Nicolas Coulombel
email: nicolas.coulombel@enpc.fr

André De Palma
email: andre.depalma@ens-cachan.fr

Variability of travel time, congestion and the cost of travel

Keywords: Value of reliability, Nash equilibrium, Bottleneck model, Scheduling, Random delay. JEL codes: C61, R41

Variability of Travel Time, Congestion, and the Cost of Travel

Introduction

Travel time reliability matters at various levels, from route choice [START_REF] Abdel-Aty | Investigating effect of travel time variability on route choice using repeated-measurement of stated preference data[END_REF] to mode choice and to the choice of departure time [START_REF] Chorus | Travel information as an instrument to change car-drivers' travel choice: a literature review[END_REF].

For [START_REF] Bates | The valuation of reliability for personal travel[END_REF], first, travel time unreliability can be detrimental to scheduling or even doing one's activities by causing early or late arrivals at destination. Missing a flight because of delays in public transit illustrates this point.

Second, people could distaste travel time variability for the stress it generates, or because it complicates the planning of activities. The user's risk aversion reflects this second phenomenon. Palma and Picard (1995) have estimated the risk aversion of drivers using a CARA representation of preferences, and have found that slightly less than one third of the drivers are "risklovers," meaning that they have negative risk aversion parameters.

Assessing the value of travel time reliability is a relatively recent topic in transportation literature. In their review, [START_REF] Bates | The valuation of reliability for personal travel[END_REF] underlined the lack of research and the absence of this issue in most national guidelines for the evaluation of road and public transport schemes. In line with the anterior but less general work of [START_REF] Noland | Travel-time uncertainty, departure time choice, and the cost of morning commutes[END_REF], [START_REF] Fosgerau | The value of reliability[END_REF] have studied the effects of the variability of travel time on the choice of departure time and on the cost of the trip. They consider the case of a single individual with (α-β-γ) preferences [START_REF] Vickrey | Congestion theory and transport investment[END_REF]Small, 1982) and facing a fixed curve of travel time, travel time being a function of departure time. [START_REF] Fosgerau | The value of travel time variance[END_REF] apply the same setting with a different utility function. In both works, the authors derive 3 the optimal departure time as well as the minimum cost of the trip; moreover, they derive a formal expression for the value of reliability (VoR), which provides a sound theoretical foundation. The "single traveler" analytical framework, however, does not consider the effect of the variability of travel time on congestion, treating travel time as an exogenous variable. As each individual adapts its departure time in response to higher or lower variability of travel time, the congestion time profile is likely to change. The single traveler framework, while valid when congestion is mild, is therefore unfit to assess the value of travel time reliability for the rush hour period.

In order to address this shortcoming, we develop a theoretical framework to study the effects of the variability of travel time on the morning commute when congestion is endogenous and reflects departure time decisions. The most salient issues are the cost for users (and the VoR) and how the unreliability of travel time changes the time profile of departures, and subsequently the congestion time profile. Our model builds on the standard bottleneck model of road congestion with peak-load demand. The variability of travel time is modelled by an additive random delay, which varies according to departure time. We are thereby in line with works which applied the bottleneck model to phenomena contributing to the variability of travel time: stochastic demand [START_REF] Palma (de) | Stochastic equilibrium model of peak period traffic congestion[END_REF][START_REF] Arnott | Does providing information to drivers reduce traffic congestion?[END_REF], stochastic capacity [START_REF] Arnott | Information and usage of free-access congestible facilities with stochastic capacity and demand[END_REF][START_REF] Arnott | Information and time-of-usage decisions in the bottleneck model with stochastic capacity and demand[END_REF][START_REF] Li | Departure time distribution in the stochastic bottleneck model[END_REF] and incidents [START_REF] Fosgerau | Trip-timing decisions with traffic incidents[END_REF]. Our work is closely related to [START_REF] Siu | Equilibrium trip scheduling in congested traffic under uncertainty[END_REF], who study a bottleneck model with a bounded random delay. We generalize their work by lifting their assumptions of boundedness and constancy of the delays over the rush Published in : Mathematical Population Studies: An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 4 hour. We also find that under some broad assumptions on the random delay, there exists a unique Nash equilibrium for the bottleneck model with an additive random delay and (α-β-γ) preferences. Compared to the case where travel times are deterministic, the variability of travel time leads to spreading of the peak: departures are spread more evenly over the rush hour, a feature which reduces congestion. This also mitigates the effect of travel time unreliability on the expected cost of the trip, leading to a lower VoR than when travel times are exogenous (as in single traveler models). When the maximum random delay is bounded and small enough, individuals' adjustments completely offsett the cost of unreliability, hence a VoR of 0. This result stems from a specific feature of (α-β-γ) preferences, which is that individuals who are sure to be early or late whatever the random delay are risk neutral. Lastly, the effect of the variability of travel time on the timing of the rush hour is non-trivial and depends on the distribution of the random delay.

Our results entail two caveats as far as cost-benefit analysis is concerned.

First, non-equilibrium models strongly overestimate the VoR for the rush hour period because they fail to capture equilibrium mechanisms, which mitigate the cost of unreliability. Second, models based on (α-β-γ) preferences likely underestimate the VoR (except in the case of "risk-lover" drivers), because utility is piece-wise linear rather than strictly convex. Utility functions featuring risk aversion, such as the CARA and CRRA functions, should be preferred when deriving the VoR.

Modeling Background

We present the standard bottleneck model of road congestion with fixed peak-load demand, from which our model derives. The bottleneck model of road congestion was firstly introduced by [START_REF] Vickrey | Pricing in urban and suburban transport[END_REF][START_REF] Vickrey | Congestion theory and transport investment[END_REF]. A thorough economic analysis of the standard model with inelastic demand can be found in [START_REF] Arnott | Economics of a bottleneck[END_REF].

The standard bottleneck model of road congestion with fixed peak-load demand

Every morning, N individuals considered as a continuum commute from point A (home) to point B (workplace). Each individual travels with his own car. A single road joins A and B, which starts with a bottleneck with fixed capacity (or service rate) s. Whenever the departure rate r(t) is greater than s, a queue develops at the bottleneck entrance if it does not exist yet. The commuting time is:

    0 .  Qt T t T s (1)
T0 is the free flow travel time between the bottleneck exit A and B. Q(t)/s is the congestion delay, equal to the ratio between the length of the queue Q(t) and the service rate. If ts is the time at which the queue starts to develop, as long as the right-hand side term remains positive, the length of the queue is given by:

     . s t s t Q t r u du s t t     (2) Q(t)
is the difference between the total number of departures from ts to t and the bottleneck outflow over the same period. In the remainder of the text, we assume that any departure rate function is right-continuous. Two functions r1 and r2 equal to one another everywhere except on a set of measure zero generate the same time profiles for both the total number of departures and for the queue. They are indistinguishable for all practical purposes, hence the assumption of right continuity. This also ensures that the function Q is continuous and that its derivative is continuous everywhere except possibly at points where r is discontinuous. At these points, Q admits at least a right derivative. The derivative of Q is given by:

                    0 if 0 and if 0 and or 0 and is discontinuous in otherwise, r Q t Q t r t s Q t r t s Q t r t s Q t r t Q t r t s                  (3)
where subscript r stands for the right-hand side derivative. Considering Eq. ( 1), all previous statements about Q hold true for T except for formulae of the derivatives, which should be divided by s.

Choice of departure time and cost of the trip

Individuals choose their departure time with the twofold objective of being on time at their destination and of minimizing their travel time. [START_REF] Vickrey | Congestion theory and transport investment[END_REF] Mathematical Population Studies: An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 7 travelling and the "schedule delay," meaning that being early or late at one's destination compared to one's preferred arrival time is costly:

            * * , C t T t t t T t t T t t             (4)
where   The cost of one minute of travel time is α; the cost of being one minute early at your destination is β and the cost of being one minute late is γ. These fixed parameters set the terms of the trade-off between travel time and schedule delay when choosing the departure time. As usual, we assume β < α (the case β  α is discussed in the Appendix of [START_REF] Arnott | Economics of a bottleneck[END_REF].

Each

Results

There exists a Nash equilibrium for the standard bottleneck model [START_REF] Smith | The existence of a time-dependent equilibrium distribution of arrivals at a single bottleneck[END_REF] and it is unique [START_REF] Daganzo | The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck[END_REF][START_REF] Lindsey | Existence, uniqueness, and trip cost function properties of user equilibrium in the bottleneck model with multiple user classes[END_REF]. At this equilibrium:

 departures are spread from tq to tq'. The departure rate is strictly positive on this interval.

 there is congestion throughout the whole rush hour (except in tq and tq') and the bottleneck always operates at full capacity. Consequently, the length of the rush hour is: tq'tq = N/s.

 congestion varies so as to equalize the cost of the trip throughout the rush hour.

This entails the conditions regarding the travel time:

    ' for , for , qp pq T t t t t T t t t t                        (5)
where tp is the congestion peak: leaving at tp means facing the biggest queue at the bottleneck.

Using Eq. ( 2) and ( 5

N t t T s N T t T T N s tt N s C T T s N t t T s                                            (7)

The Bottleneck Model with Stochastic Travel Times

After presenting the bottleneck model with stochastic travel times, we show that a user equilibrium exists and is unique.

The stochastic bottleneck model

The bottleneck model with stochastic travel times is based on the same setup as the standard model, with the difference that the travel time includes a stochastic component which represents day-to-day variability. This term takes the form of a random delay, noted εt, which may vary over the course of the day. Accordingly, the travel time T ~ is now given by:

    0 , t Qt T t T s     ( 8
)
where a tilde emphasizes the stochastic nature of a variable. The random delay being additive, the congestion level does not influence the scale of the delay. εt captures phenomena contributing to the variability of T0, including weather conditions and variations in the speed of drivers, after exiting the bottleneck. A multiplicative form where Q(t)/s would be multiplied by one plus a random noise would allow us to model the variability of Q(t)/s resulting from fluctuations in the level N of travel demand or in the bottleneck capacity s. We focus on the variability of T0 because, while the two cases are relatively similar, the algebra is much simpler with an additive random delay than with a multiplicative one.

