IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 45, NO. 2, MAY 1998 181

On the Importance of BuildingEvacuation System Components

Gunnar G. Løvas

Abstract—How can building occupants be safely evaluated inthe case of an emergency? To address this question, this paperpresents several measures of the criticality of evacuation systemcomponents, inspired by reliability theory. The escapeways in abuilding are modeled as a network, with links and nodes. Thispaper discusses how it is possible to identify the importance ofdifferent network components. The evacuees are modeled as dis-crete flow objects with certain attributes. This paper discusses theimportance of different personal attributes. Also, managementdecisions are important, e.g., establishing emergency plans andstrategies.

Index Terms—Building, emergency planning, evacuation, im-portance measure, optimal routing, queueing network, safetyengineering.

I. INTRODUCTION

W HAT IS the difference between a successful and afaulty evacuation system (ES)? Is it possible—in ad-

vance—to identify potential problems and critical parts of anES where improvements may be needed? In reliability theory,the reliability of a real-world system is often studied via so-called reliability networks. In such networks, the idea is thatthe system is okay if and only if there is a connection fromone end of the network (the source node) to the other end (thesink node). Hence, we can think of the reliability network asa special case of a transportation network. Such a network hasmany similarities to the network of escapeways in a building.For reliability networks, many different importance measuresare defined to say something about the relative importanceof the different system components (see [8] and [23]). Inanalogy with this, we will in this paper introduce measures ofimportance of the “components” in the network of escapeways.Further extensions of these ideas will also be introduced.

A. The Evacuation System

An ES is a complex system of many “components.” Thephysical environment in which the population circulates iscalled a building. The building is composed of elements(rooms, corridors, stairs), which will be calledrooms. Thelayoutof the building is determined by the way in which theserooms are interconnected bydoors. The populationwithin abuilding plays a major role in the ES. Differences in personal

Manuscript received March 4, 1994; revised October 20, 1997. Review ofthis manuscript was arranged by Guest Editors S. Tufekci and W. A. Wallace.This work was supported by the Royal Norwegian Council for Scientific andIndustrial Research.

The author is with the Department of Mathematics, University of Oslo,Oslo N-0316 Norway.

Publisher Item Identifier S 0018-9391(98)03147-X.

TABLE ITHE COMPONENTSC = fB; K; Og OF THE EVACUATION SYSTEM. THE

TABLE IS NOT COMPLETE AND SHOULD BE CONSIDERED AS ANEXAMPLE

characteristics such as sex, age, physical ability, and familiaritywith the building, safety equipment, evacuation procedures,etc. may be very important.Detectionsystems are required sodangers may be observed as early as possible, initiatingalarmwarnings, which can be understood by the building population.Managementdecisions are important in the planning of anemergency evacuation system, and managers may also play animportant role as “on-line” decision makers in a real accidentsituation. Potential hazardswithin or close to the buildingdetermine how likely it is that a rapid evacuation will beneeded. In this paper, however, the hazard will not be includedas a part of the ES. It would probably be of interest to extendthe ideas of this paper and consider the importance of differentaccident scenarios.

It will be useful to make the following rather abstractdefinitions, remembering that a component in an ES is notnecessarily a piece of “hardware.” The set of ES componentswill be called , consisting of all relevant elements of the ES.Table I shows how the set of components may be dividedinto subsets of related components: the setof buildingcomponents, the set of building occupants, and the setofmanagement-related topics. The selected “system boundary”(i.e., the definition of the ES) and the grouping of systemcomponents into subsets may be discussed. However, the pre-sentation in this paper will be closely related to the componentsubsets . We will assume that the components arenumbered in a suitable way. Associated with component,there is a parameter (or set of parameters). Let be theset of all component parameters. Defineas the “organizing

0018–9391/98$10.00 1998 IEEE

182 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 45, NO. 2, MAY 1998

function,” which relates all the components to each other intoa system.

B. Evacuation System Model

The ES includes many elements that behave “randomly.”The behavior of the building occupants is not deterministicallygiven: their initial response times, their response to accidenteffects, their wayfinding decisions, etc. are considered tobe stochastic. Some of the parameters in Table I must beunderstood as input parameters to a probabilistic model. Forexample, the “wayfinding ability” of a person determinesthe probability that he will make the “correct” wayfindingdecisions (see [20]).

Our basic model of the ES is that a population (not nec-essarily of fixed size) is located (maybe randomly) in abuilding at the time when a threat appears. After an initialdelay period, the building occupants start to move from theirorigins—hopefully toward an exit. To model the progress ofthe evacuation process fully, the following submodels areneeded: initial response model, route choice model, walkingspeed (flow) model, accident response model, and humanbehavior model. All of these are elements of Evacsim, anevacuation simulation program described in [25]. Evacsimitself is not a topic of this paper, but it will be used to calculatesome numerical measures related to the examples in this paper.

