analysis and modeling of simultaneous and staged emergency evacuations

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Analysis and Modeling of Simultaneous and StagedEmergency Evacuations

Steven I. Chien1 and Vivek V. Korikanthimath2

Abstract: Disaster response in areas of high population density is centered on efficient evacuation of people and/or goods. Developingevacuation plans suitable for different levels of urgency based on the intensity of threat is a challenging task. In this study, mathematicalmodels are developed for estimating evacuation time and delay. Evacuation time is the duration required for evacuating all vehicles froma designated region, whereas delay includes queuing and moving delays. The relationship between delay and evacuation time is investi-gated, and the impact of staged evacuation against simultaneous evacuation is analyzed. An example is provided to demonstrate theapplicability of the developed model. A numerical method is adopted to determine the optimal number of evacuation stages. Sensitivityanalysis of parameters �e.g., demand density, access flow rate, and evacuation route length� affecting evacuation time and delay isconducted. Results indicate that evacuation time and delay can be significantly reduced if staged evacuation is appropriately implemented.

DOI: 10.1061/�ASCE�0733-947X�2007�133:3�190�

CE Database subject headings: Emergency services; Disaster relief; Sensitivity analysis; Evacuation.

Introduction

Efficient evacuation can be defined as a process to safely trans-port people and goods away from a place or an area within anacceptable time period in an orderly fashion. Emergency evacua-tions are carried out in response to disasters including earth-quakes, hurricanes, nuclear power plant disasters, chemical plantleaks, terrorist attacks, etc. Due to unexpected occurrence of theseevents, it is challenging to develop a model that is practicableunder various situations. Strategies �e.g., simultaneous or staged�employed in evacuation vary over circumstantial and environ-mental conditions. For instance, an evacuation strategy for a hur-ricane event differs from that of a nuclear plant disaster. In theformer case, the event is relatively predictable, and a preventiveevacuation is possible before landfall, whereas in the latter,evacuation will need to be employed instantaneously. Thus, de-veloping a sound model that is sensitive to the urgency of thesituation and capable of providing accurate estimates of evacua-tion time and delay is desirable.

The methodology used in modeling emergency evacuationconsists of demarcating the area under threat and estimating thetime needed to evacuate people and associated delay. Evacuatingstrategies are generally classified here as simultaneous and staged.Under simultaneous evacuation, all vehicles are evacuated con-

1Professor, Dept. of Civil and Environmental Engineering, NewJersey Inst. of Technology, University Heights, Newark, NJ 07102.E-mail: [email protected]

2Research Assistant, Interdisciplinary Program in Transportation, NewJersey Inst. of Technology, University Heights, Newark, NJ 07102.E-mail: [email protected]

Note. Discussion open until August 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on February 7, 2006; approved on August 11, 2006. Thispaper is part of the Journal of Transportation Engineering, Vol. 133,No. 3, March 1, 2007. ©ASCE, ISSN 0733-947X/2007/3-190–197/
$25.00.
190 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / MARCH

J. Transp. Eng. 2007.

currently; on the other hand, in staging, vehicles are evacuated byzones in a particular sequence. Note that a zone defined here is anarea within the evacuation region from which people and goodsare expected to be evacuated in a given time period. Staging ofevacuation reduces congestion and helps in minimizing evacua-tion time �Chen and Zhan 2004�.

The model developed in this study will help to determine thetime and delay for simultaneous or multistaged evacuation. Theobjective functions formulated in this study are evacuation timeand delay, which will be minimized numerically by optimizingthe decision variables including the number and sizes of stagedzones. The impacts of staging on evacuation time and delay ana-lyzed here will provide suitable guidelines for emergency man-agement authorities in making vital decisions.

Literature Review

Evacuation planning is an important constituent of emergencyplanning. Various problems associated with emergency evacua-tion are of concern to related federal and state agencies �e.g.,compliance to evacuation warning or instructions, providing shel-ters, transit service, etc.� among which evacuation decision-making is the most significant. Thus, evacuation modeling hasgained thrust and has grown to reflect the environmentally chang-ing needs.