We do not consider the role of the provision of dynamic information regarding travel times. This excludes dynamic strategies based on such information, such as leaving home when εt appears to be low. It also implies that the dynamics of εt has no importance here, and we can assume any correlation structure for the (εt)t as long as the basic assumptions mentioned below hold. 1

As pointed out by [START_REF] Bates | The valuation of reliability for personal travel[END_REF], the most general case where the set of the distributions of (εt)t has no specific properties does not provide any significant result. We assume that:

1 If one wants to maintain some basic flow properties (such as First In First Out), some constraints must be imposed on the time dynamics of εt. In particular, assuming that the error terms are independent could lead to overtakings (if εt+δ is much lower than εt), in contradiction with the First In First Out principle.  they have the same mean μ and the same standard deviation σ.

The distribution function of the normalized random variable

  ˆ/ tt      is
denoted f and its cumulative distribution function F. Again for the sake of clarity we assume that f is piecewise continuous. In particular, there is no punctual mass in the distribution of f. This assumption is supported empirically [START_REF] Richardson | Travel time variability on commuter journeys[END_REF]Giulano, 1989).

Choice of departure time and expected cost of the trip

Travel time being stochastic, the cost of the trip becomes a random variable. As the realized values of the random delays (εt)t are not known beforehand, the choice of departure time is based on the ex-ante cost. It is given by:

                        E mt mt C t C t T t m t u f u du m t u f u du              (9) with     * t t T t mt    (10) where         s t Q T t T / 0
is the expected travel time when leaving at t and m(t) is often referred to as the normalized safety margin [START_REF] Fosgerau | The value of reliability[END_REF].

The ex-ante cost of the trip is the sum of

 

Tt  , the value of the average travel time, and of the expected cost of the schedule delay. Compared to the deterministic case, the variability of travel time mainly affects the latter term. The first term is affected only through the average delay μ, which modifies the average travel time.

12 It is unchanged if the random error has a zero mean (μ = 0).

Using

  0      du u uf
, Eq. ( 9) is rewritten as:

                    C t T t m t F m t G m t m t           (11)
where

    x G x uf u du   
is a continuous and positive function.

Equilibrium: existence and uniqueness

We establish several lemmas which are then used to prove the existence and the uniqueness of the Nash equilibrium for the bottleneck model with an additive random delay (BARD).

The function H(t,T) is:

                  ˆˆˆ, , , , , H t T T m t T F m t T G m t T m t T           (12)   with * ˆ , t t T m t T    (13)
The expected cost of the trip is related to H by:

      t T t H t C ,  (14)
H and T being continuous functions, the expected cost of the trip is a continuous function of t. Lemma 1 provides additional properties of the expected trip cost.

departure time t = t* -T -σ F -1 (δ/ β).
For any given departure time, the expected cost of the trip increases strictly with the expected travel time.

Proof

These properties stem from the two partial derivatives of H. The derivative with respect to t is:

          , * ˆ, H t T t t T F m t T F t                     (15)
The First Order Condition (FOC) is:

  , * 0 . H t T t t T F t                (16)
The first point is then straightforward. The second partial derivative is:

    , * . H t T t t T F T                (17) As β < α,   , 0 H t T T   
, hence the second point.  For the remainder of the text, we assume that the set

    / 0 x f x 

Proof

In Appendix A.  At any equilibrium solution, there is no queue at the beginning and at the end

of the rush hour, hence     '0 v v q q T t T t T     .
Moreover, the length of the rush hour is the same as in the deterministic case: '

' / v v q q q q t t t t N s     .
We can characterize the solutions further by providing another condition relative to the dynamics of the expected travel time.

Lemma 3 During the rush hour, the expected travel time solves:

                        ' 1 ' ] , [. 1 vv qq F m t F m t T t t t t F m t F m t             (18)

Proof

The derivation of Eq. ( 11) with respect to t yields:

              ' ' ' ' , C t T t m t F m t m t          ( 19
)
where

      ' 1 ' / m t T t    
. This is rewritten as:

                  ' ' . C t T t F m t F m t             (20) The equilibrium condition   v eq C t C t I    implying that   ' ' 0 ] , [
    1 / v p m t F   
. The increase and decrease in congestion are smoother than in the deterministic case. When congestion increases

    '/ Tt     ;
conversely, when congestion recedes,

    '/ Tt     .
Lastly, when σ tends toward zero, F tends toward the Heaviside step function and we find again the results of the deterministic case:

    '/ Tt     for t < tp and     '/ Tt       for t > tp.
We establish the existence and uniqueness of a Nash equilibrium in the bottleneck model with additive random delay.

                        ' ' 0 if , 1 0 if ,
is given by Eq. (X)

where

satisfie v q v v v qq v v v qq t v t v q s r t t t t F m t F m t r t t t t mt T t T r u s du t                                 0 0 ' s / , , and = / v v v v q q q q H t N s T H t T t t N s                 (21)

Proof

In Appendix B.  Again, when σ tends toward zero,

 

v rt tends toward the deterministic solution given by Eq. ( 6).

Comparative Statics

The variability of travel time changes the Nash equilibrium. To better understand why, we plot (α-β-γ) preferences as a function of travel time T (and not as a function of departure time t as it is usually done).

The cost function of the trip is piecewise linear in T with a kink in T = t*-t

(Figure 1). The slope is equal to α-β before the kink, then to α+γ. As α+γ > α-β, the cost function of the trip is convex. Individuals with (α-β-γ) preferences are riskaverse (to some extent) and thus seek to insure themselves against travel time unreliability, modifying the equilibrium. Notwithstanding, if one remains on either side of the curve (he leaves at such a time that he is sure to be either early or late at 17 his destination whatever the realized value of the random delay), the linearity of the cost of travel implies that this individual is risk-neutral. The random delay must be bounded on at least one side for this case to arise. If not, the possibility of an extreme delay means that one is never sure to be early or late, whatever the departure time.

We now study how travel time unreliability changes the main equilibrium features.

. Peak congestion level

The variability of travel time leads to reduction of the peak.

Proposition 2 For the bottleneck model of road congestion with fixed demand, the maximum level of congestion is lower with stochastic travel times than with deterministic travel times.

Proof

As          / ' t T on , vv qp tt     ,     v v v p p p q p q T t T t t t t t      . Similarly,     '/ Tt       on ' , v v p q t t     , hence     ' ' v v v p p q p q p T t T t t t t t      . Combining the two points yields     ' ' v v v p p q q q q T t T t t t t t      , which is a contradiction given that ' ' / vv q q q q t t t t N s     . 
Under (α-β-γ) preferences, convexity of the cost function is strongest around the kink. The cost of unreliability is thus highest for people leaving at the peak, who would arrive at their preferred arrival time were it not for the random delay.

Considering that people leaving at the beginning or at the end of the rush hour are less affected by the variability of travel time by being far from the kink of the cost function, peak reduction occurs as an equilibrium mechanism.

Travel time unreliability also leads to spreading of the peak. While in the deterministic case, the peak corresponds to a kink of the curve of travel time, in the stochastic case this curve is flat at the peak:  

Timing of the rush hour

The effect of the variability of travel time on the timing of the rush hour is not straightforward. Application in Section 5 will show that depending on the distribution, an increase in σ can shift the rush hour earlier or later. For a given distribution, the direction of the shift can even change depending on the value of σ.

Given the notations

      x L J x x u f u du     and       R x J x u x f u du   
, Proposition 3 provides some indication regarding the direction in which the shift occurs.

Proposition 3 If L R N N J J s s                    
, the rush hour starts earlier in the BARD than in the deterministic case, and conversely. , is always positive. This result is also conform to intuition: individuals with (α-β-γ) preferences are risk averse, at least to some extent. The VoR however is null if / v q dt d and / H   are both equal to 0. This situation arises for low values of σ when the random delay is bounded, as established by the following proposition.

Proof

Proposition 5 When the support of the random delay is bounded and when the maximum delay (in absolute value) is small enough, the equilibrium cost of the trip is independent of σ. The timing of the rush hour also remains unchanged.

Proof

In Appendix E. 

If the random delay is bounded, low levels of variability change neither the equilibrium cost of the trip nor the timing of the rush hour, but they do reduce congestion. When the maximum delay is small compared to the length of the rush hour, the first person to leave is sure to be early and the last one to leave is sure to be late. 21 beginning of this section regarding (α-β-γ) preferences) and act as "anchor points".

As long as σ remains under the threshold, they leave at the same time and their cost of commute does not change. This keeps the equilibrium cost of the trip at the same level and the timing of the rush hour unchanged. The other individuals, starting with those leaving at the peak, adjust their departure time, however, hence a decrease in congestion.

Formal derivation of the VoR is not feasible in the general case as it is for the single traveler version of this model [START_REF] Fosgerau | The value of reliability[END_REF]. It may be achieved for specific distribution functions, including the uniform case [START_REF] Coulombel | The value of reliability: an equilibrium approach[END_REF].

Application

An application illustrates the previous findings and provides an appraisal of the overestimation of the VoR for the rush hour when using the single-traveler model.

The preferred arrival time is set to 9.30 am, with an expected commute time of half an hour when there is no congestion (and including the mean random delay). As travel demand and transportation supply are of equal levels, the rush hour is one hour long. The cost parameters α, , and  are in the ratio 1.0:0.8:3.0 indicated by [START_REF] Bates | The valuation of reliability for personal travel[END_REF] as typical from the literature. We consider two distributions for the random delay, uniform and exponential, and study in each case the influence of the variability of travel time on departure rates, congestion (through the expected travel time), and the equilibrium cost of the trip.

