Closed-form Solutions of Generalized Greenshield Relations for the Social and User Optimums of a Single Entry Corridor

Closed-form Solutions of Generalized Greenshield

Relations for the Social and User Optimums of a Single

Entry Corridor

Michael Maroun

Department of Physics and Astronomy, Department of MathematicsUniversity of California, Riverside

Richard Arnott

Department of EconomicsUniversity of California, Riverside

August 14, 2013

Abstract

We present closed-form expressions relevant to the economic optimization of a singleentry positive finite length road using traffic modelled by a Lighthill-Whitham-Richards(LWR) continuous fluid model with a generalized Greenshields’ relation, γ ą 0. Theγ “ 1 solutions have been primarily addressed in DePalma and Arnott (2012) andNewell (1988). This presentation gives the quite general closed-form results for sucha model for all γ ą 0, i.e. the results are not restricted to the linear Greenshields’relation between velocity and density. We also discuss briefly, in the conclusion, thephilosophical and theoretical meaning of traffic modelled by fluid flow, such as theLWR theory and its conceptual generalizations.

1 Introduction

The Lighthill-Whitham-Richards (LWR) fluid flow, is used to model the flow of traffic (LWR-theory) in a one dimensional single entry (at x “ 0) and single exit (at x “ L) corridor oflength L ą 0. This model is not new and has been worked before but the closed form

1

relations herein are unique and not found elsewhere in the literature. In particular, theexpressions for the flow rates are determined for the generalized Greenshields’ relation,

v

v0“ 1 ´

ˆ

k

kj

˙γ

,

with the parameter gamma parametrizing an entire family of velocity-density relations. Theclassical function theory solutions are found for all γ ą 0 when they exist.

To begin the discussion of the model presented here, it is therefore desirable to start fromthe definition of the quantities of interest and the associated continuity equation governingthe LWR continuous fluid model for vehicle congestion of our single entry finite lengthroadway. We imagine the roadway as starting at x “ 0 and ending at x “ L ą 0. Furtherlet the following functions have the following meanings.

qpx, tq “ the output flow rate at point, x and time, t (1.1)

apx, tq “ the input flow rate at point, x and time, t (1.2)

kpx, tq “ the density of vehicles at point,xand time, t (1.3)

vpx, tq “ the velocity at point, x and time, t (1.4)

Qptq “ the cumulative number of cars at the point, x “ L and time t (1.5)

qptq “ dQptqdt

the output rate at x “ L (1.6)

Aptq “ the cumulative number of cars at the point, x “ 0 and time t (1.7)

aptq “ dAptqdt

the input rate at x “ 0 (1.8)

We will also use the following symbols to represent positive real numbers.

L “ the positive length of the road (1.9)

N “ the total number of cars which must arrive at x “ L (1.10)

t̄ “ the length of time of the traffic rush hour to be determined (1.11)

γ “ a parameter relating the function v to the function k (1.12)

v0 “ the free-flow velocity (1.13)

qm “ the maximum allowable flow rate of the roadway (1.14)

kj “ the jam density (1.15)

α1 “ the cost per unit time of travel (1.16)

α2 “ the cost per unit time early: α2 ĺ α1 (1.17)

σ “ 1 ´ α2

α1

the unitless cost parameter (1.18)

2

The problem is then to determine the flow rate qptq consistent with a conservation principle(the continuity equation) and compatible with an optimization in two separate cases that will,paq minimize the costs of the entire social system and pbq minimize the costs of the individualusers. To see that there is a non-trivial optimization problem at hand it is necessary to discussthe meaning of α1 and α2.

By the ‘cost per unit time of travel’, α1, one means the costs associated with making thetrip, fuel, vehicular aging etc. The longer in time a trip takes the higher the costs to theuser. This is precisely the meaning of α1. Likewise, by the ‘cost per unit time early’ onemeans the price in time lost per unit time of arriving early to one’s work, whereby one isnot yet being paid. This is denoted α2. Hence, the total economic cost of such a system canbe stated as:

t2ż

t1

!

α1 rAptq ´ Qptqs ` α2Qptq)

dt, (1.19)

where it will always be assumed that α2 ĺ α1. Incidentally, this inequality arises from thelogically obvious fact that time on the road is more restrictive than time early. In fact, ifone values time on the road inconsistent with the inequality stated, then one has always theopportunity, when arriving early, to get back on the road. But the existence of this veryoption would then force the inequality to become true. The reverse is of course not possible,namely, one cannot turn time en route into time early.

Additionally, it should be noted that the number of drivers N is conserved and that wewill not allow for late arrivals. Thus all drivers must arrive by their work start time. There isa continuity equation as well as a connection between the density and outflow rates throughthe velocity function in the LWR theory. The continuity equation is:

Bqpx, tq

Bx`

Bkpx, tq

Bt“ 0. (1.20)

The relationship assumed to hold for all x and t between density, flow rate and velocity is:

qpx, tq “ vpx, tq ¨ kpx, tq (1.21)

We study the class of velocity-density relations known as the generalized Greenshields’ rela-tion,

v

v0“ 1 ´

ˆ

k

kj

˙γ

. (1.22)

Notice that v “ 0 when k “ kj and that v “ v0 when k “ 0. Further, we have that otherthan the inequality α2 ĺ α1, there is no dependence of one cost on the other. Hence itis assumed that drivers arrive in the order of their work start times but that this is not

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mandated by our dynamics1. The reader is also reminded for convenience of the auxiliaryrelation found in equation (2.2) of Newell (1988), relating the jam density kj, the maximumflow rate qm and the free flow velocity v0 as:

qm “v0kjγ

pγ ` 1q1` 1

γ

(1.23)

Below is a graph of the velocity function, vpkq versus the density, k, followed by a graph ofthe outflow function qpkq versus the density, k, each for various values of gamma indicatedin the captions below.

0 1 2 3 4k0.0

0.2

0.4

0.6

0.8

v0=1.0

vHkL

Figure 1: In normalized units, the velocity vpkq for k ĺ 4. Values of gamma are from left to right as:

γ “ 7

24, 1

2, 4

5, 1, 2, 3.

Lastly, in order to produce graphs of various functions it is convenient to use a standard

1Meaning, individual drivers, when viewed as constituents of a quantum ensemble, may interchangethemselves in accordance with uncertainty fluctuations but that this viewpoint is neither necessary norinherent to the equations of motion. In fact, the equations of motion are of Cauchy type and thus come witha natural semi-group time evolution that enforces a time-ordering. Hence our assumption is natural.

4

0 1 2 3 4k0.0

0.5

1.0

1.5

2.0qHkL

Figure 2: In normalized units, the outflow qpkq for k ĺ 4. Values of gamma are from bottom to top as:

γ “ 7

24, 1

2, 4

5, 1, 2, 3.

normalization for the set of constants, which we shall refer to as the normalized units:

L “ 1 (1.24)

N “ 1 (1.25)

v0 “ 1 (1.26)

kj “ 4 (1.27)

qm “ 4γ

pγ`1q1`

1γ

(1.28)

α1 “ 1 (1.29)

α2 “ 1

2. (1.30)

σ “ 1

2(1.31)

Notice that in equation (1.28) that the value of qm has been chosen to depend upon thevalue of γ. Thus heretofore, qm :“ qmpγq “ 4γ

pγ`1q1`

1γin normalized units. This particular

normalization choice is such that when γ “ 1 then qm “ 1.2 In order to determine variationin γ, either kj or qm must be fixed (but not both) while the other remains free to vary asgamma changes. Since the solutions are sought for the inflow and outflow rates, the criticaljam density kj is chosen to be fixed while the maximum flow rate qm can vary with gamma.That is to say, the flow can change dynamically but not the road structure as is natural inreal life applications.

To summarize, we refer to the diagram below. Our goal is to determine the flow rates,aptq and qptq, where qpx, tq obeys the LWR equation (1.20), subject to the optimization ofeconomic costs, (1.19), for a single entry (and exit) road of length L ą 0.

