Towards realism in network simulation

Towards realism in network simulation

Terry Williams *

Department of Management Science, Strathclyde University, Graham Hills Building, 40 George Street, Glasgow G1 1QE, UK

Received 1 April 1998; accepted 1 October 1998

Abstract

The use of networks handling uncertainty to provide a temporal risk analysis of projects is now widespread.However, such analyses frequently give rise to very wide probability distributions, and thus in practice are describedas not credible. This is largely because the simulations do not re¯ect the actions that management would take to

bring late-running projects under control. These are di�cult to include in models, not because the actionsthemselves are complex, but rather because the e�ects of those actions are not well-understood. These e�ects areoften much less e�ective than expected and some are counter-intuitive. However, much work has been done in

modelling projects using system dynamics, and this work can give some useful insights into the e�ects ofmanagement actions in projects, both their behaviour and indications of their cumulative impact. This paper hasattempted to describe these indications and then to apply such lessons to network simulations, to gain the bene®t ofthe insights without losing the operational advantages of the networks. Some small illustrative models of the e�ects

are given. It is hoped that the use of such modelling can help to bring additional realism to probabilistic networkmodelling. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Networks; Simulation; Project risk; System dynamics

1. Introduction

Temporal analyses of projects are nearly always

based on the network (PERT or CPM) concept

(e.g. [34]). In recent years, much work has been done

in trying to assess the temporal risk in projects by

including uncertainty in such networks. These analyses

have become more sophisticated, but are still very fre-

quently giving results that are not believed, frequently

because they have very wide probability distributions

when the project management knows the probable

range is very much smaller.

After introducing the ideas of network simulation,

and the bene®ts it brings, this paper traces recent

developments in network simulation attempting to

enhance the realism in the modelling. The reason for

the wide probability distributions is then identi®ed as

the need to model management action and the conse-

quences thereof. Attempts to include such action are

reviewed, but these have not been successful. This is

largely because the e�ects of such actions are not well

understood. To give some insights into the e�ects of

management actions on projects, the ideas of system

dynamics (SD) are considered. SD has been used a

number of times to model project behaviour, but

nearly always at a strategic level, unrelated to oper-

ational considerations such as the project network

(hence its lack of usage by practising project man-

agers). This paper looks at the SD work and tries to

draw out some of the lessons from the modelling that

has taken place; these lessons are then used to amend

the operational network models. This can help to give

a simple model of what would actually happen in pro-

jects, leading to more believable results. This, it is

hoped, will provide some recovery of the credibility

that network simulation has lost by providing unbelie-

vable results.

Omega, Int. J. Mgmt. Sci. 27 (1999) 305±314

0305-0483/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0305-0483(98 )00062-0

PERGAMON

* Tel.: +44-141-552-4400; fax: +44-141-552-6686; e-mail:

[email protected].

The paper is written from the point of view of theauthor who was a project risk manager for a decade

before returning to academia ®ve years ago. It aims toexplore what is useful to the project management pro-fession, rather than merely what is academically inter-

esting. A recent conference similarly concluded thatwhile there could clearly be extremes of, on the onehand, academic modelers who have no impact on pro-

jects and, on the other hand, project managers whoreject all mathematical techniques to the detriment oftheir planning. Steering a middle course, development

of technique must always be driven by the needs ofproject managers and the desire to improveperformance [51].

2. Analytical methods

Network analysis is a well-established analytical tool

when the project is deterministic. When uncertaintiesare introduced into the analysis, various analyticalapproaches have been tried. The ®rst line of attack onstochastic networks is of course straight analytic sol-

ution, and there are excellent reviews of this work byAdlakha and Kulkarni [3] and Ritchie [39]. However,since there is no simple analytic solution of the dur-

ation of the stochastic network, other than for speci®ccases, e.g. exponentially-distributed networks [30],other authors (for example Dodin [17], Devroye [16]

and Kamburowski [25, 26]) have attempted to deriveexact bounds. Finally, others analytically derive ap-proximate solutions, such as Anklesaria and

Drezner [5], using stochastically dominating paths, andMehrotra et al. [33], using the commonality of activi-ties on paths and employing a Normal approximation.However, while these methods can be shown to work

on some small or specially-constructed networks, thereis no general indication of their accuracy, nor explora-tion of their use for real-life networks.