A comparison of the two distributions concludes this section.

Uniform distribution

In the case of a uniform distribution, the distribution function f, the cumulative distribution function F, and function G are given by: 2

      1 1 3 ² , , and
if 3; 3 . 2 2 3 2 3 4 3 x x f x F x G x x            (22)
Starting from σ = 0 (deterministic case), an increase in σ causes people leaving 2 Coulombel and Palma ( 2012) study the case of a uniform distribution extensively, including a computation of closed-form solutions for the equilibrium.

around the peak to change their departure time (Figure 2). Instead of the two plateaus for the deterministic case, the normalized departure rate r(t)/s decreases smoothly from α/(α-β) to α/(α+γ). As discussed previously, the unreliability of travel time is most costly for people leaving at the peak, for whom schedule delay disutility was null otherwise. For the others, the choice not to be on time partly insures them against the random delay. For instance, people leaving after the peak and who are on average late at their destination will benefit from situations where the travel time is shorter than usual. Far from the peak, people are perfectly insured against travel time unreliability, hence stable departure rates at the beginning and at the end of the rush hour (see discussion following Proposition 5). When σ exceeds a certain threshold (equal to 1/4√3 in the application), changes get more important. In our example  > , so drivers leave home earlier and earlier as variability increases in order to insure themselves against the risk of being late (Proposition 3 with f symmetrical). The departure rate is homogenized further.

While the level at the beginning of the rush hour remains the same at first, eventually the whole curve is flattened.

The congestion time profiles reflect the three main effects of the variability of travel time: reduction, spreading, and shifting of the peak (Figure 3). In the case of a uniform distribution, the random error is bounded. As indicated by Proposition 5, the peak shifts only when σ exceeds the threshold mentioned above (σ > 1/4√3).

Again, here  >  so the rush hour starts earlier and earlier as σ grows; the shift would be in the opposite direction were  to be lower than . Contrary to the shift phenomenon, reduction and spreading of the peak occur even for low values of σ.

We break down the equilibrium cost of the trip as the sum of the mean costs of uncongested travel, congestion and schedule delay. The cost of uncongested travel is a constant equal to α(T0+μ) and is independent of σ. The mean cost of congestion decreases strictly with σ as a result of the decrease in congestion (Figure 4). On the other hand, travel time unreliability puts a burden on the travelers' schedules, causing the mean cost of schedule delay to increase strictly with σ.

Exponential distribution

In the case of an exponential distribution, f, F, and G are equal to (recall that they correspond to a reduced variable):

              1 1 1 , 1
, and 1 for 1.

x x x f x e F x e G x x e x              (23)
The application leads to the same comments as for the uniform distribution, except for three points:

 shifting of the peak occurs as soon as σ > 0 (Figure 5, Figure 6);

 the whole departure rate curve is altered and becomes smoother as soon as σ > 0 (Figure 5);

 the equilibrium cost of the trip increases strictly with σ whatever the value of σ (Figure 7).

Contrary to the uniform distribution, the support of the exponential distribution is unbounded. Even for very low values of σ, everyone may be late given that the probability of sustaining an extremely high delay is not zero. It ensues that all individuals are risk averse, while in the uniform case, only a fraction of them are The effect of the variability of travel time on the timing of the rush hour is not univocal. As σ increases, the rush hour starts slightly later at first, but eventually it starts earlier and earlier (Figure 5, Figure 6). This point probably stems from the asymmetry of the exponential distribution, although we could not properly establish it.

Comparison of the two distributions

To conclude, we compare for the two distributions:

 the equilibrium and the "disequilibrium" costs, the latter being the mean expected cost of travel were people to choose the same departure time as in the deterministic case;

 the value of reliability (VoR) given by the stochastic bottleneck model with the one given by the single-traveler model exposed in [START_REF] Fosgerau | The value of reliability[END_REF].

The gap between the disequilibrium and the equilibrium costs of travel measures the extent to which individual adjustments mitigate the cost of the variability of travel time. The gap between the "equilibrium VoR" and the "F-K VoR" provides a similar indicator for the VoR (which is for reminder the derivative of the equilibrium cost of travel).

The mean expected cost of travel is lower at equilibrium than in disequilibrium for both distributions of the random delay (Figure 8). Individual adjustments are beneficial indeed, a point which was not completely obvious, even more so when uncertainty is limited. Beyond a certain threshold, the mean equilibrium and disequilibrium costs of travel both rise steadily.

For low and high values of σ, the equilibrium cost of the trip is higher for the exponential distribution, while for intermediate values of σ, it is higher for the uniform distribution. Again, this peculiar behavior could be linked to the asymmetry of the exponential distribution. Be it as it may, the spread between the two distributions remains limited, except for very high values of σ. The "equilibrium VoR" is always positive, in accordance with Proposition 4

(Figure 9). It increases with σ for both distributions.3 Like for the equilibrium cost of the trip, the two VoR are in the same order of magnitude when the variability of travel time remains moderate. The VoR is higher for the exponential distribution when variability becomes strong. The "equilibrium VoR" asymptotically converges toward the "F-K VoR" for both distributions. This point, which is not demonstrated, is intuitive: when σ is high, congestion becomes close to null and the expected travel time is quasi-constant.

On the other hand, for reasonable values of σ the VoR given by the single-traveler model is significantly higher than the VoR given by the equilibrium model, whatever the distribution. The single-traveler model substantially overestimates the value of reliability for the rush hour period. Several elements remain to be addressed. In several metropolitan areas, dynamic traffic information is available online and people can adjust their departure time based on this information. Integrating this point would increase the complexity of the model, but could provide interesting results concerning the value of this information (which could be negative, as shown by previous works).

Published in :

Mathematical Population Studies: An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 33 Secondly, one could use more detailed scheduling preferences than (α-β-γ)

preferences and include some flexibility on the preferred arrival time or a discrete penalty for being late at destination (which was empirically estimated by Small, 1982). Last but not least, the assumption of a constant volatility of the random delay is strong. Multiplicative random shocks are more realistic, considering that variability increases for larger levels of congestion [START_REF] Macdonald | Development of INCA to incorporate single carriageways and managed motorways[END_REF].

Future research will try to lift these assumptions. , I is necessarily bounded (point a).

To show b), us assume that

  0 / 0 0    t Q I t
. Because there is no congestion in t0,

,   t     0 t T t T 
. From there, using both properties of lemma 1 shows that the expected cost of the trip is strictly higher than in t0 on either ]-∞,t0[

or on ]t0,+ ∞[. Again,   v eq C t C t I    implies that if   0 / 0 0    t Q I t
, then t0 is either the lower or the upper boundary of I.

We note tq v and tq' v the lower and upper boundaries of

I. C is continuous, hence     ' v v v q q eq C t C t C 
. In addition, tq v being the start of the rush hour, we have Tt and Published in : Mathematical Population Studies: An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 37

Q(tq v ) = 0. Let us assume Q(tq' v ) > 0.   0 , '    t r t t v q   ' 1 v rq
 ' 0 v rq mt  , hence   ' v q r tt C   
, where subscript r denotes the right-hand derivative.

The expected cost of leaving a little after tq' v is lower than v eq C , in contradiction with the equilibrium condition. This proves Q(tq' v ) = 0 (point c).

To show d), let us assume

  1 ' 1 , / 0 vv qq t t t r t     
. Using the same reasoning as before, Q(t1) > 0 is absurd. This leaves the possibility Q(t1) = 0. However, it

would imply       1 ' v v v q q eq C t C t C t C    with 1' vv qq t t t  and       1 ' v v q q T t T t T t  
. This is also impossible given lemma 1, so I must be a right-open segment.

Last, based on the previous results, the queue starts at tq v and ends at tq' v .

When the bottleneck operates at full capacity, the time needed for N people to pass through bottleneck is N/s, hence point e.

Annex B -Proof of Proposition 1

Using lemma 1, one can show that the equation H(t+N/s,T0+μ) = H(t,T0+μ)admits a unique solution in t. We note tq v the solution and tq' v = tq v +N/s.

We will now show that the following departure rate function r v corresponds to a Nash equilibrium:

                      ' ' 0 if , 1 0 if , * where . v q v v v qq v v v qq t v t s r t t t t F m t F m t r t t t t t t T r u s du mt                                     (24) On ]tq v ,tq' v [, this solution solves:                             1 ' ' 0 1 v F m t F m t r t s T t C t s F m t F m t              (25)
The expected cost of the trip is therefore constant on The next step involves showing that there is no congestion in both tq v and tq' v .

This point is obvious for tq v , and    

   0 ,T t H t C v q v q
. For tq' v , demonstration is as follows:

    0 ' 0 ' , , (definition of and ) v v v v q q q q H t T H t T t t      (26)     '0 , vv qq C t H t T     (27)         ' ' 0 ' , (equilibrium condition: ) v v v v q q q q C t H t T C t C t      (28)       ' ' ' 0 , , v v v q q q H t T t H t T     (29)     ' 0 '
and 0 (using the 2nd property of lemma 1).

v v q q T t T Q t      (
implies that   [ , ] 0 ' v q v q t t t t Q    (see proof of Lemma 2). Because   0 '  v q t Q and   0 [, , ] '    t Q t t t v q v q
, we can use Eq. ( 2):

          ' ' ' ' 0 . v v q q v v q q t t v v v v v q q q t t Q t r u s du r u du s t t N          (31)
At the end of the rush hour, the total number of departures exactly adds up to N.

We have proved that the departure rate function r v a Nash equilibrium. We must now prove that it is unique.