2This choice of normalization is based on one due to Elijah DePalma

5

Laptq

x “ 0 x “ L

inputrate

qptq

outputrate

direction of trafficflow along road

Diagram of a single entry length L ą 0 corridor

2 The Continuity Equation

It is useful and convenient to examine the full space-time solution to the continuity equationfor arbitrary positive gamma. For this purpose it is simplest to solve the resulting firstorder nonlinear partial differential equation for the density function, kpx, tq. The continuityequation from (1.20) reads:

Bqpx, tq

Bx`

Bkpx, tq

Bt“ 0. (2.1)

The relation between density, kpx, tq and flow rate, qpx, tq, for positive gamma that gener-alizes Greenshields’ relation is:

qpx, tq “ qmpγ ` 1q1` 1

γ

γ

ˆ

kpx, tq

kj

˙ „

1 ´

ˆ

kpx, tq

kj

˙γ

, (2.2)

and thus one has for the partial differential equation (PDE) (2.1):

v0B

Bx

„

kpx, tq

ˆ

1 ´kγpx, tq

kγj

˙

`Bkpx, tq

Bt“ 0. (2.3)

First make the substitution, owing to (2.2) and (1.21), which is essentially the Greenshields’relation,

kpx, tq “ kj

„

1 ´vpx, tq

v0

1

γ

, (2.4)

to obtain,

rpγ ` 1q vpx, tq ´ γ v0sBvpx, tq

Bx`

Bvpx, tq

Bt“ 0. (2.5)

Finally, perform the linear translation and scaling of the velocity function as follows,

vpx, tq “ 1

γ`1rupx, tq ` γ v0s ðñ upx, tq “ pγ ` 1qvpx, tq ´ γ v0, (2.6)

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yielding,

upx, tqBupx, tq

Bx`

Bupx, tq

Bt“ 0 (2.7)

This equation is well known as Burgers’ equation. The equation has a family (parametrizedby gamma, i.e. the unspecified constants in this current context) of obvious particularsolutions pertaining to a specific form of initial conditions (to be summarized in Table 1below), given by:

upx, tq “

ˆ

ax ` b

at ` c

˙

, (2.8)

where a, b and c are arbitrary real constants. The density function, kpx, tq, is then obtainedby substituting (2.8) into the equation on the left of (2.6) and then (2.6) into (2.4). Substi-tuting this expression for the density into equation (2.2), one arrives at the expression belowfor the family of solution outflow rates qpx, tq.

qpx, tq “ qm

„

1 `1

γv0

ˆ

ax ` b

at ` c

˙ „

1 ´1

v0

ˆ

ax ` b

at ` c

˙ 1

γ

. (2.9)

If one were to impose a condition such as qpL, Lv0

q “ 0 then this would determine one of thethree constants yielding a relation such as b “ v0c, and indeed this will be the case. Thereason is that this initial condition for the outflow gives the last time t ą 0, for which qptqis still zero. The time, t “ L

v0, corresponds to the time it takes the first car to arrive, which

travels at the maximum constant free flow velocity over the length of the corridor.To determine the solution more specifically one would need to impose additional infor-

mation such as a function in time at a particular point in space, i.e. up0, tq, for example.This solution should be compared to the solution obtained for generalized Greenshields’ inthe case of the social optimum for positive gamma given in §3. One then can see how thisparticular family of solutions is ‘the’ family of solutions when the initial function is spec-ified by the social optimum. The social optimum is an extremization problem in time forthe outflow function. Thus, the output of the extremization problem is an initial conditionfunction in the sense that it specifies how the outflow must behave in time for fixed spatiallocation in order to minimize the costs. That is, there in fact exists a compatible family ofsolutions of the continuity equation consistent with the extremization problem to be shownbelow in §3. Also since qptq is the outflow, it follows that qptq “ 0 for all t ĺ L

v0. Thus for the

continuity equation, the initial conditions can be summarized as follows. Notice that thesedata are compatible with the relation q “ v k.

A more general solution of this PDE is obtained by the method of characteristics. Theinitial conditions will be the restriction of the 2-dimensional solution surface to a curve. Thismethod of characteristics follows closely that of Levandosky (2002). It is quite standard and

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k “ 0 qpk “ 0q “ 0 vpk “ 0q “ v0kpt “ 0q “ 0 qpt “ 0q “ 0 vpt “ 0q “ v0

Table 1: The initial condition for the continuity equation. The first row are the data for functions of thedensity, k, while the second row are the data for functions of time, t.

is also found in many texts concerning quasilinear3 partial differential equations. First,we perform a change of variables from the independent variables px, tq to a pair of newindependent variables pr, sq. This is tantamount to a change of space-time coordinates fora physical system. Now we let up0, tp0, sqq “: u0psq be the initial condition4. The system,(2.7) u ux ` ut “ 0, can be put in the form:

F px, t, u, ux, utq “ 0,

where ux :“ BBxu and ut :“

BBtu. The initial condition is then the restriction of upx, tq to the

specified curve u0psq. With our change of variables, we now regard our previous functions ofpx, tq as new functions depending on the new pair of independent variables pr, sq, as follows.

upxpr, sq, tpr, sqq “: zpr, sq

uxpxpr, sq, tpr, sqq “: ppr, sq

utpxpr, sq, tpr, sqq “: qpr, sq

Just as well, we will also call the initial condition functions: xp0, sq “: β1psq and tp0, sq “:β2psq. Concretely applying this method of characteristics to our system, we get:

0 “ u ¨ ux ` ut

“ F px, t, u, ux, utq

“ F px, t, z, p, qq

“ zp ` q

Our procedure has in effect created a system of ODE’s with initial conditions, as such:

d

drxpr, sq “ Fp “ z | xp0, sq “ β1psq “ 0 (2.10)

d

drtpr, sq “ Fq “ 1 | tp0, sq “ β2psq “ s (2.11)

d

drzpr, sq “ pFp ` qFq “ 0 | zp0, sq “ u0psq, (2.12)

3The Burgers’ equation is said to be quasilinear as it is linear in the first order time derivative only.4Notice that this is slightly different from the ‘usual’ initial conditions for the Burgers’ equation in the

sense that the space variable is held fixed while a function in time is what is being specified to determine asolution.

8

where Fp and Fq again stand for the partial derivatives of F with respect to p and q respec-tively. Subject to the initial conditions, one obtains as the solutions to the system of ODE’sabove,

xpr, sq “ u0psq r

tpr, sq “ r ` s

zpr, sq “ u0psq

The general solution can then be stated as:

upx, tq “ u0pspx, tqq, (2.13)

where u0psq “ xpt´sq

. This is a parametrized solution. Of special interest, as we will soon seein §3 below on the social optimum, is an initial time distribution with the functional form

u0psq “1

As ` B, (2.14)

with A,B P R. The reader can verify that this leads to an expression for upx, tq as follows,

upx, tq “Ax ` 1

At ` B. (2.15)

This equation should be compared with equation (2.8) as a valid solution of equation (2.7),subject to the restrictions of the social optimum as stated in the next section. To see thatequation (2.8) and (2.15) are indeed identical, multiply the numerator and the denominatorby 1

band redefine a

b“ A and c

b“ B.

3 The Social Optimum

Again, we use γ to denote an entire family of velocity-density relations, which generalize theGreenshields’ relation. That is, consider the relationship between velocity and density asstated in the introduction, given essentially in (2.4) or as given in equation (2.1) of Newell(1988), i.e.

v “ v0

„

1 ´

ˆ

k

kj

˙γ

, (3.1)

where v0 is the free flow velocity and kj is the value of the jam density as defined previously.Defining the following function for the change in velocity as a function of density,

∆vpkq “ v0 ´ vpkq. (3.2)

9

One can then calculate the elasticity,

E∆vpkq, k :“B∆vpkq

Bk¨

k

∆vpkq. (3.3)

From (3.1), one obtains for the elasticity,

E∆vpkq, k “ γ. (3.4)

We are interested in values of γ ą 0, for the following reasons.

