Traffic Flow Models 271 Traffic flow models and the evacuation problem Samuel W. malone Carl A. miller Daniel b. neill Duke University Durham, NC Advisor: David p Kraines Introduction We consider several models for traffic flow. A steady-state model employs model for car-following distance to derive the traffic-flow rate in terms of empirically estimated driving parameters. We go on to derive a formula for total evacuation time as a function of the number of cars to be evacuated The steady-state model does not take into account variance in speeds of vehicles. To address this problem, we develop a cellular automata model for traffic flow in one and two lanes and augment its results through simulation After presenting the steady-state model and the cellular automata models, we derive a space-speed curve that synthesizes results from both Ne address restricting vehicle types by anal rehicle speed variance To assess traffic merging, we investigate how congestion occurs We bring the collective theory of our assorted models to bear on five evac uation strategies Assumptions e Driver reaction time is approximately 1 sec Drivers tend to maintain a safe distance; tailgating is unusual All cars are approximately 10 ft long and 5 ft wide Almost all cars on the road are headed to the same destination The UIMAP Journal 22 (3)(2001)271-290. @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
Traffic Flow Models 271 Traffic Flow Models and the Evacuation Problem Samuel W. Malone Carl A. Miller Daniel B. Neill Duke University Durham, NC Advisor: David P. Kraines Introduction We consider several models for traffic flow. A steady-state model employs a model for car-following distance to derive the traffic-flow rate in terms of empirically estimated driving parameters. We go on to derive a formula for total evacuation time as a function of the number of cars to be evacuated. The steady-state model does not take into account variance in speeds of vehicles. To address this problem, we develop a cellular automata model for traffic flow in one and two lanes and augment its results through simulation. After presenting the steady-state model and the cellular automata models, we derive a space-speed curve that synthesizes results from both. We address restricting vehicle types by analyzing vehicle speed variance. To assess traffic merging, we investigate how congestion occurs. We bring the collective theory of our assorted models to bear on five evacuation strategies. Assumptions • Driver reaction time is approximately 1 sec. • Drivers tend to maintain a safe distance; tailgating is unusual. • All cars are approximately 10 ft long and 5 ft wide. • Almost all cars on the road are headed to the same destination. The UMAP Journal 22 (3) (2001) 271ñ290. 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
272 The UMAP Journal 22.3 (2001) erns density d: the number of cars per unit distance Occupancy n: the proportion of the road covered by cars Flow g: the number of cars per time unit that pass a given point Separation distance s: the average distance between midpoints of successive cars Speed v: the average steady-state speed of cars Travel Time: how long a given car spends on the road during evacuation Total Travel Time: the time until the last car reaches safety The Steady-State model Develo pmei Car-following is described successfully by mathematical models; following Rothery [1992, 4-1], we model the average separation distance s as a function of common speed u s=a+ Bu+r where a, B, and y have the physical interpretations the effective vehicle length L B=the reaction time, and the reciprocal of twice the maximum average deceleration of a following vehicle This relationship allows us to obtain the optimal value of traffic density (and speed)that maximizes flow Theorem. For q= kv(the fundamental equation for traffic Hlow) and (1), traffic How g is maximized at q*=(+21/2L12)-1, (L/7)1/2 B(/L)1 32 Proof: Consider N identical vehicles, each of length L, traveling at a steady- state speed v with separation distance given by (1). If we take a freeze-frame picture of these vehicles spaced over a distance D, the relation D=NL+Ns