However, the main problem with the analyticalapproaches is the restrictive assumptions that they allrequire, making them unusable in any practical situ-

ations; this means that currently they are not really ofinterest to practitioners. First of all, there are therestrictions of commission: most require very speci®c(and sometimes unrealistic) duration-distributions, and

are not applicable unless all the activities are of thisdistribution. Secondly, there are the very signi®cantlimitations of omission: these techniques are not appli-

cable if there are resource-constraints on the network(indeed, it shall be seen below that activity-criticalityindices are not even de®ned in this circumstance), and,

perhaps more importantly, there is the plethora ofcomplexities that the practising project modelerrequires to include in his analysis for the results to be

both relevant and credible, such as the following (seee.g. Ref. [47]):

. e�ects that operate across a range of activities and/or resources, such as third-party e�ects or common-

cause e�ects: the major risks within a project moreusually than not involve more than one activity (theassumption that activity-durations are iidrv's isenough to lose a technique credibility);

. unusual activity±duration-distributions; one obviousinstance that frequently occurs in practice is theneed to combine 0±1 and continuous uncertainties

when several risks a�ect the same activity, (particu-larly where some of those risks are epistemic (i.e.due to a lack of knowledge) and others are aleatoric

(i.e. of an underlying probabilistic nature) [49]; (asecond instance is given in Section 3);

. resource availabilities or requirements that are

uncertain (and possibly varying over time): these cansometimes be the critical uncertainties, for example,the precise timing of the availability window for avital resource can be the deciding factor in whether

a project meets its deadline;. uncertainties in the project network structure: whileall classical network methods assume that the net-

work itself is ®xed, in practice there can be probabil-istic or conditional branching;

. domain-speci®c uncertainties which can have very

particular uncertainty pro®les, such as the randomfailure of test-rigs, or weather-windows in deep-seaoil work etc;

and so on. Furthermore, of course, these complexitiesoccur not individually but severally: for example, whendesigning a military aircraft, there is a possibility that

one or more of the test-aircraft crashes: such an occur-rence would a�ect many streams of activity simul-taneously (avionics, engine and air-frame), it wouldcause the resource availability to be uncertain (so later

test-activities will take longer), it would cause individ-ual durations to have nonsimple distributions, andthese e�ects might occur at random.

It is perhaps worth noting ®nally that some analyti-cal methods (e.g. Mehrotra et al.'s [33]) provide onlycertain moments of the project duration. However, the

main purpose of this analysis in practise is to give aidin bidding, when the project manager is interested inquestions such as:

. what is the probability of meeting the project due-date?

. if there are liquidated damages, what is the expected

penalty (i.e.

�t

p�t� � LD�t�dt

T. Williams / Omega, Int. J. Mgmt. Sci. 27 (1999) 305±314306

where p(t) is the PDF of project duration and LD(t)is the damages payable for a duration of t)?.

. what is the (say) 90% con®dent project duration?

Such questions require an appreciation of the whole

duration distribution, and such distributions cannot beassumed to be any of the standard models: even for asimple stochastic network, the total duration is theresult of a combination of convolutions and taking

maxima; of course, where there are the more complexuncertainties noted above, the duration can take quitecomplicated shapes.

3. Network simulation

A number of analysts are therefore working onusing simulation to model the type of projects foundin practice. There has not been universal agreement,

obviously. Chapman [10], for example, says thatalways using Monte Carlo simulation is a simple gen-eral solution, but it is often ine�cient ùsing a sledge-

hammer to crack a nut'. More important, to continuethe analogy, it encourages an approach which makesthe walnuts very unattractive eating, and di�cult to

®nd. Similarly, Adlington and Christensen [4] feel thatthe use of sophisticated mathematics, simulation andspecialist risk managers is misplaced; they say that thefundamental aim of a risk analysis is to establish a

`picture' of the future which enables management tomake informed decisions, to grasp the opportunitiesand to control or make provision for the risks; how-

ever, even they point to the problem of dependencieswithin projects, implying a need for simulation.Despite these concerns, many practitioners have

arrived at similar solutions: network-type models,using classical network logic, within a general simu-lation framework to enable the uncertainties to be

modelled. Such authors include the following:

. Ragsdale [38] gives a good summary of the literatureof network simulation to that date.

. Williams [46, 47] describes an early such system, andapplications to real projects within the defencedomain.

. Kidd [28, 29] describes the VERT system, which has

some similar characteristics (also looking at simulat-ing cost and performance).

