ECONOMIC RELIABILITY FORECASTING IN AN UNCERTAIN WORLD

E CONOMIC R ELIABILITY

F ORECASTIN

G IN AN U NCERTAIN W ORLD

ED STOKER?and JOANNE BECHT A DUGAN?

?University of Virginia

Electrical and Computer Engineering Department

P.O.Box400473Charlottesville V A22904-4136

estoker@87892e1fc281e53a5802ffde and jbd@87892e1fc281e53a5802ffde

Abstract Substantial work has applied stochastic techniques to network reliability models.These techniques can estimate risk, variance and uncertainty values.Unfortunately,these models do not address the issues of revenues,return on investment,or the time-value of money.To address these issues,we have developed an Economic Reliability Analysis[ERA]framework at the University of Virginia that fuses reliability engineering methods with economic analysis.We combine the ERA framework with stochastic techniques to evaluate a simple network and a proposed network upgrade.We simulate key availability and ?nancial elements of both networks and apply the ERA framework to these elements.These results are compared with full path enumeration results of the same networks.This analysis provides a richer,more complete method to apply stochastic network techniques to operational network upgrades.

Index Terms Stochastic network models,Reliability and maintenance models,Stochastic simulation.

I.INTRODUCTION

Consider the following scenario.A company has a network that provides the basis for its revenue.The company must choose whether to maintain the status quo or modify the network to gain a new revenue stream.The company only has resources to choose one of these projects.The question is,which project should be implemented.The general problem,simply stated is:”How do you pro?tably operate,maintain and evolve a dependable operational network?”This general problem can be addressed by a set of smaller,more directed questions.These questions are:

1)What is the economic effect of developing and imple-

menting a network change?

2)What is the economic improvement associated with im-

proving network reliability?

3)When do the costs of improving network availability

exceed its expected bene?ts?

4)How reliable must a new network be before it becomes

operational?

These questions can be dif?cult to answer.Most organiza-tions have several different types of network components in a network,each with different associated reliability and repair cost data.In addition,different user-oriented measurements for availability and their economic impact must be understood. As such,these two costs associated with a network failure (network component repair cost and,lost revenue associated with a network failure)must be considered when modeling the economic impact of network repairs.

This article aims to extend the network reliability model tech-niques with an Economic Reliability Analysis(ERA)framework developed at the University of Virginia[7]and apply it to an operational network system.The motivation for writing this paper is threefold.First,we want to address the dilemma of picking a project that will affect the reliability and economics of an operational network system.We also want to extend the framework to provide some estimates of risk and con?dence that can be provided from the inclusion of stochastic modeling techniques.Finally,we want to illustrate the power and usefulness that simulation techniques can provide to practical business and engineering decisions.

II.RELATED WORK

Several research groups have investigated the relationship be-tween reliability and economic value.Current literature indi-cates this relationship has taken several directions. Research at British Telecom,[8],[1],[9]focused on modeling repair costs of their own telecommunications network.This directed research aimed at predicting expected costs without providing any structural insight into controlling these costs. Their system was a very large,distributed network,where the principal issue was the rapid detection,identi?cation and restoral of telephone service 87892e1fc281e53a5802ffdework design or new service offerings were not considered.

Economic models have been proposed to deal with the produc-tion and distribution of electrical power in which the reliability of the power grid,electrical production and distribution costs along with macroeconomic models are considered[10],[11], [12].Yoon and Ili`c treat electrical power as a commodity and propose a new business model for this industry.Their research aims to improve delivery of electrical power to consumers with greater economic ef?ciency.

Mitchell and Gelles[4]describe a framework for risk-value models.Research in Markov reward models[3],Petri net models[2],advanced Monte-Carlo simulation procedures[5],

and rare event simulation[6]provide insights into the use of stochastic techniques to estimate network reliability.

Current approaches do not adequately describe the monetary bene?ts(i.e.revenues)associated with operational networks or the time-value of money.Inclusion of these concepts can produce a better understanding of the economic worth of a reliable operational network.