C. Evacuation System Performance Measures

Bergman [5] states that “a reasonable requirement on animportance measure is that it indicates how important thecomponents are with respect to the chosen system performancemeasure.” It is therefore natural to start out with a choice ofa system performance measure, which can be used to tell ifthe system performs well or not. In a previous paper, Løvas[18] defined several performance measures for ES’s. Most ofthese measures are related to time in some way, based on thefact that there is limited time available for safe escape. In therest of this paper, we will therefore concentrate on measuresrelated to time.

Let us define the following stochastic variables. Let bethe time when person number leaves the building, and let

be the number of persons who have left the buildingwithin time . The times are sorted so that ,where is the size of the population, and is the time whenthe last person leaves the building. The main performancemeasures will be the expected values and distributions of

and . Let us for simplicity refer to the chosenperformance measure as. Then we have

This relationship clearly demonstrates that the performancemay be improved by reorganization , removal or

addition of components , or changes in the componentattributes .

The performance measures will normally be a functionof the hazardous event and other stochastic phenomena. Forexample, different performances will be observed if the hazardis a fire, a power blackout, or an earthquake. We will not

include the event itself in the discussions in this paper. Whendifferent hazards are realistic (as often will be the case),it will be reasonable to optimize theexpectedperformance(expectation taken over all possible hazardous events). Theoptimal system may therefore be suboptimal to any singlespecific hazard, but nevertheless be the best system for theset of all potential hazards.

D. The Evacuation Time

It will be useful to study the time development of theevacuation scenario and identify some of its basic parts. Letus therefore define the following stochastic variables. Theevacuation time (omitting the subscript , since it is of norelevance here) is the time from the outbreak of a hazardoussituation until a person is safe. This time is composed of twomain parts, the reaction time and the walking time

The reaction time will be decomposed into four subtimes,where we define as the time until the detection systemreacts, as the time before the alarm signal is given,as the time needed before the person understands what is goingon, and as the time needed before the person is ready tomove. Hence

The walking time will be decomposed into its more man-ageable subparts, with being the time needed to walk thecorrect (shortest) path from the initial position to the (safe)exit; being the extra time needed because of queueing;

being the extra time used because of the selection ofnonoptimal routes; and being the time used because ofthe accident, e.g., to walk a longer path to avoid accidenteffects. To sum up

Obviously, it is not always easy to make such a disjointdecomposition of the total time , since some of the timeconsumed may be a result of, e.g., both a nonoptimal pathchoice and queueing on this path. However, let us assume inthe following that this problem has been sorted out.

E. How to Improve Performance?

When we study ES’s, we do so because we want to see if thebuilding can be evacuated in an acceptable way. If the ES will(or may) perform unacceptably, then some corrective actionsare needed. Examples of useful actions are listed below.

• Reduce reaction time of personnel (e.g., better detectorand alarm systems).

• Reduce interpretation time (e.g., better messages to build-ing occupants).

• Reduce walking time by improving the escapeway sys-tem:

— shorter routes (e.g., new connections, new doors);

— higher flow capacities (e.g., wider doors);

LØVAS: IMPORTANCE OF EVACUATION SYSTEM COMPONENTS 183

— easy wayfinding (e.g., better marking and orienta-tion help).

• Train personnel (e.g., react correctly, walk faster, followplanned routes etc.).

• Improve organization of the evacuation process (e.g.,“optimal” routing of evacuees).

The main purpose of this paper is to define measures ofcomponent importance that can be used as a guide whenimprovement actions are assessed.

F. Historical Approaches

The most familiar way to study the importance of com-ponent is to perform a sensitivity study in which theperformance of the system is compared to the performance of asimilar system, where componenthas been slightly changed.This may be expressed as

which is exactly the definition of the Birnbaum [6] reliabilityimportance measure. Since we have decided to focus onperformance measures related to evacuation time, it willbe useful to think of as .

Several other importance measures have been defined in theliterature. These have been applied to analyze systems that arequite different from ES’s, but they are nevertheless presentingsome basic—and interesting—ideas about the problem ofidentification of critical components. The criticality measuredefines the importance of componentas the probability thatcomponent is responsible for system failure, given that thesystem is not functioning. Vesely and Fussell [14] definedtheir importance measure as the probability that component

is contributing to system failure, given that the system is notfunctioning. Butler [9] defined componentas important if ithas a critical position in the system’s structure.

