Traffic Network Optimum Principle - Minimum Probability of Congestion Occurrence
aa r X i v : . [ phy s i c s . s o c - ph ] D ec Traffic Network Optimum Principle – Minimum Probability ofCongestion Occurrence
Boris S. Kerner Daimler AG, GR/PTF, HPC: G021, 71059 Sindelfingen, Germany
Abstract
We introduce an optimum principle for a vehicular traffic network with road bottlenecks. Thisnetwork breakdown minimization (BM) principle states that the network optimum is reached,when link flow rates are assigned in the network in such a way that the probability for spontaneousoccurrence of traffic breakdown at one of the network bottlenecks during a given observation timereaches the minimum possible value. Based on numerical simulations with a stochastic three-phase traffic flow model, we show that in comparison to the well-known Wardrop’s principles theapplication of the BM principle permits considerably greater network inflow rates at which no traffic breakdown occurs and, therefore, free flow remains in the whole network. PACS numbers: 89.40.-a, 47.54.-r, 64.60.Cn, 05.65.+b . INTRODUCTION Under small enough network inflow rates, drivers move at their desired (or permitted)speeds. Usually, there are several alternative routes from an origin to a destination in a net-work for which travel times are different but close to each other. When network inflow ratesincrease considerably, traffic congestion occurs due to traffic breakdown causing a sharplyincrease in the route travel times. Thus one of the theoretical problems of traffic networks isto find an optimal feedback dynamic traffic network assignment between alternative routesthat prevents traffic breakdown under great enough network inflow rates while maintain-ing free flow in the network (see, e.g., review [1]). Traffic breakdown occurs mostly at abottleneck and leads to the emergence of spatiotemporal congested traffic patterns. Thebottleneck can result from on- and off-ramps, a road gradient, etc.An empirical feature of traffic breakdown at a bottleneck is as follows [2, 3]. Trafficbreakdown is a local first-order phase transition from free flow to synchronized flow (F → Stransition). The feature has been explained in three-phase traffic theory [2] in which thereare three phases: 1. Free flow (F). 2. Synchronized flow (S). 3. Wide moving jam (J).Synchronized flow and wide moving jam are associated with congested traffic. A widemoving jam exhibits the characteristic jam feature to propagate through bottlenecks whilemaintaining the mean velocity of the downstream jam front. In contrast, the downstreamfront of synchronized flow is often fixed at the bottleneck. It has been found that there is abroad range q (B)th ≤ q ≤ q (free B)max [4] of the link (arc) flow rate q within which free flow at thebottleneck is in a metastable state with respect to traffic breakdown (Fig. 1). The greaterthe flow rate q in comparison with q (B)th , the smaller the critical amplitude of a disturbancein free flow whose growth leads to the breakdown, i.e., the greater the probability P (B)FS ofthe breakdown occurrence during a given observation time T ob . At q < q (B)th probability P (B)FS = 0, i.e., no traffic breakdown occurs, while at the maximum flow rate q = q (free B)max traffic breakdown occurs already due to a small disturbance, i.e., with probability P (B)FS = 1.Most network optimization theories (see e.g., [1, 7–10]) are based on the application ofuser equilibrium (UE) and system optimum (SO) principles introduced by Wardrop [11]: (i) Wardrop’s UE principle: traffic on a network distributes itself in such a way that the traveltimes on all routes used from any origin to any destination are equal, while all unused routeshave equal or greater travel times. (ii)
Wardrop’s SO principle: the network-wide travel time2