If z is a departure rate function corresponding to a Nash equilibrium, using Lemma

(a) = Q(b) = 0. Equilibrium condition     a C b C  implies H(a+N/s,T0+μ) = H(a,T0+μ), hence a = tq v and b = tq' v . Then Lemma 3 implies:                             ' 1 ] , [ , ' . 1 v v v qq F m t F m t t t t T t z t r t F m t F m t               (32)
The equilibrium solution is unique indeed. Using Eq. ( 11), the expected costs of leaving at tq and tq' are:

      0 q N N N N C t T F G s s s s                                 (33)      '0 q N N N N C t T F G s s s s                                 (34)
Subtracting the two yields: Let us consider H as a function of t, T and σ instead of t and T only: where the rush hour [tq(σ), tq'(σ)] is a function of σ.

      ' 1 q q N N N N N N C t C t G G F F s s s s s s                                                               (35)       N s N s N N u f u du u f u du s s                                                       (36)   LR NN JJ ss                               (37) If                             s N J s N J R L ,     q q t C t C  ' . The equilibrium condition     v q v q t C t C '  implies tq v < tq. Conversely, if                             s N J s N J R L then     q q t C t C  ' hence tq v > tq.
                    ˆˆˆ, , , , , , , , , , * ŵith ,
The derivative of Ce(σ) is: , where M is the maximum value of the normalized random delay (in absolute terms). In the deterministic case Using this, it is equally easy to show that tq and tq' verify the condition for tq v and tq' v , hence tq v = tq and tq' v = tq'.

                                       
Using Eq.( 11), expected cost of the trip for an individual leaving at tq v is

    0 v q N C t T s      
, which is the same cost as in the deterministic case.

  and later Small (1982) have formalized this problem using what is now commonly called (αβ-γ) preferences. The overall cost of the trip values both the time spent on Published in :

  : An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 11  all (εt)t follow the same law;

  ensures that the equation F(x) = δ/β admits a unique solution. Not making this assumption leads to multiple equilibriums, but our results still hold Published in : Mathematical Population Studies: An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 14 true. Moreover, for any departure function r, we refer to the set  as the rush hour. Equilibrium solutions are denoted by a superscript v. Lemma 2 For any Nash equilibrium of the BARD, the rush hour is a right-open segment of length N/s, denoted by face congestion except for the first one, who leaves at v q t , and for the last one, who leaves at ' v q t .

Figure 1 :

 1 Figure 1: Relationship between trip cost (C) and travel time (T) under (α-β-γ)

Figure 2 :

 2 Figure 2: Effect of the variability of travel time on the normalized departure rates (uniform distribution)

Figure 3 :Figure 4 :

 34 Figure 3: Effect of travel time variability on the expected travel time profile (uniform distribution)

  : An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 27 risk averse when σ is small. Despite this difference, for low values of σ, the equilibrium cost of the trip barely increases with σ (Figure7). Again, individual adjustments almost completely offset the cost of unreliability.

Figure 5 :Figure 6 :

 56 Figure 5: Effect of the variability of travel time on the normalized departure rates

Figure 8 :

 8 Figure 8: Comparison of the equilibrium and disequilibrium (dis.) costs of the trip

Figure 9 :

 9 Figure 9: Comparison of the VoR for the stochastic bottleneck model and the

  : An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 326. ConclusionThe bottleneck model with an additive random delay predicts that the variability of travel time has three main effects on travel conditions at the rush hour: (1) departures are spread more evenly; (2) congestion decreases; (3) the peak will occur sooner or later depending on individual preferences, which are captured by the (α-β-γ) parameters, and depending on the distribution of the random delay.The risk associated with the unreliability of travel time has a cost, which is captured by the value of reliability. When all individuals adjust their departure time, the decrease in congestion greatly mitigates this cost. By treating congestion as an exogenous process, single-traveler models significantly overestimate the value of reliability for the rush hour -at least for reasonable levels of variability of travel time. On the other hand, for the two distributions (uniform and exponential) considered in our application, our model predicts a very low cost of unreliability, even for relatively high risk levels. This leads us to believe that (α-β-γ) preferences are not fully appropriate to compute the VoR. One should explicitly represent risk aversion through the use of adequate utility functions (such as CARA or CRRA).

.

  We note I the set {t / r(t) > 0} (the rush hour) and v eq C the equilibrium cost of the trip. The proof consists of five results: a) I is bounded, b) the bottleneck is congested throughout the whole rush hour, except possibly at its boundaries, c) there is no congestion in tq v and tq' v , d) I is an open-right segment and e) the length of this segment is N/s. First, Eq. (4) implies that the congestion level is bounded: The limits of the normalized safety margin m(t) are

  2, the set {t / z(t) > 0} must be of the form [a,b[ with the conditions b = a + N/s and Q

  : An International Journal of Mathematical Demography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420 41 Annex D -Proof of Proposition 4

.

  Given the previous assumption, one can easily show that F(m(tq)) = 1, F(m(tq')) = 0, and that G(m(tq)) = G(m(tq')) = 0.

  There exists a unique Nash equilibrium in the BARD. It is given by:

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	DOI: http://dx.doi.org/10.1080/08898480.2013.836420
	Proposition 1
	16

Table 1 :

 1 Parameter values for the BARD

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	DOI: http://dx.doi.org/10.1080/08898480.2013.836420

  ,If the support of f is unbounded on both sides,  

	H t T	T        			m t T F m t T 			G m t T				m t T	
		m t T			t t T  															(38)
	We have:
	 H	  , t  T	,		   			 G	  	t	*	  t		T	  		0	(39)
																												G	x		0	 x			and	 H t T  ,,   			0	.
		The equilibrium cost of the trip Ce(σ) is given by:
	e C	  		 H t	q	  	;	0 T		;  			 H t	q	'	  	;	0 T		 ; with  	t	q	'	    q t   		N s	(40)

  Mathematical Population Studies: An International Journal of MathematicalDemography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420

																																			Published in :
	  t H	 t	q	'	   ; 0 T			;			are of opposite signs. This holds true for	t H  	 t	q	   ; 0  T		;			    d dt q
	and		t H  	 t	q	'	   ;  T 0		;		     d dt q '	, so one of the two must be positive. Considering that
	the term in	  H is also positive (strictly positive if f is unbounded on both sides),
	we have	  0    d dC e	(>0 if f is unbounded), which is Proposition 4.
	Annex E -Proof of Proposition 5
	Let us assume				M		min	  	, NN ss  	  
							  d dC e			  t H			t	q			;	0 T		;	d dt q		  H	t	q	;	0 T		;
																  H t				t	q	'		;	0 T		;	dt	' d q		  H	t	q	'	;	0 T		;	(41)
	First, we have:       t t q q  '		N	/	s		    d dt q '		    d dt q	. From lemma 1 and the condition
	H	 t	q	   ;	0 T			;				H	 t	q	'	   ;	0 T			;			, we also know that	  t H	 t	q	'	   ; 0 T			;			and
																																			42

Mathematical Population Studies: An International Journal of MathematicalDemography (2014), Vol 21 (4), pp.220-242 DOI: http://dx.doi.org/10.1080/08898480.2013.836420

[START_REF] Coulombel | The value of reliability: an equilibrium approach[END_REF] show this point in the uniform case. It has yet to be established in the general case.


Nicolas Coulombel
email: nicolas.coulombel@enpc.fr

André De Palma

The marginal social cost of travel time variability

Keywords: Reliability, Congestion, Welfare, Bottleneck, Scheduling

This paper investigates the cost of travel time variability for car users at the peak hour. In particular, we derive the marginal social cost of travel time variability, which takes the feedback of travel time unreliability on the congestion profile into account. This is in contrast with the value of travel time variability, which treats congestion as an exogenous phenomenon. Congestion is modeled using the standard bottleneck model of road congestion, which we amend by adding a random delay. For individuals with ( α , β , γ ) preferences and uniformly distributed delays, the marginal social cost of travel time variability is strictly lower than the value of travel time variability. Moreover, we show that the former tends toward the latter when σ, the standard deviation of the random delay, tends toward +∞. For normally distributed delays, numerical application leads to similar conclusions. Analysis of data from the Paris area suggests that given the plausible range of σ, the marginal social cost of travel time variability is markedly lower than the value of travel time variability. When appraising the economic benefits of reliability improvements, one should prefer the marginal social cost of travel time variability for the peak period, and the value of travel time variability for the off-peak period.

Introduction

Travel time reliability is an important consideration when people make travel decisions [START_REF] Li | Willingness to pay for travel time reliability in passenger transport: A review and some new empirical evidence[END_REF][START_REF] Carrion | Valuation of travel time reliability from a GPS-based experimental design[END_REF]. [START_REF] Bates | The valuation of reliability for personal travel[END_REF] distinguish two main ways in which unreliable travel times impact people. First, it may disturb their activity schedule by causing early or late arrivals at destination. When excessive, the delay can even lead to the cancellation of an activity: for instance, an unpredicted traffic jam can cause one to arrive too late at the theater and miss the play. Second, people could dislike travel time variability per se. The unpredictability of the travel time would generate some form of anxiety, or cause some additional planning costs [START_REF] Noland | Simulating travel reliability[END_REF].

