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Building and Environment 46 (2011) 1774e1784

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Building and Environment

journal homepage: www.elsevier .com/locate/bui ldenv

A proposed pedestrian waiting-time model for improving spaceetime useefficiency in stadium evacuation scenarios

Zhixiang Fang a,b,*, Qingquan Li a,b,*, Qiuping Li a,*, Lee D. Han c, Dan Wang a

a State Key Laboratory for Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, 129 Luoyu Road, Wuhan 430079, Chinab Engineering Research Center for Spatio-Temporal Data Smart Acquisition and Application, Ministry of Education of China, 129 Luoyu Road, Wuhan 430079, ChinacDepartment of Civil & Environmental Engineering, The University of Tennessee, 112 Perkins Hall, Knoxville, TN 37996-2010, USA

a r t i c l e i n f o

Article history:Received 2 October 2010Received in revised form21 January 2011Accepted 10 February 2011

Keywords:Pedestrian evacuationSpaceetime pathMeasure of effectivenessWaiting-time modelUse efficiency

Abbreviations: MOEs, Measures of effectiveness; F* Corresponding authors. Permanent address: Tra

Wuhan University, P.O. Box C307, Road Luoyu #129Tel.: þ86 27 68778222 8110; fax: þ86 27 68778043

E-mail addresses: zxfang@whu.edu.cn (Z. Fang)liqiuping@gmail.com (Q.P. Li).

0360-1323/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.buildenv.2011.02.005

a b s t r a c t

Efficiency is a fundamental requirement in evacuation planning and operations. The “faster-is-slower”phenomenon in pedestrian evacuation has been observed and deemed a significant obstacle to evacu-ation efficiency. This paper thus focuses on two aspects of evacuation planning in the case of stadiumevacuation. The first is to define a spaceetime use efficiency measure for evaluating the utility of bothspace and time resources. The second is to propose a pedestrian waiting-time model for directingevacuees to alleviate evacuation bottlenecks. An agent-based simulation approach was employed to testthe proposed model in stadium evacuation scenarios. The results demonstrate that compelled, ormandatory, waiting time strategy generated by this model is helpful in improving the spaceetime useefficiency of network links in the evacuation process by virtue of the strategically timed movingewaitingerestarting movement pattern of evacuees. The analysis of spaceetime evacuation paths in this studyprovides a practical and insightful alternative for measuring evacuation effectiveness. Results of thisstudy compared reasonably against an existing cellular automaton based simulation both in microscopicand macroscopic perspectives. A number of future research directions were presented.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The evacuation performance of large building complexesreceived increased interest by researchers in recent years [28,39,45].Studies on this subject led to the development of several simulation-based models designed to investigate bottlenecks induced by panicbehaviour [22], pedestrian movements [7,41], and waiting time incrowded areas [9,10]. These studies employed techniques includingcellular automata models, lattice gas models, social force models,fluid-dynamicmodels, agent-basedmodels, game theorymodels, aswell as models based on animal experiments (reviewed by Zhenget al., 2009 [58]) as the basis for evacuation scenario simulations[55]. Results of a significant number of these evacuation simulationsdemonstrated a “faster-is-slower” effect [22,36], which is describedas “. trying to move faster can cause a smaller average speed of

IFO, First-in-first-out.nsportation Research Center,, Wuhan 430079, PR China.., qqli@whu.edu.cn (Q.Q. Li),

All rights reserved.

leaving.” by Helbing et al. about the bottleneck mechanism. Thiseffect implies that the utilization, in terms of both space and time, ofevacuation routes has room for improvement.

Efficient use of evacuation routes (space) and evacuation dura-tion (time) is a major challenge in evacuation planning andmanagement. To this end, one of the objectives of this paper is tointroduce a spaceetime utilization efficiency index for evaluatingthe resource usage, in terms of space and time, of alternativeevacuation routes and plans. The other objective of this paper is toestablish a pedestrianwaiting-time model, which is essential if onewants to improve evacuation efficiency by minimizing pedestrianwaiting time.

This paper is organized as follows. Section 2 discusses earlierwork related to evacuation measures as well as space and timeresources in evacuation. Section 3 presents a spaceetime use effi-ciency index to model the usage of space and time resources in anevacuation process. Section 4 introduces a pedestrian waiting-timemodel to avoid the faster-is-slower effect in evacuation operations.Section 5 introduces an agent-based simulation methodology thatincludes the route choices and waiting behaviours of evacuees.Section 6 analyses the results of computational experiments.Finally, Section 7 draws conclusions and discusses directions forfuture research.

Fig. 1. The concepts of spaceetime path and spaceetime path tree.

Z. Fang et al. / Building and Environment 46 (2011) 1774e1784 1775

2. Related work

Researchers have devised different measures of effectiveness(MOEs) for evacuation operations over the years. Among them,evacuation (clearance) time is the most often used to evaluateevacuation plans [28,29,33,45,47,50,57]. Clearance time is typicallydefined as the amount of time required for 95% of all evacuees toflee the evacuation zone. It is considered a practical andmeaningfulMOE by some studies [21,56]. However, such a simplistic constructtends to overlook certain aspects of evacuation operations.