IG. 1: Empirical (a) and simulated (b) probability of traffic breakdown at an on-ramp bottleneck.Fig. (a) is taken from Persaud et al. [5]. Fig. (b) is taken from Fig. 4.2 of [3]. In (a) the averagingtime interval for traffic variables T av = 10 min [6]. In (b) the observation time of traffic flow T ob =15 min. should be a minimum. The Wardrop’s principles reflect either the wish of drivers to reachtheir destinations as soon as possible (UE) or the wish of network operators to reach theminimum network-wide travel time (SO).However, the Wardrop’s principles do not take into account that with some probabilitytraffic breakdown occurs in the network, when the link flow rate for one of the networkbottlenecks exceeds q (B)th . This breakdown leads usually to spatiotemporal congestion prop-agation [2, 3]. Such congestion growth within the network causes the associated growth oflink travel times; as a result, under congestion conditions as has been shown by Wahle andSchreckenberg with colleagues [8] and Davis [9, 10, 12] usually no true Wardrop’s equilibriumcan be found.In this article, we introduce a network breakdown minimization (BM) principle basedon the empirical features of traffic breakdown. The application of the BM principle shouldminimize probability of congestion occurrence in the whole network. We show that the BMprinciple leads to considerably greater network inflow rates at which free flows remain in thenetwork than under application of the Wardrop’s SO and UE principles.3 I. NETWORK BREAKDOWN MINIMIZATION (BM) PRINCIPLE
The BM principle is as follows: • The optimum of a traffic network with M links and N bottlenecks is reached, whenlink inflow rates are assigned in the network in such a way that the probability P (N)FS , net = 1 − N Y k =1 (1 − P (B , k )FS ) (1)for spontaneous occurrence of traffic breakdown at one of the network bottlenecksduring a given observation time T ob reaches the minimum possible value, i.e., thenetwork optimum is reached atmin q ,q ,...,q M { P (N)FS , net ( q , q , ..., q M ) } . (2)In (1), (2), q m is the link inflow rate for a link with index m ; m = 1 , , ..., M , where M > k = 1 , , ..., N is bottleneck index [24], N > P (B , k )FS is probability that duringthe time interval T ob traffic breakdown occurs at bottleneck k . The BM principle (2)can be applied as long as free flow conditions remain in the network. In general, theBM principle (2) is devoted to the optimization of large, complex vehicular trafficnetworks consisting of a great number of links M ≫ q ,q ,...,q M { P (N)C , net ( q , q , ..., q M ) } , (3)where P (N)C , net = N Y k =1 P (B , k )C (4)is the probability that during time interval T ob free flows remain in the network, i.e., thattraffic breakdown occurs at none of the bottlenecks; P (B , k )C = 1 − P (B , k )FS .For a complete formulation of the optimization principle (2) (or (3)), link flow rates q m should be connected with network inflow rates. To reach this goal in a general caseof a dynamic traffic assignment in a traffic network, one should use a dynamic traffic flowmodel [1]. This dynamic model should calculate spatiotemporal dynamics of vehicular trafficvariables within the network under given network inflow rates that can be time-functions.4owever, for the simplicity of simulations of the BM principle (2) discussed below inSec. III, we will use here a static traffic assignment for which the following well-knownconstraints are applied [1]: X i ϕ rwi = q rw ∀ r, w, (5) ϕ rwi ≥ ∀ i, r, w, (6) q m = X r X w X i ϕ rwi δ rwm,i ∀ m, (7)where q rw is the total flow rate of vehicles going from origin r to destination w ; ϕ rwi is theflow rate of vehicles going from r to w on route (path) i ; δ rsm,i = m is on route i . (8) III. SIMULATIONS: COMPARISON OF THE BM AND WARDROP’S PRINCI-PLESA. Model
We compare the BM (2) and Wardrop’s principles through their application for a simplenetwork with only two alternative routes 1 and 2 with lengths L and L (with L > L ) forvehicles moving from origin O to destination D (Fig. 2 (a)) used often for studies of trafficcontrol with Wardrop’s principles [8–10, 12].In our model, we assume that routes 1 and 2 are two-lane roads with on-ramp bottlenecks(Fig. 2 (b)) whose on-ramp inflow rates q on1 and q on2 are given constants. Thus the networkoptimization is performed only through the assignment of a network inflow with the rate q O between links m = 1 , i = 1 , m = 1 , q , q and T , , T , ; for links m = 2 , q , q and T , , T , (Fig. 2 (a)), where q = q + q on1 , q = q + q on2 .Travel times on routes 1 and 2 are T = T , + T , and T = T , + T , , respectively. TheBM principle (2) as well as Wardrop’s UE and SO principles can be written respectively asfollows: BM : min q ,q { − (1 − P (B , ( q + q on1 ))(1 − (9) − P (B , ( q + q on2 )) } , q + q = q O , IG. 2: Explanation of model: (a, b) Sketch of a simple network with two routes 1, 2 (a) anda route model (b); bottleneck parameters are the same as those in [14–16]. (c, d) Model steadystates in the flow–density (c) and space-gap–speed planes (d); F – free flow, S – synchronized flow.