. Similar work is given by Berny and Townsend [7] in

their VISIER software.. A user-friendly fully-engineered package similarlyable to give a high degree of generality to uncertain

networks is provided by @Risk for Project [36],which applies the @Risk concept (essentially aspreadsheet, but allowing cells to be allocated ran-

dom variables) to a standard network package

(MicroSoft Project).

. Similarly, the MonteCarlo package [37] allows most

features of a network analysis to be given variability

by the user, allowing custom distribution models,

duration correlations, conditional branching, prob-

abilistic resource requirements and availabilities,

repeat-chain logic and so on.

. There are also important simulation models of

speci®c projects Ð for example, Nicolo [35] tries to

capture the whole project environment (including

nonphysical parameters), and simulates a project

network with all of these e�ects, using a scenario

generation technique, building a project GERT net-

work covering the whole environment of the project,

then using the SLAM simulation package.

Although straightforward simulation is more fre-

quently used in practice, there have been theoretical

advances in improving the information derived from

simulations, generally using either variance reduction

(e.g. [6], particularly using antithetic variables as

in [44]) or conditional simulation, choosing which arcs

to simulate (e.g. [2, 18, 31]). The aim of this is to

reduce the number of iterations required. However, the

requirement for this has decreased markedly as compu-

ter power has increased, and furthermore since these

studies are generally carried out on networks popu-

lated by subjective probability-distributions whose pro-

venance is fairly weak, the search for precise accuracy

in practical applications can be rather spurious.

These simulations are aimed not just at estimating

the project duration distribution of the project dur-

ation, but also the relative importance of the activities.

Until the late 1980's, that meant estimating the activity

criticality indices [19]. However, it is now well-known

that these can be misleading and of course they cannot

be calculated when there are the more complex model-

ling features described above (although work such

as [8] is seeking to rede®ne criticality in a resource-con-

strained setting). These problems motivated the devel-

opment of the cruciality index, de®ned for an activity

as the correlation between the (activity-duration) and

the (total project-duration), to be used in collaboration

with (not simply replacing) traditional criticality

indices [48]. This index does not depend on the critical

path. This means that the output of a network simu-

lation (even if resource-constrained) can provide cruci-

ality ®gures and can also be used to provide cruciality

indices for aspects other than activity-durations (e.g.

resource availability-levels, or the choice at a stochastic

branch-point [48]).

Finally, one further purpose for which such network

simulations have shown themselves useful is in the

rational allocation of contingency to activities within a

T. Williams / Omega, Int. J. Mgmt. Sci. 27 (1999) 305±314 307

network, a perennial problem with large, dispersed,projects [55].

4. Problems with realism

Such simulations, while popular amongst risk ana-

lysts, have been slow to gain popularity with practisingproject managers. One issue that has arisen resultsfrom the nature of estimates populating such models.

There is clearly a relationship between setting PERTestimates and setting targets (e.g. Little®eld andRandolph [32] relate PERT with management byobjectives (MBO)). A natural consequence of this is

the observation that the actual duration is rarely lessthan the target. To some extent, this is due to the ideathat activity-durations tend to be skewed Ð some esti-

mate skew to be twice as much to the right of themode as to the left [20]. However, a major cause isobviously `Parkinson's e�ect': indeed, Gutierrez and

Kouvelis [24] attempt to de®ne some models in whichthe duration of an activity is dependent upon the allo-cated duration, including mathematical statements ofèxpect all activities to be late' and ìt is the busiest

man who has time to spare'. This led Williams [50] todevelop a particular duration-distribution based onthese e�ects that has been found in some circumstances

to correspond well to managers' subjective judgementson duration-distributions.Besides this, it is still a feature of many network

simulation outputs, that the distributions are oftenvery wide. A report that gives a 95% con®dence inter-val to the project duration falling between (say) 2 and

8 years will simply be ignored. A key reason for thisproblem is that the simulations described above simplycarry through each simulation run in a dumb fashion.In reality, of course, management would take action to

control a project going out of control, but untilrecently simulations have not been able to take cogni-sance of such actions. This has, in the opinion of this

author, gravely weakened the credibility of the time-risk analyses used in common practice.A few authors have attempted to include such man-

agement actions within analytical models or complexmathematical models using simulation to reach a sol-ution. Key amongst these is Golenko-Ginzburg: [21]describes a line of work through GERT leading to

CAAN (controlled alternative activity networks),which attempts to make decisions to optimise the net-works, although the decisions do not depend on state

of project at that time. The author understands thatsuch methods are not widely used in practice sincethey have a high level of complexity while still not

incorporating su�cient generality with su�cient trans-parency for practitioner-acceptance (although see adefence of this area and subsequent discussion in [51]).