• It is remarked in Newell (1988) that multi-lane highways are modelled better by valuesof γ closer to 2 or 3 rather than 1.

• It has been noted in Inman (1978) that empirical data from real life roadways havemodest agreement with models that use values of γ less than one. This is not incontradiction to the first point above because this empirical conclusion is based onmeasurements of traffic made on single lane suburban roads.

• It seems that γ ă 1 corresponds to k ! kj , whereas γ ą 1 corresponds to k „ kj.

Consequently, it is desirable to investigate the social optimum for arbitrary values of γ ą 0.It is a straightforward matter to show that the variational problem, as defined in Newell

(1988), equation (A3) of Appendix A, is equivalent to the following variational problemwith an explicit linear time dependent function multiplying the outflow rate in place of thecumulative outflow rate. This follows from integration by parts and a corollary from thecalculus of variation5 that the function which extremizes a linear functional is unchanged bytranslations of the functional by a constant. That is,

minQPV

$

&

%

t2ż

t1

«

α2Qptq ` Lα1

9Qptq

vp 9Qptqq

ff

dt

,

.

-

“ minqPV

$

&

%

t2ż

t1

„

´α2tqptq ` Lα1

qptq

vpqptqq

dt

,

.

-

, (3.5)

where qptq :“ 9Qptq, and V is a suitable closed convex subspace of a normed vector spaceof functions.6 The function qptq can be replaced as an elementary change of variables by a

function of composition as qpvptqq “ qpvq “ qmγ`1

γpγ ` 1q

1

γ vv0

”

1 ´ vv0

ı 1

γ

, as a consequence of

5See for example Gel’fand and Fomin (1991 reprint)6We have tacitly assumed that the vector (function) space is closed under pointwise operations such as

addition and multiplication, as well as differentiation. Indeed this is reasonable and even typical of suchfunction spaces for classical solutions to PDEs.

10

substituting (2.4) into (2.2). While, the function vpqptqq “ vptq can instead be taken as theindependent variable in such a way that BI

Bq“ BI

BvBvBq. The result is that the total economic

costs can be minimized by,

minQPV

$

&

%

t2ż

t1

I rQ, q, ts

,

.

-

:“ minvPV

$

&

%

t2ż

t1

„

´α2tqpvptqq ` Lα1

qpvptqq

vptq

dt

,

.

-

. (3.6)

By the calculus of variation, we then have,

minQPV

$

&

%

t2ż

t1

I rQ, q, ts

,

.

-

ùñ ´BI

BQ`

d

dt

BI

Bq“ 0,

where the symbol ùñ implies the functional variation has been performed leading to theEuler-Lagrange like statement above. This leads to the trivially integrable differential equa-tion for vptq, i.e.,

´BI

BQ`

d

dt

BI

Bq“ 0 `

d

dt

„

BI

Bv

Bv

Bq

“ 0.

Consequently,BI

Bv“ g

Bqpvq

Bv,

where g is a real constant to be fixed by the initial conditions v´

Lv0

¯

“ v0. One then obtains

the function v, as a function of t and γ,

vptq “v0γ

γ ` 1

„

1 `Lα1

γpv0α2t ` Lα1 ´ Lα2q

. (3.7)

3.1 The Outflow and Cumulative Outflow Functions, qptq and Qptq

It is then the case from equations (2.2) and (3.1), that the outflow is given by:

qptq “ qm

„

1 `Lα1


„

1 ´Lα1

v0α2t ` Lα1 ´ Lα2

1

γ

. (3.8)

Hence, by direct integration one may obtain, for arbitrary γ ą 0, a closed form result forthe cumulative outflow function Qptq. To see this, make the substitution,

fptq “

ˆ

1 ´Lα1


˙ 1

γ

(3.9)

11

and notice that,´

t ´ Lv0

¯

9fptq “Lα1

γpv0α2t ` Lα1 ´ Lα2qfptq, (3.10)

where 9fptq :“ ddtfptq. Thus one has for the cumulative outflow:

Qptq “ qm

ż

#

„

1 `Lα1


„

1 ´Lα1


1

γ

+

dt (3.11)

“ qm

ż

”

fptq `´

t ´ Lv0

¯

9fptqı

dt (3.12)

Consequently, we obtain:

Qptq “ qm

ˆ

t ´L

v0

˙ „

1 ´Lα1

α2v0t ` Lα1 ´ Lα2

1

γ

. (3.13)

The above solution curves, in terms of γ, have been subject to the initial conditions,qp L

v0q “ 0 and Qp L

v0q “ 0, @ γ ą 0. In addition, the relation Qpt̄q “ N , with N known and

t̄ unknown, determines the length of the rush hour t̄ as follows. Making use of 1 ´ α2

α1

“ σ,we have,

Qptq “ qm

ˆ

t ´L

v0

˙

»

–

p1 ´ σq´

t ´ Lv0

¯

p1 ´ σq t ` σ Lv0

fi

fl

1

γ

. (3.14)

The length of the rush hour is then the solution to the following algebraic equation,

Qpt̄q “ N “ qm

ˆ

t̄ ´L

v0

˙

»

–

p1 ´ σq´

t̄ ´ Lv0

¯

p1 ´ σq t̄ ` σ Lv0

fi

fl

1

γ

, (3.15)

for each value of γ. For convenience let, tR “ t̄ ´ Lv0, one then has that the zeros of:

pqmqγ p1 ´ σq tγ`1

R ´ p1 ´ σqN γ tR ´L

v0N γ “ 0, (3.16)

which satisfy tR ą 0 i.e. t̄ ą Lv0, give the length of the rush hour t̄ “ t̄R ` L

v0ą L

v0.

One can consider separately each instance of equation (3.16) whenever γ P Q the set ofrational numbers, by first considering γ P N, the set of natural numbers 1, 2, 3, . . . and then

considering 1

γP N and so on. If in particular, γ P N then

´

4γ

γ`1

¯γ

P Q. Hence equation (3.16)

12

will have integer coefficients. After an obvious application of Descartes’ rule, one finds thatthere can be only one positive real zero within the allowed interval and that this is robustfor any real coefficients. Thus, it is easy to see that there will be only one real root thatsatisfies the inequality, tf ą 0 while Qptq ą 0 and subject to the optimization. Hence t̄ isunique in these cases.

Lastly, there is the technical issue of γ ą 0 but not rational. Since we are only interestedin tR ą 0, we do not have to worry about the exponentiation of negative real numbersto arbitrary real number powers, which are generally complex. Instead, we only concernourselves with the loss of polynomial behavior. Indeed, if γ is irrational, then equation(3.16) will not be a polynomial. Nonetheless, its coefficients are real and positive and onecan see that by examining the value of the function at tR “ 0 and then at tR “ 1, thelocation of at least one positive zero is easily seen to exist whenever,

´qm

N

¯γ

´ 1 ąL

v0 p1 ´ σq.

But since γ ą 0 implies γ ` 1 ą 1, one can see that the positive real exponent of equation,(3.16), is always greater than 1. This implies there will always be a positive real time tRsuch that there is at least one positive real zero of the equation (3.16), though one cannotclaim uniqueness from purely algebraic considerations. This perspective suffices to decidewhether tR will be greater or less than 1.

To complete the proof of uniqueness of the rush hour time, tR for all positive real γ,one need only consider the concavity of the curve, the first derivative and the asymptotictendency for large tR. Owing to the middle term of equation (3.16) having exponent of 1,it follows immediately then that the function has one global minimum on the set of non-negative real numbers. The curve begins negative at tR “ 0 and approaches `8 as tR tendsto `8 with no other turning points. Therefore, there is exactly one and only one positivereal zero of the function for all γ ą 0. This completes the proof of the uniqueness of therush hour time, tR ą 0.