(e) Probability of spontaneous traffic breakdown at on-ramp bottleneck as function of the flow ratedownstream of the bottleneck at q on = 1000 vehicles/h for T ob = 40 min. UE : T ( q , q on1 ) = T ( q , q on2 ) , q + q = q O , (10)SO : min q ,q { q T , + ( q + q on1 ) T , + q T , + (11)+( q + q on2 ) T , } , q + q = q O . Travel times T , , T , , T , , T , are found via probe vehicles leaving the related links.6hese travel times are used in the UE (10) and SO (11) principles for calculations of q , q as long as the probe vehicles have moved in free flows ; this explains why only the associatedtime intervals are shown in related figures below [17].For simulations, we use a discrete version [15] of the Kerner-Klenov stochastic three-phasetraffic flow model of [14] that shows the empirical features of traffic breakdown includingthe resulting flow-dependence of breakdown probability P (B)FS (Fig. 2 (e)) used in (9) [18].The model reads as follows: v n +1 = max(0 , min( v free ,n , ˜ v n +1 + ξ n , v n + aτ, v s ,n )) , (12) x n +1 = x n + v n +1 τ, (13)where n = 0 , , , ... is number of time steps, τ is a time step, x n and v n are the vehiclecoordinate and speed at time step n , a is the maximum acceleration, ˜ v n is the vehicle speedwithout speed fluctuations ξ n , v s ,n is a safe speed.The physics of this model as well as initial and boundary conditions used in simulationshave already been considered in detail in Sec. 16.3 of the book [2]. In accordance with thefundamental hypothesis of three-phase traffic theory [2, 3], steady states of synchronized flowcover a 2D-region in the flow–density plane (Fig. 2 (c)). Speed fluctuations ξ n , functions˜ v n , v s ,n , rules for lane changing and model parameters used here are taken from [16] (seeAppendix A). The one exception from the model version of [16] is that a free flow speed v free ,n rather than to be a constant depends on space gap g n to the preceding vehicle: v free ,n = v free ( g n ) , (14)where v free ( g ) = max[ v (max)free (1 − κd/ ( g + d )) , v (min)free ] , (15) κ , v (max)free are given constants, v (min)free (Fig. 2 (d)) is constant found from the equations v (free)min = g (free)min /τ, (16) v (free)min = v free ( g (free)min ) . (17) B. Critical flow rate for traffic breakdown
In simulations, we study the spontaneous occurrence of traffic breakdown at one of thebottlenecks in the network (Fig. 2 (a)) during a given observation time T ob = 40 min (where7 ob > T , T ) at given on-ramp inflow rates q on1 , q on2 under network optimization based onthe application of each of the principles (9), (10), and (11).We find that a critical flow rate q O = q (cr)O for traffic breakdown at one of the networkbottlenecks, i.e., the inflow rate q O at which the breakdown occurs with probability P (B)FS = 1on route 1 or/and 2 in the network (Fig. 2 (a)) [20], satisfies conditions q (cr , BM)O > q (cr , SO)O > q (cr , UE)O , (18)where superscripts BM, UE, and SO are related to (9), (10), and (11), respectively.Under application of Wardrop’s UE principle (10), most vehicles move on the route 1because it is shorter, i.e., q > q . This explains why traffic breakdown occurs on route1 (Fig. 3(a)). At the same flow rate q O = 4340 vehicles/h, under application of the BMprinciple (9) we find P (B , k )FS = 0 for k = 1 and 2, because for the BM principle (9) values q + q on1 and q + q on2 = 3170 vehicles/h are smaller than q (B)th ≈ q (B)th ≈ q > q ; however, thedifference q − q is not great; therefore, the critical flow rate increases (Fig. 3 (b)). Atthe same flow rate q O = 5710 vehicles/h, under application of the BM principle (9) wefind P (B , k )FS = 0.05 for k = 1 ,
2; however, even when traffic breakdown occurs, the resultingcongested patterns exists only during about 10 min dissolving later due to a return S → Ftransition (simulations made are not shown here).The greatest critical flow rate q (cr , BM)O = 6500 vehicles/h is found for the BM principle(9); in this case, traffic breakdown occurs on both routes 1 and 2 (Fig. 3 (c)) [21].Thus in comparison with Wardrop’s UE and SO principles, the advantage of the BMprinciple (9) is the smaller traffic breakdown probability at the same network inflow rateand, therefore, the greater critical network inflow rate. The disadvantage of the BM principle(9) is that more drivers move on route 2 with a longer travel time. However, this disadvantageis