The valuation of travel time variability has accordingly become an increasing concern in transport economics, to the extent that the value of travel time variability (also referred to as the value of reliability) is now along with the value of time one of the most important values obtained from travel demand studies [START_REF] Carrion | Value of travel time reliability: A review of current evidence[END_REF]. The work of [START_REF] Fosgerau | The value of reliability[END_REF] has been central in this regard. In line with [START_REF] Polak | Travel time variability and departure time choice[END_REF] and [START_REF] Noland | Travel-time uncertainty, departure time choice, and the cost of morning commutes[END_REF], the authors study the choice of departure time for a traveler facing variable travel times. Their major contribution lies in deriving the value of travel time variability (VTTV) in a case relatively general, thereby providing a sound theoretical foundation to this notion. Still, one element limits the extent of their results. In line with the others before them, [START_REF] Fosgerau | The value of reliability[END_REF] study the behavior of one traveler only and model congestion as an exogenous phenomenon. Hence, the VTTV does not capture the impact that changes in travel time variability have on congestion. Likely limited for the off-peak period, this feedback effect could be sizable at rush hour when congestion becomes a primary feature of traffic. As all individuals (and not only one) adapt their departure time in response to changes in travel time variability, the congestion time profile is likely to change. To address this shortcoming of the VTTV, we propose an alternative indicator, the marginal social cost of travel time variability (MSCTTV), which differs from the VTTV in that it does capture the equilibrium mechanism just mentioned.

This paper studies the cost of travel time variability for car users when congestion is modeled as an endogenous phenomenon. We are especially interested in the spread between the value of travel time variability and the marginal social cost of travel time variability: for a given individual, it reflects the difference between the short term and long term (i.e. before and after other individuals adjust their departure time) costs of a change in the variability of travel times. We model congestion using the standard bottleneck model of peak-load congestion, which we amend by introducing a random delay. Derivation of the Nash equilibrium in departure times yields first the equilibrium expected scheduling disutility as a function of the standard deviation of the delay, then the MSCTTV by differentiation. We focus the analysis on the case of uniformly distributed delays as it leads to closed-form solutions. This allows a direct analytical comparison of the VTTV and the MSCTTV. The more realistic case of a normal law is also considered using numerical simulation, analytical derivation of the equilibrium solution being unfeasible in this case. The primary impact of travel time variability on the Nash equilibrium is peak spreading. Departures are spread more evenly over the rush hour than under certain travel times, resulting in less congestion. As a result of this mitigating mechanism, the MSCTTV is strictly lower than the VTTV. When the scale of the random delay is small enough compared to the total length of the rush hour, the MSCTTV is even null in the case of a uniform law, and close to null in the case of a normal law. Individual adjustments regarding the departure time offset, partly or even entirely, the direct cost of an increase in travel time variability. As far as cost-benefit analysis is concerned, this means that direct uses of the VTTV, in particular in 4-step models, overestimate the benefits of policies aiming to reduce travel time variability on a road network, and likely strongly so. One should consider the effect of such policies on the departure time profile (and thus on the O-D matrix when using a 4-step model) to avoid this bias, one solution being to use the MSCTTV. Additionally, our relatively strong result that the MSCTTV is null in some cases tends to corroborate the finding of [START_REF] Börjesson | Valuations of travel time variability in scheduling versus mean-variance models[END_REF], which is that (α,β,γ) preferences only capture part of the cost that travel time variability puts on travelers. Future research might want to delve into this issue.

The layout of the paper is as follows. Section 2 reviews the prior literature. The model is presented in Section 3. Section 4 elaborates on the case where the random delay follows a uniform law, while Section 5 addresses more succinctly the case of a normal law, using numerical simulation. Section 6 discusses the results, and Section 7 concludes.

Prior literature

This paper is directly related to two main strands of the literature on the cost of travel time variability. The first one focuses on the individual and his preferences through the notion of value of reliability. The second one considers interactions between individuals and studies the impact of travel time variability on the traffic equilibrium and the cost of travel. We shortly review both.

Value of travel time variability

The value of travel time variability captures how much an individual values (in monetary terms) the cost associated with a marginal increase in travel time variability. As such, it is mainly a property of preferences. The literature on this topic is structured according to three main modeling frameworks [START_REF] Li | Willingness to pay for travel time reliability in passenger transport: A review and some new empirical evidence[END_REF][START_REF] Carrion | Value of travel time reliability: A review of current evidence[END_REF]: the mean-variance model, the mean lateness model, and the scheduling model. The mean-variance and mean lateness models follow a descriptive approach. While they assume that individuals dislike travel time variability, they do not purport to explain why. These two modeling frameworks provide valuable econometric specifications to estimate the VTTV, but lack theoretical foundations. 1Scheduling models aim to fill this gap and provide a micro-economic foundation to the value of reliability. They represent the choice of departure time for an individual who faces time constraints (e.g. work start time). Typically, they assume that a traveler has a scheduling disutility function U(t,T), which depends on the departure time t and travel time T, and study the optimal departure time and the associated minimum disutility. Building on [START_REF] Gaver | Headstart Strategies for Combating Congestion[END_REF] and [START_REF] Vickrey | Congestion Theory and Transport Investment[END_REF], [START_REF] Small | The scheduling of consumer activities: work trips[END_REF] has specified and estimated the following functional form for the scheduling disutility, which is widely used in works on the value of travel time variability:

        , * * * U t T T t t T t T t t T t                1 (1)
where (x) + = x if x is positive, 0 otherwise, and 1(x) is the Heaviside step function (equal to 1 if x  0, 0 otherwise). * t t T  is the schedule delay. It is measured relatively to a preferred arrival time t*, which usually represents the work starting time. The cost of one minute of travel time is α; the cost of being one minute early at your destination is β and the cost of being one minute late is γ. Lastly, δ is the fixed penalty for being late. These parameters set the terms of the trade-off between travel time and schedule delay when choosing the departure time. We will refer to (1) as (α , β , γ , δ ) preferences, or more simply (α , β , γ ) when the late dummy is not included (δ = 0).

To study the value of travel time variability, most authors consider the case of a single individual with (α , β , γ , δ ) preferences who faces random travel times (e.g. [START_REF] Noland | Travel-time uncertainty, departure time choice, and the cost of morning commutes[END_REF][START_REF] Bates | The valuation of reliability for personal travel[END_REF][START_REF] Batley | Marginal valuations of travel time and scheduling, and the reliability premium[END_REF].2 Travel time T is the sum of a deterministic term and of a random delay, with two central assumptions: 1) the deterministic component is an exogenous function of the departure time; 2) the distribution of the delay is independent of the departure time. In the special case where the traveler has (α , β , γ ) preferences and where the deterministic travel time is a constant T0, [START_REF] Fosgerau | The value of reliability[END_REF] show that:

1 0 * d t t T             (2)       1 1 00 , Φ d E U t T T s ds               (3)
where td is the optimal departure time and Φ the cumulative distribution function (c.d.f.) of the standardized travel time. Compared to when travel times are deterministic, the traveler takes a safety margin (which may be positive or negative), which is represented by the term σΦ -1 (γ/(β+ γ)) in (2). Travel time variability lowers the capacity of the individual to be on time nonetheless, and the cost of travel (or scheduling disutility) is increased by the integral term in (3).

Using the standard definition of the value of travel time variability as the derivative of the scheduling disutility (expressed in monetary terms) with respect to the standard deviation of travel times [START_REF] Carrion | Value of travel time reliability: A review of current evidence[END_REF], [START_REF] Fosgerau | The value of reliability[END_REF] find:

    1 1 VTTV Φ s ds        (4)
The value of reliability only depends on the schedule delay cost parameters β and γ and on the c.d.f. Φ of the standardized random delay. This result has had considerable influence at various levels. First, it shows that the mean-variance model is formally equivalent to the scheduling model (given the above assumptions). This validates prior empirical studies based on the mean-variance framework. 3 Second, it provides a simple formula for the VTTV, making it easy to integrate in national guidelines for the economic appraisal of transportation projects.

Formula ( 4) is based on the assumption that the deterministic component of the travel time is constant. [START_REF] Noland | Travel-time uncertainty, departure time choice, and the cost of morning commutes[END_REF] and [START_REF] Fosgerau | The value of reliability[END_REF] consider the more general case where the deterministic component may vary with the departure time to represent congestion. 4Notwithstanding, congestion is always treated as an exogenous phenomenon, which is consistent with the view that the VTTV should only reflect individual preferences, not how the whole system reacts when travel time variability changes. Thus, and by definition, the VTTV does not consider the feedback of travel time unreliability on congestion: it measures the marginal cost of travel time variability in disequilibrium, that is, when the other individuals do not adjust their departure times.

Equilibrium models

Yet, congestion is the sum of the trip timing decisions of all concerned individuals. If one traveler optimizes his departure time when facing random travel times, there is no reason to assume that the other travelers would not do the same. The congestion profile is likely to change depending on the level of travel time variability, and subsequently the cost of travel.

To tackle this question, several works study the impact of travel time variability on the departure time profile and the resulting traffic equilibrium. [START_REF] Noland | Simulating travel reliability[END_REF] analyze the impact of road incidents in a simulation model which combines a supply side model à la Henderson5 with a discrete choice demand model that predicts scheduling choices for morning commute trips. The bottleneck model provides an alternative representation of the supply side, and was applied to study various aspects contributing to the variability of travel time: stochastic demand (de [START_REF] De Palma | Stochastic Equilibrium Model of Peak Period Traffic Congestion[END_REF][START_REF] Arnott | Does providing information to drivers reduce traffic congestion?[END_REF], stochastic capacity [START_REF] Arnott | Information and time-of-usage decisions in the bottleneck model with stochastic capacity and demand[END_REF][START_REF] Arnott | Information and Usage of Free-Access Congestible Facilities with Stochastic Capacity and Demand[END_REF][START_REF] Li | Departure time distribution in the stochastic bottleneck model[END_REF] and incidents [START_REF] Fosgerau | On the relation between the mean and variance of delay in dynamic queues with random capacity and demand[END_REF].

However, these works do not specifically analyze the relationship between the cost of travel and the standard deviation of travel times.