To address situations where clearance time falls short, Han et al.(2007) [21] developed a four-tier MOE framework to assess theefficiency of alternative evacuation plans. The four tiers include (1)evacuation clearance time, (2) individual travel time and exposuretime, (3) time-based risk and evacuation exposure, and (4) time andspace-based risk and evacuation exposure. Overall evacuation time,which is easily obtainable from most evacuation models, does notnecessarily give a full picture of the delay endured by individualevacuees. The individual travel time and exposure time are usefulparameters in determining optimal evacuation routes for individualevacuees. The time-based risk and evacuation exposure areessential factors to consider for selecting evacuation routes undergeographically stationary hazards. The time and space-based riskand evacuation exposure account for the effects of dynamic

Fig. 2. Mergeable and unmergeable trajectories.

hazards, e.g. a radioactive plume accidentally released froma nuclear power plant. This framework is more comprehensive thanthat of most MOEs in the literature. It can be applied adequately toevacuation events of different nature and provides an assessmenttool for a wide range of evacuation scenarios. Yuan and Han (2009)[56] further presented several case studies applying thesecomplexes but useful MOEs, which consider space-based and/ortime and space-based risk exposures as well as average evacueetravel time and delay. Nevertheless, their implementation of theseMOEs has been limited to primarily vehicular evacuation cases. Inour paper, a spaceetime utilization efficiency index for pedestrianevacuation (from a major event or a large stadium) will be used asthe primary MOE.

There have been several studies designed to fully utilize spaceand time resources in an evacuation operation. One avenue of suchresearch focuses on networkflow theory [3,24,45], which addressedthree major tasks: determining the evacuation route for an evacuee[3,8,25,34,44], generating an evacuation plan based on the dynamicnature of a network [31], and simulating evacuation routings in fineor coarse networks [15,57]. These studies seldom considered thebehaviour of individual evacuees (congestion-induced waiting),especially in non-vehicular evacuation scenarios.

Another approach to evacuation focuses on space and timeorganization. For example, Løvås (1995) [33] suggested effectiveactions for improving the evacuation process, including thereduction of time required for personnel reaction, situation inter-pretation, and walking. The proposed improvement strategiesinvolve training personnel to react correctly and to follow plannedroutes, improving evacuation organization, early detection ofaccidents, and minimizing the adverse effects of such. Bakuli andSmith (1996) [4] studied the resource allocation method of resizingpassageways in emergency evacuation networks. Seyfried et al.(2009) [41] analysed the pedestrian flow and congestion bottle-necks based on trajectories. Lämmel et al. (2010) [29] adapteda traffic queue model to capture congestion bottlenecks and theevacuation time in networks with a time-dependent attribute. Caiet al. (2010) [6] introduced the concept of an efficiency factor ofcontaminant source to evaluate the performance of emergencyventilation in evacuation planning. Chow [9,10] introduced a wait-ing time index to analyse the waiting time [14,18,26,35,38,43] ofcrowded areas in order to implement appropriate fire safetymanagement. However, this type of waiting time research hasseldom incorporated factors of restart moving behaviour or thespaceetime use efficiency of downstream evacuation links orpassages. In the next section we propose a pedestrian waiting-timemodel designed to address this issue and provide a tool forassessing the spaceetime resource usage in evacuation planning.

Fig. 3. Merged and unmerged spaces.

Fig. 4. FIFO evacuation rule in evacuation congestion.

Z. Fang et al. / Building and Environment 46 (2011) 1774e17841776

3. Definition of spaceetime use efficiency

This section considers concepts related to the evaluation ofspaceetime use efficiency on the basis of evacuees’ spaceetimetrajectories. Hägerstrand’s (1970) [19] classical Time Geographyoffered a useful illustrative framework (e.g. spaceetime path,spaceetime prism, see Fig. 1) for understanding human spatialbehaviour under space and time constraints. The spaceetime pathstrack the individuals’ usage processes of space and time resources.Such a perspective on spaceetime paths (or trajectories) supportsthe identification of spaceetime usage efficiency.

Definition 1. Let STDist (traj1, traj2) denote a spaceetime distanceof spaceetime trajectories traj1 and traj2, which is the minimumdistance between the two trajectories in a three-dimensional spaceconsisting of a two-dimensional spatial (geographical) space andthe dimension of time.

Definition 2. For two spaceetime trajectories (traj1 and traj2)derived from the same road segment Rds(a), if STDist (traj1, traj2) isless than a predefined value Dd, traj1 and traj2 can be merged, orare mergeable, when evaluating the spaceetime use efficiency instadium evacuation. Otherwise, traj1 and traj2 are unmergeable.For example, trajectories 1 and 2 in Fig. 2 are mergeable, whereastrajectories 2 and 3 are unmergeable.

Definition 3. The space between two spaceetime trajectories(traj1 and traj2) is merged space if traj1 and traj2 are mergeable,whereas the space between two spaceetime trajectories (traj1 andtraj2) is unmerged space if traj1 and traj2 are unmergeable. Sm isthe area of merged space and Sum is the area of unmerged space. Forexample, Fig. 3 illustrates the merged space between trajectories 1and 2, and the unmerged space between trajectories 2 and 3. Here:

Smð1;2ÞsF; Sumð1;2Þ ¼ F;

Smð2;3Þ ¼ F; Sumð2;3ÞsF;

Definition 4. Let IUE denote the spaceetime use efficiency index.For a road segment Rds(a), its spaceetime use efficiency index,IUE(Rds(a)), in time period [ts, te] is defined as:

IUE�RdsðaÞ; t½s;e�

�¼

Pi;iþ1˛K

SmðtrajðtiÞ; trajðtiþ1ÞÞP

i;iþ1˛KðSmðtrajðtiÞ; trajðtiþ1ÞÞ þ SumðtrajðtiÞ; trajðtiþ1ÞÞÞ

; ts � ti � tiþ1 � te (1)

where K is the trajectory set on the road segment Rds(a) in timeperiod [ts, te]. Equation (1) defines an extent in which spaceetimetrajectories cover the whole spatial and temporal space, and IUE(Rds(a), t[s,e])<¼1.

