transportation science orms - queen's university belfast

TRANSPORTATION SCIENCEVol. 39, No. 1, February 2005, pp. 25–38issn 0041-1655 �eissn 1526-5447 �05 �3901 �0025

informs ®

doi 10.1287/trsc.1030.0083©2005 INFORMS

Convergence of a Discretised Travel-Time Model

Malachy Carey, Y. E. GeSchool of Management and Economics, Queen’s University Belfast,

Belfast BT7 1NN, Northern Ireland {[email protected], [email protected]}

In network models for dynamic traffic assignment (DTA), the travel time on a link is often treated as a functionof the number of vehicles on the link. Instead of applying this model to the whole link, we divide the link into

segments, apply the model (suitably adjusted) sequentially to these segments, and investigate how the solution isaffected by various levels of discretisation (as the discretisation is refined, the solution converges to the solutionof the Lighthill-Whitham-Richards (LWR) model). We also restrict the link (and segment) travel-time functionto ensure that it satisfies a first-in-first-out (FIFO) property and explore how this affects and restricts the formof the flow-density functions used in the LWR model. We numerically illustrate the solution of the discretisedmodel for various travel-time functions and patterns of inflows, for both homogeneous and inhomogeneouslinks. Subject to the above restriction on the flow-density function, the numerical results suggest that dividing“long” links into even a few segments can make the model solution closely approximate the LWR solution, whileretaining tractability in the network model. We also observe, for example, that the whole-link (undescretised)travel-time model has a “flattening” effect on the profiles of flows and travel times (this effect can be reducedto any desired extent by using discretisation); and that the travel time and outflow for an inhomogeneous linkcan be approximated very closely by treating the link as homogeneous, with capacity parameters set equal tothe average capacity from the inhomogeneous link.

Key words : traffic flow; link travel time; link flow; LWR model; first-in-first-out; dynamic traffic assignmentHistory : Received: October 2001; revision received: July 2002; accepted: January 2003.

IntroductionWe consider the “whole-link” travel-time model��t� = f �x�t��, where ��t� is the link travel time forvehicles entering a link at time t, and x�t� is the num-ber of vehicles on the link at time t. This model wasintroduced in network models for DTA by Friesz et al.(1993), and has been investigated extensively by var-ious authors, including Friesz et al. (1993), Astarita(1995, 1996), Wu et al. (1995, 1998), Ran et al. (1997),Xu et al. (1999), Adamo et al. (1999), Carey andMcCartney (2002), and Zhu and Marcotte (2000).

The model ��t� = f �x�t�� is normally applied towhole links in network models for DTA. Here, weinstead divide the link into segments, apply themodel sequentially to these segments and computethe outflow and travel-time profiles for given patternsof inflows to the link. We gradually refine the discreti-sation to see how the level of discretisation affects thesolution. As the discretisation converges to the con-tinuous limit (the segment length is reduced to zero),the solution converges to the solution of the LWRmodel (Lighthill and Whitham 1955, Richards 1956).The fact that the models converge does not in itselfindicate the speed or path of convergence, hence, weillustrate this with numerical examples. To apply theLWR model, we use its finite difference approxima-tion (Daganzo 1995). We repeat the experiments fordifferent inflow patterns and travel-time functions, in

particular, for linear and quadratic travel-time func-tions because these have been the most commonlyused in the literature. Before applying the travel-timemodel, we consider restrictions derived in the litera-ture to ensure that the model satisfies a FIFO property,and we derive implications of these for the travel-timemodel and for the flow-density function used in theLWR model.

The results are also relevant to the problem of defin-ing link lengths to be used in DTA models. Whenapplying a DTA model to a given network, usershave choices in defining the numbers and lengths oflinks. For example, should a long link be treated asseveral shorter links? This choice, and its effect onthe accuracy or quality of the DTA solutions, is usu-ally not mentioned or discussed, but is important.For example, suppose we take inflows to a homoge-neous link as given and apply the travel-time model��t� = f �x�t�� to compute the profiles of travel timesand outflows. If we then divide the link into twoor more sublinks or segments and apply the travel-time model again, sequentially to each of the sublinks,then because the link is homogeneous, it is desir-able that the results should again be the same, andbe independent of the “irrelevant alternatives” con-cerned with dividing up the link. However, we findthat the results are not the same. We investigate thisfor different levels of discretisation, or approximation,

25

Carey and Ge: A Discretised Travel-Time Model26 Transportation Science 39(1), pp. 25–38, © 2005 INFORMS

to throw light on how much difference it makes andhow much of a problem it is.

The discretised � = f �x� model presented heredivides the link into segments or cells as does thecell transmission model (Daganzo 1994). However,the models differ in the way that they treat movementof traffic over time. In the cell transmission model,traffic enters a cell at each time (clock tick) t and, typ-ically, some exits to the next cell in the same timeinterval t while some remains in the current cell. Incontrast, in the discretised � = f �x� model, traffic thatenters a segment at time t exits from it at time t+��t�,where ��t� is the time taken to traverse the segment.This is true whether we treat time as continuous ordiscretise it.

When applying the travel-time model to link seg-ments, we do not change the basic form of themodel. Hence, in existing DTA network models thatuse whole-link travel-time models, the latter can bereplaced by the discretised link model consideredhere. The rest of the network model would remainunchanged. In the references given above, theory andalgorithms have already been developed concerningusing the travel-time model � = f �x� in models fornetwork loading and for DTA. This theory and algo-rithms continue to apply when the network model isextended by discretising the links as in this paper. Inview of that, we do not set out or discuss a completenetwork model here.

1. The Travel-Time ModelFriesz et al. (1993) introduced a travel-time model foruse in DTA, particularly a linear version, thus,

��t�= a+ bx�t�� (1)

where a and b are constants, and later authors, ref-erred to above, introduced nonlinear versions

��t�= f �x�t�� (2)

wherex�t�= x�0�+

∫ t

0�u�s�− v�s��ds� (3)

x�t� is the number of vehicles on the link at time t,and u�s� and v�s� are the inflow and outflow rates,respectively, for the link at time s. For notational con-venience in what follows, we assume that the link isinitially empty (i.e., x�0�= 0), but it is trivially easy toallow x�0� > 0. Various authors (see §2) derive condi-tions to ensure that the model (2)–(3) satisfies a FIFOcondition. When (2)–(3) satisfies FIFO and flow is con-served over time, then

∫ t

0u�s�ds =

∫ t+��t�

0v�s�ds (4)

and differentiating w.r.t. t gives

v�t + ��t��= u�t�

1+ � ′�t�= u�t�

1+ f ′�x��u�t�− v�t�� (5)

where f ′�x� denotes the derivative of f (x) w.r.t. x.The link travel-time model (2)–(3) is used and

solved in the various papers on DTA for networksreferred to in the introduction, with the computationaldetails generally omitted. Though the model is statedin continuous time, it is solved by numerical meth-ods involving dividing the time horizon [0�T ] intosmall segments, with a step size chosen by the user.Hence, until §3 on numerical examples, we continueto treat time as continuous. Because we are focusingon a single link, we take the inflow u�t� to the linkand the initial loading x�0� on the link as exogenouslygiven. Also, traffic that enters at time 0 exits at time� = a+bx�0�, hence, any outflow before that time mustbe exogenously given (i.e., v�t� is exogenous for t <a+ bx�0�). All other variables are endogenous.