Thus far, we have completely specified the cumulative outflow function Qptq for all timet ą 0, since the function vanishes for t ĺ L

v0and Qptq “ N for t ľ t̄. It is also noteworthy

to mention the γ-asymptotic properties of equation (3.15). We have: t̄γÑ8ÝÝÝÑ 1

v0pNkj

` Lq and

t̄γÑ0ÝÝÑ 8.Below is a graph of the normalized function Qptq for chosen values of γ “ 7

24, 12, 45, 1, 2, 3.

The corresponding lengths of the rush hour are given approximately and respectively as,t̄ « 7.61, 4.88, 3.46, 3, 2.09, 1.803, obtained from using the normalized units in equation

13

(3.16), i.e.,1

γ ` 1

ˆ

4γ

γ ` 1

˙γ

tγ`1

R ´ tR ´ 2 “ 0.

0 1 2 3 4 5 6 7 8t0.0

0.2

0.4

0.6

0.8

N=1.0

QHtL

Figure 3: In normalized units, the cumulative outflow Qptq for t ľ 1. Values of γ are from right to left as:

γ “ 7

24, 1

2, 4

5, 1, 2, 3.

3.2 The Inflow and Cumulative Inflow Functions, aptq and Aptq

Turning our attention now to the inflow rate and its associated cumulative function, from theconservation of cars we have an equality between the input rate and the output rate of carson the road, i.e. aptq “ dAptq

dt“ dQpt1q

dt1 “ qpt1q. Here the times are such that t̄ ą t1 ą t ą 0,since the rush hour ends at t̄ and the output takes place later in time than the input fromcausality, i.e. t is the departure time and t1 is the arrivale time. The condition of the socialoptimum (3.6) places a restriction on the relationship between the times t1 and t. In general,one would expect that the relationship as found in equation (2.7) of Newell (1988) occurswhen,

dq

dk“

L

t1 ´ t(3.17)

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This expression is essentially the group velocity of the wave or the speed of the ensemble ofvehicles. It is just the ratio of the length of the road to the time difference between inputand output. This is the meaning of the expression q “ kv and the heart of the LWR theory.Thus insofar as the model is justifiable, the relationship between q and k for arbitrary γ asa result of the social optimum, give rise to the expression that:

dq

dkpvq “ upvq “ pγ ` 1qv ´ γv0 “

dq

dvpvq

dv

dkpvq. (3.18)

This expression can be obtained in our current context of the social optimum by noticing thatequations (3.7) and (3.8) imply (or alternately and more generally from equations (1.21),(1.23) and (2.2)):

qpvq “ qmγ ` 1

γ

v

v0

„

pγ ` 1q

ˆ

1 ´v

v0

˙ 1

γ

(3.19)

Equations (3.18), (3.17) together with equation (3.7) then give the relationship between t1

and t in the social optimum regime as,7

tpt1q “ σt1 ´ σL

v0, (3.20)

where we define the quantity 1 ´ α2

α1

“: σ and assume 1 ą σ ą 0, i.e. α1 is strictly greaterthan α2. It is useful to have the inverse at hand as well:

t1ptq “1

σt `

L

v0, (3.21)

In deriving these relations, it is essential that q be regarded as a function of t1 and not t, forreasons which will be clearer in the next section on the user optimum regime. Further, hadit not been for the functional restriction of the velocity function as a function of time, t1, tothe form (3.7) then the relation (3.20) would not hold and as we shall also see in the useroptimum regime such a linear relationship cannot be relied upon. The consequence is a firstorder nonlinear delay differential equation.

Next, the inflow functions are found using relation (3.21) in relation (3.8), so as to obtainaptq “ qpt1ptqq, integrating then gives the cumulative inflow. However, there is one issue andthat is when the rush hour ends at t1 “ t̄. There will be a discontinuity in the outflow, whichdepends on the time t. This means that there is a critical phenomenon at t “ σpt̄´ L

v0q. We

require the inflow and its cumulative function to remain smooth by pushing the discontinuityin q to the end of the rush hour when t “ t̄. That is to say, one demands that aptq be a

7The equalities, (3.17), (3.18), and (3.19) are true quite generally irrespective of the social optimum

15

piecewise k-smooth function during the rush hour, i.e. aptq P Ck p r 0, t̄ s q, with k P N, theset of natural numbers, i.e. t1, 2, 3, . . .u.8

Now to construct the second nontrivial portion of the inflow, aptq, one takes the non-zeropart of the function qptq as found in (3.22), sets α1 “ α2 “ 1, reflects the function aboutthe qptq vertical axis and translates to the left by t̄. Succinctly, we have aptq “ qpα1 “ α2 “1, t̄ ´ tq, where qptq has been considered a function of α1 and α2 as well as the time t andshould not be confused with the space-time outflow qpx, tq.

Lastly, we calculate the final part of the cumulative inflow Aptq. This is done by integrat-ing the expression for the second non-trivial part of aptq described above, then applying the

initial conditions A´

t̄ ´ Lv0

¯

“ N . One can verify that this expression is correct by noting

that Apσpt̄ ´ Lv0

qq “ σN . A summary of the piecewise functions qptq, Qptq, aptq and Aptqsans normalization and with σ ą 0 is given below in Table 2.

We now graph normalized functions of Aptq for the previous choices of γ “ 7

24, 12, 45, 1, 2, 3

(Figure 4), and produce a graph of both Aptq and Qptq (Figure 5). Notice in the graphbelow, Q(t) has a discontinuous derivative at t “ t̄, while Aptq is pk`1q-smooth throughoutthe rush hour (k=4)9. With all of the desired functions constructed as smoothly as possibleand specified for all times 0 ĺ t ă 8, this completes our analysis of the social optimum.

8Forcing smoothness by pushing discontinuities in the outflow to the end of the rush hour avoids thepresence of shock waves. A consequence is that one need not invoke any weak versions of the dynamics andthus in turn can avoid any extra conditions such as the so-called entropy condition as in Newell (1993), orDePalma and Arnott (2012) altogether.

9Here k is the index of the space of functions, Ck, that indicate the functions of k continuous derivatives,and is not to be confused with the density.

16

qptq “

$

’

’

&

’

’

%

0 ´8 ă t ĺ Lv0

qm

”

1 ` Lα1

γpv0α2t`Lα1´Lα2q

ı ”

1 ´ Lα1

v0α2t`Lα1´Lα2

ı 1

γ Lv0

ĺ t ĺ t̄

0 t̄ ă t ă 8

(3.22)

Qptq “

$

’

’

&

’

’

%

0 ´8 ă t ĺ Lv0

qm

´

t ´ Lv0

¯ ”

1 ´ Lα1

α2v0t`Lα1´Lα2

ı 1

γ Lv0

ĺ t ĺ t̄

N t̄ ă t ă 8

(3.23)

aptq “

$

’

’

’

’

’

’

&

’

’

’

’

’

’

%

0 ´8 ă t ĺ 0

qm

”

1 ` Lpα1´α2qγpv0α2t`Lα1´Lα2q

ı ”

1 ´ Lpα1´α2qv0α2t`Lα1´Lα2

ı 1

γ

0 ĺ t ĺ σpt̄ ´ Lv0

q

qm

”

1 ` Lv0pt̄´tq

ı ”

1 ´ Lv0pt̄´tq

ı 1

γ

σpt̄ ´ Lv0

q ĺ t ĺ t̄ ´ Lv0

0 t̄ ´ Lv0

ĺ t ă 8

(3.24)

Aptq “

$

’

’

’

’

’

’

&

’

’

’

’

’

’

%

0 ´8 ă t ĺ 0

qmt”

1 ´ Lpα1´α2qv0α2t`Lα1´Lα2

ı 1

γ

0 ĺ t ĺ σpt̄ ´ Lv0

q

N ` qm

´

t ´ pt̄ ´ Lv0

q¯ ”

1 ´ Lv0pt̄´tq

ı 1

γ

σpt̄ ´ Lv0

q ĺ t ĺ t̄ ´ Lv0

N t̄ ´ Lv0

ĺ t ă 8

(3.25)

Table 2: Summary of the solutions to the social optimum outflow function qptq, the cumulative outflowfunction Qptq, the inflow function aptq, and the cumulative inflow function Aptq, for all t P R.

