Buffering in evacuation management for optimal traffic demand distribution

Transportation Research Part E 48 (2012) 684–700

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Buffering in evacuation management for optimal traffic demand distribution Jun Duanmu a,1, Mashrur Chowdhury b,c,⇑, Kevin Taaffe b,c,2, Craig Jordan a,3 a

Virginia Modeling, Analysis and Simulation Center (VMASC), Old Dominion University, 1030 University Blvd., Suffolk, VA 23435, United States Clemson University, Glenn Department of Civil Engineering, 216 Lowry Hall, Clemson, SC 29634, United States c Clemson University, Department of Industrial Engineering, 110 Freeman Hall, Clemson, SC 29634, United States b

a r t i c l e

i n f o

Article history: Received 28 March 2011 Received in revised form 29 September 2011 Accepted 29 October 2011

Keywords: Traffic demand distribution Buffer management Evacuation Emergency management

a b s t r a c t This paper presents a new framework for managing congestion during emergency evacuations. The algorithm allows a long link of the network to be used as a buffer to keep the traffic flow moving in. Concurrently, a detour trigger time is estimated to keep the traffic under-saturated in the buffer zone and minimize the total travel time. The integration algorithm presented in this paper is an efficient mathematical solution for travel time cost calculation. A case study is presented to demonstrate the efficacy of the traffic demand buffering strategy developed in this research for managing the evacuation flow. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Distribution of evacuation traffic from an urban area under an emergency has received much attention in recent years, especially after the September 11th terrorist attacks and Hurricane Katrina, both of which resulted in great loss of life and property. Transportation systems are typically the central component in evacuation related logistics. Effective and timely traffic management and control systems are vital to a successful action plan. Though a great deal of research has been conducted in dealing with evacuations which are related to transportation, such as planning and policies (Wolshon et al., 2003; RDS, 1999), route selection (He et al., 2009; Chiu and Mirchandani, 2008), pickup location selection (Song et al., 2009) and resource optimization (Yi and Ozdamar, 2007), there has been limited traffic management research as it pertains to the demand distributions during an evacuation process. Research is particularly sparse involving distributions for evacuations of congested, long distance urban environments. This paper advances a new concept that combines traffic flow distribution and control with the aid of Wardrop’s second principle (System Optimal approach described in Section 2) of traffic assignment under the evacuation process, especially during periods of peak congestion. The methodologies presented in this paper will produce a better advanced-notice evacuation plan since the arrival rate can be roughly predicted and modeled. Specially designed solutions are derived to decompose the seemingly difficult traffic management and control into stages of different statuses. This paper begins with a brief introduction concerning ⇑ Corresponding author at: Clemson University, Glenn Department of Civil Engineering, 216 Lowry Hall, Clemson, SC 29634, United States. Tel.: +1 864 656 3313; fax: +1 864 656 2670. E-mail addresses: [email protected] (J. Duanmu), [email protected] (M. Chowdhury), [email protected] (K. Taaffe), [email protected] (C. Jordan). 1 Tel.: +1 757 686 6200. 2 Tel.: +1 864 656 0291; fax: +1 864 656 0795. 3 Tel.: +1 757 638 4463. 1366-5545/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2011.12.002

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evacuation traffic analysis. Then, a System Optimal (SO) approach to minimize travel times is applied by conducting a bottleneck analysis that uses a buffer management system in an evacuation network. Finally, a framework for applying this optimized evacuation plan is presented with a real-case study. 2. Literature review and traffic consideration in evacuation In an evacuation process, the goal is to maintain a stable and smooth traffic flow in the evacuation routes. Thus, traffic demand/capacity analysis, traffic flow distribution and traffic management are the focus in evacuation traffic management. 2.1. Travel demand forecasting in evacuation traffic management Traditional travel demand forecasting problems are solved with a classical four-step process (Papacostas and Prevendouros, 2008). However, in a complex traffic process such as the case in large-scale evacuations, the problems differ significantly, especially as they pertain to trip generation, trip distribution, and trip assignment. For trip generation, only one-way trips, those initiated by the evacuees leaving an at-risk area, are considered in evacuation. The traditional gravity model used in trip distribution seems redundant in a city evacuation model. The Federal Emergency Management Agency (FEMA) reported that evacuees will go to their relatives’ or friends’ houses, or find a motel (FEMA, 1986–2006). This makes it difficult to estimate the accurate trip numbers from an evacuated city to another specific destination. However, some states have their own evacuation routes. For example, in the South Carolina Department of Transportation evacuation plan for Charleston (SCDOT, 2011), all evacuees must follow a specific direction according to the evacuation guidelines. The traffic flow is designated toward some predefined location, regardless of the individual destinations of all evacuees. Once they have arrived at that predefined location, they can then travel to their final destination. The reason for limiting travel destinations is to ensure a relatively smooth and event-free evacuation process. The framework presented in this paper utilizes this same evacuation strategy where all evacuees must follow specific, predetermined routes. As far as trip assignment is concerned, the shortest evacuation path is not necessarily the fastest, although it normally is. The best routes are those selected by considering a trade-off between speed, capacity, risk of accident and long delays. In addition, evacuation traffic must be monitored in real-time such that, whenever a detour is required, the action or response can be executed immediately. The framework presented in this paper allows a detour to be used as an optional backup route, which has been predefined as part of the evacuation planning. 2.2. Simulation, optimization, and Wardrop’s principles A significant portion of traffic management research in evacuation is carried out by simulation; in fact, simulation is often integrated into optimization analysis. Afshar and Haghani (2008) use a mesoscopic traffic simulator in an optimization algorithm to find the system-optimal dynamic traffic assignment. Han et al. (2006) obtain an optimal destination and route assignment based on the one-destination evacuation concept, where one ‘‘super’’ destination is constructed for problem solving. Other evacuation simulation models can be found in Gangi (2009), Brown et al. (2009), and Robinson et al. (2009). In these models, a traffic assignment algorithm is typically embedded within simulation software, and these tools are then used to evaluate a designed evacuation process. In most of the models, there are no traffic distribution management processes, and it is hard to estimate if the flow distribution in the model is optimized. Optimal traffic distribution problems can also be solved by analytical approaches. In estimating the distribution of the travel demand in either static or dynamic traffic environments, the most widely used method incorporates at least one of the two principles proposed by Wardrop (1952). In addition to predicting and forecasting the traffic flow, these principles can also be used to control the distribution of the traffic. A brief introduction of Wardrop’s principles in traffic assignment is presented below. Wardrop’s first principle – User Equilibrium (UE). Wardrop’s first principle is known as the equilibrium principle, where, for a given origin–destination (OD) pair, ‘‘the travel times on all the routes actually used are equal, and less than those unused routes.’’ The underlying assumption of this principle is that all travelers will have the same travel times if they encounter the same traffic conditions provided that all travelers are also privy to the same perfect information on all possible routes throughout the network (Chow, 2007). Wardrop’s second principle – System Optimum (SO). Under the first principle (User Equilibrium), each individual attempts to minimize his or her personal travel cost, without regard to the overall total system travel cost. In economical theories, this discrepancy between the behavior of individuals and the behavior of the group across the entire system is known as the ‘‘divergence between private cost and social cost’’ (Chow, 2007). Wardrop proposed his SO principle, in which the average travel time for all users is minimized, according to this observation. In this research, the SO principle is incorporated in the process of developing a demand distribution model for city evacuation management. There are also many models that adopt Wardrop’s UE or SO theories (see, e.g., (Friesz et al., 1989; Ran and Shimazaki, 1989a,b; Janson, 1991). Chiu and Zheng (2007) presented a CTM-based model and System Optimal solution for supporting a traffic assignment and departure schedule for immediate disaster responses (i.e., without any prior notice) under dynamic, multi-priorities and multi-destination