1.1. Spatially Discretising the Model andConvergence to the LWR Model

We now consider adapting the model (2)–(3) to applyit sequentially to each segment of a discretised link.The functions (1) or (2) represent the travel time fora specific link with specific characteristics, includinga specific length L, which are embedded in the formor parameters of f �x�. Thus, f �·� cannot be appliedarbitrarily to links of other lengths, or to segmentsof the given link, so that we cannot write �� = f �x��,where the subscript refers to a segment of the link towhich (2) applies. To derive the form and parametersof a travel-time function for a segment of a link, whengiven a travel time function f �x� for the whole link,we can use various equivalent methods, one of whichis as follows. In f �x�t��, the variable x�t� can be rewrit-ten as x�t� = Lk�t�, where k�t� is the mean density onthe link at time t. Hence, the whole-link travel-timemodel (2) can be rewritten as ��t� = f �Lk�t��, i.e., asa function of the mean density. This implies that thetravel time per unit distance (= 1/speed) is

�̃�t�= f �x�t��/L= f �Lk�t��/L= f̃ �k�t�� (6)

where f̃ �k� is defined as f �Lk�/L. Applying this unittravel-time equation to a segment of length �z, thesegment travel time is

��t�=�zf̃ �k��t�� 6′�

where k��t� is the density on the segment.An equivalent method of deriving (6′) is as follows.

Because (2) takes no account of the distribution of thetraffic x�t� along the link, we can assume that at anytime t, f �x�t�� treats the distribution of traffic along

Carey and Ge: A Discretised Travel-Time ModelTransportation Science 39(1), pp. 25–38, © 2005 INFORMS 27

the link as uniform. In view of that, the travel timeon a segment of length �z is (�z/L) times the traveltime on the whole link; that is, the travel time on asegment of length �z is ��t� = ��z/L�f �k��t�L�. Thisis identical to (6′) when we recall that f̃ �k� is definedas f �Lk�/L.

Equation (6′) is written as a function of the seg-ment mean density k��t� but can, of course, be rewrit-ten as a function of the number of vehicles x��t� =k��t��z on the segment, thus, ��t�=�zf̃ �x��t�/�z�=f ∗�x��t��, which is more directly analogous to (2).However, f ∗�x��t�� has the disadvantage that its argu-ment x��t� goes to zero as �z goes to zero, and theform of f ∗ depends on �z. Hence, in what follows,we use (6′) rather than ��t�= f ∗�x��t��, because k��t�in (6′) does not go to zero as �z goes to zero, andf̃ does not depend on �z. Note that (6′) and f ∗�x��t��differ only by scale factors L and �z.

We can now use (6′) while allowing the density and,hence, travel time, to be different for each segment,whereas in the whole-link model, these were implic-itly assumed to be constant along the link. Note that(6′) ensures that if the traffic x�t� on a link is uni-formly distributed along the link and the link is thendivided into n segments, the sum of the travel times(6′) on the segments equals the travel time given by(2) for the whole link. In fact, this property could beused to define or derive the segment travel-time func-tions (6′).

Writing the segment travel time ��t� as a functionof density, as in (6′), also has other advantages. First,it makes it easier to compare with the LWR model,where flow is written as a function of density. Second,we can extend (6′) to let the segment travel time ��t�depend on the location or distance of the segmentalong the link, thus,

��t�=�zf̃ �k��t� z�� z�� 2∗�

where z is the distance from the beginning of the linkto the beginning of the segment �z. This allows “inho-mogeneous” links, whereas the whole-link model (2)implicitly assumes a homogeneous link. Similarly, wecan also extend (6′) to let the segment travel timedepend on the time t at which a vehicle arrives at thebeginning of the segment, thus,

��t� z�=�zf̃ �k��t� z�� t� z�� 2′�

where ��t� z� is the travel time on segment [z, z+�z]for vehicles entering it at time t.

Suppose that, for all time t in [0,T ], we have com-puted the travel times up to some point z on the linkand computed the (out)flows at z. These may havebeen computed by applying the model sequentiallyto segments of the link. Now, apply the model to thenext segment of length �z as follows. At time t, forsegment [z, z+�z], let

u��t� z� and v��t� z + �z� denote the inflow andoutflow rates, respectively, for the segment and

x��t� z� denote the number of vehicles on the seg-ment, so that k��t� z�= x��t� z�/�z is the mean densityin the segment.Applying (3) and (4), respectively, to the segmentgives

x��t� z�=∫ t

0�u��s� z�− v��s� z+�z��ds �3′�

∫ t

0u��s� z�ds =

∫ t+��t�z�

0v��s� z+�z�ds� �4′�

We wish to compare this discretised version of the��t� = f �x�t�� model with the LWR model. For ahomogeneous link, the latter assumes that the flowrate q�t� z� at time t at point z on the link dependsonly on the density k�t� z� at that point, and not onany earlier or later points, thus, q�t� z� = Q�k�t� z��.For a link that is inhomogeneous over space z andtime t, this becomes

q�t� z�=Q�k�t� z�� t� z�� (7)

together with a conservation equation

�q�t� z�

�z=−�k�t� z�

�t� (8)

Proposition 1. Applying the model (2)–(3) to a seg-ment �z� z+�z� of a link gives �2′�–�3′�, and FIFO implies�4′�. As �z → 0 the discretised model �2′�–�3′� convergesto the LWR model (7)–(8). That is,

(a) the travel-time Equation �2′� converges to the flow-density Equation (7) and

(b) the conservation Equation �3′� converges to �8�.

Remark. When the flow rate and speed are con-stant along a link (or link segment) and constant overtime, then (number of vehicles on the link) = (flowrate) × (link travel time). In the proof, we first showthat an analogous relationship (9) holds when thesequantities vary as defined by �2′�–�4′�.Proof. The first sentence of the proposition fol-

lows immediately from the discussion preceding theproposition. The rest is as follows.