This paper studies the cost of travel time variability in the case of endogenous congestion. In particular, we seek to compare the marginal social cost of variability and the value of travel time variability, for the rush hour period. In line with the literature using equilibrium models, the scope of our analysis is restricted to car users, and to the impact of travel time variability on the choice of departure time. We do not consider other possible reactions such as changing mode [START_REF] Chorus | Travel information as an instrument to change cardrivers' travel choices: a literature review[END_REF] or route (Liu et al., 2004). This work assumption seems sensible in a first approach, given that changing departure time is typically the easiest response to travel time variability [START_REF] Li | Willingness to pay for travel time reliability in passenger transport: A review and some new empirical evidence[END_REF]. Last, our work bears similarities with that of [START_REF] Siu | Equilibrium Trip Scheduling in Congested Traffic under Uncertainty[END_REF], who also study the impact of travel time variability when congestion is endogenous. However, the two works have different focuses: the dynamics of congestion in the case of [START_REF] Siu | Equilibrium Trip Scheduling in Congested Traffic under Uncertainty[END_REF], and the relationship between the cost of travel and travel time variability in our case.

The model

Model set-up

In the bottleneck model of road congestion with peak-load demand, a fixed number N of individuals commute every day from point A (home) to point B (work). Each individual travels with his own car, along the single road that joins A and B. The road starts with a bottleneck with fixed capacity s. Accordingly, whenever the departure rate r(t) exceeds s, a queue develops at the bottleneck entrance (if it does not yet exist).

In the standard setting, the travel time is the sum of two elements: a fixed component, representing the time necessary to go from A (or more accurately from the bottleneck exit) to B, and a variable one, representing the time spent in the queue (if any). Both terms are deterministic. In line with [START_REF] Coulombel | The variability of travel time, congestion, and the cost of travel[END_REF], we add a stochastic component to model travel time variability. This takes the form of a centered random delay, noted εt, which may vary over the course of the day. 6Accordingly, the travel time T ~ takes the following form:

    0 t Qt T t T s     (5)
where a tilde emphasizes the stochastic nature of a variable. T0 is the free flow travel time. Q(t)/s is the congestion delay, equal to the ratio between the queue length Q(t) and the bottleneck capacity s. For reminder, the queue length varies as follows:

            if 0 if 0 Q t r t s Q t Q t r t s Q t                   (6)
In order to allow a direct comparison of our results with Fosgerau and Karsltröm (2010), we assume that individuals have (α,β,γ) preferences. The scheduling disutility is given by the random variable:

        * * U t T t t t T t t T t t                     (7)
As is usual in the literature, we set the following constraint:

β ≤ α . 7
The choice of (α,β,γ) preferences implies that we disregard anxiety costs or any additional planning cost that would not be captured by this specification. While [START_REF] Noland | Simulating travel reliability[END_REF] find that (α,β,γ) preferences well fit the behavior of individuals in their survey, [START_REF] Tseng | Value of time by time of day: A stated-preference study[END_REF] argue that the assumption of time-invariant shadow prices is unrealistic, a point corroborated by their empirical analysis. Moreover, [START_REF] Börjesson | Valuations of travel time variability in scheduling versus mean-variance models[END_REF] find that (α,β,γ) preferences do not well capture the aversion of individuals to travel time variability. While acknowledging these points, the choice of (α,β,γ) preferences is driven by one of our main objectives, which is to compare our findings to those of Fosgerau and Karsltröm (2010). It leads to closed-form solutions in Section 4 to boot. 8 Also, we assume that individuals have homogeneous preferences: they all wish to arrive at the same t* and all have the same triplet (α,β,γ). True, relaxing this assumption would make the model more realistic. For instance, considering a distribution of the (α,β,γ) among the population instead of a unique value would allow to model the fact that the trip purpose may vary among travelers. 9 Still, this assumption makes the model analytically tractable, while not being central to our main findings. 10 Each individual seeks to minimize his expected scheduling disutility by adjusting his departure time. A Nash equilibrium is reached when no one has an incentive to shift his departure anymore. The equilibrium is fully characterized by the departure rate function r(t), with the condition that the expected trip cost

  Ut     E
must be minimum and constant on the set {t / r(t) > 0}.

Equilibrium solution with deterministic travel times

We start by reviewing the main findings when travel times are deterministic, i.e. when εt = 0 t, which is the standard bottleneck model of road congestion with peak-load demand. 11 First, there exists a unique pure Nash equilibrium. Its main characteristics are as follows (subscript d denoting the deterministic case as opposed to the stochastic case that will follow): (1) individuals leave home in a continuous (but not homogenous) way between 0 (3) the departures rate varies so as to equalize the scheduling disutility throughout the rush hour. This entails the following condition: 7 The case β > α leads to counterintuitive behaviors, such as preferring to waste time in the vehicle (by making circles around the destination for instance) rather than to get early at one's destination. 8 We could easily apply the same framework to another utility function. Closed-form solutions would not be guaranteed in the uniform case, however, meaning that we would have to resort to numerical analysis. 9 Heterogeneity in the bottleneck model with deterministic travel times is discussed in depth in [START_REF] Lindsey | Existence, Uniqueness, and Trip Cost Function Properties of User Equilibrium in the Bottleneck Model with Multiple User Classes[END_REF]. 10 As we will see in Section 4, the primary impact of travel time variability is that departures are spread more evenly over the rush hour than when travel times are certain. This phenomenon would still occur even if the parameters t* and (α,β,γ) were to be distributed, which is why we believe our main findings to be robust to the assumption of constant t* and (α,β,γ). 11 This case is extensively discussed in the literature (see [START_REF] Arnott | Economics of a bottleneck[END_REF]Arnott et al., , 1993, among others), among others).

    0 1 for , for , p d d p d d T t t t t T t t t t                             (8) Outside 0 1 , d d t t     ,
N t t T s N T t T T s N tt s N U T T s N t t T s                                                     (9)
where * d U is the equilibrium disutility in the deterministic case.

Equilibrium solution with stochastic travel times

As previously pointed out by [START_REF] Bates | The valuation of reliability for personal travel[END_REF], the most general case where the set of distributions of (εt)t has no specific properties does not yield any significant result. To carry the analysis further, we assume that the (εt)t follow the same law, are centered, and have the same standard deviation σ.

The assumption of a constant standard deviation is likely the most questionable. On theoretical grounds, it implies that there is no direct relationship between the bottleneck and the random delay: the random delay takes place after the bottleneck, and is not influenced by the current queue.12 On empirical grounds, while a few studies corroborate it (e.g. [START_REF] Hurdle | Characteristics of travel time and dynamic user equilibrium for travel-to-work[END_REF], more recent studies typically find a positive correlation between the congestion level and the standard deviation of travel times [START_REF] Fosgerau | On the relation between the mean and variance of delay in dynamic queues with random capacity and demand[END_REF]OECD, 2010). Still, this assumption holds several merits. First and foremost, it is consistent with our goal to compare the MSCTTV and the VTTV, seeing that the VTTV is based on the same assumption of a constant standard deviation of travel time, whatever the departure time. Moreover, assuming a constant standard deviation makes computation much easier to follow, with no evidence at this point that it strongly determines our main results. 13 The probability distribution function (p.d.f.) and the c.d.f. of the normalized random delay εt /σ are denoted φ and Φ, respectively.

Existence and uniqueness of the equilibrium solution is demonstrated in the general case (i.e. for any distribution of the random delay) in [START_REF] Coulombel | The variability of travel time, congestion, and the cost of travel[END_REF]. In particular, they show that the equilibrium solution is characterized by the following conditions (where subscript s denotes the stochastic case):

                                0 1 *
                                           E E (10)
where  

Ttis the expected travel time and m(t) the normalized safety margin. 0 s t and 1 s t mark the beginning and the end of the rush hour in the stochastic case; they are determined using (10). Last, * s U is the equilibrium disutility in the stochastic case.

Considering that there exists an equilibrium and that it is unique, we can define the marginal social cost of travel time variability as the derivative of the equilibrium expected disutility with respect to the standard deviation of travel times:

* MSCTTV s U     (11)
The definition of the MSCTTV is almost identical to that of the VTTV. The only difference lies in how the expected disutility is computed: for the MSCTTV it is computed at the Nash equilibrium, while for the VTTV it is computed at the individual optimum, assuming that the other individuals do not adjust their departure time.

We will now use (10) to analytically derive the Nash equilibrium and the marginal social cost of travel time variability in the case of uniformly distributed delays.

Uniformly distributed delays

This section examines the case where the random delay is distributed according to a uniform law. By doing so we do not aim at realism: most empirical studies find that travel times follow an either normal or log-normal law [START_REF] Richardson | Travel time variability on commuter journeys[END_REF][START_REF] Giuliano | Incident characteristics, frequency, and duration on a high volume urban freeway[END_REF]. The choice of a uniform law leads to closed-form solutions, however, which is not the case for most other distributions, including the normal and log-normal cases (to the best of our knowledge). The uniform case is thus of utmost interest as we can derive the MSCTTV analytically and directly compare it with the VTTV, using the formula provided by [START_REF] Fosgerau | The value of reliability[END_REF]. The more realistic situation where delays follow a normal law is discussed in the next section, using numerical simulation given what was just said.

For reminder, in the case of a uniform distribution, φ and Φ are given by:

    1 23 for 3, 3 . 1 2 23 x x x x               (12)

Expected scheduling disutility

Using (7), the expected scheduling disutility when leaving at t is:

                * * * * t t T t t t T t U t T t t t T t u u du t T t u t u du                                     E ( 13
)
where

    0 / T t T Q t s  is the expected travel time. Using the notation     * / m t t t T t        
for the normalized safety margin, we can rewrite (13) as follows:

                  U t T t m t m t m t m t                   E (14) where     x x u u du      is a positive function. Calculation yields   3² 43 x x   for 3, 3 x      , 0 otherwise. It ensues:                         2 if 3 3 if 3, 3 2 4 3 U t T t m t m t mt U t T t m t m t U t T t m t                                   E E E   if 3 mt           (15)
Because the random delay is bounded, people who leave with a large enough head start (   3 mt  ) are sure to be always early (m(t) > 0) or always late (m(t) < 0) at their destination, whatever the delay. For those, the expected scheduling disutility in (15) has the same formula as in the deterministic case

(remember that     * m t t t T t    
). When choosing a smaller safety margin (case   3 mt  ), one may arrive before or after t* depending on the delay one experiences, hence a different formula.