Some of the earliest work in evacuation modeling was con-cerned with hurricane evacuation �Urbanik 1978; USACE 1979�.The Three Mile Island nuclear accident in 1979 shifted the atten-tion to evacuation of nuclear sites and some of the earlier studiesprovided evacuation time estimates for nuclear plant evacuations�Urbanik and Desrosler 1981�. The occurrence of Hurricane An-drew in 1992 that proved very expensive in terms of damage andHurricane Floyd in 1999, which led to one of the greatest evacu-ations in U.S. history, moved the attention back to hurricaneevacuation. Some recent studies also focused on evacuations re-lated to suburban fire and terrorist attacks �Cova and Johnson
2002; Hamza-Lup et al. 2004�.
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Various modeling approaches were applied in previous studiesto minimize evacuation time or maximize capacity of evacuationroutes. In a study conducted by Cova and Johnson �2003�, it wasfound that traffic delays at intersections can be reduced by apply-ing lane-based routing during evacuation. Evacuation times weresignificantly reduced by channeling traffic flows at intersectionsto eliminate crossing conflicts. Malone et al. �2001� developed asteady state evacuation model based on empirically estimateddriving parameters �e.g., car length, reaction time, and the decel-eration parameter�. To address speed variance of vehicles, a cel-lular automata model for traffic flow was developed and validatedthrough simulation. It was found that evacuation time can beminimized by restricting the number and types of vehicles andregulating low speed vehicles to move on the right lane during theevacuation attempt. Sinuany-Stern and Stern �1993� applied a mi-croscopic, behavior-based simulation model with SLAM II�Simulation Language for Alternative Modeling� to analyze sen-sitivity of evacuation times in a radiological emergency situation.Results revealed that the major traffic parameters influencingevacuation time were interaction with pedestrians and intersectiondelay. It was also found that the route choice parameters affectingevacuation time were probability of evacuees choosing the short-est paths and myopic behavior of evacuees that overlooked road-way congestion. Murray-Tuite and Mahmassani �2003� studiedthe effects of household interactions �e.g., trip chains� on evacu-ation time using integer programming. A microassignment simu-lation approach was applied to capture traffic interactions, consid-ering different loading strategies. It was found that a minimum of150% of the original demand should be assumed for developingevacuation plans if the impacts of trip chains are ignored. Hanand Yuan �2005� used VISSIM, a microscopic, behavior-basedsimulation software, for a hypothetical event of nuclear powerplant accident using dynamic traffic assignment �DTA� and mostdesirable destinations �MDD� methods for routing evacuees ac-cording to traffic conditions at the time of departure and achiev-ing the shortest travel time to different destinations, respectively.The implementation of DTA and MDD led to significant improve-ment in overall evacuation time compared to using static destina-tion selection and traffic assignment. Results also showed thattraffic under police control at key intersections and reversinglanes �or called contraflow operations� on congested road sectionscan lead to reduction in delay and evacuation time.

A few studies analyzed contraflow operation, and focused onincreasing roadway capacity. Kwon and Pitt �2005� studied thefeasibility of applying Dynasmart-P for evacuating traffic indowntown Minneapolis. It was found that contraflow operationsare very effective when the capacities of key entrances to majorroadways are increased. Tuydes and Ziliaskopoulos �2004� modi-fied the cell transmission model �CTM� for better utilization ofavailable network by allowing reversibility on designated seg-ments when a drastic demand shift occurs during a disaster. Thesystem optimal solution was achieved by efficiently relocatingroadway capacity through contraflow operations. Comparison ofthe base case �original CTM� and the enhanced network showedthat evacuation time is significantly reduced when contraflow op-erations are applied.

Only a few studies analyzed evacuation staging. Chen andZhan �2004� investigated the effectiveness of simultaneous andstaged evacuation strategies using as agent-based simulation ap-proach. Results revealed that the performance of staged evacua-tion is highly related to network structure and population density.When roads are capacitated, staged evacuation helps in reducing
the total evacuation time. Malone et al. �2001� developed a cel-
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lular automata model to evaluate staggering evacuation for differ-ent coastal counties in Charleston, S.C. and found that staggeringevacuation can reduce evacuation time depending on the level anddirection of a hurricane.