(a) The flow-density equation. Adding �3′� to �4′�gives x��t� z�= ∫ t+��t�

tv��s� z+�z�ds. Using the mean

value theorem, this integral implies that there existsan (out)flow v��s� z+�z�, at some time s, t ≤ s ≤ t +��t� z�, such that

x��t� z�= v��s� z+�z��t� z�� (9)

Using �2′� to substitute for ��t� z� in (9) givesx��t� z� = v��s� z + �z�f̃ �k��t� z�� t� z��z, and usingthe mean density k��t� z� = x��t� z�/�z to substitute


for x��t� z� reduces this to k��t� z� = v��s� z + �x� ·f̃ �k��t� z�� t� z�. Rearranging gives

v��s� z+�z�= k��t� z�/f̃ �k��t� z�� t� z�� (10)

Letting �z →0 implies t + ��t� z� → t, hence, s → t,k��t� z� → k�t� z� and v��s� z + �z� → u�t� z�, whichcan be written as q�t� z�. This reduces (10) to

q�t� z�= k�t� z�/f̃ �k�t� z�� t� z�� (11)

However, (11) is of the same form as (7), which com-pletes the proof of part (a).

(b) The conservation equation. Differentiating �3′�gives �x��t� z�/�t = u��t� z� − v��t� z + �z�, and dif-ferentiating the mean density definition k��t� z� =x��t� z�/�z w.r.t. t gives �x��t� z�/�t = �z�k��t� z�/�t.Substituting the latter in the former and rearranginggives

�k��t� z�

�t=−�q��t� z�

�z� (12)

where �q��t� z� = v��t� z + �z� − u��t� z� denotes thechange in flow rate from the entrance to exit of thesegment. Letting �z→ 0, the mean density on the seg-ment k��t� z�→ k�t� z��q��t� z�/�z→ �q�t� z�/�z and(12) goes to (8). �

Remark. It is interesting to note that, as an interme-diate step in the above proof, Equations �2′� and �3′�imply discrete Equations (10) and (12) for the segment�z analogous to the continuous LWR Equations (7)and (8).

2. Implications of FIFO Conditionson � = f �x�

In this section, we consider a whole link or segmentwithout discretisation. In a travel-time function of theform � = f �x�, the variables � and x refer to the wholelink or segment and f �·� makes no reference variationalong the link or segment. Hence, we can assume thelink or segment is homogeneous, which we do in thissection.

2.1. Implications of FIFO Conditions for the Formof � = f �x�

To ensure that the travel-time functions (1) and (2)satisfy FIFO, various authors have introduced certainsufficient conditions. For the linear case (1), Frieszet al. (1993) show that FIFO holds if the inflow rateis non-negative and continuous, and Xu et al. (1999,Theorem 3.2) and Zhu and Marcotte (2000) obtainsimilar results. For the nonlinear case (2), Xu et al.(1999) prove FIFO by assuming that the inflow ratefunction is non-negative, bounded from above by apositive constant B and that f ′�x�≤ 1/B (Theorem 3.1).Zhu and Marcotte (2000) and Chabini and Kachani(1999) show similar results with slightly different

a

τ = a+bx

xJ

τJ

1/b

q = Lk/(a+bLk)

kJ

q

(a)

(b)

x

τ

k

Figure 1 q = Q�k� Implied by a Linear � = f �x�

assumptions. The Xu and Chabini papers also includeexamples showing that if the above bound is violated,then there exist some patterns of inflows that wouldcause a FIFO violation. Following the above authors,we, therefore, assume that the travel-time function� = f �x� is either linear or satisfies f ′�x�≤ 1/B, where1/B is a finite upper bound. This rules out the travel-time function in Figure 4a and allows the functions inFigures 1a and 2a.

Though a capacity limit on x is not explicitly intro-duced in the above literature on FIFO, we note thatif a given � = f �x� does not have a capacity limiton x, one can be imposed so that � = f �x� for all0 ≤ x ≤ xJ , x > xJ not permitted, and when the linkis at capacity, the travel time is � = f �xJ �. Because� = f �x� is nondecreasing, if we also assume it isconvex on [0, xJ ], then f ′�x� increases monotonicallyon [0, xJ ] and f ′�x� ≤ f ′

-�xJ �, where f ′

-�x� is the left-hand derivative of f �x� at x = xJ , hence, the conditionf ′�x� ≤ 1/B is satisfied if, and only if, f ′

-�xJ � ≤ 1/B.

Incidentally, when proving that f ′�x� ≤ 1/B is a suf-ficient condition to ensure FIFO, the above authorsdo not introduce a bound or capacity for x. However,introducing such a bound, as described above, neednot violate their assumptions, so that their results stillhold even with the above capacity. This is because wecan ensure x ≤ xJ by restricting the inflow rate whenthe bound is reached, and reducing the inflow ratedoes not affect the assumptions in the above papers.Despite the above remarks, when deriving the proper-ties of travel-time functions � = f �x� below, we do not


τBound τ = τ 0 + 1/B x

implied by f ′(x) ≤ 1/B

τ = τ c + f ′(xc)(x −xc)

τ = f(x)

xO

τ c

kO

q

xc

τ0

kc

B

(b)

(a)

kJ

xJ

τJ

q = Lk / (τ0 + B −1Lk)

Figure 2 q = Q�k� Implied by a Nonlinear � = f �x� with f ′�x�≤ 1/B

introduce a bound x ≤ xJ if that is not already implicitin the form of the function. If desired, a bound can beimposed later as described above.

2.2. Implications of FIFO Conditions on � = f �x�for the Form of Flow-Density Functions

In this paper, we compare the solutions obtained fromthe travel-time model ��t� = f �x�t�� with those fromthe LWR model. The latter assumes a flow-densityfunction at each point on a link at each point intime. To compare the solutions, we must use flow-density functions that are derived from the travel-time functions � = f �x� or vice versa. Thus, becausethe FIFO condition f ′�x� ≤ 1/B has implications forthe form of � = f �x�, it also has implications for theform of the flow-density function. We derive these inProposition 2. There, we consider four different formsof travel-time function (Figures 1a–4a) that span therange of travel-time functions that have been used inpractice. In all four forms, � = f �x� is assumed posi-tive, nondecreasing and convex. The first form (Figure1a) is linear and has been shown by Friesz et al.(1993), Xu et al. (1999), and Zhu and Marcotte (2000)to satisfy a FIFO property. The second (Figure 2a) hasa bounded gradient f ′�x� ≤ 1/B for all x ≥ 0, where Bis a finite positive constant. The third form (Figure 3a)is introduced to contrast with the second and differsfrom it only in that f ′�x� can go to + . The first threeforms are defined for all x ≥ 0 and � ≥ 0, so that thereis no upper bound on x or � . In contrast, the fourthform (Figure 4a) assumes x ≤ xJ and � = f �x� →+ as x → xJ . That is, the link jams flow when x → xJ .

xO

τc

kO

q

xc

τ0

kc

xc /τc

(b)

(a)

τ = f(x)

q = x / τ

τ = τc+ f ′(xc)(x −xc)

τ

xJ

τJ

k J

Figure 3 q = Q�k� Implied by a Nonlinear � = f �x� with f ′�x� NotBounded Above

To compare and relate the functional forms in � =f �x� and q = Q�k�, we have to apply them under thesame conditions, and for that we can consider theirrelationship when the flow and density are constantover time.

