Please Move quickly 32 Please Move Quickly and quietl to the Nearest Freeway Corey r. houmand Andrew d. pruett Adam S Dickey Wake Forest University Winston-Salem, NC Advisor: Miaohua Jiang Introduction We construct a model that expresses total evacuation time for a given route as a function of car density route length and number of cars on the route. When supplied values for the last two variables, this function can be minimized to give an optimal car density at which to evacuate We use this model to compare strategies and find that any evacuation plan lust first and foremost develop a method of staggering traffic flow to create a constant and moderate car density. This greatly decreases the evacuation time as well as congestion problems If an even speedier evacuation is necessary, we found that making 1-26 one. way would be effective Making other routes one-way or placing limits on the type or number of cars prove to be unnecessary or ineffective. We also conclude that other traffic on I-95 would have a negligible impact on the evacuation time, and that shelters built in Columbia would improve evacuation time only if backups were forming on the highways leading away from the cit Prologue As Locke asserted [1690, power is bestowed upon the government by the will of the people, namely to protect their property. A government that cannot provide this, such as South Carolina during an act of God as threatening as Hurricane Floyd and his super-friends, is in serious danger of revolutionary The UMAP Journal 22(3)(2001)323-336. @Copyright 2001 by COMAP, Inc. All rights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
Please Move Quickly 323 Please Move Quickly and Quietly to the Nearest Freeway Corey R. Houmand Andrew D. Pruett Adam S. Dickey Wake Forest University Winston-Salem, NC Advisor: Miaohua Jiang Introduction We construct a model that expresses total evacuation time for a given route as a function of car density, route length, and number of cars on the route. When supplied values for the last two variables, this function can be minimized to give an optimal car density at which to evacuate. We use this model to compare strategies and find that any evacuation plan must first and foremost develop a method of staggering traffic flow to create a constant and moderate car density. This greatly decreases the evacuation time as well as congestion problems. If an even speedier evacuation is necessary, we found that making I-26 oneway would be effective. Making other routes one-way or placing limits on the type or number of cars prove to be unnecessary or ineffective. We also conclude that other traffic on I-95 would have a negligible impact on the evacuation time, and that shelters built in Columbia would improve evacuation time only if backups were forming on the highways leading away from the city. Prologue As Locke asserted [1690], power is bestowed upon the government by the will of the people, namely to protect their property. A government that cannot provide this, such as South Carolina during an act of God as threatening as Hurricane Floyd and his super-friends, is in serious danger of revolutionary The UMAP Journal 22 (3) (2001) 323ñ336. c Copyright 2001 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
324 The umaP Journal 22.3(2001) overthrow by the stranded masses marching from the highways to the capitol Therefore, South Carolina must find the most effective evacuation program- one that not only provides for the safety of its citizens but also allows for households to rescue as many of their vehicles (read: property) as possible Pitted against the wrath of God and Nature, one can only hope the power of mathematical modeling can protect the stability of South Carolinian bureau- Since the goal is to create a useful model that even an elected official can use our model operates most effectively on the idea that government agencies are poor at higher-level math but good at number-crunching. Our model provides a clear, concise formula for weighing relative total evacuation times and likely individual trip times. This is crucial in deciding how to order a wide-area evacuation while maintaining public approval of the operation and preventing a coup d'etat. Our model shows that of the four strategies for evacuation suggested by th problem statement, staggering evacuation orders is always the most effective choice rather than simply reversing I-26. After that, applying any one of the other options, like lane reversal on I-26 and /or on secondary evacuation routes, can improve the evacuation plan. However, using more than one of these techniques results in a predicted average driver speed in excess of the state speed limit of 70 mph Of the three additional methods we find that the most effective is to make I-26 one-way during peak evacuation times. Implementing the same plan on secondary highways would require excessive manpower from law