In some situations, one has a parameter vector indexedby time. Then it is possible to define importance measuresthat identify the most important component over the wholetime interval of interest. Barlow and Proschan [4] defined thecomponent importance as the probability that component

causes a system failure. They also showed that their measurewas equal to the “expected” value of . Natvig [21] defineda component to be important if the expected reduction inremaining system lifetime due to the failure of the componentis significant. Natvig [22] also showed that his measure isa weighted average of the measure. Following some ofthese ideas, Bolandet al. [7] defined componentas importantif system performance increases much when an identicalcomponent is put in redundancy with component. Aven andØstebø [1]–[3] defined as the performance measure ofan identical system where componentoperates perfectly.They used the “improvement potential” as the basis for theirdefinition of an importance measure, saying that componentis important if the difference betweenand is large. Theexisting methods are summarized by Natvig [23] and Bolandand El-Neweihi [8].

Fig. 1. The building networkG for the standard building example. Thecircular nodes represent source and transportation nodes and the trianglesrepresent exits. The nodes are numbered 1 to 8. For easy reference, the linksare nameda to k.

Most of the measures proposed in this paper are time inde-pendent, reflecting the importance of an ES component relativeto overall system performance. An attempt to incorporate thetime-dependent behavior of the evacuation process is presentedin Section II-F.

II. I MPORTANCE OFBUILDING COMPONENTS

The building itself is obviously an important part of the ES,and it is of interest to identify weak points in the buildingdesign and areas where improvement may be needed. Tounderstand that some parts of the pedestrian circulation systemplay a more critical role than others—just think of a staircasein a multistory high-rise building in which the importanceof the stair increases as one approaches the ground level,simply because a larger proportion of the population is likelyto depend (more or less) on the stairway.

We will discuss the importance of the different links andnodes in the building network. First, however, we will definethe importance of a collection of links, and later use this to saysomething about each single link. The starting idea is that alink or node is important if its removal has great implicationson the ES or its performance.

A. The Building

A building composed of rooms and doors will bemodeled as the network , consisting of the nodes

in the set and links in the set . Wewrite if and only if there is a linkbetween nodes and (to simplify the presentation in theexamples, links will be referred to by a single letter). Thenodes represent rooms (or corridors or stairways), and thebidirectional links represent doors. The nodes are divided intothree subsets: Source nodes, transportation nodes , andexit (destination) nodes . The nodes in are absorbing, i.e.,persons will not move from a node in to a node in .We will assume that is a connected graph. The connectionbetween the real world and the networkis well documentedin [19], where it is shown how the network can be defined.Fig. 1 shows an example of a building network. We will usethis building in our examples. We will sometimes refer—inthe examples—to abuilding with only one exit, meaning thatlink is removed, leaving the evacuees with no choice butto use exit 1.

Let us define as the graph obtained when nodeis removed from the graph . Similarly, let denotethe “extended” graph where nodeis put in its “perfect”

184 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 45, NO. 2, MAY 1998

Fig. 2. Some examples of linkcuts are shown for the standard example,numbered 1 through 7.

condition. This will be done also for links, then writingor , respectively, when link has been removed orput in its “perfect” state. With “perfect state,” we will meanthe state obtained when all parameters influencing pedestrianflow characteristics have been optimized (e.g., infinite nodecapacity, infinite door width). It will also be convenient that

and may mean that either node or linkhas been added to the graph.

B. Building Network Cut Sets

1) Linkcut: A collection of links is alinkcut if the removalof these links divides the building network into separategraph components. A linkcut isminimal if it has no propersubset also being a linkcut.

Fig. 2 shows some examples of linkcuts for the buildingnetwork in Fig. 1.

2) Nodecut: A collection of nodes is anodecut if theremoval of these nodes divides the building networkintoseparate graph components. A nodecut isminimal if it has noproper subset also being a nodecut.

Since links and nodes are some kind of “dual” componentsin the ES, we will focus mainly on links in the following,remembering that many of the ideas will be applicable alsofor nodes.

C. Structural Importance of Linkcuts

1) One Exit Only: As a start, consider a building with onlyone exit.

Linkcut number is denoted , the indicating thatit is a linkcut rather than a nodecut. If the links in minimallinkcut are removed from , then appears as twoseparated subgraphs. One of these subgraphs is connected tothe exit. The other is “isolated”; call it . Obviously,if is a large graph, then it seems that is animportant linkcut. An interesting question, then, is what is alarge graph? We propose that the size of the graph could bemeasured in terms of either its corresponding population sizeor its total length of walkways. Choosing the first alternative,we say that linkcut is important if has alarge population at time zero. Obviously the population sizevaries with time, a fact that is used later to define a time-dependent measure. Hence, we define as the structuralimportance of linkcut in the following way:

Number of persons in the isolated subgraphTotal population size

assuring that . We will examplify this with

Fig. 3. The population distribution of the standard example. The number ina node indicates its population size at time zero.