Proposition 2. Let the travel-time functions � = f �x�,x ≥ 0, be positive, nondecreasing, and convex as illustratedin Figures 1a–4a. Then, the corresponding flow-densityfunctions q = Q�k� are as illustrated in Figures 1b–4brespectively. More specifically

(a) In each case (Figures 1a–4a), q = Q�k� starts at�k� q�= �0�0�.

(b) Except in the linear case shown in Figure 1a, thereis a unique tangent from �x� �� = �0�0� to the curve � =f �x�. Let (xc� �c) be a tangent point, as in Figures 2a–4a.Then, the flow-density function q = Q�k� has a peak at�kc� qc�= �Lxc� xc/�c� as in Figures 2b–4b.

(c) When � = f �x� is a straight line of slope b, as inFigure 1a, then the flow-density function q = Q�k� isq = Lk/�a + bLk�, which is asymptotic to q = 1/b as inFigure 1b.

(d) If the gradient of f �x� has a finite upper boundf ′�x� ≤ 1/B, then the flow-density function q = Q�k� isbounded below by q = x/��0 +B−1x�= Lk/��0 +B−1Lk� asin Figure 2b. (The lower bound curve q = Lk/��0+B−1Lk�starts from (0, 0) and increases asymptotically to q = B.)

(e) If the gradient f ′�x� does not have a finite upperbound, so that f �x� →+ and f ′�x� →+ as x →+


τ

xO

τc

kO

q

xc

τ0

kc

xc /τc

(b)

(a)

q = x / τ

τ = τc + f ′(xc)(x −xc)

τ = f(x)

xJ

kJ

Figure 4 q = Q�k� Implied by a Nonlinear � = f �x� with Jam xJ atWhich q = 0

(as in Figure 3a), then q → +0 as k → + (as inFigure 3b). That is, the resultant flow-density curve isasymptotic to the k axis in Figure 3b.

Proof. When the flow rate q and density k are con-stant along the link, and q, k, and � are constant overtime, x = q� , hence, q = x/� = x/f �x�. Also, x = Lk,where L is the link length, hence, q = x/f �x� gives q =Lk/f �Lk�=Q�k�, which is the flow-density equation.

(a) k = x/L and q = x/� , hence, both are zero whenx = 0.

(b) If f �x� is convex and f �0� > 0, it follows imme-diately that there is a unique tangent from �x� �� =�0�0� to the curve � = f �x�. �/x is the slope of thechord from (0,0) to any point �x� �� on � = f �x� and,as is easily seen from the figures, the chord has itsminimum slope �c/xc when it is tangent to the convexfunction � = f �x�. However when �/x is at its mini-mum, its inverse x/� is at its maximum.

(c) q = x/� , hence, when � = a+bx, q = x/�a+ bx�=Lk/�a+ bLk�. Then, as k →+ � q → 1/b.

(d) The bound f ′�x� ≤ 1/Bf ′�x� ≤ 1/B implies thatif � = f �x� passes through any point �x1� �1� on astraight line of slope 1/B, then it always remainsbelow (or cannot rise above) that straight line for allx ≥ x1. However � = f �x� starts at the point �x� �� =�0� f �0��, hence, it remains below (or cannot riseabove) the line � = f �0� + B−1x for all x ≥ 0; that is,f �x� ≤ �0 + B−1x for all x ≥ 0. But q = x/� = x/f �x�,

hence, f �x� ≤ �0 + B−1x implies q ≥ x/��0 + B−1x� forall x ≥ 0. That is, q = x/f �x� is always above a curveq = x/��0 +B−1x�. Substituting x = Lk in this, we haveq = Lk/f �Lk� = Q�k� is always above a curve q =Lk/��0 +B−1Lk�.

Finally, note that the shape of the lower boundcurve q = Lk/��0 + B−1Lk� is the same as in Figure 1b,that is; k →+ , q → B.

(e) Recall that q = x/� and let x →+ . By assump-tion, as x → + , f ′�x� = d�/dx → + , and the lat-ter implies � → + is infinitely faster than x → + ,hence, x/� →+0. Thus, x →+ implies q = x/�→+0.Recall that x = Lk, hence, q = x/� → 0 as k→+ .Also, recall that q = x/� = x/f �x� = Lk/f �Lk� =Q�k�, hence, q = Q�k� is asymptotic to the k axis ask →+ . �

Note that when x, the number of vehicles on thelink, is less than xc, the tangent point in Figures 2a–4a,this correspond to the upward sloping part of a flow-density curve (Figures 2b–4b). Similarly, any x beyondthe tangent point in Figures 2a–4a correspond to flowson the downward sloping (congested or unstable)part of a flow-density curve. That does not seem tohave been noted in previous discussion of using the��t�= f �x�t�� model in dynamic traffic assignment.

When inflows are varying over time, the model(2)–(3) does not exclude inflow rates larger than qc =xc/�c, and, indeed, in the existing literature, inflowrates greater than this are commonly assumed. How-ever, we have seen in Proposition 2 that, when flowsare constant over time, the maximum possible flow onthe link is qc. Because this applies all along the link,it applies at the entrance, hence, may be taken as themaximum permitted (capacity) inflow rate for the linkeven when flows are varying over time. If potentialinflows larger than this arrive at the entrance of thelink, they can be prevented from entering the link byassuming a queue just before the entrance to the link,with queue outflow capacity equal to the link inflowcapacity qc. If the arrival rate at the link exceeds qc,the excess is then held in the entrance queue.

3. Numerical ExamplesIn this section, we use numerical examples to illus-trate how the results are affected by the fineness ofthe discretisation. In §§3.1 and 3.2, we assume homo-geneous links, to separate the effects of discretisationfrom those of inhomogeneity, and in §3.3, we illustratethe effects of introducing inhomogeneity. Throughoutthis section, we assume linear and quadratic formsof the � = f �x� model, because these are the formsmost widely used in the literature, and apply theseto given inflow functions. We divide space (the linklength) into n segments, and apply the f �x� model toeach link segment in succession as described in theprevious section.