In[7],Stoker and Dugan de?ne an Economic Reliability Anal-ysis methodology to evaluate the economic worth of a reliable network.The general strategy behind the ERA framework is to collect and use availability and?nancial data about a system from within an organization rather than build”yet another reliability/?nancial model.”The ERA framework provides a means to determine how changes in component reliability, service pricing,and component/task dependencies in?uence system availability,return on investments,and design.

Step Function

1Determine the duration,size and scope of the analysis.

2Build network reliability models for all design choices.

3Map network reliability information into

component and task failure data.

4Calculate revenue vectors for all design choices.

5Calculate lost revenue vectors for all design choices.

6Calculate recurring cost vectors for all design choices.

7Calculate other cost vectors for all design choices.

8Calculate capital cost vectors for all design choices.

9Calculate[ERV]for all design choices.

10Analyze and interpret results of evaluation.

TABLE I

ERA F RAMEWORK A LGORITHM

Table I provides a summary of the operational set of processes performed by the ERA framework.These processes will be illustrated in the following example.

III.STOCHASTIC RELIABILITY EXAMPLE

Economic and reliability processes will be simulated using stochastic methods,sampled and evaluated with the ERA frame-work.The results of this simulation will be compared to results using deterministic methods.See[7]for a complete description of the Economic Framework,network solutions and exact re-sults.These simple simulations will allow us to easily compare the impact of uncertainty on expected system reliability and economic worth.In addition,these simulations will provide an expected range of parametric values for both networks.Fi-nally,comparisons that account for normal variances between networks can be made.

Figure1shows a current network(Network A)and a pro-posed network(Network B).The proposed change will be to add a node and move two links(A3and A6)to connect between Nodes s and 87892e1fc281e53a5802ffdework B is more reliable than Network A,

Network A Network B

Fig.1.Example Networks

assuming that only edges fail and that edge reliability metrics are identical in both Networks.

IV.QUESTIONS TO ANSWER

Stochastic network reliability models can provide con?dence intervals on reliability parameters by simulating when com-ponents components fail and repair rates.These models also allow one to examine the impact of uncertainty on network availability.Both of these elements can signi?cantly alter net-work design choices.Below are a set of questions that can be answered using a stochastic network model.

1)What is the range of expected availability for a given

network?

2)What is the range of expected Economic Reliability Val-

ues for a given network?

3)What is the impact of uncertainty on expected network

availability?

4)What is the impact of uncertainty on network Economic

Reliability Value?

5)How does component reliability variances affect net-

work availability?

6)How does component reliability variances affect net-

work Economic Reliability Value?

A.Model Assumptions

1)A Net

A

$1500and A Net B$1550.

2)Network failure cost is$100per 87892e1fc281e53a5802ffdework failure

duration is2hours.

3)Edge MTTR is2hours,edge availability is.99,average

edge repair cost is$10/repair and the average edge repair rate is$0/hour for all edge repairs and applies to both networks.

4)F Net

A

$0and F Net B$10.

5)G Net

B

$1000for the initial time period else,G Net B

$0.G Net

A

$0.

6)Discount rate process(DR)is1%per month for the

duration of the analysis.

7)Investment period is24months.This is used to limit the

size of the economic vectors.

8)All revenues and expenses are estimated on a monthly

basis.

9)Only edges fail.Nodes do not fail.

10)All edges fail identically and independently in all time

periods.

11)The analysis only deals with the two-terminal (s -t)

network availability.

12)Performance failures and costs are ignored in this anal-ysis.This limits the size and complexity of the analysis.

B.Uncertainty

We will now add uncertainty to stochastic model assumptions (1,2and 3)by incorporating stochastic rather than determin-istic revenue,cost,and component failure functions.We will also assume that all network failures in both networks are detected and solved.Revenue uncertainty is usually treated as receiving less than expected (or contracted)payments for services.Typically,accountants will assign a ’reserve’for bad credit extended to customers.It is important to model revenue stochastically rather than as a weighted average to account for the time value of the revenue vector.The same reasoning applies to modeling network and component costs.