true at small enough network inflow rates only. At greater network inflow rates, becauseof traffic congestion resulting from traffic breakdown under application of the Wardrop’sUE and SO principles, we find a quick growth of travel time on the shorter route 1. Thegreater network inflow rate exceeds the critical rate, the shorter the mean time delay oftraffic breakdown and the quicker the growth of congestion.For an example shown in Fig. 4, under application of Wardrop’s UE principle (10) due8o congestion on route 1 travel time on this route becomes as long as under application ofthe BM principle (9) [17].Above we have used symmetric bottleneck parameters q on1 = q on2 for which under appli-cation of Wardrop’s principles traffic breakdown occurs always on the shorter route 1 (Figs. 3and 4). Under asymmetric bottleneck parameters, we find the effect of change in route onwhich traffic breakdown can occur (Fig. 5): When q on1 ≪ q on2 , traffic breakdown occurs onthe longer route 2 (Fig. 5(a)), whereas at considerably greater flow rates q on1 traffic break-down occurs on route 1 (Fig. 5(b)) as that in Figs. 3 and 4. Thus for a given q on2 , there isa single flow rate q on1 = q (ch)on1 ( q on2 ) for which P (B , = P (B , , i.e., q (cr , SO)O = q (cr , BM)O = 6900vehicles/h; however, for all other flow rates q on1 condition (18) is valid. IV. BM PRINCIPLE AND TRAFFIC OPTIMIZATION AT SINGLE BOTTLE-NECK
Breakdown probability at any single bottleneck exhibits no minimum: the breakdownprobability is always a monotonously increasing flow rate function (Fig. 2 (e)). For thisreason, the minimization of breakdown probability P (B)FS for a single bottleneck is not possi-ble. However, the minimization of breakdown probability P (N)FS , net (1) for a traffic network ispossible, as formulated in the BM principle of Sect. II.To understand the sense of this conclusion, we consider the simple network shown in Fig. 2(a). There are two different bottlenecks in this case and, therefore, traffic assignment in thenetwork changes breakdown probabilities for both bottlenecks. For this reason, althoughbreakdown probability for each of the bottlenecks separately has no minimum, there is aminimum in breakdown probability P (N)FS , net (1) for the network (Fig. 6 (a, b)).Thus the BM principle for the optimization of a traffic network is conceptionally dif-ferent in comparison with known traffic optimization approaches at a single bottleneck, inparticular, with on-ramp metering.Figures 6 (a, b) correspond to symmetric bottleneck parameters in the network shownin Fig. 2 (a); this explains why the minimum of breakdown probability in the network isrelated to the condition q = q (Fig. 6 (a)). However, breakdown probability at the on-ramp bottleneck P (B)FS depends on the on-ramp inflow rate considerably (Fig. 6 (c)). For thisreason, under asymmetric bottleneck parameters the minimum of breakdown probability9 (N)FS , net in the network shown in Fig. 2 (a) is usually related to condition q = q (Fig. 6 (d,f, g)). V. CONCLUSIONS
1. The network breakdown minimization (BM) principle introduced in the article statesthat the network optimum is reached, when link flow rates are assigned in the networkin such a way that the probability for spontaneous occurrence of traffic breakdown atone of the network bottlenecks during a given observation time reaches the minimumpossible value; this is equivalent to the maximization of probability that traffic break-down occurs at none of the network bottlenecks. We have shown that the maximumnetwork inflow rate at which free flows still remain in the network is considerablygreater under application of the BM principle than that under application of theWardrop’s UE or SO principles.2. A traffic network optimization that is consistent with the empirical features of trafficbreakdown of Sect. I can consist of the stages:(i) The minimization of traffic breakdown probability in the network based on theBM principle introduced in this article.(ii) A spatial limitation of congestion growth, when traffic breakdown has neverthelessoccurred at a network bottleneck, with the subsequent congestion dissolution atthe bottleneck, if the dissolution of congestion due to traffic management ina neighborhood of the bottleneck is possible. An example of this stage is theANCONA on-ramp metering method [2, 3].A further development of this approach could be an interesting task for future inves-tigations.