(This work is continued in, for example, Golenko-Ginsberg and Gonik [22]).

Therefore, work is ongoing to attempt to includemanagement actions within project simulations.Tollersrun [45] makes a ®rst step towards this with

Statoil's TOPPS system by allowing a degree of correc-tive action during the project, based on the currentoutcome; for him, a project is de®ned as a dynamic de-

cision process. Also from Norway is TerreMar'sDynRisk package described by Skogen andHuseby [43], which again attempts to include decisions

within the simulations.However, there is a key problem with such work. It

has indeed recognised the need to incorporate manage-ment control into the models and has gone a little way

in this direction. However, these are simple controlactions with simple results, whereas in practice theresulting e�ects of such actions are often not obvious:

there are multiple in¯uences that combine to producethe resulting e�ects (often counter-intuitive). Thesemultiple in¯uences are not taken into account, and

thus not modelled, in current network simulationmodels.How can we learn about, and thus try to model, the

e�ects that occur when management take actions tocontrol projects? One area of work, which has studiedsuch actions and their resulting e�ects, has used theidea of system dynamics.

5. System dynamics

This alternative approach has ignored discrete-eventsimulation and project networks altogether and con-

centrated on the dynamics within the management de-cision-making process, in particular using systemdynamics (SD). Such work was begun by authors par-

ticularly such as Brooks [9] and Cooper [12], and thereis now a considerable body of work using SD tomodel the temporal ¯ow of projects (see a review byRodrigues and Bowers [40]). The SD modelling tech-

nique has four important bene®ts.

. SD shows the compounding e�ect of combinationsof in¯uences (the `2 + 2 = 5 rule').

. It is able to explicate the e�ects of feed-back on theproject.

. It is helpful in modelling the soft e�ects involved in

project management (schedule pressure, motivation,etc.).

. SD's original design was to model the e�ects of the

information ¯ows; this helps to allow decision-rulesto be built in to try to mimic the actions that man-agement would take.


These advantages of SD have enabled it to be used

to give interesting, and sometimes somewhat counter-

intuitive, insights into the behaviour of large projects.

This is particularly so when it has been used to expli-

cate past project behaviour in post-mortem

analysis [52]. In particular, it has been shown that

management actions, taken to try to achieve di�cult

schedules, or particularly to try to regain schedules

that have slipped, will often have much less e�ect than

expected, due to a number of interrelated e�ects, for

example:

. more workers have to be taken on than originally

planned, and/or extra shift-patterns and more over-

time have to be worked in order to keep to the sche-

dule; this exacerbates the feedback loops to cause

increasing ine�ciency; this well-known phenomenon

is commonly referred to as the $2000 hour, a term

originally coined by Cooper [14];

. the lack of system freeze, combined with a tight

time-constraint, forces management to work on pro-

ject-elements for which the surrounding system is

not yet frozen (which has to be reworked if there are

changes in the as-yet unfrozen surrounding system);

. such action has secondary e�ects, such as disincenti-

vising the design sta� as they work with unclear par-

ameters and knowing that their work may turn out

to be nugatory;

. when a concurrent manufacturing phase is con-

sidered, there are additional e�ects, both because de-

sign activities ®nish later and thus increase

concurrency, but also because items begin manufac-

ture and are then changed, which leads to retro®t,

degradation of manufacture learning and so on.

Cooper [15] gives a good overview of the modelling

of such e�ects in complex projects.

However, this SD modelling takes an overall, sys-

temic view of the whole project. It puts aside ideas

such as activities in a network and work breakdown

structures, and thus loses much of the operational in-

formation about the project. This means that, while it

has shown interesting insights in post mortem analyses,

and is able to give some useful guidelines before the

start of a project, it has been virtually ignored at the

operational project management level which monitors

and controls ongoing projects.

One answer to this is to try to combine the classical

simulation methods with SD and gain a management

structure in which each informs each other; this has

been tried on at least one real large project by

Rodrigues and Williams [41] and Rodrigues is further

developing formalised methodologies for its implemen-

tation. This is certainly a possible way forward,

although needing more work to operationalise the

method, and a lot of work to sell it to practising risk

managers. An alternative is to seek to gain lessonsfrom the SD work and apply it to the current network

simulation models, which can then be used in a moreintelligent way than at present.