17

0 1 2 3 4 5 6 7t0.0

0.2

0.4

0.6

0.8

N=1.0

AHtL

Figure 4: In normalized units, the cumulative inflow Aptq for t ľ 0. Values of gamma are indicated from

right to left as: γ “ 7

24, 1

2, 4

5, 1, 2, 3.

18

0 1 2 3 4 5 6 7 8t0.0

0.2

0.4

0.6

0.8

N=1.0

QHtL and AHtL

Figure 5: In normalized units, the cumulative flow curves Qptq (solid) and Aptq (dotted) for t ľ 0. Values

of gamma are indicated from right to left as: γ “ 7

24(orange), 1

2(black), 4

5(blue), 1 (violet), 2 (green), 3

(red).

19

4 The User Optimum

The primary dynamics of the user optimum is governed by the relation between the cumu-lative in and out rates. This equation was used implicitly throughout the social optimumanalysis but is most prominent in the user optimum. By ’user optimum’ we mean the optimalbehavior of an individual driver so as to minimize that individual’s costs of commute. Asa result, no driver will benefit from altering their departure time. This also will imply thateach user then necessarily has the same cost of commute, since if this were not true eachcommuter would then have an incentive to try to alter their commute so as to attempt torealize the minimum or at least a suitable infimum. In terms of calculation, this optimizationis accomplished by constraining the dynamics of the LWR flow to satisfy this user constraint.The result is a functional delay equation on the outflow qptq, as will be shown below. Thusto summarize, the user optimum is the equilibrium of an equal trip-price condition and thedynamics will be ordered such that the first users in are the first users out, a.k.a. we assumeFIFO ordering.

It is important to recall that in the social optimum section, none of the solutions are

analytic10 through the entire rush, t̄ ą t ą 0. In fact, the solutions in (3.22), (3.23), (3.24)and (3.25) are only analytic in small sub-intervals contained in the rush. This comes as nosurprise because the functions, upon inspection, behave as fpxq „ 1

px´aqp, p ą 0, a P R,

which clearly is not differentiable even in a neighborhood around the point x “ a. Onthe subintervals mentioned, the functions clearly have non-zero radii of convergence and areanalytic on those subintervals. We bring this up because we should expect no less from theuser optimum, namely that analytic solutions need not exist for arbitrary γ ą 0. Nonethelessas we shall see below for certain values of γ, there indeed does exist analytic solutions to theuser optimum.

Consequently, the plan is as follows: first eliminate erroneous solutions, then for γ “ 1and by assuming there exists an analytic solution for the outflow qptq- find said solution, andfinally by formal power series for γ ą 0, find the remaining solutions. One could instead takeup the mathematical question of existence and bypass the need for assuming the existenceof certain types of solutions, but we must find the functional form of the solution, so thatnumerical methods can be applied for purposes of empirical modelling. Finding solutions bysupposition will help generate actual functional forms for solutions, as will be seen below.

To start, we give the primary dynamic relation, which governs the user optimum regime.

10By analytic here one means the precise mathematical definition. That is functions that are infinitelysmooth with convergent Taylor series on all of the interval r0, t̄s. So, one can see that none of the solutionsare analytic through the entire rush (including end points hence the use of the brackets for the interval)because they are not even elements of the vector space of infinitely smooth functions, C8pr0, t̄sq, let alonehave convergent Taylor coefficients.

20

It is,

Aptpt1qq “ Qpt1q ´ qpt1q

«

Ldq

dk

´L

v

ff

, (4.1)

with the time relation for tpt1q given by equation (3.17) but restated here in terms of thefunction,

wptq :“v0

dq

dkptq

, (4.2a)

tpt1q “ t1 ´Ldq

dk

“ t1 ´L

v0wpt1q. (4.2b)

Above, it is worth noting that the function wpt1q is the unitless multiplicative inverse of thegroup velocity and so it gives the time delay t1 ´ t ą 0 between the arrival time, t1, and the

start time, t. Note for example that when t1 “ Lv0, one has that w

´

Lv0

¯

“ 1 and consequently,

t “ 0. This simply says that the first user starting their trip at t “ 0 travels at the maximumfree flow velocity and thus arrives at time t1 “ L

v0. As traffic accumulates, wpt1q, increases

from unity, dilating the time it takes to traverse the corridor.In order to relate the cumulative functions, we know that at the start of the traffic

flow, initially there are no drivers, i.e., Ap0q “ Qp Lv0

q “ 0. And in general, we expect

A´

fpt1 ´ Lv0

q¯

“ Qpt1q for some relation f . This is expected because of the following two

facts. First, the number of cars is conserved. Second and lastly, the minimum time delayfor the ensemble of drivers must be, no less than and precisely equal to, the minimum timefor a commuter to traverse the roadway at free flow velocity, which is precisely L

v0, the ratio

of the length of roadway, L, to the free-flow velocity, v0.Now, in order to maintain both causality and one to one correspondence between time

of departures and cumulative counts, it is necessary that f be an affine relation of the formfpt1q “ at1 ` b, for some suitable real constants a and b. This is because the drivers arrive inthe order of their work start times. But more so, at the user optimum, no user should havea desire to alter the order of their departure. It is tantamount to the following proposition.Suppose you are a commuter. Then, no one who departs after you will arrive to work beforeyou and no one who departs before you will arrive after you. Furthermore, we accomplishthis by demanding that the costs of each user be uniformly the same, i.e. an uniform costrate. This constraint requires the constants a and b to be linear in the unitless cost parameterσ :“ 1 ´ α2

α1

with α1 ą α2 ą 0 implying 1 ą σ ą 0.

Thus fpt1 ´ Lv0

q “ σ ¨ pt1 ´ Lv0

q and consequently, Aptq “ Qp 1

σt` L

v0q is the constraint. This

21

constraint, together with equation (4.2b), turns equation (4.1) into,

A

ˆ

t1 ´L

v0wpt1q

˙

“ Q

ˆ

1

σ

ˆ

t1 ´L

v0wpt1q

˙

`L

v0

˙

“ Q

ˆ

1

σt1 ´

1

σ

L

v0wpt1q `

L

v0

˙

“ Qpt1q ´ qpt1qL

v0

„

wpt1q ´v0

vpt1q

.

Now, equating the last two equalities gives a single equation for the cumulative outflow Qpt1qas,

Q

ˆ

1

σt1 ´

1

σ

L

v0wpt1q `

L

v0

˙

“ Qpt1q ´ qpt1qL

v0

„

wpt1q ´v0

vpt1q

. (4.3)

But now how to proceed? If the social optimum greatly simplified by writing its primarydynamic equation, (2.5), in terms of the velocity function v, en lieu of the outflow q, thenone might naively hope that a similar simplification will occur for the user optimum. Recallthat in the social optimum regime the algebraic form of the function, qpvq, in contrast to thefunction, vpqq, led to a simplified equation.