Appendix A: Discrete Version of Kerner-Klenov Stochastic Three-Phase TrafficFlow Model and Model Parameters
A traffic flow model used in this article (Tables I–VIII) is a discrete version [15] ofthe Kerner-Klenov stochastic three-phase traffic flow model of Ref. [14]: rather than the10
ABLE I: Discrete version of stochastic model v n +1 = max(0 , min( v free , n , ˜ v n +1 + ξ n , v n + aτ, v s ,n )), x n +1 = x n + v n +1 τ ,˜ v n +1 = max(0 , min( v free , n , v s ,n , v c ,n )) ,v c ,n = v n + ∆ n at g n ≤ G n , v n + a n τ at g n > G n , ∆ n = max( − b n τ, min( a n τ, v ℓ,n − v n )) ,v free ,n = v free ( g n ), g n = x ℓ,n − x n − d , τ = 1; a and d are constants; the lower index ℓ marks variables related to the preceding vehicle. continuum space co-ordinate, a discretized space co-ordinate with a small enough value of thediscretization cell δx is used. Consequently, the vehicle speed and acceleration (deceleration)discretization intervals are δv = δx/τ and δa = δv/τ , respectively, where time step τ = 1 s.Because in the discrete model version discretized (and dimensionless) speed and accelerationare used, which are measured respectively in the discretization values δv and δa , the value τ in all formulae below is assumed to be the dimensionless value τ = 1. Explanations of thephysics of vehicle motion rules in this model can be found in Sect. 16.3 of [2].A choice of δx in the discrete model version determines the accuracy of vehicle speedcalculations in comparison with the initial continuum in space stochastic model of [14]. Wehave found that the discrete model exhibits similar characteristics of phase transitions andresulting congested patterns at highway bottlenecks as those in the continuum model at δx that satisfies the conditions δx/τ ≪ b, a (a) , a (b) , a (0) . (A1)11 ABLE II: Functions in model I: Stochastic time delay of acceleration and deceleration a n = a Θ( P − r ), b n = a Θ( P − r ), P = p if S n = 11 if S n = 1 , P = p if S n = − p if S n = − ,S n +1 = − v n +1 < v n v n +1 > v n v n +1 = v n ,r = rand(0 , z ) = 0 at z < z ) = 1 at z ≥ p = p ( v n ), p = p ( v n ), p is constant.TABLE III: Functions model II: Model speed fluctuations ξ n = ξ a if S n +1 = 1 − ξ b if S n +1 = − ξ (0) if S n +1 = 0 ,ξ a = a (a) τ Θ( p a − r ), ξ b = a (b) τ Θ( p b − r ), ξ (0) = a (0) τ − r ≤ p (0) p (0) < r ≤ p (0) and v n >
00 otherwise ,r = rand(0 , a (a) = a (a) ( v n ), a (b) = a (b) ( v n ); p a , p b , p (0) , a (0) are constants.TABLE IV: Functions in model III: Synchronization gap G n G n = G ( v n , v ℓ,n ), G ( u, w ) = max(0 , ⌊ kτ u + a − φ u ( u − w ) ⌋ ) ,k ( k >