We commented earlier (just before §1.1) on theexogenous variables and on numerically solving themodel over time for a single link or spatial segment.We make only a brief further comment, because weare not presenting a new solution method. At eachtime t, from t = 0 to t = T , we use (3) to compute x�t�and (2) to compute ��t�. The key to solving the model(2)–(3) is to note that, given u�t∗�, v�t∗�, and x�t∗� attime t∗, then (5) yields v�t∗ + ��t∗��; that is, yields thevalue of v�·� at a time step ��t∗� further ahead. Hence,at each time t = t∗ + ��t∗�, the outflow v�t� is alreadycomputed, hence, (3) can be applied to find x�t� and(2) can then be applied to find ��t�. For the computa-tion, we also introduce further notation, for example,an integer index j = 1� � � � �m, to denote time intervals,each of length �t, so that tj = �j − 1��t is the start ofthe jth time interval. Because the travel-time function��t� is continuous, for a vehicle entering the link seg-ment at time tj , the exit time tj + ��tj �, j = 1� � � � �m,is continuous, hence, can fall inside one of the timeintervals j = 1� � � � �m. Hence the traffic that enters asegment in the jth time interval can exit from the seg-ment spread over one or more of the m time intervals,but from this, the outflow rate in each time intervalcan be easily computed by interpolation.

We assume the link is empty at the initial time t = 0.The results are shown in Figures 5–7. In these figures,the inflow u�t� is the inflows to the first segment ofthe link at time t, and the outflow v�t� is the out-flows from the final segment of the link at time t. Thetravel time ��t� is the time taken to travel from thebeginning of the first segment to the exit of the finalsegment.

3.1. Using a Linear � = f �x� FunctionIn the following examples, we assumed linear travel-time functions.

� = a+ bx (13)

We assume a = 1�1 min, b = 0�02 min/veh, and alsoassume a link length L = 1�2 km (see the appendixfor a justification of these parameter values). For sim-plicity of exposition, the units will be omitted in thelater presentation. To obtain the corresponding flow-density function, for the LWR model, substitute x =Lk and � = x/q = Lk/q in (13) and rearrange, thus,

q = Lk/�a+ bLk�� (14)

As illustrated in Figure 1b, this function starts atthe origin and increases asymptotically to qc = 1/b =50 veh/min.Example 1. With plateau shaped inflow profile.In this example, we assumed inflows rise over time

from zero to a peak at 32, remain there for a time, andfall off again asymptotically to 20. The inflow profile

0 4 8 12 16 20 24

0

5

10

15

20

25

30

35

u(t)

v(t) generated by the LWR model

v(t) generated by τ = f(x) on the 1-segment link




u(t)

,v(t

) (v

eh/m

in)

t (min)

(a) Outflow Profiles

0 4 8 12 16 20 24

1.1

1.3

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

τ (t) generated by τ = f(x) on the 1-segment link




τ (t) generated by the LWR model

τ(t

) (m

in)

t (min)

(b) Travel-Time Profiles

Figure 5 How Discretisation Affects Solution Profiles in Example 1

is chosen, as described in the appendix, to avoid sharpchanges in the inflow rate. More specifically, we let

u�t�=

32 sin�"t/10� 0≤ t < 532 5≤ t < 1020+ 12 sin5�"�t + 4�/28� 10≤ t ≤ 24

(15)

in veh/min. We applied model (13) to these inflowsand computed the link travel times and outflowsat each point in time, using a fine discretisationof time (dividing the time horizon [0, 24] into12,000 time intervals). (Any finer discretisation oftime gives almost identical results.) We then dividedthe link into n = 2 identical segments and appliedthe model again, sequentially to each segment, asdescribed in §1 above. We repeated this using n =


0 4 8 12 16 20 24

0

5

10

15

20

25

30

35

u (t)

v (t) generated by the LWR model

v (t) generated by τ = f (x) on the 1-segment link




u(t

),v

(t)

(veh

/min

)

t (min)


0 4 8 12 16 20 24

1.1

1.3

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

τ (t) generated by τ = f (x) on the 1-segment link





τ(t)

(m

in)

t (min)



2�10�50�100� � � � �550 segments, and in each case,computed the profiles of outflows and travel times.(Note that the maximum feasible number of link seg-ments can be determined as follows, to ensure thatthe free flow travel time (fftt) on a segment does notexceed the length of one time interval. The fftt for thewhole link is a = 1�1 time units, but the time horizon[0, 24] is divided into 12,000 time intervals, hence, thefftt for the whole link consists of 1�1 × 12�000/24 =550 time intervals. If there are n link segments, the ffttfor a link segment is then 550/n in time interval unitsand, because we need this to be greater than or equalto 1 time interval, we have 500/n≥ 1, hence, n≤ 550.)

As the number of link segments n increases, thelink outflow and travel-time profiles converge to limitprofiles, which are also the LWR solution profiles,

0 4 8 12 16 20 24

0

5

10

15

20

25

u(t)

v(t) generated by the LWR model





u(t

),v(t)

(veh

/min

)

t (min)


0 4 8 12 16 20 24

1.1

1.3

1.5

1.7

1.9

2.1

2.3

2.5

2.7






τ(t)

(m

in)

t (min)



as shown in Figures 5a and 5b. To avoid clutter,only a few graph lines are displayed, correspond-ing to n = 1�2�10�550. In Figures 5a and 5b (andalso in later Figures 6 and 7), the lines correspond-ing to n = 550 link segments are not visible becausethey coincide with the LWR graph lines. The solu-tion profiles become very close to the LWR limit pro-files long before the maximum number of segments(550) is reached. This shows that applying the whole-link model � = f �x� sequentially to each segment ofa discretised link can be used to approximate thetraffic characteristics associated with the LWR mod-els. These characteristics include the following. If theinflow u�t� is increasing with time t, the vehicle trajec-tories, and density waves, “fan out,” so that the out-


flow rate v�t� increases more gradually. The converseoccurs if u�t� is decreasing with t, so that the out-flow rate decreases more quickly, and a discontinuityin outflow can occur (a shock wave, Newell 1988). InFigure 5a, we can see such a discontinuity in outflowsat about time t = 19.