As we will see, this guess is not entirely true. But in the course of pursuing this avenue, wewill in fact obtain a greatly simplified expression, by making use of a simultaneous expressionin terms of both w (rather than v) and q. First for ease of manipulation and brevity, let usagree to name a new function τ of t1:

τpt1q :“

„

1

σt1 ´

L

σv0wpt1q `

L

v0

. (4.4)

Furthermore, let ddtwptq “: 9w. Since equation, (4.3), is now a first order delay differential

equation in solely the variable t1, we shall drop the primes on the variable t1 and let un-primed t be the independent variable of (4.3). With our definition of τ , equation (4.3) canbe written more compactly as,

Qpτq “ Qptq ´ qptqL

v0

„

wptq ´v0

vptq

. (4.5)

We now take a first time derivative with respect to (primed) t of equation (4.5) arriving at,

qpτqdτ

dt“ qptq ´ 9qptq

L

v0

„

wptq ´v0

vptq

´ qptqL

v0

„

9wptq ´d

dt

ˆ

v0

vptq

˙

(4.6)

Now to clearly rewrite the expression (4.6), it is first necessary to note that:

22

dτ

dt“

1

σ

ˆ

1 ´L

v09w

˙

(4.7)

qptq “ qm

ˆ

1 `1

γwptq

˙ ˆ

1 ´1

wptq

˙ 1

γ

(4.8)

d

dtqptq “ ´ qptq

pγ ` 1q 9w

pγw ` 1q γw p1 ´ wq(4.9)

v0

vptq“

pγ ` 1qwptq

γwptq ` 1(4.10)

d

dt

ˆ

v0

vptq

˙

“pγ ` 1q 9w

pγ w ` 1q2. (4.11)

Equation (4.7) is the time derivative of equation (4.4); (4.8) follows from the definition of was wptq :“ v0

dq

dkptq

and the global relation between q and k in equation (2.2). Lastly, equation

(4.10) is a consequence of the definition of v; recalling v :“ q{k.At any rate, substituting the above expressions into equation (4.6) yields the greatly

simplified expression,

rσ qptq ´ qpτq s

„

1 ´L

v09w

“ 0. (4.12)

This is our significantly simplified relation. It would appear that there are two separateconditions from (4.12). But this is not so. The only relevant condition is

σ qptq “ qpτq. (4.13)

Immediately after the first arrival time, t “ Lv0, the first derivative of qptq must be positive

real and increasing to correspond to the obvious facts about the outgoing traffic. It then canbe shown that the condition,

1 ´L

v09w “ 0,

is an erroneous solution to the user optimum11. This is relegated to the appendix.

11That is to say, economically irrelevant solutions due to mathematical spuriousness.

23

4.1 Solutions for Greenshields’ Relation γ “ 1

In the γ “ 1 Greenshields’ case, one can directly construct term by term the analytic solutionto equation (4.13). Thus setting γ “ 1 in equation (4.2a), we have an explicit form for wptq,

wptq “1

b

1 ´ qptqqm

.

Starting from the initial condition for the outflow, q´

Lv0

¯

“ 0, and following from the

definition of each of the functions below, we get the line of logical implications:

q´

Lv0

¯

“ 0 ñ w´

Lv0

¯

“ 1 ñ τ´

Lv0

¯

“ Lv0.

Note that by taking the time derivative of (4.13), while requiring 9q´

Lv0

¯

‰ 0, implies:

9τ´

Lv0

¯

“ σ ñ 9w´

Lv0

¯

“ v0L

p1 ´ σ2q ñ 9q´

Lv0

¯

“ 2qmv0L

p1 ´ σ2q.

It can be checked that allowing the first derivative to vanish will generate the trivial analyticsolution, qptq “ 0.

Continuing to take higher-order time derivatives of (4.13), as well as, of the relationbetween w and q, will define the power series for the γ “ 1 solution q1,σptq “: qσptq of theform,12

qσptq “ q1,σptq “8ÿ

n“1

qpnqσ

´

Lv0

¯

n!

ˆ

t ´L

v0

˙n

,

with,

9qσ

ˆ

L

v0

˙

“2qmv0L

p1 ´ σ2q

:qσ

ˆ

L

v0

˙

“´6qmv

20

L2

p1 ´ σ2q2p1 ` σq

p1 ` σ ` σ2q

;qσ

ˆ

L

v0

˙

“6qmv

30

L3

p1 ´ σ2q3

p1 ` σ ` σ2q24 ` 8σ ` 3σ2 ` 8σ3 ` 4σ4

p1 ` σ2q

qp4qσ

ˆ

L

v0

˙

“´30qmv

40

L4

p1 ´ σ2q4p1 ` σq

p1 ` σ ` σ2q34`12σ`4σ2`19σ3 ` 30σ4`19σ5`4σ6`12σ7`4σ8

p1 ` σ2qp1 ` σ ` σ2 ` σ3 ` σ4q

12From here on subscripts denote names and parametric dependence and no longer partial derivatives.

24

That this function qσptq is a solution to the continuity equation in the case γ “ 1, is equivalentto requiring for fixed x “ L the initial time distribution,

u0pspL, tqq “L

v0pt ´ sq“

d

1 ´qσptq

qm.

That this function qσptq is analytic in a positive radius of convergence can be checked viaapproximation methods for the radius of convergence when only a small finite number ofcoefficients are known.

To summarize this subsection, in the user optimum regime for γ “ 1, we are seekingsolutions of the equation of continuity that are compatible with the constraint (4.13). The

requirement that 9qσ

´

Lv0

¯

‰ 0 was to avoid the trivial solution qptq “ 0. This forced a

condition on 9τ and hence produced the higher order time derivative coefficients in order togenerate the power series for qσptq :“ q1,σptq. But as we will see, this process will not alwaysyield linearly independent conditions with which to uniquely define the analytic series forqptq or may over determine the system, each a potential issue for arbitrary γ ą 0.

4.2 Solutions for Generalized Greenshields’ Relation

After an examination of other values of the parameter γ one sees that the procedure abovecannot always be used. Fortunately, this does not stop us from solving the system for arbi-trary γ ą 0. We will accomplish this by first assuming there exists an analytic solution thatwe wish to elucidate and demanding that the first derivative not vanish in a neighborhoodof the initial condition for the purpose of inverting power series. We proceed as follows.

The following is a calculation based on an essentially similar method in the case of γ “ 1due to Danqing Hu and shown to the first author by Elijah DePalma. Iterate by compositionand replace t with τ repeatedly in equation (4.13), a total of n times,

σqpτq “ q

ˆ

1

σrτ ` σ

L

v0´

L

v0wpqpτqqs

˙

“ σ2qptq (1st iteration)

“ q

ˆ

1

στptq `

L

v0

„

1 ´1

σwpσ qptqq

˙

,

25

σ2qpτq “ q

ˆ

1

στpτq `

L

v0´

1

σ

L

v0wpσ qpτqq

˙

“ σ3qptq (2nd iteration)

“ q

ˆ

1

σ2τptq `

L

v0

„

1

σ´

1

σ2wpσ qptqq ` 1 ´

1

σwpσ2 qptqq

˙

,

σ3qpτq “ q

ˆ

1

σ2τpτq `

1

σ

L

v0´

1

σ2

L

v0wpσ qpτqq `

L

v0´

1

σ

L

v0wpσ2 qpτqq

˙

“ σ4qptq (3rd iteration)

“ q

ˆ

1

σ3τptq `

L

v0

„

1

σ2´

1

σ3wpσ qptqq `

1

σ´

1

σ2wpσ2 qptqq ` 1 ´

1

σwpσ3 qptqq

˙

,

and so on. After n such iterations one arrives at,

σnqptq “ q

˜

1

σnτptq `

L

v0

n´1ÿ

k“0

„

1

σk´

1

σk`1wpσn´kqptqq

¸

(4.14)

We are interested, now, in solutions such that the inverse function q´1 exists on a non-zero

interval contained with in the interval”

Lv0, t̄

ı

. In this case, (4.14) becomes,

σnq´1 pσnqptqq “ τptq `L

v0

n´1ÿ

k“0

“

σn´k ´ σn´k´1wpσn´kqptqq‰

(4.15)

Now, we are free to re-order our finite summation in (4.15). This is equivalent to letting theindex of summation k Ñ n ´ k ´ 1. Thus, equation (4.15) can be re-expressed as,

σnq´1 pσnqptqq “ τptq `L

v0

n´1ÿ

k“0

“

σk`1 ´ σkwpσk`1qptqq‰

(4.16)

