.

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Overall, biasing method 2 gives results similar to those of method 1, with a slight improvement for k values in the range [0.4, 0.7].

Conclusions

Two methods have been proposed for improving the robustness of the two-terminal unreliability estimate of complex multi-state network systems by biased Monte Carlo simulation. Both methods are governed by only one parameter whose choice determines the number of biased arcs and the magnitude of the bias, according to their performances. The effectiveness of the methods has been analyzed with respect to different values of the biasing parameter. A reference network of literature has been taken as case study. Ranges of the values of the biasing parameter can be identified for which the biased simulations are more effective than the analog simulation, with respect to a figure of merit which accounts for the reduction in the variance of the unreliability estimate and for the computational time.

A drawback in relying on only one parameter to control the biasing could be that the number of biased arcs is directly connected to the magnitude of the bias of the natural probabilities, with a slight loss of flexibility for a more general rule of bias. On the other hand, having to set only one parameter reduces the arbitrariness of the method, thus allowing to achieve more stable results.

Further developments of interest regard the merging of these biasing methods with some technique of splitting to reduce the total weight of unfavoured network configurations. Moreover, knowledge of the MMCVs is required to evaluate whether the sampled network is capable or not of meeting the required demand: analytical methods for MMCV identification and M2TRd evaluation can be used only on very simple systems. In this respect, it seems worthwhile to investigate the possibility of coupling the proposed simulation method with numerical methods for assessing the transmission flow through the network nodes, e.g. by path flow algorithms or Cellular Automata.

Acknowledgements

The authors wish to express their appreciation for the fruitful discussions with Prof. Claudio Rocco of the Universidad Central de Venezuela and Dr. Luca Podofillini of the Paul Scherrer Institute, Switzerland.

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