For a road Rd(m), its spaceetime use efficiency index, IUE(Rd(m)),in time period [ts, te] is defined as:

IUE�RdðmÞ; t½s;e�

�¼

Pi˛M

IUE�RdsðiÞ; tes

�LengthðRdsðiÞÞ

Pi˛M

LengthðRdsðiÞÞ (2)

where M is a road segment set of road m. Length(Rds(i)) indicatesthe length of road segment i.

For an evacuation plan P that consists of all roads in an evacu-ation network, its spaceetime use efficiency is defined as:

IUEðPÞ ¼

Pm˛E

Ztmax

t0

IUE�RdðmÞ; t½t;tþDt�

�dt � LengthðRdðmÞÞ

Pm˛E

LengthðRdðmÞÞ (3)

where E is a road set of an evacuation network, tmax is themaximum clearance time of P and Length(Rd(m))is the length ofroad m. Equation (3) will be used to evaluate the spaceetime useefficiency of evacuation plan P in the evacuation network in thefollowing section.

4. Pedestrian waiting-time model

A pedestrian waiting-time model based on the concepts ofspaceetime usage efficiency and a first-in-first-out (FIFO) queuediscipline is presented herein.

4.1. First in, first out (FIFO) movement rule in stadium evacuation

The queue discipline of evacuees is often considered to be first-in-first-out, or FIFO [29]. This is especially true for evacuees ona narrow path with limited room for passing or overtakingmanoeuvres.

Fig. 4 shows the effect of the FIFO queue discipline in an evac-uation case. P(i) and P(i þ 1) are two evacuees at time t0 and Ds isthe distance between the two evacuees at time t0. Evacuee P(i)moves a distance L(i) in a duration of Dt1 while evacuee P(i þ 1)moves a distance L(i þ 1) in a duration of Dt2. After Dt1 or Dt2, P

(iþ 1) stays behind P(i) in the queue. This study uses the FIFO queuediscipline, which has three restrictions: (1) the length of time anevacuee can be on a particular road link is limited; (2) the link flowcapacity limits its outflow; and (3) the link storage capacity limits

Table 1The parameters and evacuation comparisons in three scenarios.

Scenario S1 S2 S3

Parametersw1 5 0 0w2 3 2 2w3 0 1 1w4 1 3 3

Evacuation performancesTime (s) 1581 860 675Length (km) 2670.92 1583.74 1547.53IUE (P) 0.731762 0.951530 0.959682

Waiting timeTotal evacuation time (s) 13877646 9960579 7438002Cumulative wait time (s) 3484434 1664990 1064915Compelled wait time (s) 0 0 684214Wait time index (WTI) 0.25 0.17 0.14

Change of spaceetime use efficiencyPercentages of Links S2 vs. S1 (%) S3 vs. S2 (%)Increasing 65.92 19.73Reducing 19.73 13.90Unchanged 14.35 65.02

Fig. 5. The study environment and experimental data: (a) the stadium; (b) the stadiumevacuation network.

Z. Fang et al. / Building and Environment 46 (2011) 1774e1784 1777

the number of evacuees on the link and, hence, its inflow. Theserestrictions provide the basis for the simulation of various scenariosof pedestrian evacuation.

The FIFO queue discipline is a realistic and reasonable assumptionfor stadium evacuation scenarios, because evacuees on foot oftenform queues in narrow passages of the stadium, where passing andovertaking manoeuvres are limited. In open space evacuees mayfollow similar pedestrian dynamics where lines of evacuees form inmultiple elongated, but often indiscernible, homocentric areasextending from the exits to the open space. Pedestrians can movefrom one homocentric area to an adjacent homocentric area fromtime to time. This extended rule is helpful for addressing the evac-uation efficiency problem in more complex evacuation scenarios.

4.2. Pedestrian waiting-time model

Before proposing a pedestrian waiting-time model, this sectionintroduces two concepts that are helpful in modelling a pedestrianevacuation queue; namely, the minimum movement gaps distanceand the pedestrian density.

The minimum movement gap distance l in an evacuationbottleneck was defined by Fruin’s function (1971) [14]:

l ¼ 1�bp þ 0:1

�r� dp (4)

where r is the pedestrian density, bp is the width of road and dp isthe body depth [14] of an evacuee. This minimum movement gap

distance is used as the minimum acceptable value to constrainadjacent evacuees in queues of links.

The pedestrian density in the area between A and B beforepedestrian P(i) in Fig. 4 is defined as:

dðnt ; atÞ ¼ ntat

(5)

where nt is the number of evacuees ahead of P(i) on the same linkand at is the area between locations A and B on this link.