The evacuation models briefly discussed earlier tended tomaximize throughput and/or minimize evacuation time; howeverthe discussion of associated evacuation delay was generally ig-nored. The model developed in this study estimates the impact ofcritical factors �e.g., demand density, access rate, route length,and free flow speed� on evacuation time and delay. Some studies�Chen and Zhan 2004; Malone et al. 2001� provided groundworkfor the investigation of evacuation staging, but a comparativeanalysis of simultaneous and staged evacuation is limited. Mostevacuation studies applied microscopic simulation approach forestimating travel time and delay during an evacuation event.However, simulation might be costly in terms of computationtime to cover a full range of parameters �e.g., demand distribu-tion, street patterns, etc.�. This study aims at developing an ana-lytical model, which is generic, simple, and can be readily appliedto different situations.

Model Formulation

This section focuses on formulating evacuation time and delay fora region with one evacuation route under a general and simplifiedsetting. The objective is to optimize the number of evacuationstages that minimizes evacuation time and the associated delay.The evacuation time is expressed as a function of the time forvehicles to be discharged under an achievable rate, the averagetransit time for vehicles traveling on the evacuation route, and thesetup or preparatory time between subsequent stages. Evacuationdelay is comprised of two components namely, the queuing delayincurred by evacuees accessing the evacuation route and the mov-ing delay while traversing the evacuation routes.

System Assumptions

The following assumptions are made to formulate the problem:1. Number of vehicles to be evacuated denoted as Q, in the

Fig. 1. Configuration of the studied area

studied region shown in Fig. 1, is uniformly distributed over

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the evacuation route. Demand density, denoted as Q̄, is Qdivided by the length of the evacuation route denoted as L,from S to E.

2. The number of evacuation stages is denoted as N, whoselengths are assumed to be identical and equal to L /N. Theaccumulated flow increases as distance increases, beforereaching the maximum discharge rate, i.e., the accumulatedflow at the end of a staged zone of length x is qx, whereq=access flow rate per mile �see Fig. 2�.

3. Full compliance of residents to evacuation instructions is as-sumed �deterministic behavior�.

All the above-noted assumptions can be relaxed to enhance thedeveloped model to be applicable in a more realistic setting. Forinstance, the demand distribution discussed in Assumption 1 maybe replaced by an actual one if data are available. Assumption 2might be adapted to have staged zones with different lengthsbased on the relaxation of Assumption 1. Assumption 3 will nothold in most of the cases as evacuee behavior is difficult to pre-dict, which would certainly impact the results of analysis. How-ever, Assumption 3 can be relaxed by approximating an actualdemand-loading curve derived from field data.

Evacuation Time

Fig. 3 shows staged evacuation zones of the studied area withinwhich demand is uniformly distributed. The evacuation time isprimarily estimated by the number of vehicles to be evacuated ineach staged zone, the discharge flow rate at the end of the evacu-ation route, the average transit time for vehicles exiting the evacu-

Fig. 2. Maximum discharge rate and accumulated flow rate on theevacuation route

Fig. 3. Staged evacuation zones



ation region, and the required setup or preparatory time betweensubsequent stages. Thus, the total evacuation time TE is

TE = TD + TR + TS �1�

where TD, TR, and TS=discharge time, transit time, and setuptime, respectively.

Discharge Time „TD…

Discharge flow rate of an evacuation route is determined by thespeed-density relationship of Edie’s discontinuous exponentialmodel, which applies the Greenberg’s model for congested re-gime, for example kx�50 vehicles per mile �vpm� such that

ux = uc ln� kj

kx� �2�

and Underwood’s model for uncongested regime, for examplekx�50 vpm, expressed as

ux = ufe�−kx/kc� �3�

where ux=average travel speed; kx=average density; uc=criticalspeed �speed at maximum discharge rate�; kc=critical density�density at maximum discharge rate�; uf =free-flow speed �speedachieved under very low density�; and kj =jam density �maximumdensity at which all vehicles are stopped� for vehicles originatingfrom a staged zone.