One can now exploit the fact that 0 ă σ ă 1 by taking the limit n Ñ 8 and causing the lefthand side of equation (4.16) to vanish, hence arriving at,

τptq “L

v0

8ÿ

k“0

“

σkwpσk`1qptqq ´ σk`1‰

“L

v0

8ÿ

k“0

σkwpσk`1qptqq ´L

v0

σ

1 ´ σ

“1

σt ´

L

σv0wptq `

L

v0,

26

the last equality being simply the definition of τptq from equation (4.4). Now, the last twolines of equality can be re-worked into the following important and beautiful expression,

v0

Ltpqq `

σ

1 ´ σ“

8ÿ

k“0

σkwpσkqq (4.17)

By a series expression for w in terms of q and an interchange of summation, which is alsovalid as each series converges provided we are working inside its radius of convergence, wearrive at a second series representation for the function τptq. We accomplish this by firstwriting the function qpwq obtained from (4.8) namely,

qpwq “ qm

ˆ

1 `1

γw

˙ ˆ

1 ´1

w

˙ 1

γ

. (4.18)

Now around the vicinity of the initial condition, one has,

Nÿ

n“0

p´1qn

p1 ` nγq βpn ` 1, 1 ` 1

γq

pw ´ 1qn` 1

γNÑ8ÝÝÝÑ qpwq,

which we write as,

qpwq “8ÿ

n“0

p´1qn

p1 ` nγqβpn ` 1, 1 ` 1

γq

pw ´ 1qn` 1

γ . (4.19)

Here the function βpx, yq :“ Γpxq ΓpyqΓpx`yq

is the beta function and the Gamma function, Γpxq, isgiven by,

Γpxq :“

8ż

0

tx´1 e´tdt.

We can now invert the series (4.17) in w of the function qpwq, to obtain a series in q for thefunction wpqq leading to,

wpqq “ 1 `

ˆ

γ

γ ` 1

q

qm

˙γ 8ÿ

n“0

´

γ

γ`1

q

qm

¯n γ

γ pn ` 1q pγ ` 1qnβpγpn ` 1q, n ` 1q

, (4.20)

“ γFγ´1

´

1

γ`1, 2

γ`1, . . . , γ

γ`1; 1

γ, 2

γ, . . . , γ´1

γ;

´

q

qm

¯γ ¯

, γ P N

27

where again the beta function appears and nFmpa1, a2, . . . , an; b1, b2, . . . , bm; xq is the hy-pergeometric function defined as,

nFmpa1, a2, . . . , an; b1, b2, . . . , bm; xq :“8ÿ

k“0

Γpa1 ` kq . . .Γpan ` kq ¨ Γpb1q . . .Γpbmq

Γpa1q . . .Γpanq ¨ Γpb1 ` kq . . .Γpbm ` kq

xk

k!.

Substituting equation (4.18) into equation (4.17) and interchanging summations as suggestedearlier yields,

v0

Ltpqq ´ 1 “

8ÿ

n“1

pγ ` 1qΓ pn pγ ` 1qq´

q

v0kj

¯γ n

n! γ Γpγ nq p1 ´ σγ n`1q, (4.21)

where qm (from (4.18)) has been replaced with its value in terms of γ, v0, and kj . Next, wemust invert the analytic function on the left to obtain the desired result, qptq for γ ą 0. Ifwe define,

0 ĺ x :“q

v0kj,

then (4.21) can be rearranged to be,

L

v0

«

1 `8ÿ

n“1

pγ ` 1qΓ pn pγ ` 1qq pxγqn

n! γ Γpγ nq p1 ´ σγ n`1q

ff

“: φσ,γpxq. (4.22)

We have now defined a special function φσ,γpxq whose inverse function is what we seek.Actually, this special function is not new it is a mix of the hypergeometric function definedabove and what is known as the Heine basic hypergeometric function or q-hypergeometricfunction. This function has applications in number theory and the combinatorics of grouptheory as pointed out in Gasper and Rahman (2004).

In general, the function is not a function. Rather, it is a multi-valued relation. But forthe restricted domain that we are interested in, 0 ĺ q ă qm, it indeed defines a single valuedfunction for which an inverse makes sense and is well-defined. The quantity we seek is thusgiven by,

qptq “ v0kj“

φ´1

σ,γ ptq‰ 1

γ (4.23)

The cumulative outflow function is then,

Qptq “

tż

Lv0

qpt1q dt1 “ v0kj

tż

Lv0

“

φ´1

σ,γ pt1q‰ 1

γ dt1, (4.24)

28

and the time, t̄, for the rush hour to end is then,

N “ Qpt̄q “

t̄ż

Lv0

qpt1q dt1 “

t̄ż

Lv0

“

φ´1

σ,γ pt1q‰ 1

γ dt1 (4.25)

4.2.1 Analysis of the User Optimum for the Generalized Greenshields’ Relation

Several comments are in order. First, the inversion of the series is justified since we knowthat composition of continuous functions is again a continuous function and in particular,C1pra, bsq, the vector space of continuous functions whose first derivatives are also continuousfunctions. Recalling that the inverse function theorem says that locally the relation (4.18)defines a function wpqq, it then follows that our estimate is reliable since we know that thefirst derivatives do not vanish at the point where the analytic series is being inverted.

Second for values of q ľ qm, the series expression will not converge. In which case,one may at best expect, for example, only a function qptq P C1p r L

v0, t̄ s q. That is to say

the function ceases to be analytic but this does not mean it does not exist. Continuousfunctions can be quite pathological, and thus the solution may exist as an element of thespace C1p r L

v0, t̄ s q but have no convergent power series. In this case the series diverges by no

surprise but the series expression is not worthless but rather gives an asymptotic estimatefor the differentiable but non-analytic would-be solutions. Considering the nonlinear delaynature of our original equation, this comes as more of a serendipity than a snag because itimplies numerical analysis is still possible. This is the next point.

Third, while this analysis may seem cumbersome, it tells us that we can get arbitrarily ac-curate numerical estimates for non-analytic solutions through approximation by polynomialsof the form,

qnptq „n

ÿ

k“1

rkpγ, σq

ˆ

t ´L

v0

˙k

,

where rkpγ, σq is a strictly positive rational function of γ ą 0 and 1 ą σ ą 0. One needonly take up the issue of rapid convergence for the purposes of numerical efficiency. Thisresult is guaranteed by the Stone-Weierstrass approximation theorem for continuous (oncedifferentiable in our case) functions.

Finally, the divergence of the series in the second point above can also be caused bythe failure of the condition N ă Bpγ, σq ¨ L ¨ kj, where Bpγ, σq ą 0 is a constant thatdepends upon only γ and σ. For fixed γ, σ, and kj, the number of vehicles can exceed therequirements for the analytic dynamics to exist. This is because q cannot increase withoutbound but is asymptotically (in time) confined by qm, which itself depends upon the critical

29

jam density, kj, the road geometry, γ, and the maximum free flow speed, v0. This situationfor the existence of analytic solutions forces a queue to start. The length of the queue timesare related to the time it takes to clear the excess vehicles from the road and allow thenumber of vehicles to decrease below the value of the critical number, BLkj.

Below are the graphs of the outflow, qptq, as well the cumulative inflow, and cumulativeoutflow functions, Aptq and Qptq, in the user optimum regime for the same range of samplegamma’s as in the social optimum. The outflow, qptq, increases asymptotically in time, whichis possible so long as N ă BLkj . The flow will stop normally whenever the time has elapsedto the value t̄ such that Qpt̄q “ N . But if N ľ B Lkj then the flow stops abruptly prior tothe time to end the rush, t̄. This causes a queue to form and the wait time in the queue isthe time it takes the number of vehicles to drop below the threshold number, BLkj .

The following formula, which connects the integral of a function to its composition inversevia a change of integration variables is extremely useful in this analysis.