To measure convergence to the limit profile (theLWR profile), we computed the mean percentagedeviation (MPD) of the outflow profile from the limitprofile. More formally, let vn�t� denote the outflowscomputed using n link segments and vl�t� denote theoutflow computed in the limit using a large numberof link segments (or computed using the LWR model).Then, let

MPD1= 100T

T∑t=1

∣∣vn�t�− vl�t�∣∣

vl�t��

In the summation, we do not include vn�t� for the ini-tial zero outflows, to avoid a zero denominator. Wealso computed the mean deviation from the LWR pro-file as a percentage of the mean outflow, instead ofthe LWR outflow, and denote this MPD2. Similarly,we computed the MPD for travel times, thus,

100T

T∑t=1

∣∣�n�t�− �l�t�∣∣

�l�t��

The results in Table 1 show that only a small frac-tion of the maximum number of segments is neededto obtain a solution that is within, say, 1% or 2% fromthe limit profile, and this is generally less than theerror already inherent in the given data. When usingthe ��t�= f �x�t�� model in dynamic traffic assignmentfor a network, dividing all links into multiple seg-ments increases the computation, and it is desirable touse as few segments as possible, while having enoughsegments to give an acceptably accurate solution.Example 2. With hill-shaped inflow profile.In the example above, the extended flat peak

allowed the outflow and travel-time profiles to conv-erge to the flat peak. In the current example, weremove the flat peak, by simply letting the inflowstake longer to build up to the peak. More specifically,we let

u�t�={

32 sin�"t/20� 0≤ t < 1020+ 12 sin5�"�t + 4�/28� 10≤ t < 24�

(16)

Table 1 Mean Percentage Differences from the Limit (LWR)Solutions in Figures 5a and 5b

Number of Segments n 1 2 10 550

Outflows t = �0�24� MPD1 8�0948 5�5121 1�6435 0�1487MPD2 6�8970 4�8219 1�5531 0�0386

Travel times t = �0�24� 10�4243 5�9722 1�2992 0�0134

Table 2 Mean Percentage Differences from the Limit (LWR)Solutions in Figures 6a and 6b

Number of Segments n 1 2 10 550

Outflows t = �0�24� MPD1 6�2186 4�2348 1�3278 0�1037MPD2 6�0050 4�1953 1�4154 0�0238

Travel time t = �0�24� 8�7229 5�033 1�1511 0�0131

We repeated the same experiments as in Example 1,using the same linear travel-time model. The resultingoutflows and travel times are shown in Figures 6a and6b. These again show that, as the number of link seg-ments n increases, the solution converges to a solutionthat is also the solution of the LWR model. A mea-sure of convergence; that is, of closeness to the limitor LWR solution, is given in Table 2.

We also experimented with several other inflowprofiles and found similar results and speeds ofconvergence.

3.2. Using a Quadratic � = f �x� FunctionIn the above examples, we assumed linear travel-timefunctions. To show that similar convergence resultsare obtained using a nonlinear travel-time function,we here assume a quadratic function

� = a+ bx+ cx2� (17)

As before, we assume a = 1�1 min, b = 0�02 min/veh, and link length L= 1�2 km. and also assume c =10−4 min/veh2 (see the appendix). To obtain the cor-responding flow-density function for use in the LWRmodel, substitute x = Lk and � = x/q = Lk/q in (17),and rearrange, thus,

q = Lk/�a+ bLk+ cL2k2�� (18)

As illustrated in Figure 3b, this function starts at q = 0when k = 0, increases to a peak at

�kc� qc� =(�1/L�

√a/c�1/�2

√ac+ b�

)≈ �55�2�24�4� (19)

and then bends downward (with negative gradient)to approach the horizontal k axis asymptotically. Thefunction, thus, has two values of k for each value ofq �0 < q < qc�, but a single value of q for each k.Example 3. We assume that the inflow profile for

the link is

u�t�=

22 sin�"t/10� 0≤ t < 522 5≤ t < 1010+ 12 sin5�"�t + 4�/28� 10≤ t < 24

(20)

in veh/min. We repeated the same experiments as inExamples 1 and 2, using the inflows and travel-timefunction and the resulting outflows and travel times


are shown in Figures 7a and 7b. These again showthat, as the number of link segments n increases, thesolution converges to a solution that is also the solu-tion of the LWR model.

3.3. Numerical Experiments with InhomogeneousLinks

We could repeat the above experiments, with inho-mogeneity introduced in the discretised link, butthat would not necessarily give much insight intothe behaviour of the model. Instead, we conducteda series of experiments that illustrate the effects ofintroducing inhomogeneity, and report here on twothat capture the nature of the results obtained. Inthese numerical examples, we assume inhomogeneityover space and not over time. One reason for this isfor comparability with existing related literature, inwhich the travel-time functions are assumed homoge-neous over time.

In the whole-link travel-time model (2)–(3), the linkflow parameters (maximum flow capacity, free-flowspeed or travel time) are included in the form of thefunction f �·�, and implicitly are assumed to be con-stant along the link. In practice, these flow parametersmay vary along a link and, in that case, the parame-ters in the whole-link travel-time model are presum-ably an average of the parameter values along thelink. To see what effect this averaging along the linkhas on the computed profile of travel times and out-flows from the link, we conducted a series of exper-iments. In these, we discretised the link into a largenumber of equal length segments, i = 1� � � � �n, and letf̃i�·� denote the travel-time function for each segment:the functions f̃i�·� were obtained from the whole-linktravel-time function (2) as in (2∗) above. We then com-pared three cases, as follows.Case 1. Let the parameters of f̃i�·� remain constant

along the link, so that f̃i�·� reduces to f̃ �·�. (This, ofcourse, implies that the flow capacity and the free-flowtravel time are the same for all segments.)Case 2. Choose the parameter values of f̃i�·� for

each segment i = 1� � � � �n, so that the flow capacityincreases from segment to segment along the link, butthe mean flow capacity remains equal to the constantflow capacity assumed in Case 1. (The flow capacityis computed as in (19).)Case 3. Use the same parameter values as in Case 2,

but in reverse order, from segment n to 1 instead of1 to n. Thus, the mean flow capacity decreases fromsegment on successive segments along the link, andthe mean flow capacity is again equal to the constantflow capacity assumed in Case 1.

Thus, in the three cases, the mean flow capacityis the same along the link, but the distribution ofthe flow capacity is different. For the three cases, weassumed everything else is the same, including the

inflow profile and number and length of segments.To our surprise, the results (the profiles over timeof link travel times and outflows at the end of thelink) were identical or almost identical in Cases 2 and3. We experimented with linear travel-time functionsand quadratic travel-time functions, and with higherinflows and lower link flow capacities, so that inflowsbecame close to capacity. In all cases, the resultingprofiles at the end of the link were identical or almostidentical in Cases 2 and 3.

What is happening in the above experiments is asfollows. In Case 2, on successive segments along thelink, the flow capacity increases, hence, the speedincreases and the segment travel time decreases. Thereverse is true in Case 3, so that the segment traveltimes start at a lower level than in Case 2 and increasealong the link. The profile of segment travel timesexperienced by a vehicle traversing the link in Case 3is approximately the reverse of that in Case 2, so thatthe mean, over all segments, of segment travel timesis identical or almost identical in both cases.

We have noted that results (the profiles over time oflink travel times and outflows at the exit of the link)were the same or almost the same for Cases 2 and 3.However, these results change, at least a little, if wechange the range over which the flow capacities areassumed to vary in Cases 2 and 3. This is illustratedin Figure 8. Note that Case 1 is simply a special casein which the range of variation in the flow capacity iszero. In Figure 8, we present results for three differ-ent examples, based on three ranges of segment flowcapacities, namely, qc held constant at 27.0 along thelink, qc varying linearly from 31.0 to 23.0 along thelink, and qc varying linearly from 32.0 to 22.0. It isinteresting to note, in Figure 8a, that as the range ofthe segment flow capacities increases, the “fall-off”in outflows just after the peak becomes steeper, andwhen the range is 32.0 to 22.0, the fall-off forms ashock wave.