Edie’s model describes the entire range of speeds and densi-ties. A saturated flow condition occurs when the accumulated flowfrom a staged zone �qx� exceeds capacity q* and the correspond-ing density reaches 80–100 vpm �TRB 2000�. Although it is bestto work with sets of roadway data in which all three variableshave been measured and no estimation is needed, the Edie’smodel is used as an alternative to determine the speed-densityrelationship of the evacuation route. Thus, a density of 100 vpmis assumed for saturated conditions in the numerical example dis-cussed later. Capacity has been defined as the maximum flow ratethat can be achieved under prevailing roadway conditions �Roesset al. 1998�. The discharge rate for each staged zone with a lengthof x is the product of flow density kx and its corresponding speedux. Thus

qx = uxkx if qx � q* �4�

According to the Underwood model

qx = ufe�−kx/kc�kx �5�

kx can be obtained from Eq. �5� and ux can then be obtained bysubstituting kx into Eq. �3�. The maximum discharge rate q* isformulated as

q* =kcuf

e�6�

The required discharge time for vehicles in a staged zone is

tD =Q̄x

uxkx�7�

where Q̄x=number of vehicles to be evacuated per zone.
Thus, the discharge time for an N-stage evacuation is
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TD =NQ̄x

uxkx�8�

As xN=L and Q̄=Q /L, TD can be derived as

TD =Q

uxkx�9�

Transit Time „TR…

The transit time for a staged zone denoted as tR is the averagevehicle travel time from the zone to the end of the evacuationroute shown in Fig. 3. tR can be formulated as

tR =�N − i +

1

2�x

ux�10�

where �N− i+1/2�x=average distance traversed by vehicles fromzone i on the evacuation route. ux can be obtained from Eq. �3�.Note that the average distance traveled considered in Eq. �10� isbased on Assumption 1, and can vary depending of the demand

Fig. 4. Flow rate versus time

travel times under free-flow and congested conditions.

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distribution of the evacuation region. Thus, the average transittime for an N-stage evacuation is

TR =

�i=1

N �N − i +1

2�x

N� 1

ux� �11�

Setup Time „TS…

Setup time is the preparatory time required between subsequentevacuation stages to clear barricades or initiate appropriate trafficflow. It is assumed to increase linearly with the number of stages

TS = �N − 1�ts �12�

where ts=setup time per stage. According to the discharge time,transit time, and setup time derived in Eqs. �9�, �11�, and �12�,evacuation time TE is

TE =Q

uxkx+

�i=1

N �N − i +1

2�x

Nux+ �N − 1�ts �13�

where xN=L. By substituting for ux obtained from Eqs. �3� and�2� into Eq. �13�, the required evacuation times under free-flow

Fig. 5. Queue length versus time

and congested regimes can be approximated using

TE =� Q

ufe�−kx/kc�kx

+

�i=1

N �N − i +1

2�L

ufe�−kx/kc�N2 + �N − 1�ts

if qx � q* �14�

Q

uckx ln� kj

kx� +

�i=1

N �N − i +1

2�L

uc ln� kj

kx�N2

+ �N − 1�tsif qx � q* �15� �

Delay Estimation

Delay discussed here consists of queuing delay incurred by ve-hicles accessing the evacuation route and moving delay as ve-hicles transit the route. The queuing delay is computed based onqueue length in each staged zone and the accumulated flow rate,whereas the moving delay is based on the difference of vehicle

Queuing Delay „DQ…

According to Assumption 2, the accumulated flow rate at the endof a staged zone is qx. Queuing delay occurs when qx is greaterthan the discharge rate uxkx. Fig. 4 shows that the discharge timeis determined by the accumulated flow rate and discharge rate,where the queue formation time t� is Q̄x divided by qx. Thus,
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tD� =Q̄x

qx�16�

However, if tD� is not realized �i.e., if all vehicles in the stagedzone are not discharged in tD� � owing to uxkx�qx, a queue isformed. Consequently, the queue length nq at time tD� is

nq = �qx − uxkx�tD� �17�

As shown in Fig. 5, the queue is fully discharged at time tD,and the queuing delay denoted as dq for a staged zone is

dq = 12 tDnq �18�

By substituting for tD, tD�, and nq, from Eqs. �7�, �16�, and �17�,respectively, the queuing delay for an N-stage evacuation is