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environments. In addition, Ng and Waller presented a CTM-based SO model for evacuation that accounts for demand and capacity uncertainty (Ng and Waller, 2010). The following SO formulation is adopted from Peeta (1994) and Carey (1986). The parameters and decision variables used in the formulation are listed as below: xta: hta(xta): mta: dta: Itn : Otn : B(n): ga(xta):

C(n):

The number of vehicles on link a at the beginning of interval t. The cost incurred (in terms of disutility such as delay, travel time and transportation cost) when link a contains xta vehicles at the beginning of time interval t. The number of vehicles exiting link a in interval t. The number of vehicles entering link a in interval t. The number of vehicles generated or joining the network at node n in the time interval t. The number of vehicles reaching their destination node n in interval t. The link traffic flow leaving node n. This is the exit function related traffic conditions on the link. This functions is assumed to be a (i) continuous, (ii) non-decreasing and (iii) concave function and represents the maximum number of vehicles that can exit from link a at time t. Link traffic flow entering node n.

Minimize zðxÞ ¼

XX t

Subject to :

X

hta ðxta Þ

ð1Þ

a

tb

d ¼

X

mtc þ Itn  Otn

8t; n;

c 2 CðnÞ; b 2 BðnÞ

mta 6 g a ðxta Þ 8t; a mta ¼ xta  xtþ1a þ d xta P 0;

mta P 0;

ð2Þ

c

b

ð3Þ ta

8t; a ta

d P 0 8t; a

ð4Þ ð5Þ

Eq. (2) represents the node balance conditions. That is, the total number of leaving vehicles (entering set b) is equal to the total number of entering vehicles (leaving set c), minus those vehicles entering and are absorbed plus those vehicles generated from within. Eq. (3) is the exit capacity limit, and Eq. (4) is used for the update of link balance. Peeta (1994) also mentions that no first-in first-out (FIFO) constraint is defined in the formulas. Because the objective function represents the result in a series of time intervals, a traffic behavior changing with time can be considered by defining different time slots. The ‘‘congestion buffer’’ idea presented in this paper originated from these formulations and will be discussed later in details. Under System Optimal conditions, an ideal assumption is that everyone follows the System Optimal paths. Some travelers may be directed to routes that have higher costs even though the less expensive routes are also accessible to them. Such higher cost routes are selected because the additional costs incurred by those travelers will be outweighed by the savings gained by other travelers using the quicker routes. The SO model is very helpful in forecasting traffic flow during a mass evacuation if an active control of the traffic is considered. In the next sections, the detailed description about analytic methods in evacuation management is presented. Four aspects of optimal traffic distribution are discussed: (1) link performance and bottleneck analysis; (2) exit flow estimation; (3) Travel cost calculation; and (4) optimization in evacuation.

3. Demand distribution framework in an evacuation logistics system 3.1. Link performance analysis When the link density is changed due to a different arrival rate, the entering and exiting flow rates, as well as the travel speed, will also change. Thus, the travelers should select their routes accordingly. In his reviews of the measurement and formulation of link capacity functions, Branston (1976) depicts the relationship between entering and exiting flow in a link (or segment). Since both flows have set capacities, they might not be analogous. Assuming that the entry capacity is higher than the exit capacity in daily traffic fluctuations, the entry flow first keeps increasing and then decreasing until settling to the level of the exit capacity, at which point the entire link reaches a balance of flow (Branston, 1976). Indeed, when the traffic is oversaturated, it is very possible that because the higher arrival rate causes an immediate change in travel speed and congestion, the inflow sometimes is much lower than the exit capacity, resulting in an unstable traffic condition. One possible effective method for keeping the inflow equal to the outflow in the balanced state involves detouring some traffic flow at a specific time to avoid congestion. However, Branston’s theory of flow

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fluctuation can assist researchers in forming new traffic management concepts to further improve a city evacuation process. First, the transportation stakeholder allows a higher traffic arrival rate for a specific amount of time and uses the link as a ‘‘buffer’’ to absorb as much traffic as possible. If the high traffic rate continues until the density reaches a predefined level, a detour order will then be issued. Thus, congestion is avoided, and the best link is still utilitized to its maximum potential. The original link is also kept at an inflow rate which is equal to or less than the outflow rate since the additional arrivals will be forced to detour. Consequently, Branston’s theory is realized. In addition, an optimized demand distribution is needed to maximize the utilization of the best route and reach the minimum total travel time in a system level. In this paper, it is assumed that the best route is the shortest path. A detour that is ordered too early may cause a waste of the best resource; meantime, any delay in flow distribution management may cause long time delays and congestion.

3.2. Maximum flow at merge points and exit points The following subsection presents an analysis of merging points within the evacuation network. A merging point is the exit point where vehicles leave from several upstream roads and join together to start the next travel segment; it is also the starting point of a merged downstream segment.

3.2.1. Traffic variability in an evacuation process During actual evacuation events, the arrival rate of traffic first increases and upon reaching a peak value will then begin to decrease. Consequently, in a short time, vehicles will overwhelm the entire route, resulting in extremely high traffic densities. After lengthy delays, these long queues will ultimately be dissipated. Simultaneously, the arrival rate will decrease as most of the people (and their vehicles) will have left the evacuation zones. Fig. 1 provides an example of an arrival rate/response time curve in a 24-h evacuation time window. The vertical axis represents the portion of total demand arriving in the past hour. Eq. (6) is a typical accumulative arrival function in evacuation models (Tweedie et al., 1986). Eq. (7) is the arrival rate function when Equation (6) is differentiated; this is also the arrival rate function depicted in Fig. 1. a =bi Þ

F i ðtÞ ¼ 1  eðt i AðtÞ ¼

ð6Þ

ai ai 1 ðtai =bi Þ t e bi

ð7Þ

In Eq. (6), Fi(t) represents the cumulative density function of vehicle departures, and ai and bi are parameters estimated for each evacuation area i. The area below the curve in Fig. 1 is 100% of the total demand in a 24-h evacuation time window. According to previous research results (FEMA, 1986–2006), during a 24-h evacuation process, 80% of the evacuees will begin their journey within a 10-h interval, from 7:00 am to 5:00 pm. To ensure a more constrained time window, this research has condensed the time interval from 10 h to 8 h. Therefore, it is assumed that approximately 10% of the evacuees will arrive prior to 8:00 am, and 80% of the evacuees will arrive during the next 8 h (from 8:00 am to 4:00 pm). Using these values in Eq. (6) (i.e., F(8) = 0.1 and F(16) = 0.9), parameters a and b are found to be 4.45 and 99309.8, respectively, for a single evacuation entrance.

Fig. 1. Arrival rate curve.