6. A modi®ed network simulation

We have established that network simulations lackrealism and credibility because they lack modelling oftwo features:

. management actions in response to a late-runningproject;

. the e�ects resulting from such actions; clearly, sim-

plistically modelling the fact that management act tobring a late-running project into line would implythat projects all end up on time Ð however, the sys-

tem dynamics work provides some useful lessonsabout the e�ects of such actions, which could besimpli®ed and included in our network simulations.

It is not the aim of this paper to give an exhaustiveaccount of all possible management actions, nor amodel of the e�ects of these actions generally appli-

cable to all projects. The examples below aim to indi-cate the type of network simulation models that can bedeveloped drawing upon the lessons from systemdynamics.

There are two generic types of action which appearto be most important:

1. Bringing an activity forward. If activity i has a stan-dard ®nish±start dependency (or dependencies) asshown in Fig. 1a, this action would start activity iearlier than would normally be the case, resulting

(theoretically) in the dependency shown in Fig. 1b.2. Reducing an activity's duration: thus if activity i

has duration ti, this action would change its dur-

ation to lti (where l is a crashing proportion, suchthat 0 < l< 1). Of course, this idea of crashing ac-tivity durations is well-known and there is a fair

amount of literature on ®nding the optimum projectduration when each of a subset of activities inde-pendently has crashing potential. However, this is ahypothetical situation and it is not clear to what

extent the ideas are used in practice;Kamburowski [27] described an optimum algorithmrecently, but in the subsequent discussion

Chapman [11] comments that work on the proper-ties of exact project scheduling network modelsìs . . . of little practical interest'. One reason for this

is because the down-stream e�ects of crashing oneactivity are not included in the model; we can lookto SD to gain insight into what these e�ects are.


What are the e�ects of such management actions?

Typical papers that discuss these e�ects include the fol-lowing.

1. Rodrigues and Williams [42] discuss (inter al) the

e�ects of reducing an activity's length (based onwork in the software development domain), show-

ing that it:

Ðleads to more downstream problems due toquality problems (errors in code)

Ðleads to more problems in parallel activities due

to increased parallelism between activities.

2. Cooper [13] similarly uses quality as a criticalmetric, being de®ned as the fraction of work being

executed that will not require subsequent rework.

3. Williams et al. [54] describe a series of interlinkedfeedback loops (based in a manufacturing domain)

where bringing activities earlier (thus doing workbefore engineering judgement said it should be

done) increased rework, which increased delays,which led to more parallelism and thus increased

cross-relations between activities, hindering a systemfreeze and hence more work done before it should

be done.

4. Graves [23] discusses the e�ect of compressingR&D project times, giving a time±cost curve

suggesting that if duration is reduced to lt (where0 < l < 1) then cost becomes divided by l k where

typical ®gures of k ranging from 0.9 to 2 are given.

5. Cooper [14] (referring to Brooks [9]) discusses thefeedback loops involved when extra personnel are

brought in to reduce activity-times, leading again toincreased durations and increased costs.

6. Abdel-Hamid and Madnick [1] give full details ofsome SD models designed to investigate such

e�ects.

How should such indications be modelled in our net-

work simulations? Looking at the two managementactions above, simple models could include the follow-ing, where activity i has nominal duration ti and costci:

1. Bringing activity i forward by t (where t is a timesuch that 0 < t < ti). This will have the e�ect that

Ðeven being optimistic, the remaining activity-dur-ation after the dependency, theoretically tiÿt,will in fact be a ®gure greater than that, say

tiÿat, where a is a factor such that 0 < a < 1;Ðthe new cost, rather than ci will be ci (1ÿ b(t))

where b is a function of t such that 0 < b(t) < 1;

Ðboth parallel and down-stream activities willhave their nominal durations increased from tjto tj(1 + d(j, t)), where d(j, t) is a nonnegativefunction of t, that decreases exponentially as ac-

tivity j moves towards the end of the project.

2. Reducing an activity's duration from ti to lti.3. the work described above gives a crashing time±cost

function more realistic than a simple linear func-tion, where the cost is divided by l k;

4. parallel activities j will be a�ected: that is, their dur-

ations will become tj(1 + g(l)) (where g(l) is a non-negative function of l);

5. down-stream activities will have their nominal dur-

ations increased from tj to tj(1 + e(j,l)), whereagain e(j,t) is a nonnegative function of l thatdecreases exponentially as activity j moves towards

the end of the project.