1)A Net A $1500occurs with a probability of 0.95;A Net A

$1300occurs with a probability of 0.05of the time when one or more customers do not pay.This amounts to a $200loss of revenue in the month that it occurs.A Net B $1550occurs with a probability of 0.95;A Net B $1300occurs with a probability of 0.05of the time when one or more customers do not pay.This amounts to a $250loss of revenue in the month that it occurs.

2)The network failure cost function for Net A and Net B

exhibits a uniform pdf between $75per failure and $125per failure with an average of $100per failure.The network MTBF function has a uniform pdf between 5667hours per failure and 7667hours per failure with an average of 6667hours per failure.

3)The component failure cost function for Net A exhibits a uniform pdf between $5per failure and $15per fail-ure with an average of $10per failure.The component MTBF function has a uniform pdf between 168hours per failure and 228hours per failure with an average of 198hours per failure.

V .QUESTIONS ANSWERED

A simulation was run on both networks using the assump-tions described above.The simulation consisted of 10,000runs for each month for both networks.Minimum,mean,median,and maximum along with the 5th and 95th quantile values for revenues,lost revenues and component repair costs were captured.The ERV s for both networks were also solved using full path enumeration to get a deterministic set of values.This analysis is used to answer the questions raised earlier.1.What is the range of expected availability for a given network?

Monthly availability for Network A ranges from a minimum of 09962in month 3to a maximum of 09983in month 4with an average of 09970over the 24month duration.These values compare with the exact monthly availability value for Network A of 09997.The monthly availability for Network B ranges from a minimum of 09996in months 01121to a max-imum of 10in months 151722with an average of 09998over the 24month duration.These values compare with the exact monthly availability value for Network B of 099998.

2.What is the range of expected Economic Reliability Val-ues for a given network?

Table II compares the exact and stochastic ERV s for both networks over a 24month period.The ?rst observation to note is that,for the duration of the analysis,the stochastic Network A model always has a greater ERV than stochastic Network B model.This is a different result than is obtained from solving an analytic model.The analytic choice over a 24month period is Network B .

Time Analytic

Analytic

Stochastic

Stochastic

Net A Net B Net A Net B 0119691247361184582211312381971482372357691430952355529270515351868262769347170039158246684538124645867205114515806764986875700602630133693282614930681335674764080491173005979249948639839153638440158103552697917510247239569169114496410932261133045106862810125331912061481240232117916311136060113179521346329128863612146682014286481451458139709113157198815382491555511150438714167611416467651658521161063215177921017542061760511171589816188128518605841861553182004017198234919659081961543192321118208241320701892060620202540219218148621734382158705212648620227957722756652255730222654821237669823768792351855232568422247285724770922447017242384023

2568064

2576312

2541169

2521039

TABLE II

A NALYTIC /S TOCHASTIC ERV T ABLE

The discrepancies between the stochastic and analytic models occur in the revenues .They are lower in the stochastic models and most notably in the network costs which are higher in the stochastic models.These differences disappear as one narrows the range of the costs and the MTBF values to the average .The net result shows the difference in the analytic ERV s decreases more quickly than the stochastic ERV s.

3.What is the impact of uncertainty on expected network availability?

Allowing uncertainty into availability calculations(in the form of a probablistic function for component MTBF),produced a slightly lower average network availability over the two year forecast for both networks(see Table III).The proportional error difference for the expected network availability of these networks is027%for Network A and0019%for Network B.

Network Min.Avg.Exact Max.

Avail.Avail.Avail.Avail.

Net A09962099700999709983

Net B999699989999810

TABLE III

A VAILABILITY C OMPARISON T ABLE

4.What is the impact of uncertainty on network Economic Reliability Value?

Uncertainty has a slightly greater impact on the economic elements that form the ERV than it has on network availability. This is due in part to the asymmetric nature of uncertainty leading to lower revenue and higher failure costs.This impact can be seen in Table IV.