Travel time dynamics

Based on (10), the expected travel time satisfies:

              01 if 3 2 2 3 , if 3, 3 2 2 3 ss T t m t mt t t t T t m t mt Tt                                                  if 3 mt               (16)
Again, when the safety margin is large enough (in absolute value), the expected travel time varies like in the deterministic case. In the case of a smaller margin, the travel time derivative is smoothed compared to when travel times are perfectly reliable. Indeed, when m(t) varies from 3  to 3 ,   t T shifts continuously from /(α-) to -/(α+), and not instantly and discontinuously as in the deterministic case.

The differential equation corresponding to   3 mt  can be integrated, which yields:

      2 23 * if 3, 3 2 2 3 T t t t A t m t                                              ( 17
)
where A is a constant to be determined using boundary conditions.

Equilibrium solutions

Given the above elements, there are four classes of solutions based on the value of σ and the relative magnitude of  and .

The general intuition is as follows: the greater the travel time variability, i.e. the greater σ, the greater the number of people who adjust their departure time. At first, only a small fraction of the N drivers change their departure time, those who leave around the peak (case 1). Adjustments remain minor and perfectly offset the cost of travel time variability. At some point, either the first (case 2) or the last (case 3) driver to leave is impacted, and the rush hour period starts to shift (later in case 2, earlier in case 3). Adjustments become more important and the equilibrium disutility increases. Eventually, σ is so important that all drivers may be early or late at their destination depending on the delay they experience (case 4). Individual adjustments become less and less efficient, and the cost of travel time variability markedly increases. ), similarly the last one is sure to be late (condition  

1 3 s mt 
).

People leaving near the peak can arrive before or after t* depending on the delay they experience.

The equilibrium solution verifies:

0 0 * 0 * s s N t t T s N U T s                    (18)
The timing of the peak period and the equilibrium disutility are the same as in the deterministic case. On the other hand, and as we will observe in the numerical application in subsection 4.5, departure rates are different, leading to changes in the congestion profile. Compared to case 1, travel time variability is such that the first commuter to leave arrives after t* if the random delay is too high. The last commuter remains always late.

The equilibrium solution is now characterized by:

0 0 * 0 * 3 2 3 3 2 3 s s N t t T s N N U T s s                                  (19) Case 3:  >  and     min , 1 1 min , 3 4 3 N N s s         
Case 3 is analogous to case 2 with the difference that  > . Accordingly, it is the last commuter who may be early or late while the first one is always early. The equilibrium solution satisfies: . In this set of solutions, all individuals may be early or late depending on the delay they are subjected to. Those leaving first are naturally more likely to be early, and conversely. Last, we have:

0 0 * 0 * 3 2 3 3 2 3 s s NN t t T s s N N U T s s                                   (20)
0 0 2 * 0 2 1 * 3 2 3 16 3 s s N t t T s N U T s                             (21)

Marginal social cost of travel time variability

Using the formulae above and then regrouping cases 2 and 3 in a single expression, we can derive the marginal social cost of travel time variability, which for reminder is defined by ( 11):

          2 22 min , 1 MSCTTV 0 if 3 min , min , 1 1 1 MSCTTV min , 3 1 if min , 3 3 4 3 MSCTTV 3 16 3 N s N N N s s s N s                                               1 if min , 43 N s                 (22)
MSCTTV is null when the standard deviation of travel times is below the threshold corresponding to case 1, which depends on the length of the rush hour N/s and on the preference parameters β and γ. Two elements account together for this counterintuitive result: 1) the choice of (α , β , γ ) preferences; 2) the boundedness of the random delay. The scheduling disutility function U(t,T) corresponding to (α , β , γ ) preferences is linear in T on ]-∞,t*-t] as well as on [t*-t , +∞[, with a kink in T = t*-t. Individuals leaving with a margin large enough so that they are sure to be always early (positive margin) or always late (negative margin) stay on one linear part of the disutility function; they are de facto risk neutral. In the case 1 of our bottleneck model, the first driver always arrives before t*, whatever the delay, similarly the last driver always arrives after t*. Considering what was just said, these two drivers are risk neutral. 14As long as they remain so, they act as "anchor points" and the equilibrium disutility does not change. To conclude, the specific properties of (α , β , γ ) preferences regarding risk aversion explain why for low values of σ, the cost associated with travel time variability and subsequently the MSCTTV are null.

It is straightforward to show that the MSCTTV increases with σ using ( 22), strictly so when

  min , 1 3 N s      
, and that it converges toward 3     when σ tends toward +∞. Using (4), one finds that the VTTV is also equal to 3     when the random delay is uniformly distributed. This means that the MSCTTV is strictly lower than the VTTV, and that the former converges toward the latter when σ tends toward +∞.

The fact that the MSCTTV is strictly lower than the VTTV results from the sum of all individual adjustments. As will be clear in the applied case, when σ increases, departures are more and more uniformly spread over the rush hour. This reduces congestion (compared to the deterministic case) and mitigates the cost of unreliability. However, as σ increases, individual adjustments become less efficient, and the MSCTTV strictly increases. Ultimately, when σ tends toward +∞, the MSCTTV is equal to the VTTV. The macroscopic equilibrium mechanism, i.e. the decrease in congestion, becomes ineffective, and everything happens as if congestion were exogenous and individuals no longer interacted with one another.

Applied case

We develop an applied case to illustrate and discuss at greater length the main results of our model. The values of the various parameters are summarized in Table 1.

Table 1: Parameter values for the applied case

Type of parameter Value

Schedule preferences

α = 1.2,  = 1,  = 3 Preferred arrival time t* = 9
Expected free flow travel time T0 = 0.5

Travel demand N = 2 000

Transportation supply s = 1 000

This set of parameters intends to represent a somewhat plausible situation. The preferred arrival time is set to 9 am, with a free flow travel time of half an hour. Travel demand is twice the transportation supply, resulting in a rush hour that is two hours long. 15 Finally, the preference parameters α,  and  are close to the ratio 1.0:0.8:3.0 indicated by [START_REF] Bates | The valuation of reliability for personal travel[END_REF] as typical from the literature. 16Keeping these values fixed, we study how the equilibrium solution changes when travel time variability, which is measured by σ, increases. While the parameter values influence the magnitude of the various phenomena that we will observe, all results can be considered as general to the case  > . The case  <  resembles the case  > , with similar patterns for the MSCTTV. The only noteworthy difference is that the rush hour shifts in the opposite direction (it starts later and later as σ increases).

Therefore, the case  <  is not considered.17

Travel time variability has three main effects on the congestion time profile: peak reduction (smaller top travel time), peak spreading and peak shifting. Peak shifting only occurs when travel time unreliability exceeds a certain threshold, though. These points are illustrated in Figure 2. As long as σ ≤ 1/2√3 (condition associated with case 1), the timing of the rush hour period stays unchanged, but the congestion peak is flattened when σ increases. Beyond this threshold, the rush hour period starts earlier and earlier as variability increases, and congestion keeps decreasing. The analysis of cumulative departures sheds light on these results. Starting from σ = 0 (deterministic travel times), an increase in σ induces adjustments before and after the peak (Figure 3). Indeed, before adjustments occur, travel time variability is most costly for people leaving at the peak, for whom schedule disutility was null under deterministic travel times. For the others, the choice not to be on time to avoid congestion insures them against the random delay to a greater or lesser extent. For instance, people leaving before the peak and arriving before t* under deterministic travel times will suffer less from a positive delay (it will increase their travel time but reduce their schedule delay). As a result, the cumulative departure curve is only changed around the peak at first. It is drawn to the right, meaning that a certain share of departures is delayed. It is interesting to note that this prediction is the opposite of what the model with exogenous congestion predicts. As a matter of fact, applying (2) to the uniform distribution and  > , the latter model predicts that travel time unreliability causes the traveler to leave earlier than under deterministic travel times, and not later as happens in our bottleneck model. When σ increases further, the equilibrium switches to the regime corresponding to case 3. Changes become more significant. People disliking being late more than being early ( > β), they start leaving earlier, and the whole curve shifts to the left. The curve is flattened at the same time toward a single straight line, meaning that the departure rate becomes more and more homogenous. All in all, the first commuters leave earlier than in the deterministic case (this time in accordance with the model with exogenous congestion), but past a certain time they leave later. Eventually, when σ 2/√3 , the equilibrium solution switches to case 4. Unreliability is so important that the whole curve is flattened. The peak period keeps moving earlier and earlier.

We now turn our attention to the crux of this paper: the influence of σ on the marginal social cost of travel time variability. As discussed in 4.4, for low values of σ (in the interval corresponding to case 1), the MSCTTV is null (Figure 4). Then starting from σ  1/2√3, the MSCTTV strictly increases, with a small kink at σ  1/√3 (marking the transition between cases 2 and 4). It asymptotically converges towards the VTTV, which is for reminder given by (4). Obviously, the central question is where we are on the curve. If in practice σ is in the interval corresponding to case 4 (right part of the curve in Figure 4), VTTV is a good approximation of MSCTTV. If σ is in the interval corresponding to case 1, MSCTTV is null and using VTTV as a substitute for MSCTTV is completely unfit (given the assumption of uniformly distributed delays).

Before attempting an answer, we turn to the case where the delay follows a normal law to check the robustness of our results.