Furthermore, these secondarily lengthened activities

will in general themselves be crashed, to stop thembecoming critical, so their costs will increase, and thee�ects described above will be repeated in a secondary

wave of knock-on e�ects.

7. Example

Modelling of these e�ects has to depend upon theproject being modelled. Given here is the very simplest

possible example, to indicate the type of modelimplied, and the type of result. Suppose that a projecthas an important mid-point mile-stone; we will com-

bine together all of the activities before this point intoa single activity A. Suppose further that there is a sub-sequent activity B that management can control. We

Fig. 1. (a) Standard ®nish±start dependency. (b) Finish±start

dependency where management takes the decision to start ac-

tivity i early.


will combine together all of the other activities into

two: activity C, which represents activities parallel to B

and activity D, representing down-stream activities.

The resulting simple network is shown in Fig. 2.

Suppose the activity-durations have distributions as

shown in Table 1 (where the unit is months).

Assuming the activities are independent, this is

obviously simple to simulate. Suppose that the ®nal

date for completion of the project (beyond which liqui-

dated damages are payable) is 35 months; simulation

shows a 90.2% probability of achieving this.

But suppose that the milestone by which A is sup-

posed to be ®nished is at month 23, and management

knows that if this slips, the ®nal target date of 35

months is unlikely to be met. If activity A slipped then

management would take two actions:

. if activity A completed after month 23, part of ac-

tivity B would be brought forward, so that two

months work would be done in parallel to A. As

always, Fig. 2 represents the planned activities, but

management usually has the option of bringing for-

ward part of an activity if the project is running

late, for example setting engineers to work on de-

signs even though the concept has not been ®nalised,

or taking o� detailed designs that are not fully com-

pleted (so not really ready for release) and beginning

construction of the basis of them. Practising project

personnel will recognise these occurrences from their

own experience. However, due to the e�ects dis-

cussed above:

Ðthe remaining duration of activity B would be

reduced only by 1.5 months (i.e. a= 0.75);

Ðthe cost of activity B would be increased by 6.5/6;

Ðactivity C (parallel to B) would be increased by0.5 months;

Ðactivity D(downstream from B) would beincreased by 0.5 months.

. if activity A completed after month 24, activity B(that element which was not brought forward tobefore the milestone) would be crashed, so that itslength would be reduced by 20%. However, again:

Ðthe cost would be increased by 25% (i.e. k= 1)Ðthe duration of activity C would be divided by 0.9Ðthe duration of activity D would be divided by

0.95.

This network has been simulated 1000 times using

the @Risk package [36], both the initial version (withno management control) and the latter version. Figure3 shows the probability of the total project duration

exceeding certain times for both the crude simulationand this latter controlled version; in particular, thecontrolled version now has a 95.2% chance of achiev-

ing the deadline. Furthermore, the cruciality ofActivity A (i.e. the correlation between its durationand the duration of the whole project) is reduced from

95 to 93% in going from the crude to the controlledversion, showing that management can take someaction to reduce the e�ect of activity overrun, but can-not simply catch up completely (implying a determinis-

tic project duration and thus a cruciality index ofzero).Thus we have shown:

. the e�ects of management control, restricting thelonger project durations;

. the resulting project duration which is less optimistic

than might be ®rst thought of by management, sincethe time saved by bringing an activity forward orcrashing it is lost to a certain degree by the e�ects of

the project dynamics;. additional costs: since the ®rst of the above actionsoccurred in 14.9% of iterations, and the second in

8.7% of iterations, the cost of activity B wasincreased by 3.2%.

8. Conclusion

The use of networks handling uncertainty to providea temporal risk analysis of projects is now widespread.However, the results of such analyses are frequently

not regarded as credible, partly because of the resultingwide probability distributions; this is because themodels do not re¯ect the actions that management

Fig. 2. Example network.

Table 1

Activity Distribution Minimum Mean Maximum

A beta 15 20 30

B beta 5 6 10

C beta 2 36

D beta 4 5 8


would take to bring late-running projects under con-

trol. These are di�cult to include in models, notbecause the actions themselves are complex, but ratherbecause the e�ects of those actions are not well-under-

stood. However, the work in system dynamics model-ling of projects can give some indications of whatthose e�ects would be. This paper has attempted to

describe these indications and give some small illustra-tive models of the e�ects. It is hoped that the use ofsuch modelling can help to bring additional realism to

probabilistic network modelling.

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Towards realism in network simulation