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Considering different evacuation time windows, the shapes and parameters will be changed accordingly. Duanmu et al. (2011) gave a detailed analysis on the properties of the arrival rate definition in different evacuation scenarios. The shape of the arrival rate will not directly change the principles of the algorithms presented in this paper. 3.2.2. Merging area capacity Traffic can increase quickly at the merging points within an evacuation traffic network, overwhelming its merging capacity. Such centers of merging should be considered as bottlenecks in determining the real performance of an exit point. In this case, two branches will merge together with a major road. We assume that a capacity of 2, 500 vehicles/h is applied for the merging point. The merging points for these three roads may not be exactly the same but will be very close to one another. To simplify the calculation in this paper, we assume two branches join the main road at the same place. Thus, a relatively higher value of capacity for the merging lane is applied compared to the simple one merging point situation. In this example, when the inflow rate is higher than this capacity value, it is very likely that the traffic will become unstable. Normally, the outflow rate shouldn’t exceed the capacity; consequently, the road density will increase, resulting in an even lower level of service (Highway Capacity Manual (2000)). 3.3. Travel time cost calculation While most SO expressions are clearly understandable, they are often quite difficult to solve in real case studies, especially in a congested situation, and consequently give limited contribution to actual traffic forecast or management. Part of this reason is that it is hard to generate the cost algorithm in the SO model. In this paper, the calculation of travel time cost under a congested traffic situation is based on queuing theory (May, 1990). The formula is shown below.

C¼

Z

TQ

½Ar ðtÞ  Ad ðtÞdt

ð8Þ

0

Eq. (8) is a general cost function (May, 1990), where Ar(t) is the arrival vehicle per hour; Ad(t) is the departure vehicle per hour; C is the travel time cost; and TQ is the duration of the queue time. This calculation is very complex, and a computer program might be needed (May, 1990). The calculation is divided into different time stages, from the forming of queues until the end of dissipation of these queues. The queuing theory focuses on the study of queue length and waiting time, where the research objects are the vehicles that wait in the queue. It does not consider the traffic flow in a link as a whole system. In other words, it does not matter if the vehicle is in the queue or still flowing toward the queue. During the evacuation, the length of time a queue exists is very long. The logistics team needs to select the best link to utilize and, at the same time, minimize congestion. Therefore, it is very possible that they change the arrival or departure flows while the queue is still forming. They cannot wait for the queue to dissipate by itself, but instead, they must manually distribute the travel demand to gain a System Optimal travel time cost. The new generated algorithm in this paper not only considers the waiting time in the formed queue, but it also integrates the travel time cost based on the observation at the entrance and exit points in a link. The travel time cost can be derived at any required time point. In the next section, the algorithms that are used to calculate the travel time cost in the middle of a queuing process as well as the travel time cost when a vehicle is traveling in a congested link are shown in detail. When the detour time is a decision value in the system optimization process, these cost values are used to solve the SO model, and the solution helps balance the demand and capacity during the evacuation process. 3.4. An optimization expression This section presents an optimization model specifically designed for flow distribution during evacuations, closely analogous to Peeta’s (1994) SO expression presented in Eqs. (1)–(5). Peeta showed the trip assignment for specific time intervals t. Similarly, in the evacuation model, two time controls are added: (1) t0, the time to start the calculation; and (2) tstop; the time to change the status of the traffic situation, for example, the time to trigger the detour when the link reaches its defined saturation level. Thus, by combining other time stages, an optimized traffic flow control model is obtained. Some node flow generation or absorption expressions are neglected since most of the traffic in the evacuation network is determined by the beginning and ending points. There are no additional entities generated within the link, and all vehicles complete their trips at the end node. Therefore, the model becomes much easier to solve, and a SO evacuation flow control procedure is achieved in a simple but effective way. Eqs. (9)–(13) provide a model tailored for an evacuation network SO distributing process. The following notation is required for the model: For all link a: i: Xai: xa = fi(t)a:

The different time stages in the evacuation process. The total number of vehicles that arrived at link a in time stage i. The number of vehicles at time t on stage i in link a.

J. Duanmu et al. / Transportation Research Part E 48 (2012) 684–700

ha(Xi): Ari(t)a: Adi(t)a: B(n): C(n): Ca: ti:

The cost incurred by vehicles arrived in link a during time stage i. The arrival rate onto link a in time stage i. The departure rate of vehicles exiting link a in time stage i. Link traffic flow leaving node n. Link traffic flow into node n. The exit capacity. The time at which stage i begins, in a defined time stage, it is named as t0.

Minimize z ¼

XX a

Subjectto : Adi ðtÞa 6 C a

a

h ðX i Þ

ð9Þ

i

XZ

Ari ðtÞdt ¼ i

b

Z

689

XZ c

Adi ðxÞdt

c 2 CðnÞ; b 2 BðnÞ;

8i; a;

Ad ðtÞa dt ¼ ððfi ðt i ÞÞa  ðfiþ1 ðtiþ1 ÞÞa Þ þ

i

Assumption :

8i; n;

ð10Þ

i

ð11Þ Z

Ar ðtÞa dt

8i; a

ð12Þ

i

Ar ðtÞa P 0; Ad ðtÞa P 0

ð13Þ

In the above equations, the objective of function (9) is to minimize the total travel time cost incurred on roadway links in a specific time stage. Eq. (10) represents the node balance constraints (i.e., inflow equals outflow for any particular node); it actually controls the bottleneck behavior in a buffer link model. Eq. (11) confirms that each link’s outflow cannot exceed the link’s capacity. Eq. (12) is the link balance condition. That is, for each time stage, the total number of departed vehicles is the sum of the conserved and entered vehicles minus the new conserved vehicles at the beginning of the next time stage. When using ti in Eq. (12), it is implied that fi(ti)a is the number of conserved vehicles in link a at the beginning of time stage i. The last line shows the non-negativity assumption on several variables. The value of z is normally obtained by integration as will be described in Section 4. In general, each vehicle’s travel time in the congested link is calculated based on the elapsed time from entering the link until all vehicles ahead of the one in question have left the link. Therefore, the elapsed time is calculated in the same sequence as the entering vehicles’ sequence, and the FIFO principle is automatically confirmed. The integration method is very helpful in solving the flow distribution problem since complex queuing calculations are not required, yet the flow automatically follows the FIFO principle in congested traffic. The buffer analysis and case study in Sections 4 and 5 will show the process of calculating a congested travel time cost in a buffer link. In this group of optimization functions, there is only one increasing and decreasing arrival rate cycle (see Fig. 1). If the arrival rate behaves in a more complex mode in which there is more than one cycle, the solution approach remains the same. However, the set of i will include more time stages or intervals. 3.5. Network impact of the optimization process Section 3.4 provides optimization equations that can be used in evacuation analysis. In this section, a network impact of the optimization strategy will be discussed. Unlike general traffic network, where most of the segments are two way links and vehicles are traveling on either direction on the links, the evacuation network are mostly one-way flow network. Consequently, the buffering strategy is more straight forward and easier to apply. The following principles can be considered in the flow distribution strategy: (1) The distribution strategy needs to keep the major road fully utilized; thus minimizing the total travel costs. (2) The detour routes should have enough capacities in the anticipated peak hours; therefore, although some vehicles need to detour, these detoured vehicles will be able avoid heavy congestion on the detoured route. Fig. 2 shows an example of the major roadways in an evacuation network. The solid lines are the major evacuation roadways, while the dashed lines are possible detour routes. For a link in the network, a bottleneck forms when its exit point reaches the capacity. This generally happens when several links merge together and the entrance point for the downstream link cannot accept more traffic. From upstream to downstream, the merging points 5, 6, and 7 are the possible bottlenecks. When a bottleneck reaches its capacity, the corresponding upstream links becomes buffers. At that time, detouring might be considered. The traffic conditions of detour routes may vary significantly; therefore, the detour trigger times need to be calculated based on the actual road condition. Each of the bottlenecks can have their own detour trigger times. Consequently, different detour strategies will cause different system travel costs. The time a bottleneck forms and the time a detour triggers can be defined as a series of events. In addition, the detour trigger time can affect the formation of bottleneck conditions at