enforcement officials, and regulating the passenger capacity is too difficult a venture in a critical situation We also find that, given the simplifications of the model, I-95 should have a negligible effect Furthermore, the construction of shelters in Columbia would facilitate the evacuation only if the highways leading away from Columbia were causing backups in the cit Analysis of the problem To explain the massive slowdowns on 1-26 during the 1999 evacuation, our team theorized that substantially high vehicle density causes the average speed of traffic to decrease drastically. Our principal goal is to minimize evacuation times for the entire area by maximizing highway throughput, thatis, the highest speed at which the highest density of traffic can travel. Given this fact, we seek to find the relationships between speed, car density, and total evacuation time
324 The UMAP Journal 22.3 (2001) overthrow by the stranded masses marching from the highways to the capitol. Therefore, South Carolina must find the most effective evacuation programó one that not only provides for the safety of its citizens but also allows for households to rescue as many of their vehicles (read: property) as possible. Pitted against the wrath of God and Nature, one can only hope the power of mathematical modeling can protect the stability of South Carolinian bureaucracy. Since the goal is to create a useful model that even an elected official can use, our model operates most effectively on the idea that government agencies are poor at higher-level math but good at number-crunching. Our model provides a clear, concise formula for weighing relative total evacuation times and likely individual trip times. This is crucial in deciding how to order a wide-area evacuation while maintaining public approval of the operation and preventing a coup dí¥etat. Our model shows that of the four strategies for evacuation suggested by the problem statement, staggering evacuation orders is always the most effective choice rather than simply reversing I-26. After that, applying any one of the other options, like lane reversal on I-26 and/or on secondary evacuation routes, can improve the evacuation plan. However, using more than one of these techniques results in a predicted average driver speed in excess of the state speed limit of 70 mph. Of the three additional methods, we find that the most effective is to make I-26 one-way during peak evacuation times. Implementing the same plan on secondary highways would require excessive manpower from law enforcement officials, and regulating the passenger capacity is too difficult a venture in a critical situation. We also find that, given the simplifications of the model, I-95 should have a negligible effect. Furthermore, the construction of shelters in Columbia would facilitate the evacuation only if the highways leading away from Columbia were causing backups in the city. Analysis of the Problem To explain the massive slowdowns on I-26 during the 1999 evacuation, our team theorized that substantially high vehicle density causes the average speed of traffic to decrease drastically. Our principal goal is to minimize evacuation times for the entire area by maximizing highway throughput, that is, the highest speed at which the highest density of traffic can travel. Given this fact, we seek to find the relationships between speed, car density, and total evacuation time
Please Move quickly 325 Assumptions To restrict our model, we assume that all evacuation travel uses designated evacuation highways We assume that traffic patterns are smoothly and evenly distributed and that drivers drive as safely as possible. There are no accidents or erratically driving"weavers"in our scenario. This is perhaps our weakest assumption, since this is clearly not the case in reality, but it is one that we felt was necessary to keep our model simple Our model also requires that when unhindered by obstacles, drivers travel at the maximum legal speed. Many drivers exceed the speed limit; however, we do not have the information to model accurately the effects of unsafe driving speeds, and a plan designed for the government should avoid encouraging o. As suggested by the problem, we simplify the actual distribution of popu- lation across the region placing 500,000 in Charleston, 200,000 in Myrtle Beach and an even distribution of the remaining approximately 250,000 people Multiple-lane highways and highway interchanges are likely to be more complicated than our approximation, but we simplify these aspects so that our model will be clear and simple enough to be implemented by the government Distribution of traffic among the interstate and secondary highways in our model behaves according to the results of a survey, which indicates that 20% of evacuees chose to use 1-26 for some part of their trip The model We We begin by modeling the traffic of 1-26 from Charleston to Columbia, as believe that understanding 1-26 is the key to solving the traffic problems We derived two key formulas, the first s(p) describing speed as a function of car density and the second e(p) describing the total evacuation time as a function of car density 5280 Lp+N The constants are k= braking constant, l= average length of cars in feet, and b= buffer zone in feet The variables are p=car density in cars/mi