TABLE IITHE STRUCTURAL IMPORTANCEJLC(m) OF THE SEVEN LINKCUTS SHOWN IN

FIG. 2. RESULTSARE GIVEN FOR THE TWO POPULATIONS (a) AND (b) FOR THE

STANDARD BUILDING EXAMPLE, WITH ONE OR TWO EXITS “A VAILABLE ”

two different population distributions: (a) there are equallymany persons in all nodes or (b) the persons are located asshown in Fig. 3.

The importance of the linkcuts is clearly different in thesetwo cases, even though the ordering remains the same, asshown in Table II’s middle supercolumn. The table shows theimportance of the seven linkcuts in Fig. 2, assuming that exit2 is removedfrom the network. As expected, then, linkcut 7is much more important than linkcut 1.

2) Standard Example:With the termstandard examplewewill mean the building in Fig. 1, where the persons aredistributed as in Fig. 3.

3) More Exits: Now, let us return to the general buildingwith severalexits.

The importance of a linkcut has been defined relative toa single exit. When there are more exits, a new definitionmight be needed, or at least, the interpretation of the structuralmeasure must be different. Let us examplify this with thelinkcuts shown in Fig. 2. It is not easy to know the relativeimportance of the linkcuts. Is linkcut 7 still (when both exitscan be used) more important than linkcut 1?

With inspiration from graph theory, we could approach thisproblem by introducing a common exit node, representingthe “safe” area (normally outside a building). All links tonormal exits can then be “redirected” to this common exit,and the remaining graph has thus only one exit. Even thoughthis may seem a convenient solution, it will perhaps be morefruitful to keep the network with many exits and introduceother simplifications that are more relevant to the evacuationproblem. We will try this in the following.

Let us assume that each person in the building prefersa specific exit, either because it is the nearest or becausehe is instructed to use this exit. As before, if the linksin linkcut are removed from , then appearsas two separated subgraphs. Each of these subgraphs has acorresponding population, and in both of these populations,

LØVAS: IMPORTANCE OF EVACUATION SYSTEM COMPONENTS 185

some persons might be isolated from the preferred exit. Letbe the number of isolated persons. A reasonable

and general definition of the structural importance of linkcutis then

Total population size

Assuming that the relative distances are correctly visualizedin Fig. 2 and that the persons prefer the nearest exit, we can usethe above definition and find numerical values to the structuralmeasure. This has been done, and the results are shown inTable II’s last supercolumn. The results comply well with“engineering intuition,” showing that the importance increasesin the neighborhood of exits.

D. Structural Importance of Links

1) Based on Linkcuts:We have considered the structuralimportance of linkcut and will use this tosay something about the structural importance of link

. Nonminimal linkcuts are not very informative, since itis impossible to know the relative importance of the differentlinkcut members, some of which might be totally unimportant,even for an important linkcut. Furthermore, since there existan enormous amount of linkcuts, we will restrict our consid-erations tominimal linkcuts in the rest of the paper. In thefollowing, a linkcut is a minimal linkcut.

Obviously, a link is important if it is a member of animportant linkcut. Inspired by Butler [9], we say that a linkis more important if it is a member of a small linkcut than ifit is member of a large linkcut. The size of linkcut

equals the number of links in the cut. We thereforepropose the following measure of (structural) link importance:

This model corresponds to saying that all links in a linkcutare equally important (with respect to this linkcut). In manycases, this might seem “unfair,” and it could be possible to“split” the total linkcut importance in another way, e.g., bytaking into account the different (flow) capacities of the links.

The described method relies on the existence of suitablealgorithms for identification of linkcuts. Such solution methodsexist (see, e.g., Ford and Fulkerson [13] or Jarvis and Tufekci[16]), but they are time consuming, and it isnot recommend-able to search for all linkcuts in a large building network. Ifall linkcuts up to size have been found, then one knows thatthe maximum contribution from larger linkcuts is ,which rapidly decreases with. Based on this fact, it will besufficient to find all linkcuts of “small” size. A simple iterativealgorithm can be implemented to assure that is foundwith minimal work.

2) Another Approach:It is also possible to say somethingabout a link’s importance in the network, without having toconsider linkcuts. We will present an approach wheredistancesare used.