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Fig. 2. Evacuation network.

other links. According to the optimization equations presented in Section 3.4, the evacuation time can be divided into different time stages while events occur. The entire network cost is determined by the event times. If the estimated data on upstream arrival rates and entrance capacities are available, the total travel cost can reach an optimized value based on a set of selected detour event times. The computational process will be described in the following. As an example, in Fig. 2, Lij is the name of the link starting from node i and ending at node j. t0ij is the time a bottleneck forms on link Lij, and tsij is the time to trigger a detour on link Lij. Arij is the arrival rate to Lij, and Adji is the departure rate of node j from link Lij. In addition, Adj is the departure rate of node j formed by the total flow-in rates of node j and Arj is the total arrival rate on node j. The bottleneck forming time is the function of arrival and departure rate related to that node, and the time to form a bottleneck and the time to trigger a detour are decided by the upstream arrival rates and other nodes’ events. For link L15, we have:

t015 ¼ f ðAr15 ; Ar25 Þ ts15 ¼ f ðAr15 ; Ar25 Þ The arrival rate of the downstream link is related to the upstream event times and arrival rates. For example, the arrival rate of link L57 is calculated by the departure rate of L15 and L25, and it can also be affected by the trigger time of the detour for the downstream links as represented in the following expression:

Ar57 ¼ Ad5 ¼ f ðt 015 ; t 025 ; ts15 ; ts25 ; Ar15 ; Ar25 Þ As a direct network impact, when a bottleneck forms, the downstream arrival rate becomes constant at the entrance capacity. A detour decision can relieve the congestion on the upstream links and thus, can support a smooth evacuation process. Not all the merging points may become bottlenecks. For example, in link L7x, if the capacity of entrance rate Ar7x is higher than the sum of the two downstream links’ departure capacity, there will be no detour requirement for node 7. Under that circumstance, we only need to consider the congestion of its upstream links. However, if the capacity of Ar7x is lower than the sum of capacity Ad75 + Ad76, the detour must be considered. In addition, the detour time can be estimated according to actual cost calculations on a network level, instead of only on local consideration. Flowcharts shown in Fig. 3 present some possible detour strategies.  The left option shows that node 7 reaches its capacity earlier than node 5 and 6. It does not mean the detour on node 7 occurs right at the time when capacity is reached since link L57 and L67 can be used as buffer. After that, node 6 and node 5 might reach their capacities also. The optimum cost calculation may suggest detouring on node 7 first and then node 5 and 6.

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Ad5 and A d6 increase

Ad5 and A d6 increase

Ad5 and A d6 increase

Ar7 reaches its capacity

Detour flow on node 6 first even it has not reached the capacity

Detour on node 7 when necessary

Ad5 and A d6 increase and may reach the capacity

Ad5 and A d7 increase and may reach the capacity

Ar5 and A r6 reach their capacities

Detour on node 5 or node 6 when necessary

Detour on node 5 or node 7 when necessary

Detour on node 5 and node 6 when it is necessary

Detour on node 7

Detour the rest one when it is necessary

Ar7 reaches its capacity

691

Fig. 3. Example of different detour decisions.

 The flowchart in the middle of Fig. 3 shows another possible result. When node 7 reaches its capacity and link L57 and L67 act as buffers. The optimization may suggest detour on node 5 or 6 first and then node 7.  Another option is shown in the rightmost flowchart of Fig. 3. We can trigger the detour on node 6 earlier and lower the flow of L67 to node 7 to prevent link 7 reaching its capacity in a short time. Therefore, node 5 or node 7 may reach their capacities later than the detour on node 6. These different strategies shown as examples are event driven processes and the costs related with each time stage can be calculated. Therefore, the total network travel costs related with those events are different concerning the different strategy choices. Some strategies may save considerable amount of time and the evacuation planning team can carefully evaluate the detour strategies to obtain the best results. The optimized detour arrangements can be identified by various heuristic algorithms, such as genetic algorithms. The travel costs are calculated based on pre-defined road densities, estimated arrival rates and departure rates. The optimization result can also be used as a quick reference for traffic management decisions. Sections 4 and 5 will present the detailed algorithm, as well as an example, for the travel cost calculations based on the functions shown in Section 3.4. 4. Buffered travel time cost analysis in a roadway segment In this section, the algorithm to calculate the object function in Eq. (9) under congested evacuation traffic management process is explained in detail. This algorithm is used to calculate the travel cost of vehicles generated from the beginning of congestion until the detour trigger time tx, or tstop. The total cost can be optimized to reach its minimum value. This algorithm can also be used in other similar congested situations. Before the flow distributing and management process can be presented, it is necessary to explain in detail the concept of buffer analysis in traffic flow management. In an evacuation process, the key to avoiding an incident and to having a

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successful evacuation is to keep the traffic moving at a stable rate. Thus, a minimum acceptable travel speed must be estimated at bottleneck points where several highway links merge together, since this may well be the earliest point reaching capacity. In addition, this speed will be used to decide the acceptable density in that link, and such points can then be used to derive the exit flow rate for a link. Hereafter, the traffic situation is classified into three categories according to the arrival rate and flow conditions: Under capacity with low density: The traffic is smooth and stable. According to Highway Capacity Manual (2000), this corresponds to a level of service of A or B, where the exit rate from the link is the same as the arrival rate. Congested with no detour: With an increase in the arrival rate, travel speed decreases and the links become increasingly congested; however, the travel time cost on the route is still less than the travel time if alternative links were used. Congested with detour: When some part of the network reaches its capacity and bottlenecks form, traffic controllers estimate a time to enact a partial detour (i.e., a portion of the arrival rate is forced to another route). This decision can also be made according to the real density on the road in which sensors are installed to collect the real-time data. As discussed in Section 3.1, when the arrival rate (denoted as Ar(t)) is higher than the departure rate (denoted as Ad(t)), a high arrival flow that enters into the segment can only be maintained for a certain period, and after some time, the entry rate will equilibrate to the same level as the exit rate. This is an ideal situation in which no surge or stop occur. In actual traffic conditions, the shockwave is a most natural result when the arrival rate is higher than the exit rate in a link without any control systems. To maximize the utilization of the road, traffic management personnel should allow the entering flow with high arrival rates to enter the link and then force part of the entering flow to detour, thus avoiding the main cause of traffic jams – the uncontrollable accumulation within queues. The link behaves as a buffer to absorb the first arrivals as much as possible, and if the real-time buffer quantity reaches a certain threshold which is the boundary condition, traffic managers must begin to deflect the remaining vehicles via detours in order to avoid congestion. This type of System Optimal is also based on the ideal assumption in Wardrop’s second theory. A well optimized detour trigger time can both avoid oversaturation and also minimize the average travel time across the system. The key aspect for optimizing the system is to determine this detour trigger time, tx. It is also the decision value in Eq. (9). Here, we define a time stage i for our discussion, and we assign the start time t0 as the time when the departure rate reaches its capacity. The initial number of conserved vehicles in the link is ci, which represents the same meaning as x = f(t0), and it takes tclear to let ci vehicles leave the link. From time t0 = 0 to the time when the density reaches a defined safety level, a detour has to be triggered. We denote this time as tstop. There are then two cases to explore based on whether all ci have left the link when the detour begins: tclear 6 tstop or tclear > tstop. C, which is expressed in (vehicleh), is denoted as the summation of travel time cost for all related vehicles. The followed algorithms decompose the two ‘‘C’’s into several parts and gives each of them a clear physical meaning. 4.1 Case 1: tclear 6 tstop From t0 to tstop, the departure rate is denoted as a constant Ad(t) = Ad. During that time, the total number of vehicles without ci that leave the buffer link is Qleft:

Q left ¼ Ad  ðt stop  t 0 Þ  ci

ð14Þ

According to Eq. (12), the final number of conserved vehicles, Qconserve, is the difference of the integrated result between arrived and departed vehicles plus those initially conserved. So,

Q conserv e ¼

Z

t stop

Ar ðtÞdt 

t0

Z

t stop

Ad ðtÞdt þ ci

ð15Þ

t0

Considering the FIFO principle, it will need tform to form the last conserved vehicles. This time represents the elapsed time from when the first vehicle entering the link in the final conserved group until tstop. Mathematically, the elapsed entering time can be expressed as:

tenter ¼ t stop  t form þ e

ð16Þ

where e is a very small amount of time. In addition, based on Eq. (12), the number of conserved vehicles on link i at any time t is:

Rt Rt x ¼ fi ðtÞ ¼ t0 Ar ðtÞdt  t0 Ad ðtÞdt þ ci    t  Ad ðt  t 0 Þ ¼ ArT FðtÞ t0    t a  Ad ðt  t 0 Þ þ ci ¼ ArT ð1  eðt =bÞ Þ t0

ð17Þ

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ArT is the total expected arrival vehicles in that link. When a vehicle enters the link, it will need tdc to leave the link. This is the time needed to evacuate the leading vehicles (those in front of the entering one).

tdc ¼

x Ad

ð18Þ

Based on the above analysis, the total travel cost related with the arriving vehicles from t0 to tstop can be represented by the following five components: (1) The cost to form the final conserved quantity Qconserve:

C1 ¼

Z

tstop

Ar ðtÞðtstop  tÞdt

ð19Þ

t stop t form

(2) The total travel time cost for the conserved vehicles (Qconserve) to leave the buffer link can be shown as:

C2 ¼

Z

tleav e

Ad ðtleav e  tÞdt

0

ð20Þ

where tleav e ¼ Q conserv e =Ad represents the required time that the last conserved vehicle leaves the buffer link. (3) The time cost for the Qconserve vehicles to complete their travel, given that the remaining travel time after these vehicles leave the buffer link is t2, will be:

C 3 ¼ Q conserv e t 2

ð21Þ

(4) The total travel time cost for the departed vehicles (Qleft) to leave the buffered link is:

C4 ¼

Z

tstop t form

ðAr ðtÞt dc Þdt

ð22Þ

t0

where tdc is obtained through Eqs. (17) and (18). (5) The time for all of the vehicles that were evacuated from the buffer link to finish their remaining trip is:

C 5 ¼ Ad ðt stop  t 0 Þt 2

ð23Þ

In this process, those initially conserved vehicles’ travel cost Cci should be removed from C5 since they actually stayed in the link before t0 and should not be considered as newly arrived vehicles’ travel time in the objective function.

C ci ¼ ci t2

ð24Þ

Then, the total travel time cost CCase1 can be formulated as: 5 X

C Case1 ¼

C i  C ci

ð25Þ

i¼1

4.2. Case 2: tclear P tstop In this case, the following components comprise the total travel time cost for new arrived vehicles in time stage i. Denoting the newly entered vehicles’ total entering travel cost as C6, we have:

C6 ¼

Z

tstop

Ar ðtÞðtstop  tÞdt

ð26Þ

t0

Similar to Eq. (22), the total travel time cost for the newly arrived vehicles to leave the buffered link is:

C7 ¼

Z

tstop

ðAr ðtÞt dc Þdt

ð27Þ

t0

Finally, these vehicles require the following travel time cost to complete their remaining trips, given that the remaining travel time after these vehicles leave the buffer link is t2:

C8 ¼

Z

t stop

 Ar ðtÞdt d2

ð28Þ

t0

Then, the total travel time can be represented by:

C Case2 ¼

8 X

Ci

ð29Þ

i¼6

Since in most cases all of the initial vehicles will leave the buffer at time tstop, Case 2 is not the common situation in an evacuation scenario. However, under certain extreme conditions, such as during an accident or traffic jam, Case 2 might occur.

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5. Case study: demand distribution in an evacuation of Charleston, South Carolina This section depicts an actual case study conducted for the I-26 Corridor, which is the primary evacuation route out of Charleston, South Carolina. According to the evacuation guidance issued by SCDOT, a major part of the evacuation will be carried out via Highway I-26 and its two branches called I-526, as shown in Fig. 4. All of these routes will lead the evacuees toward a final destination of Columbia (or any point beyond I-95, the interstate that runs parallel to the coastline but 60+ miles inland). The case study includes the estimation of the maximum acceptable traffic speed and flow rate, together with the calculation of CCase1 in a link of an evacuation network. 5.1. Traffic on the main I-26 corridor In Fig. 4, the number of vehicles shown in each entering point named ‘‘En-’’ is the total vehicles arrived in a 24-h window per lane. There are 3 lanes for each branch. The arrival rate curve in a 24-h period is shown in Fig. 1. By a rough estimation, 13% of the total evacuees will arrive at each link in the peak evacuation hour. Typical capacity for each lane of interstate highway is about 2, 200 vehicles/h. Given that there are three lanes in the direction of Columbia after the merging point, the total capacity is about 6, 600 vehicles/h. According to Fig. 4, the total demand from the three upstream flows is about 22, 740 per lane in the evacuation time frame. It can be estimated that the merging point will exceed the capacity at some time. If the upstream traffic keeps increasing, the traffic becomes unstable, necessitating the use of a control or management method to prevent congestion. In order to simplify the calculation, all of the vehicles entering from different points are assumed to experience the same travel time to the merging point as the vehicles entering from the furthest entrance because the link is relatively short. Based on actual road tests, the average travel times without congestion are shown in Table 1.

Fig. 4. Merging flows of traffic evacuation from Charleston, S.C.