Figure 8 shows only the profiles, over time, of linktravel times and outflows at the link exit. While this isimportant, it is easier to see what is happening if wealso consider points along the link. Though the pro-files at the link exit were identical or almost identicalin Cases 2 and 3, that is not true at points along thelink. To show this, in Figure 9, we present the profilesof link travel times and link flows at the mid-point ofthe link.

In Figure 9, we see that the (profile of) travel timesto the midpoint of the link is lower in Case 3 thanin Case 2. That is what we would expect. In Case 3,the link flow capacities on the first half of the linkare higher than in Case 2, which allows the traffic totravel faster than in Case 2. On the second half ofthe link, the reverse is the case so that, by the endof the link, the differences have cancelled out and the


0 4 8 12 16 20 24

0

4

8

12

16

20

24

u(t)

v(t) using quadratic τ = f (x) on discretised inhomogeneous link

constant capacity at qc = 27.0

capacity varies from 31.0 to 23.0 or 23.0 to 31.0


u(t

), v

(t)

t (min)


0 4 8 12 16 20 24

0.70

0.99

1.28

1.57

1.86

τ(t) using quadratic τ = f (x) on discretised inhomogeneous link

constant capacity at qc = 27.0



τ(t)

t (min)


Figure 8 How Exit Profiles Are Affected by the Range of Flow CapacityVariation Along the Link in Example 4

travel-time profiles at the exit are the same as shownin Figure 9.

For replicability of the results, in Figures 8 and 9,we note that the data below were used to generatethese. However, we also note that in all the otherexperiments, not reported here, the results had verysimilar properties as discussed above. We chose topresent the particular examples in Figures 8 and 9only because the distances between the lines in thesegraphs were slightly larger and more visible.

For Figures 8 and 9, the whole-link travel-timefunction used was the quadratic form (17). As before,we assume b = 0�02 min/veh and c = 10−4 min/veh2.The link length is given L = 1�2 km but the free-flowtravel time changes as the capacity varies. The inflow

0 4 8 12 16 20 24

0

4

8

12

16

20

24

u(t)

v(t) using quadratic τ = f(x) on discretised inhomogeneous link

v(t) at middle of link in Case 2

v(t) at middle of link in Case 3

v(t) at exit of link in both Cases 2 and 3

u(t)

, v(t

)

t (min)


0 4 8 12 16 20 24

0.45

0.90

1.35

1.80

2.25

2.70

τ (t) using quadraticτ = f (x) on discretised inhomogeneous link

τ (t) at middle of link in Case 2

τ (t) at middle of link in Case 3

τ (t) at exit of link in both Cases 2 and 3τ(

t)

t (min)


Figure 9 Profiles at the Beginning, Middle, and End of the Link inExample 5

profile used was the same as in Example 3, i.e., (20).As before, time was treated as “continuous,” whichmeant dividing the 24-minute time span into 12,000time intervals. The link length of 1.2 km was dividedinto 222 segments.

From Equation (19), the link flow capacity is givenby qc = 1/�2

√ac + b�, hence, to vary the flow capac-

ity for successive segments (in Cases 2 and 3), wecan vary parameter a or b. In the initial experiments,we varied b, but in later experiments (including thosereported here), we varied the free-flow travel-timeparameter a, because it seemed that this generated alarger difference between the results for Cases 2 and 3.Hence, the free-flow travel-time function for each seg-ment i was 222−1 × �1/qi

c − b�2/�4c�, where qic denotes


the capacity of segment i. We let the capacity varylinearly, say, from 23.0 to 31.0, which results in theparameter a in (17) varying from a= �1/qc −b�2/�4c�=1�38 to 0.38.

4. Concluding RemarksWe applied the travel-time model � = f �x� to a linktreated as a whole link or divided into segments.The resulting solution profiles (for travel times andoutflows) approximate the solutions obtained fromthe LWR model. As the number of link segments isincreased, the solution profiles converge to a limit thatis also the LWR solution, as illustrated in Figures 5–7.Measures of closeness to the LWR solution are givenin Tables 1 and 2.

For many purposes, the solutions are even betterthan are implied by the measures in Tables 1 and 2for the following reason. In Figures 5–7, even whenthe outflow profile is a few percent away from thelimit profile, it still captures the shape of the limit pro-file, which may be more important than a small offsetin time. For example, consider two outflow profilesthat are identical, except that one has a few minutestime lag relative to the other. As a result of the timelag, the computed percentage difference between thetwo outflow profiles can be large. However, this maybe misleading, depending on the purpose for whichthe outflow profiles are obtained. If outflow profilesare being computed to estimate infrastructure capac-ity needs, then even if outflows are shifted slightlyover time, this may not matter.

In the discussion and numerical examples in thispaper, we assumed that time is treated as continuous,because in the DTA literature where the travel-timemodel ��t� = f �x�t�� is used, time is treated as con-tinuous. Of course, when solving the model numeri-cally over time, time is divided into small steps anda numerical solution method used because continu-ous analytic solutions generally do not exist or are toocomplex to derive.

Instead of treating time as continuous and treatingthe link as a whole link, or as discretised into seg-ments, we can discretise both time and link length. Weconsider that elsewhere (Carey and Ge 2002), wherewe find that the solutions are closer to the LWR solu-tion, if the link segments and the time steps are coor-dinated as follows. Choose time steps of length equalto the free-flow travel time for the link segment. If thetime steps and segment lengths are not coordinatedin this way, then it takes a larger (often much larger)number of time steps or space segments to achievethe same closeness to the limit solution (LWR solu-tion). This result is of interest in the DTA literaturebecause it is counter to the usual intuitive assump-tion in that literature that a continuous-time model is

more “accurate” than a discrete-time formulation ofthe same model.

In the results (the time profiles of link travel timesand outflows) for a single link, we found that ineach case, the results deviate from the LWR resultsin the same “direction.” For example, compared tothe LWR results, any peaks tend to be “flattened” abit. This flattening decreases as the discretisation isincreased. Consider a series of links, for example, aroute from an origin to a destination. If some flatten-ing of the solution profiles occurs at each link in theseries, then there is a cumulative flattening over theseries of links. In view of this, the larger the distancefrom the origin to the destination, the finer should bethe discretisation if the same level of accuracy is to bemaintained at the destination. Note that this accuracyproblem is not due to any discretisation introducedin this paper: the “flattening” and inaccuracy, relativeto the LWR model, is even greatest when using thewhole-link travel-time model (that is, with undiscre-tised links).