DQ =NQ̄2x

2quxkx�qx − uxkx� �19�

As xN=L, and Q̄=Q /L, DQ is derived as

DQ =Q2

2Lquxkx�qL

N− uxkx �20�

Moving Delay „DM…

The moving delay is defined as the difference between traveltimes at the operating speed ux and the free-flow speed uf. Forinstance, the moving delay for vehicles originating from a staged
zone i is given by dmi �see Fig. 3�, such that
the evacuation route is 2,000 vph. The notations used here with



dmi = �N − i +1

2�x

ux−�N − i +

1

2�x

uf�Q̄x ∀ i � N �21�

where �N− i+1/2�x is the average distance traversed by vehiclesfrom zone i on the evacuation route. Thus, the moving delay foran N-stage evacuation is

DM = �i=1

N �N − i +1

2�Q̄x2� 1

ux−

1

uf� �22�

Since xN=L and Q̄=Q /L, DM is derived as

DM = �i=1

N �N − i +1

2�QL

N2 � 1

ux−

1

uf� �23�

Total Delay „DT…

The total delay defined here is the sum of DQ and DM. Thus

DT =Q2

2Lquxkx�qL

N− uxkx + �

i=1

N �N − i +1

2�QL

N2 � 1

ux−

1

uf��24�

By substituting for ux from Eq. �3� into Eq. �24�, the total delaysunder free-flow and congested regimes are derived as Eqs. �25�
and �26�, respectively,
DT =�Q2�qL

N− ufkxe

�−kx/kc�2Lufqkxe

�−kx/kc� + �i=1

N �N − i +1

2�QL

N2 � 1

ufe�−kx/kc� −

1

uf� if qx � q* �25�

Q2�qL

N− uckx ln� kj

kx�

2Lquckx ln� kj

kx� + �

i=1

N �N − i +1

2�QL

N2 1

uc ln� kj

kx� −

1

uf� if qx � q* �26� �
Numerical Example
The objective of this example is to demonstrate the applicabilityof the developed model in determining the optimal number ofstages that minimizes evacuation time. The tradeoffs between theevacuation time and delay for both simultaneous and stagedevacuations are investigated. The length of each staged zone isdetermined by dividing the length of the evacuation region by thenumber of stages. Eqs. �14�, �15�, �25�, and �26� are applied tocompute the evacuation time and associated delays of the studiedevacuation route for both free-flow and congested regimes.

In this numerical example, the optimal staging for evacuationof a given region with a length of 15 mi is analyzed. The evacu-ation demand is assumed to be 20,000 vehicles and is uniformlydistributed over the studied route. The access flow rate, q, is as-sumed to be 350 vph per mile. The maximum discharge rate on

their baseline values are summarized in Table 1. These values areintended to demonstrate the application of this model rather thanto represent any specific case. Note that 1 mi=1.6 km is appliedto the discussion of this example.

Results and Discussion

Fig. 6 shows the estimated evacuation time at a various numberof stages for the base case �L=15 mi, q=350 vph, Q=20,000vehicles� without considering setup time between stages. Theevacuation time is a convex curve, and yields a minimum solutionat N=3.

The evacuation time is primarily influenced by the dischargetime depending on the flow rate on the evacuation route. Eithersimultaneous evacuation �N=1� or a two-stage evacuation
�N=2� causes congestion on the evacuation route due to the ac-
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cumulated flow rate rising beyond the available capacity. Thisresults in jam densities and considerably reduces the speed onthe route. Thus, the discharge rate obtained under these conditions�N=1 or 2� is less than the discharge rate for N=3 �1,750 vph�.However if N�3, evacuation time increases because of reduceddischarge rates �at lower densities with higher speeds�. Althoughtransit time decreases as the number of stages increases due tohigher speeds �45–50 mi/hr�, it is less than the increase in dis-charge time and does not significantly help to reduce the evacu-ation time.