Due to the FIFO queue discipline, the waiting time and restartescape velocity (the escape velocity after the wait behaviour of anevacuee) of P(iþ 1) are entirely dependent on P(i). P(1), the evacueenearest to the exit of the current link, has an escape velocity of:

vði ¼ 1Þ ¼ minðvmax;1=dðnt ; atÞÞ (6)

where nmax is the maximum escape velocity for evacuees. It takesevacuee P(1) a duration of t(1) to travel a distance of L(1). The nextevacuee in line, P(2), waits Dt, in the process of following P(1), andtakes a duration of t(2) to travel a distance of L(2). The relationshipsamong these variables in the subsequent cases (i ¼ 2, 3, .) can bedetermined by Equations (7)e(9):

tðiÞ ¼ LðiÞ=vðiÞ; tðiþ 1Þ ¼ Lðiþ 1Þ=vðiþ 1Þ (7)

tðiþ 1Þ ¼ tðiÞ � Dt (8)

Lðiþ 1Þ=vðiþ 1Þ ¼ LðiÞ=vðiÞ � Dt (9)

where n(i þ 1) can be calculated by Equation (6). Considering theminimum gap distance between evacuees, this study limits thedistance variable as follows:

LðiÞ þ Ds� Lðiþ 1Þ � l (10)

�Ds

vðiþ 1Þ þ Dt�vðiÞ � Ds � l (11)

Equation (10) states that the difference of the distancestraversed by P(i) and P(i þ 1) is no less than the sum of theminimummovement gap [14] l and the initial distance Ds betweenthem at time t0. Equation (11) states that the restart distancebetween P(i) and P(i þ 1) is no less than the minimum movement

Fig. 6. The evacuation spaceetime paths of all evacuees in three scenarios.

Z. Fang et al. / Building and Environment 46 (2011) 1774e17841778

gap l after P(i þ 1) waits for a duration of Dt. According to Equation(10), the maximum value of L(i þ 1) is:

MaxðLðiþ 1ÞÞ ¼ LðiÞ þ Ds� l (12)

The minimum value of waiting time Dt is defined as:

MinðDtÞ ¼ lþ DsvðiÞ � Ds

vðiþ 1ÞDs (13)

Therefore, the relative waiting time difference between P(i) andP(iþ 1) and the restart escape velocity of P(i þ 1) can be defined as:

Dt0 ¼ lvðiÞ þ

Ds2

vðiÞðvðiÞtðiÞ � lÞ (14)

vðiþ 1Þ ¼ vðiÞtðiÞ � lvðiÞtðiÞ � l� Ds

vðiÞ (15)

Equations (14) and (15) reveal the dependency of relativewaiting time and restart velocity of P(i þ 1) on P(i). These twoimportant equations are combined and used to model the pedes-trian waiting time in the process of evacuation.

Fig. 7. The evacuation spaceetime paths of an evacuee in three scenarios.

5. Simulation methodology

Simulation is a practical and commonlyaccepted tool for assessingthe effects of evacuation policies and plans. Agent-based simulationapproach is an important technique in modelling the interactions,decision-makings, and status of pedestrians. This approach has beenemployed by several important pedestrian dynamics studiesincludingAdamatzky’swork (2005) [1] on characterizing spaceetimedynamics of crowd-minds using an agent-basedmodel [5] andWas’sreview (2009) [54] on the characteristics of some pedestrians agent-basedmodels including DijkstraeTimmermanseJessurunmodel [11],Alpsim [16], Social Distances Model [53], SBBPedis [27], DynamicsNavigation [48], Situated Cellular Agents [40].

The study presented in this paper employs an agent-basedsimulation approach to model the evacuation process. In thissimulated evacuation process, each evacuee is represented by anagent. All agents are subject to the aforementioned FIFO queuediscipline on the links of the stadium network. Pedestrian flow,

which is implemented on the basis of a queue structure, is describedby three parameters, namely flow capacity, free flow speed, andstorage capacity that were defined by Lämmel (2010) [29].

It should be noted that this study considers two crucial aspects ofevacuation, i.e. route choice strategy andwaiting time rules, to assurean adequate evacuation process. Before introducing these aspects,this section addresses the status and actions of an agent [42] who isinitially allocated to a seat in a stadium. Let AgentðaÞ ¼ fSðaÞ;ActðaÞgrepresent the status SðaÞ and behaviour Act(a) of an agent a.SðaÞ ¼ flr ; vr; qr ; drg , where lr is the current location of agent a onlink r, with an escape speed of nr, at an angle of qr between his currentmoving direction and the direction towards an exit (see Fig. 1),and at a distance of dr to the end of the current link r.ActðaÞ ¼ fmovea;waita; chooseag , where movea, waita and chooseaare the moving, waiting, and route-choosing behaviours of agent a.

5.1. Route choice strategy of an agent

In this study, an agent, i.e. evacuee, selects a route by taking intoaccount the accessibility of the network, the pedestrian trafficstatus of the links, and the pedestrian conformity [17,23]. In orderto make proper route choice decisions in the simulation process,each agent’s status S(a) and the flow and capacity status of all linksare updated for every time period tp. The candidate routes for agentawho is currently on link i, see Fig. 1, can be identified by using anaccessible spaceetime shortest path tree in a spaceetime prism(see Yu and Shaw, 2009). All candidate routes are available to theagent as the “next” or “downstream” evacuation links.