Table 1 Travel times. Arrival link start point

Travel time (min)

Operation speed (miles/h)

En59 En38 En310

8 8 13

65 55 60

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This information can then be used to calculate the dynamic arrival rates at the merging point coming from the three flows – West (Flow 1), South (Flow 2), and East (Flow 3). According to Eq. (7) and the results shown in Section 3.2.1, we obtained the following expressions before a bottleneck level is reached: The arrival rate from the west is:

 4:45 8 Ar1 ðtÞ ¼ 3  ð3120Þð99309:8 Þðt  60 Þ



4:451

ðt 8 Þ4:45 60 99309:8



e ðt0:133Þ4:45

¼ 0:419  ðt  0:133Þ3:45  e  99309:8

The arrival rate from the south is:

 8 4:45  ðt Þ

 60 8 4:451 Ar2 ðtÞ ¼ 3  ð3120 þ 4800 þ t  60  e 99309:8 ðt0:133Þ4:45

¼ 1:645  ðt  0:133Þ3:45  e  99309:8 4:45 4320Þð99309:8 Þ

The arrival rate from the east is:



4:45 99309:8



Ar3 ðtÞ ¼ 3  ð3360 þ 1320 þ 2700Þ t ðt0:217Þ4:45

¼ 0:922  ðt  0:217Þ3:45  e  99309:8

13 4:451 60

 e



ðt13Þ4:45 60 99309:8



Fig. 5 shows the theoretical arrival rate at the merging point. It can be observed that at t = 10.2 h, the road reaches a flow rate of 6, 600 vehicles/h, or 2, 200 vehicles/h per lane. It is impossible for the exit flow rate to increase any more. More importantly, the time at which the entering ramp reaches its capacity must also be estimated. As we mentioned before, because the left and right branch will merge into I-26 in very close proximity to each other, they can be considered together as a single merging lane. By denoting Ad1 and Ad3 as the departure rate of the east and west branches, the expression can be shown as:

Ad1 ðtÞ þ Ad3 ðtÞ 6 2500

ð30Þ

It can be shown that the combined arrival rate reaches 2, 500 (vehicles/h) at t ¼ 9:65 h. Considering the complex situation in an evacuation route, the bottom line is to maintain a constant flow. A conservative minimum stable speed without surging and stopping, based on estimation through observation on general high-density traffic, is about 20–25 miles per hour. Beginning at t = 9.65, the two branches’ outflow rate is limited to the upper bound, where Ad1 becomes a constant value of 800 (vehicles/h) and Ad3 becomes 1, 700 (vehicles/h). The flows are allocated according to the approximate ration for each branch (Ad1 and Ad3), which are 800 vehicles/h and 1, 700 vehicles/h, respectively. Meanwhile, the flow of Ad2 is 3, 072 (vehicles/h). Though the arrival rates are still under the capacity of 2, 200 vehicles/h per lane (given three lanes), the merging point has reached the upper limit. With traffic continuing to increase, the east and west branches will now act as buffers until the density finally reaches the upper bound level for traveling at about 20–25 miles per hour.

Fig. 5. Arrival rate after merging.

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5.1.1. Latest time for inflow control The maximum number of vehicles conserved in the left branch can be estimated using the safe following distance value and the total length of the road, L1. Given a road length of L1 = 7.0 miles and assuming the average car length is 16 feet, the maximum occupancy, Dsafe, becomes approximately 55 feet/car (Papacostas and Prevendouros, 2008). Here, we have a conservative maximum number of L1 =Dsafe ¼ 7:0  5280=55  669 vehicles. This is a simple estimation since the ending part can merge together with a density of about 55 feet/car, and it is simply extended throughout an entire link by a single lane. Actually, the total number can have a relatively higher value considering there are three lanes in the west branch. Since it took 8 min to travel from the entrance to the exit, the entering cars before t = (9.65–8/60) = 9.517 h have already R 9:65 left the link at t = 9.65 h. Thus, the real conserved vehicle number at t0 is: c1 ¼ f ð9:65Þ ¼ 9:517 Ar1 dt ¼ 107ðvehicleÞ. The latest time a detour should be initiated is also the time when the conserved value reaches 669, which is the maximum buffer size or the saturation level. This time is denoted as tlatest. Considering the definition of tstop, we have: tstop 6 tlatest. tlatest is calculated as follows:

Z

t latest

Ar1 dt  800ðtlatest  9:65Þ þ 107 ¼ 669

9:65

) t latest  12:1 At t = 12.1 h, even though the ingress is still permitted to arrival vehicles, the whole system experiences a progressively higher risk of instability—a detour must take place. To simplify the expression, the new Dt = tlatest = 12.1–9.65 = 2.45 h is set. Case 1 can then be used to calculate the total travel time of the vehicles from t0 = 9.65 to tlatest = 12.1. After calculating, tform = 0.58 h, the travel time cost to form the final conserved quantity Qconserve is:

C1 ¼

Z

t stop

Ar ðtstop  tÞdt ¼ 151 ðvehicle  hÞ

t stop t form

According to Eq. (20), since the conserved vehicles require t drc ¼ 669 ¼ 0:836 h to be emptied, the associated travel time cost 800 is:

C2 ¼

Z

0:836

Ad ð0:836  tÞdt ¼ 280 ðvehicle  hÞ

0

Here, the remaining travel time after the buffer link is t2. Given that there is no congestion and by letting t2 = 1.25 (h), the travel time cost for C3 is:

C 3 ¼ 669  1:25 ¼ 836 ðvehicle  hÞ According to Eqs. (22) and (18), the buffer link travel cost for Qleft is obtained as follows. According to Eq. (6), at t0 = 9.65 h, the total number of vehicles that arrived and left the buffer link is 2, 013. Based on Eq. (18), after t = 9.65 h,

Rt tdc

¼

x Ad

¼

0

Rt

Ar ðtÞdt2013

t0

Ad dtþci

800

¼

ArT FðtÞ2013dðtt 0 Þþci 800

¼ 9360ð1e

ðt4:45 =99309:8Þ Þ2013A

d ðtt 0 Þþc i

800

C4 is then calculated as follows:

C4 ¼

Z

t stop t form

ðAr  t dc ÞdtL

9:6

¼

Z

11:52

9:65

  4:45 ðt0:133Þ4:45

9360  ð1  eðt =99308:8Þ Þ  2013  800  ðt  9:65Þ þ 107 0:419  ðt  0:133Þ3:45  e  99309:8  dt 800

¼ 554:61 ðvehicle  hÞ The time for the vehicles in item 4 to finish the remaining trip is:

C 5 ¼ Ad t stop t2 ¼ 800  ð12:1  9:65Þ  1:25 C 5 ¼ 2450 ðvehicle  hÞ According to Eq. (25), a total of about 134 (vehicleh) can be removed. Therefore, the total number of arrived vehicles takes Ccase1 to finish this trip.

C Case1 ¼ 151 þ 280 þ 836 þ 555 þ 2450  134 ¼ 4138 ðvehicle  hÞ In conclusion, after calculating from t = 9.65 h to t = 12.1 h, a total of 2, 523 vehicles enter the system. If these vehicles take the left branch as a buffer link and travel through it, they will spend a total of around 4, 138 vehicle-h to finish the trip.

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Fig. 6. Total travel cost with different trigger time.