1) and φ are constants, ⌊ z ⌋ denotes the integer part of a real number z . ABLE V: Functions in model IV: Safe speed v s ,n v s ,n = min ( v (safe) n , g n /τ + v (a) ℓ ) ,v (a) ℓ = max(0 , min( v (safe) ℓ,n , v ℓ,n , g ℓ,n /τ ) − aτ ) ,v (safe) n = ⌊ v (safe) ( g n , v ℓ,n ) ⌋ is taken as that in [22], which is a solution ofthe Gipps’s equation [23] v (safe) τ safe + X d ( v (safe) ) = g n + X d ( v ℓ,n ),where τ safe is a safe time gap, X d ( u ) = bτ (cid:18) αβ + α ( α − (cid:19) , α = ⌊ u/bτ ⌋ and β = u/bτ − α are the integer and fractional parts of u/bτ ,respectively; b is constant. IG. 3: Traffic breakdown under application of UE (10) (a), SO (11) (b) and BM principles (9)(c), respectively. Speed in time and space in the right lane on route 1 (left) and route 2 (right). In(a) q = 3250, q = 1090, q (cr , UE)O = 4340 vehicles/h. In (b) q = 3250, q = 2460, q (cr , SO)O = 5710vehicles/h. In (c) q (cr , BM)O = 6500 vehicles/h. T (B)FS is a random time delay of traffic breakdownlabeled by arrows F → S: T (B)FS = 30 (a), 21 (b), 13 (route 1) and 11 (route 2) (c) min. q on1 = q on2 =1000 vehicles/h; road location of on-ramp bottleneck x on = 15 km; L = 20, L = 25 km. IG. 4: Comparison of travel times under application of Wardrop’s UE (10) and BM (9) principlesat q O = 4680 vehicles/h: (a) Speed in the right lane in space and time for route 1 (left) and 2(right) under application of (10) ( q = 3360, q = 1320 vehicles/h). (b) Time-dependences of traveltimes on route 1 for (10) (dashed curve) and on route 2 for (9) (solid curve). T (B)FS = 15 min. Underapplication of (9), P (B , k )FS = 0 , k = 1 ,
2. Other parameters are the same as those in Fig. 3. IG. 5: Change in route on which breakdown occurs with probability P (B)FS = 1 under asymmetricbottleneck parameters and application of SO principle (11): (a) Traffic breakdown on route 2. (b)Traffic breakdown on route 1. q on2 = 1050 vehicles/h. ( q on1 , q (cr , SO)O ) = (60, 7200) (a), (800,6320) (b) vehicles/h. q (ch)on1 ≈
350 vehicles/h. L = 23 km. T (B)FS = 22 (a) and 15 (b) min. Underapplication of (9), P (B , k )FS < k = 1 ,
2. Other parameters are the same as those in Fig. 3. IG. 6: Comparison of BM principle and traffic optimization at single bottleneck: (a, b) Probabilityof traffic breakdown P (2)FS , net in the network with two bottlenecks shown in Fig. 2 (a) (i.e., whenin Eq. (1) the value N = 2) as a function of the flow rates q and q (a) and a function of the flowrates q and q O = q + q (b) for symmetric bottleneck parameters q on1 = q on2 = 1000 [vehicles/h].(c) Probability of traffic breakdown P (B)FS at a single on-ramp bottleneck as a function of theflow rate downstream of the bottleneck for on-ramp inflow rates q on = 1000 [vehicles/h] (curve1 that is the same as that in Fig. 2 (e)) and q on = 500 [vehicles/h] (curve 2). (d, e) P (2)FS , net asfunctions of q and q (d) and of q and q O = q + q (e) for asymmetric bottleneck parameters q on1 = 1000 and q on2 = 500 [vehicles/h]. (f, g) P (2)FS , net as a function of q for different given values q O = q + q = 6200 (f) and 6650 (g) [vehicles/h] associated with figure (e). ABLE VI: Lane changing occurring with probability p c from the right lane to the left