Let Rða; tpÞ represent a set of candidate routes to agent a in thetime period tp. Parameter numðr; tpÞ is the number of evacueescurrently on link r and aatðr; tpÞ is the accessibility of link r to thisparticular agent. Here, aatðr; tpÞ is calculated based on the length oflink r in the shortest spaceetime path tree. Parameter distðr; tpÞ isthe distance between the end point of link r in the shortestspaceetime path tree and the exit. The candidate routes of an agentare shown in Fig. 1:

R�a; tp

� ¼ fr; r1; r2gIn addition, exit selection is based on the shortest route

distances to all exits of the stadium [56].

Fig. 9. The evacuation curves of the three scenarios and STEPS.

Fig. 8. The waiting patterns in spaceetime evacuation paths.

Z. Fang et al. / Building and Environment 46 (2011) 1774e1784 1779

The pedestrian conformity of an agent a choosing link j in theevacuation simulation is calculated as:

fpcða; jÞ ¼ Desire num�j; tp

�P

j˛RðaÞDesire num

�j; tp

� (16)

where Desire numðj; tpÞ is the number of evacuees who want to fleethrough candidate link j in time period tp. For this study, the agentchooses the link with the highest value of pedestrian conformityamong all candidate links.

The other three factors affecting the route choice are:

faatða; jÞ ¼ aatðj;tpÞPj˛RðaÞ

aatðj;tpÞ

fdistða; jÞ ¼ distðj;tpÞPj˛RðaÞ

distðj;tpÞ

fqða; jÞ ¼ qjðtpÞPj˛RðaÞ

qjðtpÞ

(17)

where faatða; jÞ; fdistða; jÞ and fqða; jÞ each gauges the influence ofaccessibility of the network, travel distance, and travel orientationon agent a’s decision of choosing link j. Taking all factors intoconsideration, an agent, a, on link j, wants to choose a candidatelink, j, with the highest probability value Piða; tpÞ, as the next link inhis escape route.

Pi�a; j; tp

� ¼ fpcða; jÞw1faatða; jÞw2

fdistða; jÞw3fqða; jÞw4 (18)

where w1, w2, w3 and w4 are the weights associated with theinfluence of pedestrian conformity, accessibility in the network,travel distance, and travel orientation on choice route, respectively.All w coefficients should be no less than zero. Subsequently, theagent’s choice behaviour can bemodelled as: choosea ¼ fjjPiða; j; tpÞ� Piða; q; tpÞ; q˛RðaÞ; qsjg.

5.2. Waiting rules in an agent evacuation queue

In order to simulate an evacuation process in a realistic fashion,this section introduces some rules related to waiting behaviour ina queue of evacuees (agents).

Rule 1. The number of evacuees exiting from a link within a shorttime period tp is dependent on the flow capacity of the current linkand the storage capacity of the next link:

N�i; tp

� ¼ min�FCðiÞ � tp3600

; SCðjÞ � FCðjÞ�

(19)

where FC(i) is the flow capacity of link i, FC(j) and SC(j) are the flowcapacity and storage capacity of link j, where link j is the linkimmediately downstream of link i. The maximum number ofevacuees exiting link i within tp is:

FCðiÞ � tp3600

If there is only one link immediately downstream of link i, thenumber of agents flet within time tp can be calculated directly usingEquation (19).

Rule 2. If there is more than one link immediately downstream oflink i, the number of fled agents from the link i within a short timeperiod tp dependents on route-choosing and -moving behaviour:

ff ðmovea; chooseaÞ f ðmoveb; choosebÞ;.gof the agents. This number is defined by Equation (20):

N�i; tp

� ¼ min

0@FCðiÞ � tp

3600;Xa˛Ai

f ðmovea; chooseaÞ1A (20)

where Ai is the number of evacuees on link i. Equation (21)describes the movement of an agent that has already entered thenext link.

f ðmovea; chooseaÞ ¼�1 movea ¼ 1; choosea > 00 else

(21)

Table 2The waiting time distribution of all evacuees in three scenarios.

Wait time of each evacuee Number of evacuees

S1 S2 S3

No waiting 9250 11853 4685(0,100] 6642 6402 14789(100,200] 2448 2413 2881(200,300] 1349 1669 1431(300,400] 1037 1197 374(400,500] 944 581 19(500,600] 755 64 0(600,700] 623 0 0(700,800] 399 0 0(800,900] 286 0 0(900,1000] 216 0 0(1000,1100] 173 0 0(1100,1200] 56 0 0(1200,1300) 1 0 0

Z. Fang et al. / Building and Environment 46 (2011) 1774e17841780

Rule 3. As Nði; tpÞ evacuees exited the link i, the remainingagents stay in a queue against the exit end of the link. The lengthof the queue depends on the number of agents remaining onthe link.

Rule 4. The waiting time and restart moving speed of an agent inqueue are determined by Equations (14) and (15) in an FIFO order.In other words, thewaiting time of an evacuee is constrained by theevacuee immediately ahead of him in the queue; the restart timeand moving speed also depend on the clearance of the evacueeahead. Here:

waita ¼ fwtimea; rvagwtimea ¼ wtimea�1 þ Dt0 (22)

Dt0 is the relative waiting time between agents a and a � 1,which is calculated by Equation (14). rna is the restart speed definedby Equation (15) after agent a has waited for a period of wtime.