Month Min.5th Q Mean Median95th Q Max.

Annuity Annuity Annuity Annuity Annuity Annuity

0952.061111.671184.581190.691244.941301.74

1930.111113.391184.841190.601245.501319.61

2942.991113.841184.321190.071244.951305.40

3949.171116.691184.601190.401245.311298.22

4950.341113.211184.531190.671244.991299.44

5962.091109.981183.501189.801244.261296.89

6960.241119.121184.971190.051245.411299.84

7966.441110.771184.191190.151246.201312.84

8948.551107.641184.211190.021244.211301.29

9946.441109.431184.701190.241244.931300.53

10921.131108.021184.021190.191245.171296.77

11943.771112.361183.691190.001245.151312.01

12960.121111.441184.621191.041244.831323.03

13944.081111.831184.221189.781244.071301.74

14949.721106.151184.071190.031244.661308.01

15949.221111.661184.071190.281245.471302.65

16893.831114.041184.801190.361245.061300.81

17949.101108.181184.181189.741245.721313.72

18946.721110.851185.111191.041244.791302.16

19929.601112.141184.971191.391245.681309.00

20925.561103.781183.881190.401245.171310.42

21940.281114.161184.641190.441244.481300.13

22953.941114.661184.491189.931243.061299.00

23948.041106.081183.641189.771243.491302.75 Average944.311111.301184.371190.301244.901304.92

TABLE IV

N ETWORK A M ONTHLY A NNUITY M ETRICS T ABLE

The exact monthly annuity for Network A is119396.The percentage error for the median stochastic annuity and the exact annuity is031%,which compares with the relative net-work availability error of027%.The percentage error of the expected stochastic ERV from the expected closed form ERV is032%for Network A and089%for Network B.Figure2 plots the relative ERV error for both networks over time.The comparatively large relative error in Network B in the?rst month is caused by the small size of Network B ERV in the ?rst month.

VI.SUMMARY AND CONCLUSIONS

87892e1fc281e53a5802ffdework A&Network B relative error over time

Stochastic reliability techniques have long been used to esti-mate network availability.Even simple stochastic models of networks can provide reasonable estimates of both economic ef?ciency and network availability with greater behavioral re-alism than comparable analytical methods.We have applied these techniques to estimate the expected economic impact of network and component availability and validated the results against a closed form solution.

The initial results are encouraging.We have taken our Eco-nomic Reliability Analysis framework and incorporated stochas-tic methods into it with satisfactory results.Further research and application is planned.Current plans are to apply the methods described in this paper to an operational network.

R EFERENCES

[1]P.Bell,R.Walling,and J.Peacock,“Costing the access network-an

overview,”British Telecom Technology Journal,vol.14,no.2,pp.128–132,April1996.

[2]Y.Dutuit,E.Ch?a telet,J.P.Signoret,and P.Thomas,“Dependability

modelling and evaluation by using stochastic petri nets:application to two test cases,”Reliability Engineering and System Safety,vol.55,pp.

117–124,1997.

[3]K.Goseva-Popstojanova and K.Trivedi.(2001,November9)Stochastic

modeling formalisms for dependability,performance and performability.

[Online].Available:link.springer.de/link/service/series/0558/bibs/ 1769/17690403.htm

[4] D.W.Mitchell and G.M.Gelles,“Risk-value models:Restrictions and

applications,”European Journal of Operational Research,vol.145,pp.

109–120,2003.

[5]H.J.Pradwarter and G.I.Schu¨e ller,“On advanced monte carlo

simulation procedures in stochastic structure dynamics,”International Journal of Non-Linear Mechanics,vol.32,no.4,pp.735–744,1997.

[6]P.Shahabuddin,“Rare event simulation in stochastic models,”in

Proceedings of the27th conference on Winter simulation.New York, N.Y.,United States:ACM Press,1995,pp.178–185.[Online].