8

If vehicles travel in uncongested traffic conditions, in total, all of the vehicles will require only C total ¼ 60 þ t 2  2523 ¼ 3490 ðvehicle  hÞ to complete the evacuation. Considering a detour route that requires approximately 3.5 h to complete at a free flow speed, if all vehicles take the detour at time 9.65, they will use C det our1 ¼ 3:5  2523 ¼ 8830 vehicle-h to complete their trips. In this case, the use of the merge lane as a buffer provides a travel time savings over the detour. Another option is to use the buffer as an evacuation route and keep the flow-in rate at the capacity while remaining traffic can be detoured to the by-pass route. This result is discussed below. To reach to an optimal level, we don’t have to use up all of the buffer capacity until the link reaches its saturation level, that is tstop = tlatest. At this time, we must set a time point tx, or tstop, to start the detour in order to minimize the total travel time. tx should be less than tlatest. Fig. 6 is an example of how the total travel time changes with different detour trigger times from t = 9.65 to t = 12.1. As shown in Fig. 6, from t = 9.65 to t = 12.1, if the detour travel time is not very long (about 1.7 h), the best time to trigger a detour is t  10.95. If we choose an incorrect detour trigger time, we might waste several hundred hours in total time. Thus, we can use the SO model described in Sections 3.4 and 3.5 to find the optimized trigger time tx to achieve the maximum amount of time saved. For example, we can divide the time stage as: (1) from t = 9.65 to t = tx; (2) from t = tx to t = trecovery, where trecovery is the time the arrival rate drops back to less than 800 vehicles/h. If the network is very complex, we can only take into consideration a partial network, or we can divide the big system into small, local problems in order to obtain a good solution to our congestion problem. The model can also combine all of the links in the system together to reach system optimization. However, considering that congestion is occurring at different times, a local solution is more straightforward and practical. At the same time, a global optimization solution for each event time in the network is still available under ideal assumptions. 5.1.2. Case 2 study If the originally embedded vehicles move very slowly and still stay within the link when the link has reached its saturation level, the Case 2 algorithms can be applied. Case 2 can be regarded as an extreme situation of Case 1 when the departure rate is very low. In some cases, a sudden increase of arrival rate may also cause the sharp congestion in a link. Under this circumstance, any new arrived vehicles would experience longer waiting times than tclear. Therefore, if the cost of the detour route is less than tclear, a detour should be triggered immediately. However, if the detour road takes a very long travel time, it is still worth the calculation to find a best trigger time that can utilize the congested link as a buffer. Technically speaking, Case 1 and Case 2 have the same approach to reach the optimal solution. 5.1.3. Discussion about convexity It is possible that the total travel time has a minimum value concerning different detour trigger times. Typically, the detour travel time is longer than the free flow travel time on the main route. For example, here it is estimated that the main route travel time is 8 ðminÞ þ 1:25 h  1:38 ðhÞ. At the beginning, when the exit point reaches the bottleneck, vehicles may still travel faster on the main route than the detour route. So, if we don’t trigger the detour at that time and delay it till a later time, the total travel cost can be saved compared to the case when detour is triggered at the beginning time. Therefore, in Fig. 6, a later trigger time provides less total travel cost in the beginning, but with the increase of traffic on the main route, travel time increases and the main route may become slower than the detour route. Under that circumstance, if we still delay the trigger time, the vehicles within the main route (buffer link) will experience a long queue, and the total travel cost will increase. Therefore, with the shifting of the triggering from the time the bottleneck is formed until the moment the buffer link is saturated, the total travel cost first decreases and then increases. Consequently, the shape of total cost is a convex line as shown in Fig. 6. An optimized trigger time will give the minimum travel time cost of the system. The cost of the whole network will be affected by an individual link’s trigger time; consequently, a globalized value for each link exists. The globalized individual trigger times are also the system-wide optimization values.

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In addition, the average travel time within the buffer link is based on a defined level of saturation. A higher density that is permitted in the buffer link might cause a longer average travel time. Therefore, if the condition of the detour road is not ideal and a long delay might occur in the detour route, it is better to set a higher level of density threshold in the buffer link. 5.2. Summary and discussions 5.2.1. Buffer concept is useful for congestion analysis The buffer concept was found to be a useful tool for traffic flow management during an evacuation to minimize the total travel time as well as to lower the risk of congestion. Previous traffic assignment efforts have yielded a variety of models and algorithms for use in determining the flow rate on each link. However, these models and algorithms are difficult to implement, particularly in a large scale evacuation process with complex road networks and high traffic volumes. 5.2.2. Arrival rate and departure rate functions In this research, the arrival rate is defined as an exponential function. However, the buffer calculation is not sensitive to the definition of the arrival rate. The physical concept within the buffer model is that once the arrival rate is higher than the departure capacity, the link starts to absorb the additional vehicles and acts like a buffer. In contrast, the departure rate definition is based on some experienced estimation or calculation. In Daganzo (1997), it is approximated as a bottleneck value. In the buffer model, the arrival rates and departure rates are used as the input for the integration method. Therefore, the only requirement is that the curve of arrival and departure should be continuous (however, not necessarily be smooth since there might be some turning points at the event time) and can be integrated at each time stage. As a result, the researchers can use very flexible equations to simulate arrival and departure behaviors and hence increase the accuracy of the calculation. 5.3. Simulation analysis VISSIM, which is a widely accepted and validated microscopic traffic simulator, was used in modeling the west branch of our study network. A 7-mile link with three lanes was modeled, and the arrival rate and speed distribution are set according to the conditions of the west branch. It is then merged at the exit area with a capacity of 800 vehicles/h after t = 9.65. The travel cost for vehicles passing through the buffer link from t = 9.65 to t = 12.1 is recorded. The simulated costs in Case 1 are C4 and C2 in this model, where C4 is the travel cost for those vehicles that left the buffer prior to t = 12.1 and C2 is the travel cost for the conserved vehicles at t = 12.1 to leave the buffer link. C2 and C4 are directly related with the buffering behavior and demonstrate the key part of queuing time calculation. The remaining travel costs are not directly related with the buffer model and therefore are not recorded for analysis. All other parameters are set to the default values in VISSIM. We performed multiple simulation tests with different replication numbers from 20 to 45. The output shows very consistent mean values for the two cost measures, and the standard deviations of these two measures were very low. To ensure that the mean values were representative of the true mean values, we calculated the half-width of the confidence interval. Assuming our simulation can provide a result within ±half-width, this would provide the necessary support that the simulation adequately represents the system. The following equation shows the half-width calculation with 95% confidence and 40 replications:

S h ¼ t 39;0:975 pffiffiffi n where h is the half width with 95% confidence interval; s the sample standard deviation; n is the number of replications. We targeted a half-width value within 1% of the estimated mean value, which is less than 6 vehicle-h for C4 and less than 3 for C2. This target value was reached after 40 replications. The new obtained mean and standard deviation are:

C 2 ¼ 387 ðvehicle  hÞ; SC2 ¼ 5:7ðv ehicel  hourÞ C 4 ¼ 511 ðvehicle  hÞ; SC4 ¼ 4:5 ðvehicle  hÞ Therefore, we have:

ffiffiffiffi  1:9 ðvehicle  hÞ hC2 ¼ 2:02  p5:7 39 ffiffiffiffi  1:5 ðvehicle  hÞ hC4 ¼ 2:02  p4:5 39 The mean value of C2 with 95% confidence is 387 ± 1.8 (vehicleh), and the mean value of C4 is 511 ± 1.5 (vehicleh). For C4, the simulated value and calculated value are very close (less than 10% apart). For C2, however, the conserved vehicles take more time to leave the buffer link in the simulation model. The reason is due to the violation of FIFO in general driving behavior, especially in roads with multiple lanes. It was observed in the simulation that some drivers wait too long on the merging part, which causes additional delays. This situation is more obvious in a highly congested situation. Therefore, additional delays occur in a highly congested condition when a long queue has been formed.