lane ( R → L )and from the left lane to the right lane ( L → R ) and safety conditions for lane changing [14]Incentive conditions for lane changing: R → L : v + n ≥ v ℓ,n + δ and v n ≥ v ℓ,n , L → R : v + n > v ℓ,n + δ or v + n > v n + δ .In conditions R → L and L → R , the value v + n at g + n > L a and the value v ℓ,n at g n > L a are replaced by ∞ , where L a is constant.Safety conditions for lane changing:rules ( ∗ ): g + n > min( v n τ, G + n ), g − n > min( v − n τ, G − n ), where G + n = G ( v n , v + n ), G − n = G ( v − n , v n ), or rule ( ∗∗ ): x + n − x − n − d > g (min)target with g (min)target = ⌊ λv + n + d ⌋ ,the vehicle should pass the midpoint point x (m) n = ⌊ ( x + n + x − n ) / ⌋ between two neighboring vehicles in the target lane, i.e., x n − < x (m) n − and x n ≥ x (m) n or x n − ≥ x (m) n − and x n < x (m) n . Speed after lane changing: v n = ˆ v n , ˆ v n = min( v + n , v n + ∆ v (1) ),in ˆ v n the speed v n is related to the initial lane before lane changing.Vehicle coordinate after lane changing:Vehicle coordinate does not changes under the rules ( ∗ )and it changes to x n = x (m) n under the rule ( ∗∗ ). λ , δ , ∆ v (1) are constants; superscripts + and − in variables, parameters,and functions denote the preceding vehicle and the trailing vehiclein the “target” (neighbouring) lane, respectively;the target lane is the lane into which the vehicle wants to change. G ( u, w ) is given in Table IV. ABLE VII: Models of vehicle merging at on-ramp bottlenecks that occurs when a safety rule ( ∗ ) or a safety rule ( ∗∗ ) is satisfied [14] Safety rule ( ∗ ): g + n > min(ˆ v n τ, G (ˆ v n , v + n )) , g − n > min( v − n τ, G ( v − n , ˆ v n )) , ˆ v n = min( v + n , v n + ∆ v (1) r ) , in ˆ v n the speed v n is related to the initial lane before lane changing,∆ v (1) r > ∗∗ ): x + n − x − n − d > ⌊ λ b v + n + d ⌋ ,x n − < x (m) n − and x n ≥ x (m) n or x n − ≥ x (m) n − and x n < x (m) n ,λ b is constant. Parameters after vehicle merging: v n = ˆ v n . Under the rule ( ∗ ): x n maintains the same,under the rule ( ∗∗ ): x n = x (m) n .Speed adaptation before vehicle merging v c ,n = v n + ∆ + n at g + n ≤ G ( v n , ˆ v + n ), v n + a n τ at g + n > G ( v n , ˆ v + n ) , ∆ + n = max( − b n τ, min( a n τ, ˆ v + n − v n )) , ˆ v + n = max(0 , min( v free , n , v + n + ∆ v (2) r )) , ∆ v (2) r is constant. ABLE VIII: Model parameters used in simulationsVehicle motion in road lane: τ safe = τ = 1, d = 7 . /δ x, δx = 0.01 m, v free ( g ) = max[ v (max)free (1 − κd/ ( g + d )) , v (min)free ], v (max)free ≈ . − /δv ( v (max)free = 140 km/h), v (min)free ms − /δv is constant found from the system of equations: v (free)min = g (free)min /τ and v (free)min = v free ( g (free)min )( v (min)free ≈
70 km/h), b = 1 ms − /δa , δv = 0 .
01 ms − , δa = 0 .
01 ms − , k = 3, p = 0.3, φ = 1, p b = 0 . p (0) = 0 . p ( v n ) = 0 .
48 + 0 . v n − v ), p ( v n ) = 0 .
575 + 0 .