Fig. 10. The spaceetime use efficien

6. Computational experiments

6.1. Experimental design

The stadium at the Wuhan Sports Centre in China was used asthe test bed for mass-evacuation in this study (Fig. 5(a)). Thestadium has 42 bleachers (tiers of seats) distributed on all 3 floorsand has 10 exits (Fig. 5(b)) for evacuation; 24,197 spectators areallocated to the 42 bleachers randomly. The evacuation network inthis stadium (Fig. 5(b)) has 223 links and 157 nodes. Each noderepresents one of the bleachers, stairs, exits and passages able tohold evacuees.

Three evacuation scenarios (S1eS3) were defined to evaluatethe evacuation performance in this study. In S1, each agent chooseshis evacuation route according to the pedestrian flow and queuelengths on the links. The value of pedestrian conformity for a link isderived from Equation (16). In this scenario, the agent is notfamiliar any of the stadium exits but is physically located near theexits. The major characteristic of this scenario is the agents’pedestrian conformity, or following the movement of other agents.

In S2, the agent does not follow others in terms of pedestrianconformity. Each agent chooses the next link to evacuate accordingto the accessibility of the network, distance to travel, and move-ment orientation towards the target exit defined by Equation (18).In addition to the factors considered in S2, the third scenario (S3)requires agents to wait a duration of wtimea where

wtimea ¼ wtimea�1 þ Dt0 (23)

according to Rule 4 in Section 5.2, before restarting at a relativelyhigh walking speed to traverse the bottleneck links. This scenario isdesigned to address the faster-is-slower effect for the purpose ofimproving the overall spaceetime use efficiency of the stadiumevacuation process. Detailed scenario parameters of S1eS3 areprovided in Table 1.

To our knowledge, most commercially available simulationprograms, e.g. VISSIM [46] and Simulation of Transient Evacuation

cy in the evacuation network.

Fig. 11. The spaceetime use efficiency distribution on each link.

Fig. 12. The pedestrian flows and densities in two queues on link #15.

Z. Fang et al. / Building and Environment 46 (2011) 1774e1784 1781

and Pedestrian movements (STEPS) [37], do not report complexdetails such as spaceetime trajectories and spaceetime use effi-ciency. Therefore, the evacuation simulation framework wasimplemented in a visual Cþþ 6.0 environment and executed on anIntel Core Duo based platform.

In this study, bp and dp derived from research by Lin et al. (2004)[32] are assigned as 0.5 m and 0.25 m respectively. The parameterDd in Definition 2 is 1. Themaximumwalking speed nmax¼ 1.66 m/srefers to work by Lämmel et al. (2010) [29]. The pedestrian densityparameter is calculated as:

r ¼ 1=LOSperson ¼ 1=0:46 ¼ 2:17 persons=m2 (24)

where the level of service is LOSperson ¼ 0.46 m2/person whenpedestrian flow is unstable but not at breaking point as defined byTransportation Research Board (2000) [49].

For comparing the results between our proposed model andother models, this study simulated the pedestrian evacuationprocess in the same stadium by using the STEPS software, whichemploys an agent-based approach based on cellular automata (CA)theory [2], to model pedestrian dynamics under both normal andemergency conditions. Themaximumwalking speed of pedestriansin STEPS is 1.66 m/s, same as the value in our program.

6.2. Result analysis

This study examines five different aspects of the simulatedresults, namely the spaceetime paths, evacuation curves, waitingtime, spaceetime use efficiency, and comparison between ourproposed model and the STEPS software.

6.2.1. Spaceetime pathsFig. 6 shows the spaceetime paths of evacuation solutions of the

three scenarios. The highest spaceetime paths are in the orderS1 > S2 > S3. These highest values, which are the time the lastagents exiting the stadium and equivalent to the maximum clear-ance evacuation time (t1 ¼ 1581 s, t2 ¼ 860 s and t3 ¼ 675 s), aregiven in Table 1. The average heights of the spaceetime paths in thethree scenarios present a similar trend in favour of S3 over S2 andthan S1.

Fig. 7 shows three distinctly different spaceetime paths for thesame randomly-chosen sample evacuee in the three scenarios.Parameters t1, t2 and t3 indicate the time that this evacuee arrivesat the exit (destination). The three spaceetime paths show anidentical response time, which is the start time for this evacuee tocommence evacuation. This is caused by the initial evacuationqueue formed around its original location.

Table 1 gives the total evacuation distances, i.e. the sum of thedistance travelled by all evacuees, in each of the three scenarios.The total evacuation distances in S2 (1583.74 km) and S3(1547.53 km) are within 2e3% from each other, but the totalevacuation distance (2670.92 km) in S1 is much greater than that inS1 and S2.

Fig. 8 illustrates the regular movingewaitingerestart movingpattern derived from spaceetime paths of evacuees. The timedimension is completely filled by the spaceetime paths without

Fig. 13. The pedestrian flows and densities in two models on link #15.

Z. Fang et al. / Building and Environment 46 (2011) 1774e17841782

pedestrian bottlenecks. The vertical segments of these spaceetimepaths represent either (a) the compelled waiting time as per ourproposed model or (b) the waiting time when agents are in queue.The advantages of using a waiting-time model are examined in thefollow sections.