Fig. 7 shows the estimated evacuation time at a various num-

Table 1. Explanation of Variables

Variable DescriptionBaseline

value

dq Queuing delay for a staged zone �vhr� —

DQ Queuing delay for N stages �vhr� —

DM Moving delay for N stages �vhr� —

DT Total delay for N stages �vhr� —

kx Density of vehicles from staged zoneon evacuation route �vpm�

—

L Length of the evacuation route 15 mi

nq Number of vehicles queued from a zoneat time tD�

—

N Number of staged evacuation zones —

q Access rate per mile 350 vph

q* �Capacity� maximum dischargeon the evacuation route

2,000 vph

qx Accumulated flow rate on theevacuation zone �vph�

—

Q Total demand in the evacuation region 20,000 veh

Q=Q /L Demand density over the evacuation route 1,334 veh/mi

tD Discharge time for a staged zone �h� —

tR Transit time for a staged zone �h� —

tS Setup time per stage �h� —

tD� Queue formation time for a staged zone �h� —

TD Discharge time for N stages �h� —

TR Transit time for N stages �h� —

TE Total evacuation time for N stages �h� —

TS Setup time for N stages �h� —

ux Speed of vehicles from staged zoneon evacuation route �mi/hr�

—

uf Free flow speed 55 mi/h

x Length of a staged zone �mi� —

Note: 1 mi=1.6 km=1,600 m.

Fig. 6. Evacuation time versus number of staged zones �excludingsetup time�

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ber of stages for the base case �L=15 mi, q=350 vph, andQ=20,000 vehicles� while setup time between stages �ts=0.5 h�is considered. Although the evacuation time increases for Stages2–5, when compared with the previous case �Fig. 6�, the evacua-tion time is still minimum at N=3.

Fig. 8 shows delays at various values of N. The moving delayon the evacuation route primarily depends on the speed of theevacuating vehicles. This speed is 13 mi/hr when N�2 is em-ployed. Consequently, the time taken for vehicles traversing theevacuation route increases as N decreases. Queuing delay ismainly influenced by the queue length over time �from 0 to tD� asshown in Fig. 5�. The queue length for simultaneous evacuation issignificantly higher than that of a multistaged evacuation, due to agreater difference between the access and discharge rates. As aresult, the queuing delay is the highest under simultaneous evacu-ation, and it decreases as N increases. The queuing delay is zero ifqx is less than uxkx.

In this regard, based on the urgency of the evacuation, thedecision of a simultaneous or multistaged evacuation could beundertaken. For instance, evacuations for predictable events �e.g.,hurricanes� allow a relatively longer preparatory time to evacuatepeople and goods from the areas under threat. Therefore, usingmultistaged evacuation reduces delay but at increased evacuationtime. In contrast to this, for evacuations that do not allow muchflexibility in time �e.g., terrorist attacks and nuclear plant disas-ters� the evacuation strategy that yields the minimum evacuationtime is recommended.

Sensitivity Analysis

Results of sensitivity analysis are discussed here to illustrate therelations among variables and identify the relative importance offactors that contribute to them. The parameters that are considered

Fig. 7. Evacuation time versus number of staged zones �with setuptime�

Fig. 8. Delays versus number of staged zones


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to be most sensitive to the evacuation time and delay are accessflow rate q, demand Q, and the evacuation route length L.

Fig. 9 illustrates the variation in TE at varying q �L=15 mi,Q=20,000 vehicles�. The number of stages at which the totalevacuation time is minimum, increases while q increases. As ac-cess rate reduces �e.g., q=150 vph�, simultaneous evacuation ispreferred as it yields the minimum evacuation time althoughcongested conditions prevail �qx�q*�. This is because theachievable discharge rate for simultaneous evacuation at q=150�uxkx=1,300 vph� is greater than the discharge rates attained fornoncongested conditions at higher values of N. But, as q increases�e.g., q=450 vph�, higher discharge rates are attained at highervalues of N, thereby increasing the number of stages at whichevacuation time is minimum �e.g., N=4 achieves the least TE forq=450 vph�.