272 The UMAP Journal 22.3 (2001) Terms Density d: the number of cars per unit distance. Occupancy n: the proportion of the road covered by cars. Flow q: the number of cars per time unit that pass a given point. Separation distance s: the average distance between midpoints of successive cars. Speed v: the average steady-state speed of cars. Travel Time: how long a given car spends on the road during evacuation. Total Travel Time: the time until the last car reaches safety. The Steady-State Model Development Car-following is described successfully by mathematical models; following Rothery [1992, 4-1], we model the average separation distance s as a function of common speed v: s = α + βv + γv2, (1) where α, β, and γ have the physical interpretations: α = the effective vehicle length L, β = the reaction time, and γ = the reciprocal of twice the maximum average deceleration of a following vehicle. This relationship allows us to obtain the optimal value of traffic density (and speed) that maximizes flow. Theorem. For q = kV (the fundamental equation for traffic flow) and (1), traffic flow q is maximized at q∗ = (β + 2γ1/2L1/2) −1, v∗ = (L/γ) 1/2, k∗ = β(γ/L)1/2 − 2γ β2 − 4γL . Proof: Consider N identical vehicles, each of length L, traveling at a steadystate speed v with separation distance given by (1). If we take a freeze-frame picture of these vehicles spaced over a distance D, the relation D = NL + Ns
Traffic Flow Models 273 must hold, where s is the bumper-to-bumper separation. Since s=s-L,we obtain k=N/D=N/(NL+NS=1/(L+s=l/s We invoke(1)to get k +Bu+r This is a quadratic equation in v; taking the positive root yields (k)= Applying q= ku, we have 9(k)k 21/41+(2-4D) Differentiating with respect to k, setting the result equal to zero, and wading hrough algebra yields the optimal values given. Interpretation and Uses We can estimate q*,k*, and u* from assumptions regarding car length(L), reaction time(B), and the deceleration parameter(). If welet L= 10 ft, B= ls and? x023 s /ft(a typical value [Rothery 1992]),we obtain 0.510cars/s,*=20.85ft/s,k=0.024cars/ft A less conservative estimate for y is ?=2(af -a1 ) where af and aL are the average maximum decelerations of the following and lead vehicles Rothery [1992]. We assume that instead of being able to stop instantaneously(infinite deceleration capacity), the leading car has deceleration capacity twice that of the following car. Thus, instead of y= 1/ 2a=023 s/ft, we use the implied value for a to computer=i(a-1-2a)=37=0.0115 s2/ ft and get 9=0.596 cars/s, v*=29.5 ft/s N 20 mph, k*=.020 cars/ft Going 20 mph in high-density traffic with a bumper-to-bumper separation of 40 ft is not bad The 1999 evacuation was far from optimal. Taking 18 h for the 120-mi trip from Charleston to Columbia implies an average speed of 7 mph and a bumper-to-bumper separation of 7 ft Limitations of the steady-State Model The steady-state model does not take into account the variance of cars' speeds. Dense traffic is especially susceptible to overcompensating or under- compensating for the movements of other drivers A second weakness is that the value for maximum flow gives only a first order approximation of the minimum evacuation time Determining maximum flow is distinct from determining minimum evacuation time
Traffic Flow Models 273 must hold, where s is the bumper-to-bumper separation. Since s = s − L, we obtain k = N/D = N/(NL + Ns )=1/(L + s )=1/s. We invoke (1) to get k = 1 α + βv + γv2 . This is a quadratic equation in v; taking the positive root yields v(k) = 1 2γ 4 γ k + (β2 − 4γL) − β 2γ . Applying q = kv, we have q(k) = k 2γ 4 γ k + (β2 − 4γL) − kβ 2γ . Differentiating with respect to k, setting the result equal to zero, and wading through algebra yields the optimal values given. Interpretation and Uses We can estimate q∗, k∗, and v∗ from assumptions regarding car length (L), reaction time (β), and the deceleration parameter (γ). If we let L = 10 ft, β = 1 s, and γ ≈ .023 s2/ft (a typical value [Rothery 1992]), we obtain q∗ = 0.510 cars/s, v∗ = 20.85 ft/s, k∗ = 0.024 cars/ft. A less conservative estimate for γ is γ = 1 2 (a−1 f − a−1 l ), where af and al are the average maximum decelerations of the following and lead vehicles Rothery [1992]. We assume that instead of being able to stop instantaneously (infinite deceleration capacity), the leading car has deceleration capacity twice that of the following car. Thus, instead of γ = 1/2a = .023 s2/ft, we use the implied value for a to compute γ = 1 2 a−1 − 2a−1 = 1 2 γ = 0.0115 s2/ft and get q∗ = 0.596 cars/s, v∗ = 29.5 ft/s ≈ 20 mph, k∗ = .020 cars/ft. Going 20 mph in high-density traffic with a bumper-to-bumper separation of 40 ft is not bad. The 1999 evacuation was far from optimal. Taking 18 h for the 120-mi trip from Charleston to Columbia implies an average speed of 7 mph and a bumper-to-bumper separation of 7 ft. Limitations of the Steady-State Model The steady-state model does not take into account the variance of carsí speeds. Dense traffic is especially susceptible to overcompensating or undercompensating for the movements of other drivers. A second weakness is that the value for maximum flow gives only a firstorder approximation of the minimum evacuation time. Determining maximum flow is distinct from determining minimum evacuation time.