125 min (1 , v n /v ), a (b) ( v n ) = 0 . a ++0 . a max(0 , min(1 , ( v − v n ) / ∆ v ), a (0) = 0 . a , κ = 1 . a (a) = 0, v = 12 . − /δv , ∆ v = 2 .
778 ms − /δv , v = 10 ms − /δv , v = 15 ms − /δv , a = 0.5 ms − /δa .Lane changing: δ = 1 ms − /δv , L a = 150 m /δx , p c = 0 . λ = 0 .
75, ∆ v (1) = 2 ms − /δv .On-ramp bottleneck model (see Fig. 16.2 of the book [2]): λ b = 0.75, v free on = 22 . − /δv ,∆ v (2)r = 5 ms − /δvL r = 1 km /δx , ∆ v (1)r = 10 ms − /δv , L m = 0.3 km /δx .
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Introduction to Modern Traffic Flow Theory and Control. (Springer, Berlin, NewYork, 2009).[4] The flow rates q (B)th and q (free B)max can depend considerably on bottleneck characteristics andtraffic parameters like the percentage of long vehicles, weather, etc. In empirical observations, q (free B)max − q (B)th ≈ ÷ , 64–69 (1998).[6] When in empirical observations of traffic breakdown the averaging time interval for trafficvariables T av is longer than a characteristic time of the breakdown development (about 1 min),then one can apply T ob = T av in the definition of breakdown probability (see explanations inSec. 10.3.1 of the book [2]).[7] K. Lee, P.M. Hui, B.-H. Wang, N.F. Johnson, J. of the Phys. Soc. Japan
236 (2002); T. Roughgarden, J. of Comp. andSys. Sc.
341 (2003); W.-X. Wang, B.-H. Wang, W.-C. Zheng, C.-Y. Yin, T. Zhou, Phys.Rev. E
669 (2000).[9] L.C. Davis, Phys. A q th ≤ q ≤ q (free)max traffic breakdown occurs on this link spontaneously with probability P FS >
0, however, ata random location within the link [2]. In (1), this link can also be considered containing abottleneck with the threshold flow rate q (B)th = q th and maximum flow rate q (free B)max = q (free)max .
14] B.S. Kerner, S.L. Klenov, Phys. Rev. E T of a probe vehicle between the time instant of trafficbreakdown and the time instant at which flow rates q and q begin to change through theapplication of (10) or (11) under congestion conditions.[18] The application of a three-phase traffic flow model is explained as follows. As shown inSect. 10.3 of [2], the first-order phase transition in traffic flow models that belong to theGeneral Motors (GM) model class reviewed in [19] is associated with an F → J transition. Incontrast, in empirical observations rather than the F → J transition, an F → S transition governstraffic breakdown at a bottleneck in vehicular traffic.[19] D. Chowdhury, L. Santen, A. Schadschneider. Physics Reports , 199 (2000); D. Helbing.Rev. Mod. Phys. , 1067–1141 (2001); T. Nagatani. Rep. Prog. Phys. , 1331–1386 (2002);K. Nagel, P. Wagner, R. Woesler. Operation Res. , 681–716 (2003).[20] To find probability of traffic breakdown P (B)FS , a study of traffic breakdown in the network isrepeated for 40 different realizations for each given flow rate. In these 40 realizations modelparameters are the same, however, the initial conditions for random model fluctuations aredifferent.[21] It should be noted that the application of the BM principle (9) has a sense for free flowsonly; therefore, the development of congested patterns after traffic breakdown has occurred isshown in Fig. 3 (c) only for the illustration of emergent congestion under unchanged modelparameters.[22] S. Krauß, P. Wagner, C. Gawron. Phys. Rev. E .[23] P.G. Gipps. Trans. Res. B q th ≤ q ≤ q (free)max traffic breakdown occurs on this link spontaneously with probability P FS >
0, however, at a random location within the link [2]. In (1), this link can also be consideredcontaining a bottleneck with the threshold flow rate q (B)th = q th and maximum flow rate q (free B)max = q (free)max ..