Let us define as the shortest distance in the graphfrom node to an exit. Define as the number of persons

TABLE IIILINK IMPORTANCE(STRUCTURAL) FOR ALL LINKS SHOWN IN FIG. 1, IN THE

CASE WHEN THE POPULATION DISTRIBUTION IS AS IN FIG. 3. LINK k IS

REMOVED WHEN WE CONSIDER THEBUILDING WITH ONLY ONE EXIT

in node . Some interesting parameters are then

where equals one if is true and zero otherwise.The measures to be proposed in the following arenot

normalized to lie in the interval [0, 1], but they are positive andincrease with growing importance—as before. It is assumedhere that a link is important if its removal (or blocking)forces the building occupants to walk longer—perhaps muchlonger—routes to reach a safe area

Increase in total distanceNormal total distance

Increase in maximum distanceNormal maximum distance

Maximum increase in distanceNormal maximum distance

It is generally not possible to say that one of these isuniformly better than one of the others. Their usefulness willdepend upon the actual situation to which they are applied.However, the consequences on the evacuation time when a linkis “removed” from the network are easily recognized:is closely related to the increase in average evacuation time,

tells us a lot about the increase in maximum evacuationtime; and the last measure, , gives us an idea of themaximum increase in evacuation time for a person.

3) Short Discussion:For the standard example in Fig. 1,we obtain the numerical values for link importance shown inTable III. The first “supercolumn” refers to the building withone exit, whereas the last column is related to the buildingwith two exits.

Which of the links, , are the most important? Andwhich of the measures are the best? Each link is described byfour numbers. If these are put in order— , , ,

186 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 45, NO. 2, MAY 1998

—then it is possible to sort the links in lexicographicorder (sort by first number; if ties exist, sort these out bythe next number, etc.). This is done because the authorbelieves—highly subjective—that measure 2 is the most in-formative in this example, measure 1 the second best, etc. Forour standard example, with two exits, we find from Table IIIthe following ordering (the first being the most important):

where link is judged so important since its functioning ishighly critical for eight persons.

Obviously, any other multiple criteria decision rule couldhave been applied instead of the lexicographic ordering usedin this example. It can also be noted from Table III that theordering (for some of the links) is rather robust to differentchoices of decision criteria. For example, linkdominateslink (which again dominates link ) for all the suggestedimportance measures.

The structural measures of link importance are useful andhave a great advantage in that their values are easily calculable.However, such measures are not related to the actual (orexpected) performance of the system. This is a drawback thatmotivates other definitions of link importance, based on alink’s criticality for system performance.

E. Performance Based Importance of Links

When we are able to say something about the actualbehavior of the ES, this should be used to identify importantlinks. For example, a link that isused much (by manyevacuees) can be said to be important or—at least—to havea high “business” level. But how important? It is not a goodidea to follow this proposal because there is not necessarilya strong correlation between a link’s business level and thelink’s importance with respect to system performance.

Three possible definitions of the importance of linkare given below. All of them are functions of the actualperformance measure that is used to describe the evacuationprocess. We assume thatis related toevacuation time, so alow value for corresponds to good performance (i.e., lowevacuation time). Let be the performance of the ESwhen its evacuation network equals. Remember also that

can be regarded as a summary statistic for thestochasticevaluation process.

The first measure, denoted , will be called “bot-tleneck potential”

and the second, denoted , will be called “improve-ment potential”

A more traditional approach could be to findas a functionof the link width , or to find the derivative of with respectto the link width. Obviously, if the performance is sensitive

TABLE IVLINK IMPORTANCE(PERFORMANCEBASED, BOTTLENECK POTENTIAL)

FOR ALL LINKS SHOWN IN FIG. 1 IN THE STANDARD EXAMPLE

to changes in , then link is important because it is apotential bottleneck. For later reference, we therefore define

Even though all of these measures “should be” positive,there is actually no guarantee that the real world is so “simple.”An example, in relation to : making the linkperfect by putting its flow capacity infinitely high will probablylead to improved performance. However, one could easilyimagine situations in which a link is preferred to have lowflow capacity so as to reduce downstream queueing problems.

To evaluate these measures, one must be able to find. This is not an easy task, and its degree of difficulty

depends highly on the chosen performance measure underconsideration. Some analytical solution methods have beenproposed, primarily focusing on rather “simple” performancemeasures (see, e.g., Chalmetet al. [10], Choi et al. [11], andHamacher and Tufekci [15]). For somewhat more “complex”performance measures, several simulation programs exist, ofwhich some are briefly described in Løvas et al. [17]. In thispaper, we will use the simulation program Evacsim, which isdesigned to simulate evacuation from many types of buildings(see Drageret al. [12]).

Assuming that the persons walk the shortest way to theexit with a log-normal walking speed ( m/s and

m/s), and reacting immediately, we will make somecalculations with Evacsim. Table IV shows the performance-based importance of the different links in the standard examplewith two exists. We have chosen , (the median evac-uation time) and (the longest evacuation time) as twointeresting performance measures, meaning that the contentsof the table’s columns are and ,respectively.