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6. Conclusions This paper presents a simple but effective method to analyze, distribute and control evacuation traffic by utilizing a traffic demand buffering strategy. The contributions of this paper is the development of a system that provides (i) an event-driven framework to achieve an optimal travel demand distribution, and (ii) a new travel time cost algorithm with an integration method and a buffer concept to help solve the optimization problem. The earlier models for optimal travel demand distributions are based on optimization principles. This model, however, is based on the buffer concept, which provides a practical solution to optimizing the evacuation traffic flow distribution. The earlier model met some difficulties in defining the time stages, especially in a congested traffic network. This optimization framework is an event-driven model, and the time stages are defined and optimized by the time of events in an evacuation process. The buffer concept and the integration algorithm presented in this paper provide a clear traffic behavior explanation in a congested link and also provide an efficient solution for travel cost calculation. Therefore, this framework is not only a novel traffic demand distribution concept, but also a solvable and feasible solution for real-world application. This algorithm for cost calculation can also be used in other congested traffic situations and bottleneck-related traffic flow analyses. The microscopic traffic behavior in congested roadways that often leads to complicated traffic flow analysis can be simplified by only considering the performance in the bottlenecks. In addition, the total travel time in the system is converted into a simple integration calculation based upon the flow rate at the entrance and exit points of the evacuation routes. FIFO holds because the travel time is obtained through an integration method to calculate the total processing time of the vehicles that are ahead of the vehicle that just entered into the buffer link. Therefore, the vehicles’ travel time is calculated one after the next based on each one’s entering sequence. As an example, from the point of a traffic incident, the queue accumulates upstream and the outflow and inflow rate can be used to assess the entire link’s performance. The location and time of a detour can also be derived from the optimization algorithm presented in this paper. The best application of this demand distribution methodology is to consider pre-calculated traffic demand estimations (resulting from an emergency event) as a quick reference for evacuation management. This algorithm can be programmed by a mathematical algorithm to conduct a quick estimation with real-world data. In addition, in an evacuation sketch analysis, a skilled evacuation management team must determine the pivotal location in each link as the bottleneck in which maximum density is determined by considering the risk of unstable traffic. By applying different levels of risk criteria, the outflow of the bottleneck may be defined at various levels. The traffic diversion trigger time, tstop, may also differ. For example, if the arrival rate increases at a relatively low speed within a short duration, we may apply a relatively high density to determine the buffer size since the risk of traffic congestion is not as high as a quickly increasing arrival rate. Consequently, we may calculate the number of conserved vehicles from the corresponding traffic density. Once the conserved vehicle number is obtained, the optimization function can be solved easily. With the increased application of real-time traffic monitoring technologies through Intelligent Transportation Systems (ITS), traffic management professionals may apply the proposed method directly using on-line traffic data and pre-calculated reference values to make a system-wide estimation of detour trigger time. In a real-time traffic monitoring scenario when the network density and exit flow rate data are collected and the upstream demand can be derived from the obtained data, the detour trigger time can be obtained automatically from the system optimization framework presented in this paper.

References Afshar, A. M., Haghani, A., 2008. A heuristic framework for optimization hurricane evacuation operations. TRB Annual Meeting. Washington, DC. Branston, D., 1976. Link capacity functions: a review. Transportation Research 10, 223–236. Brown, C., White, W., Slyke, C.V., Benson, J.D., 2009. Development of a strategic hurricane evacuation dynamic traffic assignment model for the Houston region. TRB Annual Meeting. Washington, DC. Carey, M., 1986. A constraint qualification for a dynamic traffic assignment model. Transportation Science 20, 55–58. Chiu, Y.C., Mirchandani, P.B., 2008. Online behavior-robust feedback information routing strategy for mass evacuation. IEEE Transactions on Intelligent Transportation Systems 9, 264–274. Chiu, Y.C., Zheng, H., 2007. Real-time mobilization decisions for multi-priority emergency response resources and evacuation groups: model formulation and solution. Transportation Research Part E 43, 710–736. Chow, A.H.F., 2007. Trip Assignment – A Literature Review. California PATH, UC Berkeley. (accessed on 20.11.09). Daganzo, C.F., 1997. Fundamentals of Transportation and Traffic Operations. Pergamon–Elsevier, Oxford, UK. Duanmu, J., Chowdhury, M., Taaffe, K., 2011. A simulation framework for community-wide evacuation planning. Journal of Transportation Security 4, 1–18. FEMA, 1986–2006. Public Response. Federal Emergency Management Agency (FEMA). (accessed 25.03.11). Friesz, T.L., Bernstein, D., Mehta, N.J., Tobin, R.L., Ganjalizadeh, S., 1989. Dynamic network traffic assignment considered as a continuous time optimal control problem. Operations Research 37, 893–901. Gangi, M.D., 2009. Planning Evacuation by Means of a Multi-modal mesoscopic Dynamic Traffic Simulation Model. Geocomputation and Urban Planning, SCI176. Springer, Berlin/Heidelberg. Han, L.D., Yuan, F., Chin, S., Hwang, H., 2006. Global optimization of emergency evacuation assignments. Interfaces 36, 502–513. HCM, 2000. Highway Capacity Manual, US Customary Version, Transportation Research Board. He, S., Zhang, L., Song, R., Wen, Y., Wu, D., 2009. Optimal transit routing problem for emergency evacuations. TRB Annual Meeting. Washington, DC. Janson, B.N., 1991. Dynamic traffic assignment for urban road network. Transportation Research 25B, 143–161. May, A.D., 1990. Traffic Flow Fundamentals. Englewood Cliffs, Prentice-Hall, New Jersey. Ng, M., Waller, S.T., 2010. Reliable evacuation planning via demand inflation and supply deflation. Transportation Research Part E 46, 1086–1094. Papacostas, C.C., Prevendouros, P.D., 2008. Transportation Engineering & Planning. Pearson Education.

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Peeta, S., 1994. System Optimal Dynamic Traffic Assignment in Congested Networks with Advanced Information Systems. Doctoral dissertation. The University of Texas at Austin. Ran, B., Shimazaki, T., 1989a. Dynamic user equilibrium traffic assignment for congested transportation network. In: The Fifth World Conference on Transport Research. Ran, B., Shimazaki, T., 1989b. A general model and algorithm for the dynamic traffic assignment problems. In: The Fifth World Conference on Transport Research. Yokohoma, Japan. RDS, 1999. Executive Summary of a Socioeconomic Hurricane Impact Analysis and a Hurricane Evacuation Impact Assessment Tool (Methodology) for Coastal North Carolina: A Case Study of Hurricane Bonnie. Greenville, NC, Regional Development Service (RDS), Department of Sociology, and Department of Economics, East Carolina University. Robinson, R.M., Khattak, A.J., Sokolowski, J.A., Foytik, P., Wang, X., 2009. Role of Traffic Incidents in Hampton Roads Hurricane Evacuations. TRB Annual Meeting. Washington, DC. SCDOT, 2011. Evacuation Route – Detailed Written Directions. South Carolina Department of Transportation (SCDOT). (accessed 20.03.11). Song, R., He, S., Zhang, L., 2009. Optimum transit operations during the emergency evacuations. Journal of Transportation Systems Engineering and Information Technology 9, 154–160. Tweedie, S.W., Rowland, J.R., Walsh, S., Rhoten, R.R., Hagle, P.L., 1986. A methodology for estimating emergency evacuation times. The Social Science Journal 23, 189–204. Wardrop, J., 1952. Some theoretical aspects of road traffic research. In: Proceedings of Institute of Civil Engineers, Part II, vol. 1, pp. 352–362. Wolshon, B., Urbina, E., Levitan, M., 2003. National review of hurricane evacuation plans and policies. In: Institute of Transportation Engineers (ITE) Technical Conference and Exhibit. Institute of Transportation Engineers Yi, W., Ozdamar, L., 2007. A dynamic logistics coordination model for evacuation and support in disaster response activities. European Journal of Operational Research 179, 1177–1193.