Qptq :“

tż

Lv0

qpsq ds “ q ¨ q´1pqq ´

qż

0

q´1pxq dx.

0 1 2 3 4 5 6 7 8t0

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00qHtL

Figure 6: In normalized units, the outflow curves qptq for t̄ ľ t ľ 1. Values of gamma are indicated from

bottom to top as: γ “ 7

24(black), 1

2(long dash black), 4

5(short dash black), 1 (blue), 2 (red), 3 (green).

30

0 1 2 3 4 5 6 7 8t0.0

0.2

0.4

0.6

0.8

N=1.0

QHtL

Figure 7: In normalized units, the user optimum (UO) cumulative outflow curves Qptq for t̄ ľ t ľ 1. Values


24(black), 1


5(short dash black), 1

(blue), 2 (red), 3 (green).

1 2 3 4 5 6 7t0.0

0.2

0.4

0.6

0.8

N=1.0

AHtL

Figure 8: In normalized units, the user optimum (UO) cumulative inflow curves Aptq for t̄ ľ t ľ 0. Values


24(black), 1


5(short dash black), 1

(blue), 2 (red), 3 (green).

31

0 1 2 3 4 5 6 7 8t0.0

0.2

0.4

0.6

0.8

N=1.0

AHtL and QHtL UO vs. SO for Γ=7��24

Figure 9: In normalized units for only γ “ 7

24, the cumulative inflow curves Aptq are all dashed. The graph

shows curves for both the user optimum (UO) in black, and the social optimum (SO) in red. The cumulative

outflow curves Qptq are solid.

0 1 2 3 4 5t0.0

0.2

0.4

0.6

0.8

N=1.0






32

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0t0.0

0.2

0.4

0.6

0.8

N=1.0






0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5t0.0

0.2

0.4

0.6

0.8

N=1.0

AHtL and QHtL UO vs. SO for Γ=1

Figure 12: In normalized units for only γ “ 1, the cumulative inflow curves Aptq are all dashed. The graph



33

0.0 0.5 1.0 1.5 2.0 2.5t0.0

0.2

0.4

0.6

0.8

N=1.0

AHtL and QHtL UO vs. SO for Γ=2




0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00t0.0

0.2

0.4

0.6

0.8

N=1.0

AHtL and QHtL UO vs. SO for Γ = 3




34

5 Conclusion

First, a few words regarding validity and possible modifications to the LWR-theory is inorder.

At the very heart of the matter one must realize that the discrete nature of vehicles on aroadway precludes a continuum model being arbitrarily accurate in predicting the behaviorof such systems. However as is the case with all phenomena, an issue of resolution anduncertainty lie at the center of arguments for and against the usefulness of all models. Itis very much the case that a system comprised of a large number of discrete vehicles hasparallels in its modelling with that of physical systems governed by, what one might call,the statistical mechanical ensemble.

The theory of the mechanics of statistical ensembles has, as its central challenge theinterplay between the continuum and the discrete. It is then natural to adopt here whatone invokes there as a resolution of the conflict between the continuum nature of the modelsand the discrete nature of the phenomena being modelled. Once one accepts that the causesand origins of phenomena such as traffic cannot be determined precisely, then a fundamentaluncertainty is inherent. Consequently, one adopts the viewpoint that fluid models of thecontinuum are automatically statements of approximation and as well a statement thatstochastic13 fluctuations are unavoidable for such systems. In short, randomness plays acentral role.

It will be stated without proof, the belief of the first author, that any approach involvingthe modelling of traffic on a one-dimensional roadway, which takes into account stochasticfluctuations, in either a quantum statistical mechanical way or by the ad hoc insertion ofstochastic behavior, will ultimately be equivalent to a model of the creation and destructionof bottlenecks on the roadway. However, this equivalence between a quantum statisticalmechanical approach and the reduction to bottlenecks will not be true in more than onedimension.

This dimensional dependence is similar to the contrast between one dimensional andtwo dimensional random walks, when the spatial evolution in the continuum is second orderlinear, i.e. a bona fide diffusion process. To see why this could be true, one must invoke aquantum field theoretic approach where bottlenecks are created and destroyed like interactingparticles. In one dimension, the field theory is completely convergent and straightforward tocalculate. But more detail would be beyond the scope of our topic. Hence no proof is givenbut it is worth mentioning for those interested in generalized modelling of traffic.

13Fluid systems are regarded as stochastic in several different regimes. First, they are random becausethey may be turbulent and hence chaotic. They may also be random because they are governed by a diffusiveprocess, which governs Brownian motion, or finally they are random because they obey a transport equationderived from statistical considerations, such as Boltzmann transport phenomena.

35

Acknowledgments

The first author wishes to thank Elijah DePalma for his shared insights and extensive researchand expertise on the LWR theory, as well as the second author, professor Richard Arnottfor his generous professional and technical advice. This author is also indebted to professorArnott for allowing him the opportunity to work on the project. Many thanks also goes to Dr.Jonathan Sarhad for his many interesting and instructive conversations on the philosophyand rigorous mathematics of the theory of PDE. This research was partially funded by theEdward J. Blakely Center for Sustainable Suburban Development (CSSD) at the Universityof California, Riverside.

Appendix A Elimination of Erroneous Solutions

The initial condition, qp Lv0

q “ 0, implies wp Lv0

q “ 1. This initial condition for wptq, togetherwith the first order ordinary differential equation below,

1 ´L

v09wptq “ 0

give uniquely,

wptq “v0

Lt.

Thus, we get the outflow rate,

qptq “ qm

ˆ

1 `L

γ v0t

˙ ˆ

1 ´L

v0t

˙ 1

γ

,L

v0ĺ t ĺ t̄, (A.1)

with the cumulative outflow given by Qptq “tş

Lv0

qpt1q dt1 and the fact that Qp Lv0

q “ 0,

Qptq “ qm

ˆ

t ´L

v0

˙ ˆ

1 ´L

v0t

˙ 1

γ

,L

v0ĺ t ĺ t̄, (A.2)

However, this family of solutions is not a solution to the user optimum. In fact, thesesolutions upon inspection are independent of the unitless cost parameter, σ! So what hashappened? This particular family of solutions is analogous to the critical points of a function,fpxq, which satisfy, by definition of critical points, the equation f 1pxq “ 0 but do not corre-spond to any extremal value, local or otherwise. Hence this resolves the issue of identifyingand removing erroneous solutions and explains the origin of the erroneous nature.

36

References

Bleistein, N., Handelsman, R., 1986. Asymptotic Expansion of Integrals, Dover, New York.

DePalma, E., Arnott, R., 2012. Morning commute in a single-entry traffic corridor with no latearrivals. Transportation Research Part B 46(1), 1-62.

Fetter, A., Walecka, J. D., 2003. Theoretical Mechanics of Particles and Continua, Dover, NewYork.

Gasper, G., Rahman, M., 2004. Basic Hypergeometric Series 2nd ed. in Encyclopedia of Mathe-matics 96, Cambridge University Press, New York.

Gel’fand, I., Fomin, S., 1991. Calculus of Variations, Dover, New York.

Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S., 2009. Optimization with PDE Constraints inMathematical Modeling: Theory and Applications 23, Springer, Berlin.

Inman, R., 1978. A generalized congestion function for highway travel. Journal of Urban Eco-nomics 5(1), 21-34.

Levandosky, J., Fall 2002. Lecture course: partial differential equations of applied mathematics,Stanford University, Stanford, CA.

Newell, G. F., 1988. Traffic flow for the morning commute. Transportation Sciences 22(1), 47-58.

Newell, G. F., 1993. A simplified theory of kinematic waves in highway traffic, part I: generaltheory. Transportation Research Part B 27(4), 281-287.

Pinch, E., 1993. Optimal Control and the Calculus of Variations, Oxford University Press, Oxford.

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Documents

Closed-form Solutions of Generalized Greenshield Relations for the Social and User Optimums of a Single Entry Corridor