Since our aim is to evacuate the whole population safely,the longest evacuation time is a more meaningful performancemeasure than the median evacuation time. If we sort the linksaccording to importance, based on the longest evacuation time

in Table IV, we find the ordering

LØVAS: IMPORTANCE OF EVACUATION SYSTEM COMPONENTS 187

Fig. 4. Time-dependent link importance for two links in the standard example.

We observe many similarities to the ordering presented inSection II-D3. From this, we may hope that the structuralmeasures proposed earlier will be able to tell almost the samestory as the performance-based measures. The big advantageof the structural measures is that they can be calculated withminimal effort compared to the performance-based measures.

F. Time Dependence of Importance Measures

Until now, the measures have been static, correspondingto an assumption that a link or node is “removed” from thenetwork at time zero (e.g., an explosion might have destroyedthis building element). In practice, however, it will often be ofinterest to consider a situation in which an accident developswith time (e.g., a fire that grows and “blocks” off more andmore building elements). Another problem is the fact that,in this case, building elements are not blocked at the sametime for different evacuees. Research about human behaviorin fires suggests that people have different tolerance limits,causing some people to consider an escapeway unuseable andothers to use it.

Let us define a time-dependent graph in thefollowing way:

ifif

This graph corresponds to a building where linkis removedat time . The time-dependent importance of link couldthen be expressed as

Importance of in

e.g.,

In this way, one could find for all times ofinterest. Hence, we have a time-dependent measure of linkimportance. Such a measure could be quite useful, since itis able to take into account the fact that a link may be veryimportant in a small time interval and less important in the rest.From our example in Fig. 1, we see that linkis important in

the beginning, but after some time, when all evacuees have leftnode 1, it is not important at all. This is in contrast to linkinthe example, which is important for a longer time. Normally,one would assume that it is worse if a link is closed early thanlate. However, this assumption is not true if one assumes thatthe persons will be informed (e.g., by a loudspeaker system)immediately after—andnot before—a link is closed. Fig. 4shows the time-dependent link importance for two of the linksin our standard example, calculated with Evacsim. The figureshows that the measure will be approaching zerofor large values.

III. I MPORTANCE OF THEPOPULATION

The population in the building is important, and its safetyis our main concern. We will therefore study certain aspectsrelated to the population and see how these influence theES performance. We will focus on the population’s sizeand localization, initial response, and walking speed. Manyother characteristics of the population are important too, e.g.,the evacuees’ wayfinding ability (see Løvas [20]), and thepresentation here is by no means complete. The sensitivitystudies in the following are inspired by the studies of Urbaniket al. [24].

A. Localization of Population

Obviously, the localization of the population members isimportant. If the persons are relocated closer to the exits, thennormally they will have shorter evacuation times. How canthis be measured?

As a basis for the following discussion, let us define a sortof reference localization rule. Define as the probabilitythat a person is in node at time zero. For the referencerule, let , meaning that all nodes are equallyimportant. (This choice might be substituted with anotherstandardized rule, e.g., area of node divided by totalbuilding area.) All members of the reference population arethen “placed” in the building according to this probability.The performance of the ES with this population can then

188 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 45, NO. 2, MAY 1998

be found. By comparing the actual performancewith thereference performance , it is possible to say somethingabout the importance of the actual localization of persons. Ifcomparison with an artificial reference feels unnatural, onecould instead compare two (or more) alternatives against eachother.

If one considers a population that normally circulates quitefreely in the building, the “uncertainty” in the persons’ po-sitions (at time zero) will often be a great contributor tothe total uncertainty in the results. Let be the standarddeviation of when each person is randomly located inthe building [with probabilities ]. Similarly, let bethe standard deviation of when the population is locatedat fixed positions (proportion of the persons in node).Normally, . An interesting measure of the localizationrule’s contribution to the total uncertainty is

For the standard example, with the proportions/probabilitiesgiven (indirectly) by Fig. 3, Evacsim calculates

and , giving .

B. Size of Population

Since the building occupants interact and compete forlimited resources, it is obvious that the population size is animportant parameter of the ES. From transportation theory, oneknows that—for large population sizes—the evacuation time

is almost a linear function of the size due to queueing.For smaller populations, the evacuation time does not increasemuch with , since queueing is then no longer a problem.However, an increase in the evacuation time is still expectedas grows, because it is more likely that a larger populationcontains some slow-reacting, slow-walking persons.