In the whole-link model ��t� = f �x�t��, variousparameters are embedded in f �·�. In calibrating themodel, presumably one either assumes that the linkis homogeneous, or uses parameter values that reflectsome type of mean or average of the parameters atvarious points along the link. For example, observethe mean of the (maximum) flow capacities along thelink and choose the parameters for f �x� to ensure thatf �x� reflects this flow capacity (see end of §3.3). Isusing a mean flow capacity in this way a significantsource of error, as compared to treating the link as aseries of segments each with its own flow capacity? Toinvestigate this, in numerical experiments, we com-pared three scenarios in which the link flow capacity(a) remains constant, (b) decreases, and (c) increasesalong the link, with the mean flow capacity along thelink being the same in each case. In each case, wedivided the link into segments, computed the flowcapacities and, hence, parameters for each segment,and applied the travel-time model to the successivesegments. As expected, we found that, at points alongthe link, the three scenarios yielded different flowrates, and different travel time up to that point onthe link. However, we found that, at the link exit, thetime profiles of link outflows and link travel timeswere almost identical for the three scenarios. This isbecause of letting the mean flow capacity along thelink be the same in each case. Whether the capacitywas increasing, decreasing, or constant along the linkmade little difference at the exit: the effects almostcancelled out. This is reassuring for applying ��t� =f �x�t�� to inhomogeneous links, using appropriatelychosen parameter values reflecting average capacitiesalong the link.


The computation involved in using the link travel-time models in a DTA network model is addressed inseveral of the papers referred to in the introduction.Here, we need only note that discretising each link ina network into, say, n segments yields an expandednetwork with n times the number of links, and n− 1times the number of nodes. However, for each newnode, the conservation equation is simple: at eachtime t, the inflow from the single preceding segmentequals the outflow to the single succeeding segment.Computing flows on this series of n sublinks is muchsimpler than for a similar number of links in a gen-eral network. Hence, the n-fold increase in number oflinks in the network does not increase computationsas much as would an n-fold increase in the numberof ordinary links.

AcknowledgmentsThe authors would like to thank the anonymous refereesand an associate editor for their helpful comments andsuggestions. This research was supported by U.K. Engi-neering and Physical Science Research Council (EPSRC)Grants GR/L/80904 and GR/R/70101, which are gratefullyacknowledged.

Appendix. Travel-Time Functions and Inflow Ratesfor Numerical Examples

In §3, we assumed particular linear and quadratic travel-time functions and inflow rate functions. Because thebehaviour of the model depends on the parameter values,these values should be realistic, reflecting behaviour thatcan occur in practice, hence, we briefly explain the choiceof parameters and their interpretation.

The Linear Travel-Time Function. We assumed that thelinear travel-time function � = a + bx has parameters a =1�1 min, b = 0�02 min/veh, and assumed a link length L =1�2 km. From Equation (14), we note that as k →+ , q →1/b, hence, 1/b is the maximum feasible flow rate for thelink (the link capacity). For single-lane traffic, the inverseof the maximum flow rate (in veh/min) is the minimumheadway (in min/veh). Hence, b represents the minimumheadway; that is, the time from the front of a vehicle passesa point until the front of the next passes the same point.A typical value of minimum headway is 1.2 sec/veh (e.g.,see Gartner et al. 1997, ch. 10, p. 10), which is equivalent to0.02 min/veh. Hence, we let b = 0�02.

The parameter a in � = a + bx is interpreted as the free-flow travel time for the link, because � → a as x → 0. Welet a= 1�1 min, explained as follows. The travel-time model��t� = f �x�t�� does not explicitly take account of the distri-bution of traffic along the link, hence, it implicitly assumesthe distribution is uniform or approximately uniform. Intu-itively, this approximation is more reasonable or acceptableif the link length is not “long” and the rate of inflow tothe link is not changing (usually increasing or decreasing)rapidly over time. We, therefore, chose a link length L =1�2 km rather than, say, a freeway of several kilometers. Fora link of length 1.2 km, we assume a free-flow travel timeof a= 1�1 min, which implies a free-flow speed of 65.45 kmper hour (40.6 miles per hour).

The Quadratic Travel-Time Function. We used the sameparameters a and b for the quadratic travel-time function� = a + bx + cx2 as for the linear function and the samelink length L. We chose the parameter c as follows. From(19), the quadratic travel-time function implies a maximumlink flow qc = 1/�2

√ac + b�. If c = 0, this reduces to the

maximum flow rate qc = 1/b for the linear travel-time func-tion. Hence, the maximum flow rate for the quadratic caseis always smaller than for the linear case (assuming thesame value of b). To ensure that introducing the quadraticterm has a substantial impact, so that the results for thequadratic case are substantially different than those for thelinear case, we chose a value of c so that the capacity inquadratic case is about half that in the linear case. Choosingc = 10−4 min/vec2 gives

qc = 1/�2√

ac+ b� = 1/�2√

1�1× 10−4 + 0�02�

≈ 24�4 veh/min�

which is about half the maximum flow rate qc = 1/b =1/0�02= 50 for the linear case.

It is also interesting to give an interpretation of theparameter c, and the quadratic term, in the quadratic travel-time model. From (19), the density at the peak flow rateis kc = �1/L�

√a/c, hence, c = a/�Lkc�

2. However Lkc = xc isthe number of vehicles on the link when the flow rate is atits peak. Hence, c = a/x2

c , so that cx2 can be replaced witha�x/xc�

2. This equals 0 when x = 0 and equals a when x = xc .If x exceeds the xc associated with the peak flow rate (seeFigure 2 or 3), then the a�x/xc�

2 term adds more than a tothe travel time.

Inflow Rate Profiles. As noted above, the � = f �x� modelgives a better approximation if the inflow rate is not chang-ing sharply over time. Hence, to give a “fair” illustrationof the model, as used in the existing literature, in Exam-ple 1, we chose an inflow profile that takes approximatelyfive times the free-flow travel time to build up from nearzero to a peak (i.e., 5×1�1 min). Similarly, we let the inflowrate decline from the peak, to near zero, over a time spanof approximately 10 times the fftt (i.e., 10× 1�1 min).

As already noted, the linear travel-time function has aflow capacity 1/b = 1/0�02 = 50 veh/min. To not exceedthat, we assume a maximum inflow rate of 32 veh/minin Examples 1 and 2. From Equation (19), the flow capac-ity implied by the quadratic travel-time function is qc ≈24�4 veh/min, hence, we can assume that this is the capac-ity inflow rate. The inflow rate used in Examples 1 and2 (with a linear travel-time function) has a maximum at32 veh/min, hence, is not suitable here. We, therefore,assumed a maximum inflow rate of 22 veh/min.

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