Available:87892e1fc281e53a5802ffde/10.1145/224401.224460

[7] E.J.Stoker and J.B.Dugan,“A framework for economic reliability

analysis,”University of Virginia,Charlottesville,V A,United States, Technical Report TR-ES2003,February2003.

[8]M.R.Thomas,P.Bell,C.A.Gould,and J.Mellis,“Fault rate analysis,

modelling and estimation,”British Telecom Technology Journal,vol.14, no.2,pp.133–139,April1996.

[9]J.Tindle,S.J.Brewis,and H.M.Ryan,“Advanced simulation and

optimisation of the telecommunications network,”British Telecom Tech-nology Journal,vol.14,no.2,pp.140–146,April1996.

[10]Y.T.Yoon and M.D.Ili`c,“Independent transmission company(itc)

and markets for transmission,”in Power Engineering Society Summer Meeting,vol.1.Los Alamitos,CA,United States:IEEE Press,2001, pp.229–234.

[11]——,“Price-cap regulation for transmission:objectives and tariffs,”in

Power Engineering Society Summer Meeting,vol.2.Los Alamitos, CA,United States:IEEE Press,2001,pp.1052–1057.

[12]——,“A possible notion of short-term value-based reliability,”in Power

Engineering Society Winter Meeting,vol.2.Los Alamitos,CA,United States:IEEE Press,2002,pp.772–778.

Ed Stoker(born1951)received his B.A.,M.A.,and M.B.A in1975,1980and 1981respectively from the University of Pittsburgh,Pittsburgh PA,USA and has worked in computer network engineering since that time.He is currently a PhD.candidate in Computer Engineering at the University of Virginia. Joanne Bechta Dugan(F’00)received the B.A.degree(1980)in mathematics and computer science from La Salle University,Philadelphia,PA,and the M.S. and PhD.degrees in1982and1984,respectively,in electrical engineering from Duke University,Durham,NC.

She is Professor of Electrical and Computer Engineering with the Univer-sity of Virginia.She has performed and directed research on the development and application of techniques for the analysis of computer systems that are designed to tolerate hardware and software faults.Her research interests in-

clude hardware and software reliability engineering,fault tolerant computing and mathematical modeling using dynamic fault trees,Markov models,Petri nets and simulation.

Dr.Dugan was an Associate Editor of the IEEE T RANSACTIONS ON R ELI-ABILITY for10years,and is Associate Editor of the IEEE T RANSACTIONS ON S OFTWARE E NGINEERING.She served on the USA National Research Council Committee on Application aof Digital Instrumentation and Control

Systems to Nuclear Power Plant Operations and Safety.APPENDIX

Symbol De?nition

BPR Business Process Re-engineering

DCF Discounted Cash Flow

ERA Economic Reliability Analysis

ERV Economic Reliability Value

NPV Net Present Value

A Revenue process as a function of network

design and?nance

A Revenue vector produced by A

B Lost revenue process as a function of network failure

B Lost revenue vector produced by B

T F n Task n failure vector

T C n Task n repair cost per failure vector

PF n Process n failure vector

PC n Process n repair cost per failure vector

C Lost revenue process as a function of QoS failure

C Lost revenue vector produced by C

LR Lost revenue process:B C

LR Lost revenue vector:B C

D Component repair cost process as a function

of network failures

D Component repair cost produced by D

E Component repair cost process as a

function of QoS failures

E Component repair cost vector produced by E

F Other recurring cost process based on

normal network operations

F Other recurring cost vector produced by F

OC Other recurring cost process unrelated to

normal network operations

OC Other recurring cost vector produced by OC

R C Recurring Cost process:D E F OC

R C Recurring cost vector:D E F OC

G Capital cost process as a function of

G network design and?nance

G Capital cost vector produced by G

H Annuity process as a function of reliability

H Annuity vector produced by H

DR Discount rate process

DR Discount rate vector produced by DR

EC ERV Contribution process

EC ERV Contribution vector produced by EC

TABLE V

N OTATION

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