1) No Queueing Problems:Let us for a while assume thatqueueing is no problem, i.e., that the building has sufficientcarrying capacity. Reusing the variables of Section I-D wewrite the evacuation time of person as

. Both and are stochastic according to somedistribution. Findings in the literature (see, e.g., [25]) suggestthat both the walkingspeedand the reaction time can bemodeled as log-normally distributed. If the speed of walking islog-normally distributed, so is thetimeneeded to walk a certaindistance. Let us therefore assume that both and arelog-normally distributed. The distribution of is hence given(we need no explicit convolution-produced formula here). Letus sample values of from this distribution and let

max . We repeat this times and get differentobservations of . In this way, we are able to establish theempirical distribution of . A reasonable summary statisticis . This whole procedure can be carriedout for different values of ; as a result, we have the expectedtime until the last person has exited, , as a function ofthe population size .

Fig. 5 shows the relationship of evacuation time and popula-tion size—assuming that queueing doesnot occur—calculatedas described above for our standard example. The followingcoassumptions are used to produce the result. The last person

Fig. 5. Simulated evacuation time as a function of population size for thestandard example. It is assumed that queueing does not occur. ForK > 10,a suitable regression line isTK 33 + 4:4 lnK. Simulated with BLSS.

who leaves the building started from node 1 (reasonable sincethe persons in node 1 have the longest distance to walk andsince queueing is assumed to be no problem); reaction time is

s and s; walking speed is m/s andm/s; distances are correctly visualized in Fig. 2; and

the length of link is 10 m. Fig. 5 shows that with ,as in the standard example, one has s. As thepopulation size grows, grows in a logarithmic fashion,growing very slowly for large values. With unrealistic

, one gets the even more unrealistics. The simulations and calculations have been performed withthe statistical package BLSS, with .

2) Queueing: As the population size grows, sooner or laterqueueing will become a problem. Normally, the main queuingproblems occur at (or close to) the exits. Assuming that alllinks in our example have width m, it is reasonableto assume that the building’s two exiting links (and ) havea total flow capacity of approximately persons/s (seeLøvas [19]). If 100 persons want to leave the building, weknow that they need at least 100/5 seconds plus the time

needed before the queue builds up. A simple relationshiptherefore exists: , approaching equalityas (this is called a hydraulic flow model; seeLøvas et al. [17]). The discussion related to Fig. 5 (when noqueueing effects are present) shows that lnapproaching equality as decreases. As a summary, we have

max ln (1)

Fig. 6 shows obtained with Evacsim under the follow-ing assumptions: The normal linkwidth is m; the per-sons are located in the network in the same fixed “proportions”as in the standard example; reaction time, walking speed, anddistances are as before. For mediate population sizes,

, Fig. 6 shows that (1) clearly underestimates theevacuation time. It is not easy—beforehand—to identify where(1) is a poor estimate, and eventually how poor. Therefore, andsince most buildings normally have a mediate population size,we see that “exact” models are needed, also incorporating theeffects of queueing.

LØVAS: IMPORTANCE OF EVACUATION SYSTEM COMPONENTS 189

Fig. 6. Simulated evacuation timeETK as a function of population size for the standard example. Queueing occurs for largeK values. Simulated withEvacsim. The lines describe the “asymptotic” behavior ofETK , as discussed in the text.

C. Initial Response

The persons’ “reactiveness” is an important determinant forsuccessful evacuation. Unnecessary delays in the initial stagesof the evacuation process seriously reduce the safety of thebuilding occupants. The initial response of the evacuees ispoorly modeled in most of the existing simulation programs,mainly because knowledge about the cognitive processes of theevacuees is not very well understood. Many simulation pro-grams assume—quite optimistically—that the evacuees startto move immediately.

Evacsim models the response time of person in thefollowing way. Define the “individual” reaction time of person

as . In Evacsim, it is assumed that is log-normallydistributed, with mean and standard deviation . Let

be the time when accident effects are “visible” for person. Let be the time when the person is warned, either by

an alarm system or by another person who has “seen” theaccident. All the ’s are stochastic, and we define

min

Several other aspects of the reaction process could have beenincluded, “simply” by adding extra ’s.

Table V shows the sensitivity of the evacuation timeto changes in and for the standardexample. Since no accident is modeled here, both and

are infinite, so that . When reading thetable, one should remember that theexpectedwalking timefor the slowest walking evacuee is approximately 36 s. Theresults of Table V clearly indicate that the deviation is veryimportant. An implication of this could be that it is importantthatall the building occupants are warned in an understandableway—even if this means that the warning has to be (a littlebit) delayed.

D. Walking Speed

Another important factor is the walking speed of the oc-cupants. We assume that the persons have their individual

TABLE VTHE EXPECTED EVACUATION TIME ETK FOR DIFFERENT

REACTION-TIME PARAMETERS. STANDARD BUILDING EXAMPLE

TABLE VITHE EXPECTED EVACUATION TIME ETK FOR DIFFERENT

WALKING- SPEEDPARAMETERS. STANDARD BUILDING EXAMPLE

characteristic walking speed , which—subjected to situa-tional factors—results in an actual walking speed. In Løvas[19], it is argued that may be considered log-normallydistributed with mean and standard deviation . Wewill study the importance of and for the expectedevacuation time .