Please Move Quickly 325 Assumptions To restrict our model, we assume that all evacuation travel uses designated evacuation highways. We assume that traffic patterns are smoothly and evenly distributed and that drivers drive as safely as possible. There are no accidents or erratically driving ìweaversî in our scenario. This is perhaps our weakest assumption, since this is clearly not the case in reality, but it is one that we felt was necessary to keep our model simple. Our model also requires that when unhindered by obstacles, drivers travel at the maximum legal speed. Many drivers exceed the speed limit; however, we do not have the information to model accurately the effects of unsafe driving speeds, and a plan designed for the government should avoid encouraging speeding. As suggested by the problem, we simplify the actual distribution of population across the region placing 500,000 in Charleston, 200,000 in Myrtle Beach, and an even distribution of the remaining approximately 250,000 people. Multiple-lane highways and highway interchanges are likely to be more complicated than our approximation, but we simplify these aspects so that our model will be clear and simple enough to be implemented by the government. Distribution of traffic among the interstate and secondary highways in our model behaves according to the results of a survey, which indicates that 20% of evacuees chose to use I-26 for some part of their trip. The Model We begin by modeling the traffic of I-26 from Charleston to Columbia, as we believe that understanding I-26 is the key to solving the traffic problems. We derived two key formulas, the first s(ρ) describing speed as a function of car density and the second e(ρ) describing the total evacuation time as a function of car density: s(ρ) = 1 k 5280 ρ − l − b , e(ρ) = Lρ + N ρs(ρ) . The constants are: k = braking constant, l = average length of cars in feet, and b = buffer zone in feet. The variables are: ρ = car density in cars/mi
326 The UMAP Journal 22.3 (2001) L =length of highway in feet, and n= number of cars to be evacuated The method is to maximize e(p) for a given N and L, which gives us an optimal car density Derivation of the model The massive number of variables associated with modeling traffic on a micro such factors as a d. v complex and difficult problem. Ideally, one could consider basis leads te mood, the condition of the mode of transportation, whether his or her favorite Beach Boys song was currently playing on the radio, etc. Then one could use a supercomputer to model the behavior of several hundred thousand individua interacting on one of our nation,'s vast interstate highways. Instead, our model analyzes traffic on a macro basis The greater the concentration of cars, the slower the speed at which the individuals can safely drive. What dictates the concentration of cars? Well, the concentration is clearly related to the distance between cars, since the greater the distance, the smaller the concentration, and vice versa. On any interstate highway, drivers allot a certain safe traveling distance between their car and the car directly in front of them, to allow time to react. Higher speeds require the same reaction time but consequently a greater safe traveling distance. How do we determine what the correct distance between cars at a given speed? The braking distance d of a car is proportional to the square of that cars speed v That is, d=ky2 for some constant k. The value for k is 0.0136049 ft-hr2mi2 we derive this value by fitting Ind=In k+bIn u with data from Dean et al. [2001] the fit has r2=.99999996 However the distance between cars is an awkward measurement to use Our goal is to model traffic flow. With our model, we manipulate the traffic flow until we find its optimal value. The distance between cars is hard to control but other values, such as the concentration or density of the cars in a given space, are much easier to control How do we find the value of the car density? To start, any distance can be subdivided into the space occupied by cars and the space between cars. The space occupied by cars can be assumed to be a multiple of the average car length The space between cars is clearly related to the braking distance, but the two are not necessarily the same. The braking distance at low speeds(< 10 mph)is less than a foot. However, ordinary experience reveals that even at standstill traffic, the distance between cars is still much greater than a foot; drivers still leave a buffer zone in addition to the safe breaking distance. Then each car has a space associated with it, given by d+l+b, where b is the average buffer zone in feet. Since this expression is of the form of 1 car per unit distance, this is in itself a density. We can also convert this to more useful units, such as cars per