Fig. 10 demonstrates the variation in total delay DT, withvariation in q �L=15 mi, Q=20,000 vehicles�. DT decreases as Nincreases at all values of q. For a particular N, DT increases whileq increases, primarily due to an increase in the queuing delaycaused by a large difference between the access flow rate anddischarge flow rates on the evacuation route. But, the movingdelay on the evacuation route remains a constant despite variationin the access rate as the average speeds on the route are unaf-fected by variation in q.

Fig. 11 shows the variation in TE with varying Q �L=15 mi,q=350 vph�. The variation in demand is an indication of variationin demand density for a given region. Although the evacuationtime increases with increase in demand for any N, the minimumevacuation time is yielded by N=3 for various Q. This suggeststhat staging of evacuation could be applied to areas of both low

Fig. 9. Total evacuation time versus number of staged zones forvarious flow rates

Fig. 10. Total delay versus number of staged zones for various flowrates



and high densities, but the time savings are more pronounced athigher densities.

Fig. 12 shows the total delay DT with variation in Q�L�15 mi, q=350 vph�. DT decreases as N increases at all valuesof Q. Also,DT increases as Q increases for any value of N, due toan increase in volume entering the route from the staged zones.However, increase in delay at higher values of N �e.g., N=5� isonly due to the occurrence of moving delay as discharge ratesattained are less than capacity �qx�q*�, which results in noqueues.

Fig. 13 shows TE with variation in L �q=350 vph, Q=20,000vehicles�. Demand of the evacuation region varies proportionallywith the length of the evacuation route at a constant demand

density of Q̄. This analysis helps in determining evacuation sce-narios for affected areas of various sizes. The optimal number ofstages at which the evacuation time is minimum, increases as Lincreases. For a short L �e.g., L=5 mi�, the minimum evacuationtime is when N=1 �simultaneous evacuation�, and staging doesnot further reduce evacuation time. However, as L increases, con-gested conditions prevail at lower values of N, thereby reducingthe discharge rates, which increases evacuation time. Thus, theminimum evacuation time is achieved by increasing N for higherL. Fig. 14 illustrates the variation of total delay DT with variationin L �q=350 vph, Q=20,000 vehicles�. DT decreases as N in-creases at all values of L. Both queuing and moving delays in-crease while L increases.

Conclusions

The model developed in this study optimizes the number ofevacuation stages for minimum evacuation time at reasonable

Fig. 11. Evacuation time versus number of staged zones for variousdemands

Fig. 12. Total delay versus number of staged zones for variousdemands

2007

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delay. The benefits of the model could be realized by applying itin various emergency evacuation situations such as hurricaneevacuations, nuclear plant disasters, and terrorist attacks.

The proposed modeling technique demonstrates how the num-ber of evacuation stages can be optimized, considering criticalfactors �i.e., demand in the evacuation region, the evacuationroute length, access flow rate, capacity, etc.� that affect the deci-sions of emergency management authorities. When urgency isneeded, the optimal number of stages should minimize the evacu-ation time, whereas the importance of evacuation delay reduces.On the other hand if an evacuation allows flexibility in time,employing a multiple-stage evacuation can significantly reducethe delay. With user-specified input parameters, the developedapproach is simple to apply for optimizing an evacuation consid-ering various conditions, and can serve as a tool for generatingguidelines to facilitate emergency management authorities inmaking critical decisions during evacuations.

Although the developed model is yet practical, it could beenhanced as a complicated one to address the impact of traffic

Fig. 13. Evacuation time versus number of staged zones for variousroute lengths

Fig. 14. Total delay versus number of staged zones for various routelengths

control �e.g., contraflow�, evacuees’ behavior �e.g., compliance

JOURNAL O


and demand loading patterns�, and multiple evacuation routes.This enhancement would be the immediate extension of thisstudy.

References

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