For a given population in a given building, one knows thatthe evacuation time will normally decrease if the walkingspeed increases. One also knows that if queueing surely occurs,then an increase in the characteristic speedwill not helpmuch. Our standard example is a sparsely populated building(only 30 persons), and there will be (almost) no queueing inthis example. With the same simplifying assumptions as inthe previous subsection, we can use Evacsim to obtain somesimple estimates of the importance of and . The resultsare shown in Table VI.

190 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 45, NO. 2, MAY 1998

TABLE VIISEVERAL PERFORMANCEMEASURESARE REPORTED FOR THE

FOUR ROUTING PLANS. STANDARD BUILDING EXAMPLE

WITH 150 PERSONS.RESULTS SIMULATED WITH EVACSIM

The effects of congestion will of course be stronger when thepopulation size increases, as already mentioned in Section III-B. With , one would find that an increase inwould not lead to a “corresponding” reduction in evacuationtime. This can be seen from (1) and Fig. 6, where now

, where the last term is dominating andindependent of the walking speed.

IV. I MPORTANCE OFMANAGEMENT ISSUES

Unless the design of a building is inadequate regardingits evacuation potential, management issues are at least asimportant as the physical design of an evacuation system.Løvas [18] presents a case study from a Norwegian offshoreoil production platform, where the objective of the study wasto identify the best of two alternative evacuation strategies.The strategies corresponded to two different evacuation routingplans, and they also had a different plan on how to inform thepersonnel during the evacuation process. Typical managementissues are related to questions like the following: What kindof plans exist in the case of an emergency? Are the personsexpected to approach a certain exit? Are they supposed to walkor run? What kind of messages are they given?

Many evacuation models have “optimal routing of evac-uees” as their main objective. Some of these models are basedon a set of quite unrealistic assumptions (e.g., that evacueesreact immediately at time zero). Let us nevertheless assumethat four alternative routing plans have been proposed for ourstandard example.

1) All persons walk the shortest way to the nearest exit(base case).

2) The persons in node 7 walk to exit 1. Otherwise as basecase.

3) The persons in node 2 walk to exit 1. Otherwise as basecase.

4) Of the persons in node 1, route 3/8 of them to exit 1.Otherwise as base case.

The last three alternatives try to route the persons so as toget a more equal utilization of the exits and to reduce possiblequeueing problems at exit 2. We have studied the effects of therouting plans with Evacsim for our standard example, wherethe population size has been increased to (to have apopulation size in the “area” where queueing starts to becomea problem; see Fig. 6). The results are shown in Table VII,where some other performance measures are also reported.Remember that is the median evacuation time and that

is the number of evacuees that has left the building attime .

Table VII shows that rule 4) is the best with respect toall the measures in the table, a result that fits well withengineering intuition. A drawback of the rule is its rather“awkward” nature, planning to split the persons in node 1 intotwo different groups. Is it reasonable to believe that this planwill be followed in an emergency situation? The table alsoshows that plans 2) and 3) perform worse than the base case,a fact that might seem strange. However, if the population sizewas greatly increased, it is likely that plan 2) would be betterthan the base plan.

V. CONCLUSIONS

This paper has introduced several new measures of theimportance of escapeway elements. Through the examples, ithas been illustrated that some of these measures can easilyidentify the most important components. Further research isobviously required to make the ideas more mature. Some ofthe proposed measures can be modified and improved, andother measures can be invented.

In some years, hopefully, a few importance measures willbe recognized as useful by the evacuation experts. Then it willbe a simple task (from a technical point of view) to implementcalculation algorithms in standard software used by architects,civil engineers, or other evacuation planners. The importancemeasures can then give valuable advice (with little effort) todecision makers at an early stage of the planning process.

ACKNOWLEDGMENT

The author wishes to thank Prof. T. Aven and Prof. B.Natvig for their valuable and inspiring comments on this work.Their experience with reliability importance measures has beenof great help. Many thanks to J. Wiklund, Quasar Consultants,Oslo, Norway, who contributed many good ideas.

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Gunnar G. Løvas received the M.Sc. degree fromthe Norwegian Institute of Technology in 1990 andthe Ph.D. degree from the Statistics Department ofthe University of Oslo, Norway, in 1994.

He combined his Ph.D. studies with work atQuasar Consultants, Oslo, where he participatedin the development of the evacuation simulationprogram Evacsim. At present, he is working in theNorweigian power industry, where he studies thereliability of transmission networks (using manyideas from evacuation network models).