6.2.2. Evacuation curvesThe evacuation curves resultant from the three scenarios (Fig. 9)

were subsequently examined. In S3, 95% of evacuees exited thestadiumwithin 10 min (or 600 s) after the evacuation commences.This is significantly superior to the performance observed of S1 andS2. The evacuation clearance time is w1581 s in S1, w860 s in S2,and w675 s in S3. The significant improvement achieved in S3 isexplained below. At first sight, this result, which is from the orga-nization of spaceetime paths of evacuees in Fig. 6, appearsimpossible; however, it is possible if all evacuees are drilledaccording to the calculated plan (see the website for the report ofthe US Department of Homeland Security, 2008 [51]).

6.2.3. Waiting timeWaiting time of all agents on all links is recorded in each of the

three scenarios. Scenario S3 also includes the compelled waitingtime based on the proposed pedestrian waiting-time model.Waiting time index (WTI) proposed by Chow and Ng (2008) [10] foridentifying the bottlenecks in the evacuation network of thestadium is applied in this study. Table 1 gives the total evacuationtime, cumulative waiting time, compelled waiting time, and wait-ing time index for the three scenarios.

S1 has the longest total evacuation time (13,877,646 s), cumu-lative waiting time (3,484,434 s) and waiting time index (0.25)while S3 has the shortest in comparison (total evacuation time7,438,002 s, cumulative waiting time 1,064,915 s, and waiting time

index 0.14) for the evacuation process from beginning to comple-tion. A very interesting observation is the compelled waiting timein S3 (684,214 s) based on our proposed waiting-time model is64.25% of the cumulative waiting time, while S2 saw no suchcompelledwaiting time at all. Yet, thewaiting time index is reducedfrom 0.17 in S2 to 0.14 in S3.

Table 2 presents the waiting time distribution of all evacuees inthe three scenarios. There are 9250 evacuees in S1, 11,853 evacueesin S2, and 4685 evacuees in S3 who did not experience any waitingin their evacuation process. But 14,789 in S3, far more than those inS1 and S2, experienced a waiting time between 0 and 100 s. In S3,(4685 þ 14789)/24179 or 80.54% of all evacuees experienceda waiting time of less than 100 s, and all evacuees waited less than500 s. However, in S1, 34.27% of all evacuees experienced a wait oflonger than 100 s, and 230 evacuees still drift about in the stadium1000 s after the evacuation order was given. This phenomenonindicates that high conformity, following other evacuees, in S1could lead to longer waiting time than the proposed waiting-timemodel in S3. The waiting time data in Tables 1 and 2 demonstratethat the proposed waiting-time model has the ability of betterorchestrating spaceetime paths to alleviate bottlenecks, and thatthe few seconds of strategically compelled waiting are very helpfulin facilitating the whole evacuation process.

6.2.4. Spaceetime use efficiencyThe spaceetime use efficiency index, IUE, values are presented in

Table 1. The value for S3 (0.959682) is higher than those for S1(0.731762) and S2 (0.951530). Fig. 10 illustrates the spaceetime useefficiency in the evacuation network of the stadium. Fig. 10(aec)shows the spaceetime use efficiency of network links in S1, S2 andS3, respectively. Most links in S1 have lower efficiency than those inS2 and S3. Although most links in S2 and S3 have similar efficiencyvalues, several links have higher index values (red links) in the top-right area of S3 than those of the same links (orange links) in S2.This demonstrates that the proposed waiting-time model hasimproved the spaceetime use efficiency in that area. (For inter-pretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

Fig. 10(def) illustrates the spaceetime use efficiency of all linksover time. Although most links start off efficiently, build-up ofqueues tend to take a toll on their performance as time goes on.Because of the overall efficiency of the compelled waiting strategyin S3, most links stayed quite efficient in the evacuation process andcompleted the entire operation in just over 600 s; see Fig. 10(f).While S2 is also moderately efficient, movements were boggeddown on some of the links, which is evident on the decline inefficiency on these links, see Fig.10(e). Many of the links in S1 nevercould perform efficiently; even those that were efficient early onalso saw decline in efficiency over time (Fig. 10(d)). In these figures,a steep drop to zero means all evacuees have passed through thislink. On the other hand, slow declines typically mean link build-upand performance problems. Long drown-out ridges of a steady levelof efficiency are likely bottlenecks operating at a fixed throughputchoking up the evacuation process.

Fig. 11 illustrates the statistical spaceetime use efficiencydistribution on each link. Fig. 11(a) shows the index values of linksin the three scenarios. The index value of zero in area A representslinks never used for evacuation purposes for all three scenarios,while the high index values in area B indicate links with highoperational efficiency during the evacuation process. It is clear thatsymbols representing S2 and S3 are typically found in high effi-ciency areas while those representing S1 are distributed across thewhole efficiency spectrum.

Fig. 11(b) and (c) shows the differences in efficiency between S1and S2, and between S2 and S3 for each link. A positive value in

Fig. 14. The CA-based evacuation simulation in STEPS.