326 The UMAP Journal 22.3 (2001) L = length of highway in feet, and N = number of cars to be evacuated. The method is to maximize e(ρ) for a given N and L, which gives us an optimal car density. Derivation of the Model The massive number of variables associated with modeling traffic on a micro basis leads to a very complex and difficult problem. Ideally, one could consider such factors as a driverís experience, his or her psychological profile and current mood, the condition of the mode of transportation, whether his or her favorite Beach Boys song was currently playing on the radio, etc. Then one could use a supercomputer to model the behavior of several hundred thousand individuals interacting on one of our nationís vast interstate highways. Instead, our model analyzes traffic on a macro basis. The greater the concentration of cars, the slower the speed at which the individuals can safely drive. What dictates the concentration of cars? Well, the concentration is clearly related to the distance between cars, since the greater the distance, the smaller the concentration, and vice versa. On any interstate highway, drivers allot a certain safe traveling distance between their car and the car directly in front of them, to allow time to react. Higher speeds require the same reaction time but consequently a greater safe traveling distance. How do we determine what the correct distance between cars at a given speed? The braking distance d of a car is proportional to the square of that carís speed v. That is, d = kv2 for some constant k. The value for k is 0.0136049 ft-hr2mi2; we derive this value by fitting ln d = ln k + b ln v with data from Dean et al. [2001]; the fit has r2 = .99999996. However, the distance between cars is an awkward measurement to use. Our goal is to model traffic flow. With our model, we manipulate the traffic flow until we find its optimal value. The distance between cars is hard to control, but other values, such as the concentration or density of the cars in a given space, are much easier to control. How do we find the value of the car density? To start, any distance can be subdivided into the space occupied by cars and the space between cars. The space occupied by cars can be assumed to be a multiple of the average car length l. The space between cars is clearly related to the braking distance, but the two are not necessarily the same. The braking distance at low speeds (< 10 mph) is less than a foot. However, ordinary experience reveals that even at standstill traffic, the distance between cars is still much greater than a foot; drivers still leave a buffer zone in addition to the safe breaking distance. Then each car has a space associated with it, given by d + l + b, where b is the average buffer zone in feet. Since this expression is of the form of 1 car per unit distance, this is in itself a density. We can also convert this to more useful units, such as cars per
Please Move quickly 327 p= cars=cars x 5280 ft=5280 x cars=5280 cars mi d+b+ Solving for v gives 5280 At this point we can substitute k=0.0136049 ft. h2/mi2,I=17 ft(from researching sizes of cars), and b= 10 ft(from our personal experience)and graph speed as a function of density(Figure 1) 70 00 406080100120140160180200 Figure 1. Speed as a function of density Note the maximum density at 195.6 cars/mi. To understand why a maxi- mum density exists, consider the case b=0; as v-0, the distance between the cars approaches zero. We now determine how long it takes this group of cars to reach their desti- nation. For now, we say that the group reaches its goal whenever the first car arrives. This is a simple calculation: We divide the length of the road L by the average speed of the group t(p)= 5280
Please Move Quickly 327 mile: ρ = cars mi = cars ft × 5280 ft mi = 5280 d + b + l × cars mi = 5280 kv2 + b + l cars mi . Solving for v gives s(ρ) = v = 1 k 5280 ρ − l − b . At this point we can substitute k = 0.0136049 ft · h2/mi2, l = 17 ft (from researching sizes of cars), and b = 10 ft (from our personal experience) and graph speed as a function of density (Figure 1). Figure 1. Speed as a function of density. Note the maximum density at 195.6 cars/mi. To understand why a maximum density exists, consider the case b = 0; as v → 0, the distance between the cars approaches zero. We now determine how long it takes this group of cars to reach their destination. For now, we say that the group reaches its goal whenever the first car arrives. This is a simple calculation: We divide the length of the road L by the average speed of the group t(ρ) = L s(ρ) = L 1 k 5280 ρ − l − b