274 The UMAP Journal 22.3 (2001) Minimizing evacuation Time with the steady-State Model Initial Considerations The goal is to keep evacuation time to a minimum, but the evacuation route must be as safe as possible under the circumstances. How long on average it takes a driver to get to safety( Columbia)is related to minimizing total evacu ation time but is not equivalent A General Performance measure A metric M that takes into account both maximizing traffic flow and mir mizing individual transit time T is M=W+(1-W) where< w< I is a weight factor, D is the distance that to traverse, I is the number of lanes, and n is the number of cars to evacuate. This metric assumes that the interaction between lanes of traffic(passing)is negligible, so that total flow is that of an individual lane times the number of lanes. Giver w, minimizing M amounts to solving a one-variable optimization problem in either v or k. Setting W= l corresponds to maximizing flow, as in the preceding section Setting W=0 corresponds to maximizing speed, subject to the constraint u< cruise, the preferred cruising speed; this problem has solution M=D/cruise. The model does not apply when cars can travel at cruise Setting W=1/2 corresponds to minimizing the total evacuation time The evacuation time is the time D/v for the first car to travel distance d plus the time N/lq for the n cars to flow by the endpoint. To illustrate that maximizing traffic flow and maximizing speed are out of sync, we calculate the highest value of W for which minimizing M would result in an equilibrium speed of cruise. This requires a formula for the equilibrium value u* that solves the problem minimize M(u)=WW(L +Bu+u2) subject to0<v≤ Cruis
274 The UMAP Journal 22.3 (2001) Minimizing Evacuation Time with the Steady-State Model Initial Considerations The goal is to keep evacuation time to a minimum, but the evacuation route must be as safe as possible under the circumstances. How long on average it takes a driver to get to safety (Columbia) is related to minimizing total evacuation time but is not equivalent. A General Performance Measure A metric M that takes into account both maximizing traffic flow and minimizing individual transit time T is M = W N lq + (1 − W) D v , where 0 ≤ W ≤ 1 is a weight factor, D is the distance that to traverse, l is the number of lanes, and N is the number of cars to evacuate. This metric assumes that the interaction between lanes of traffic (passing) is negligible, so that total flow is that of an individual lane times the number of lanes. Given W, minimizing M amounts to solving a one-variable optimization problem in either v or k. Setting W = 1 corresponds to maximizing flow, as in the preceding section. Setting W = 0 corresponds to maximizing speed, subject to the constraint v ≤ vcruise, the preferred cruising speed; this problem has solution M = D/vcruise. The model does not apply when cars can travel at vcruise. Setting W = 1/2 corresponds to minimizing the total evacuation time N lq + D v . The evacuation time is the time D/v for the first car to travel distance D plus the time N/lq for the N cars to flow by the endpoint. To illustrate that maximizing traffic flow and maximizing speed are out of sync, we calculate the highest value of W for which minimizing M would result in an equilibrium speed of vcruise. This requires a formula for the equilibrium value v∗ that solves the problem minimize M(v) = W N(L + βv + γv2) lv + (1 − W) D v subject to 0 < v ≤ vcruise
Traffic Flow Models 275 The formula for M(u) comes from (1),q= ku, and k= 1/s. Differentiating with respect to u, setting the result equal to zero, and solving for speed yields Cruise, L For cruise to equal the square root, we need (uZruisey-L) Using N= 160,000 cars, D=633, 600 ft(120 mi), I=2 lanes, cruise =60 mph 88 ft/s, y=0115 s /ft, and L=10 ft, we obtain W N 1/11. Thus, minimizing evacuation time in situations involving heavy traffic flow is incompatible with allowing drivers to travel at cruise speed with a safe stopping distance Computing Minimum Evacuation Time rom the fact that T= 2M when W=1/2, we obtain "=ku V=IL+ DU/N B1/2L+DN]-1/2-27+乙+N DL/N] B2-4D-7+D/N1 The minimum evacuation time is lq Predictions of the Steady-State Model a For N large, evacuation time minimization is essentially equivalent to the ow maximization( Figure 1), and it can be shown analytically that Nim How(N) The predicted evacuation time of 40 h for N= 160,000 seems reasonable We can evaluate the impact of converting I-26 to four lanes by setting l=4 in the equation for minimum evacuation time, yielding T N 23 h. For the steady-state model, this prediction makes sense, since the model does not deal with the effect of the bottleneck that will occur when Columbia is swamped by evacuees. The bottleneck would be compounded by using four lanes instead of two. On balance, however, doubling the number of lanes would lead to a net decrease in evacuation time
Traffic Flow Models 275 The formula for M(v) comes from (1), q = kv, and k = 1/s. Differentiating with respect to v, setting the result equal to zero, and solving for speed yields v∗ = min vcruise, 1 γ L + (1 − W) W · Dl N . For vcruise to equal the square root, we need W = 1 + N Dl v2 cruiseγ − L −1 . Using N = 160,000 cars, D = 633,600 ft(120 mi), l = 2 lanes, vcruise = 60 mph = 88 ft/s, γ = .0115 s2/ft, and L = 10 ft, we obtain W ≈ 1/11. Thus, minimizing evacuation time in situations involving heavy traffic flow is incompatible with allowing drivers to travel at cruise speed with a safe stopping distance. Computing Minimum Evacuation Time From the fact that T = 2M when W = 1/2, we obtain q∗ = k∗v∗, v∗ = 1 γ [L + Dl/N], k∗ = βγ1/2[L + Dl/N] −1/2 − 2γ [L+ 1 2 Dl/N] [L+Dl/N] [β2 − 4γL] − γ [Dl/N]2 [L+Dl/N] . The minimum evacuation time is T ∗ = N lq∗ + D v∗ . Predictions of the Steady-State Model For N large, evacuation time minimization is essentially equivalent to the flow maximization (Figure 1), and it can be shown analytically that lim N→∞ Tflow(N) Tmin(N) = 1. The predicted evacuation time of 40 h for N = 160,000 seems reasonable. We can evaluate the impact of converting I-26 to four lanes by setting l = 4 in the equation for minimum evacuation time, yielding T ≈ 23 h. For the steady-state model, this prediction makes sense, since the model does not deal with the effect of the bottleneck that will occur when Columbia is swamped by evacuees. The bottleneck would be compounded by using four lanes instead of two. On balance, however, doubling the number of lanes would lead to a net decrease in evacuation time.