Z. Fang et al. / Building and Environment 46 (2011) 1774e1784 1783

these figures means improvement in link efficiency in the latterscenario, and vice versa. Fig. 11(b) clearly shows improvement inspaceetime use efficiency in majority of the links when comparingS1 and S2. Fig. 11(c) provides less telltale signs of superiority interms of link efficiency between S2 and S3. A point to be notedthough is when link efficiency drops to zero, with two cases ofefficiency dropping from 1.0 to 0.0 (IUE(3) e IUE(2) ¼ �1.0) as rep-resented by the two points at the bottom part of Fig. 11(c), it simplymeans those links are not utilized at all in the new scenario. This isnot necessarily undesirable.

Table 1 provides information on the percentages of links withincreasing, reducing, and unchanged efficiency between differentscenarios. In the case of S2 vs. S1, 65.92% of all links saw an increasein spaceetime use efficiency, which clearly indicates S2 to bea more efficient strategy. For the case of S3 vs. S2, 19.73% of all linkssaw an increase in efficiency with 65.02% of all links staying aboutthe same. Once again S3 is more efficient than S2, which attests tothe advantage of the only strategic difference between S3 and S2:the compelled waiting concept.

In depth examination was performed on the utility of space-etime resource on several links, including link #15, to further studyevacuation queues with the proposed compelled wait strategy andnon-compelled wait behaviours. On link #15, there were originally230 agents to be evacuated with a maximum walking speed of1.66 m/s. The overall loading time of the 230 evacuees is 60 sfollowing a normal distribution. Fig. 12(a) illustrates the evacueeflow over time in the two cases, while Fig. 12(b) depicts the evacueedensity over time on this link. Fig. 12(a) shows that the compelledor mandatory wait scenario enjoys a relatively stable and higherflow during the evacuation process than that of non-compelledwait scenario.

Fig. 12(b) shows that the density value in compelled wait case isequal to or less than that in non-compelled wait case. The densityon this link is relatively low when t < 50 s. During this period, thetrends of evacuee flow and density are both increasing. Whent ¼ 60 s, the link becomes congested evacuee density has reached2.27 persons/m2, which is more than the initial parameter r ¼ 2.17.The congestion of this link in non-compelled wait case continues toget worse and finally the flow reduce to zero when t ¼ 80 s. But thecongestion in the compelled wait case was in-check so a seriousbreakdownwas avoided. The clearance time for link #15 is 330 s fornon-compelled wait scenario and 240 s for the compelled waitscenario.

6.2.5. Comparison of proposed model and STEPSWe also compared the evacuation performance of the proposed

model and the CA-based simulation model in STEPS from bothmicroscopic and macroscopic perspectives. For microscopiccomparisons, evacuation process was simulated on link #15 inSTEPS according to the same conditions implemented in our study.That walking space is divided into grid cells of 0.5 m� 0.5 m in size.Each pedestrian is assigned to one and only one grid cell, which isessential for CA simulation. STEPS was employed to model theevacuation operations on link #15 under compelled waiting andnon-compelled waiting scenarios for multiple times to account forstochastic effects. Fig. 13 shows the average evacuee flows andaverage densities resultant from the two simulation models on link#15.

The trends of evacuee flows and densities in our proposedmodel and the cell automata model in STEPS are similar. Bothmodels yielded an average clearance time of about 260 s, which isremarkable. The maximum evacuation flow in the CA model isslightly larger than that in the proposed model, while themaximum pedestrian density in the proposed model is larger thanthat in the CA model. The values in our proposed model appear tobe more stable than those in the CA model of STEPS.

For macroscopic comparisons, we compared the overall pedes-trian evacuation curves in Fig. 9 between all three scenarios and theCA model-based simulation result in STEPS. An instance of STEPSsimulation of our test bed is shown in Fig. 14. The maximumevacuation clearance time is 731 s in the CA model of STEPS, whichis slightly greater than that of S3, but smaller than that of S1 and S2.

7. Conclusions

This paper introduces a concept of spaceetime use efficiencyfor gauging the usage of space and time resources in an evacuationprocess. This provides an alternative approach to establishedmeasures the effectiveness (MOEs) of evacuation operations.Based on this concept, a pedestrian waiting-time model wasdevised to improve evacuation performance by avoiding orreducing the faster-is-slower effect. This study demonstrates thatthe compelled waiting time derived from the proposed model canimprove overall spaceetime use efficiency. It is also establishedthrough the comparisons of the various scenarios that theproposed approach is suitable and effective for space and timeassignment of evacuees from a stadium, and perhaps othercomplex build environments.

Future directions for this study include extending the proposedevacuee waiting-time model to optimize evacuation plan usingseveral approaches; for example, developing a practical locationtechnique to track evacuees’ physical locations in the stadium (e.g.the radio-frequency identification (RFID)-based system describedby Vanem and Ellis, 2010 [52]), extending the pedestrian servicelevel concept [12] to model the evacuation level of service ina stadium environment, and modelling the interactions [7]between evacuees and predicting the time-varying patterns ofevacuating agents for real-time operational management [20]. Ourresearch will focus on the spaceetime accessibility analysis ofevacuees in supporting the exit-choosing behaviour [13] and pre-dicting waiting time and evacuation clearance time [30].

Acknowledgements

This research was supported in part by the National ScienceFoundation of China (grants #40701153, #40971233, #40830530,#60872132), and LIESMARS Special Research Funding. The authorswould also like to thank Dr. Shih-Lung Shaw and the anonymousreviewers for their valuable comments and suggestions.

Z. Fang et al. / Building and Environment 46 (2011) 1774e17841784

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