Normally distributed delays

This section considers the more realistic case where the random delay follows a normal distribution. In this case we have:

    ²/2 ²/2 2 1 2 x x u e x x e du                 (23)
The differential equation in ( 10) no longer has a closed form solution (to the best of our knowledge). We resort to numerical simulation to address this case, and consider the same applied case as in Section 4.

The time profiles of congestion (Figure 5) and of cumulative departures (Figure 6) vary with σ in a way extremely similar to the case of the uniform distribution, hinting that the main results weakly depend on the distribution. The same three mechanisms are observed (peak reduction, peak spreading and peak shifting), and the peak reduction is in the same order of magnitude. There are two minor differences: peak shifting occurs even for low values of σ, but is at the same time less pronounced. The fact that peak shifting occurs as soon as σ >0 is linked to the fact that the random delay is unbounded. Because extreme delays (positive and negative) may occur, whatever his safety margin, one is never sure to be early or late at his destination. Attempting a comparison with the uniform case, we would always be in case 4 (all individuals may be early or late). This being said, the peak shifting phenomenon is barely perceptible for low values of σ. The MSCTTV curve is similar to the uniform case (Figure 7) Likewise, it increases with σ and converges asymptotically toward VTTV. 18 The MSCTTV strictly increases as soon as σ > 0, however, even if only slightly, which is again linked to the fact that the normal distribution is not bounded. Convergence occurs at the same pace as for the uniform distribution.

Discussion

Given that the MSCTTV curves are relatively close for the uniform and normal cases, the remaining question is where we are approximately on these curves. We can attempt answer in the case of the Paris metropolitan area by considering the time profile of morning departures for car commuters (Figure 8). By analogy to the road bottleneck model, the peak period starts when departure rates have higher values than usual, and ends when they become close to null. For the Paris area, a conservative estimate of the peak period would be from 7h30 to 9h30, hence being approximately two hours long. In our applied case, the length of the rush hour is also 2, meaning that we can directly read σ in hours. While in practice travel time variability may substantially vary from one route to another, we can realistically assume the standard deviation of travel times to be below 0.5 (i.e. half an hour), or even just below 1 (i.e. one hour) to be very conservative. Based on Figure 4 and Figure 7, this implies that the MSCTTV is well below the VTTV. We have just shown that the assumption of exogenous congestion can lead to greatly overestimate the cost of an increase in travel time variability. At rush hour, the fact that individuals adjust their departure time in response to travel time variability reduces congestion (compared to the situation where travel times are certain), which mitigates the cost caused by unreliability. However, one should not infer that travel time variability is a good instrument to struggle against congestion. The adjustments predicted by our model are unlikely to be as efficient in real life, be it only because of the difficulty for people to access good data on travel times, and in particular on their variability. Even more importantly, our model, like that of Fosgerau and Karlström, is based on the assumption that (α,β,γ) preferences correctly represent the behavior of travelers. As mentioned in subsection 3.1, this point is controversial. And, taking some perspective on our results, our finding that the MSCTTV is null for low values of σ (provided that the random delay is bounded) does seem relatively strong. Therefore, our work tends to corroborate the idea supported by Börjesson et al. (2012) that (α,β,γ) preferences capture only part of the cost of travel time variability.

There are several possible interpretations to the "missing cost". As mentioned in the introduction, it could be related to the anxiety caused by uncertainty or to additional planning costs. Another rationale is that there exist schedule constraints at the origin (e.g. having to drop one's kids at school), which are not modeled by (α,β,γ) preferences and which limit the capacity of individuals to adjust their departure time. In all cases, taking this "missing cost" into account in the utility function would likely reduce the gap between the MSCTTV and the VTTV. Notwithstanding, the basic mechanism brought to light by our model (departures are more spread, which decreases congestion and thereby the cost of unreliability) would still exist, and the MSCTTV would still remain strictly lower than the VTTV.

Conclusions

We have extended the theory on the valuation of travel time variability to the case of endogenous congestion, thereby taking into account interactions between individuals and equilibrium mechanisms. On a theoretical note, we have established that the marginal social cost of travel time variability asymptotically converges toward the value of travel time variability for uniformly distributed delays. Numerical simulation leads to similar conclusions for a normal distribution, but with no formal proof. On a more empirical note, analysis of travel data for the Paris area strongly suggests that the marginal social cost of travel time variability is substantially lower than the value of travel time variability. For all practical purposes, when appraising economic benefits linked to reliability improvements, one should use the relevant indicator depending on the period considered (MSCTTV for the peak period, VTTV for the off-peak period).

Our analysis is subject to a number of caveats. First, we considered homogeneous (α,β,γ) preferences. Heterogeneity regarding the (α,β,γ) parameters and, in another direction, alternative scheduling preferences (e.g. the Vickrey-Tseng-Verhoef preferences) should be considered to test the robustness of our findings. Second, and as mentioned before in subsection 3.3, the assumption of a constant standard deviation is unrealistic. Again, the robustness of our results could be put to a test by linking the standard deviation to the congestion level. Third, the convergence of the marginal social cost of travel time variability toward the value of travel time variability was only proved for the uniform distribution. While numerical simulations incline us to reckon this result as general, a formal demonstration of this point has yet to be achieved. Finally, we only consider the impact of travel time variability on the departure time choice. Extending the model to include the mode choice as well as the route choice seems a promising way toward a better understanding of how travel time variability influences the whole trip making process.

  the bottleneck always operates at full capacity on this interval. Consequently, the length of the rush hour is:1 d t -0 d t = N/s;

  there is no congestion but the cost of travel is greater than inside 0 schedule delay costs. During the first part of the rush hour 0 , departure rate is greater than s. Queue length increases and reaches its maximum at the peak p d t . Waiting time is also greatest at p d t , conversely the schedule delay is minimal and null. People leaving before the peak are thereby early, while those leaving after the peak are late.These various results are illustrated in Figure1. As we can see, cumulative departures rise quickly during the first part of the rush hour 0 , grow at a slower rate, until everybody has left at 1 d t . Because it has a limited capacity, the bottleneck acts as a filter and people arrive at their destination at a constant rate s, which one can observe on the curve of cumulative arrivals. The horizontal distance between the two curves gives the total travel time, which is the sum of the free flow travel time T0 and the queuing time. Again, we see that the queuing time is null for people leaving at the beginning or at the end of the rush hour, while it is maximum for people leaving at p d t .

Figure 1 :

 1 Figure 1: The standard bottleneck model

  (equal to √3σ in absolute value) is small enough compared to the length of the rush hour N/s. The first commuter to leave is sure to be early at his destination whatever the delay he experiences (condition  

  implies β > ).

Figure 2 :

 2 Figure 2: Impact of travel time variability on the expected travel time (uniform law)

Figure 3 :

 3 Figure 3: Influence of travel time variability on the cumulative departure curve (uniform law)

Figure 4 :

 4 Figure 4: Comparison of the MSCTTV and the VTTV (uniform law)

Figure 5 :

 5 Figure 5: Impact of travel time variability on the expected travel time (normal law)

  19

Figure 7 :

 7 Figure 7: Comparison of the MSCTTV and the VTTV (normal law)

Figure 8 :

 8 Figure 8: Home-work trips made by car (counted in number of cars), 2001 (Source: EGT, DRIEA)

As pointed out later,[START_REF] Fosgerau | The value of reliability[END_REF] provided a theoretical grounding to the mean-variance formulation a posteriori, using the scheduling model. The mean-variance model remains a descriptive model per se nonetheless.

A few works consider other utility functions than the standard (α , β , γ , δ ) preferences (e.g.[START_REF] Siu | Equilibrium Trip Scheduling in Congested Traffic under Uncertainty[END_REF][START_REF] Fosgerau | The value of travel time variance[END_REF], including the Vickrey-Tseng-Verhoef preferences popularized by[START_REF] Vickrey | Pricing, Metering, and Efficiently Using Urban Transportation Facilities[END_REF] and[START_REF] Tseng | Value of time by time of day: A stated-preference study[END_REF]. The general framework remains the same.

This result is actually controversial, not on a theoretical ground but when trying to put it to practice. While[START_REF] Noland | Simulating travel reliability[END_REF] support it,[START_REF] Börjesson | Valuations of travel time variability in scheduling versus mean-variance models[END_REF] find that the valuations of travel time variability given by the mean-variance approach and the scheduling approach are inconsistent.

[START_REF] Fosgerau | The value of reliability[END_REF] find that doing so only marginally changes the VTTV and that (4) still applies in first approximation.

In his model,[START_REF] Henderson | The economics of staggered work hours[END_REF] treats time as a discrete variable and represents congestion using the standard speed-flow curves. See[START_REF] Chu | Endogenous Trip Scheduling: The Henderson Approach Reformulated and Compared with the Vickrey Approach[END_REF] for a comparison of the bottleneck model with Henderson's approach.

Strictly speaking, εt encompasses all phenomena leading to travel time variability other than those related to fluctuations in demand (N) or in the bottleneck capacity (s). This includes weather conditions, variations in the driving speed after exiting the bottleneck and the parking time at destination among other things. Furthermore, the role of the provision of dynamic information regarding εt is not considered in this paper. This excludes dynamic strategies based on such information (like choosing to leave when travel times are lower than usual). The time dynamics of εt has therefore no importance here, and any behavior can be assumed as long as the basic assumptions mentioned further hold.

However, the existence of the random delay does lead to changes in the departure times of the various individuals and thus to changes in the queue dynamics.

Preliminary analyses corroborating that the assumption of a constant standard deviation does not play a central role in our main results regarding the MSCTTV are available upon request from the authors.

In case 1, not all individuals are risk neutral: only those who are sure to be early or sure to be late are, i.e. the first and the last individuals to leave. Drivers leaving in the middle of the rush hour are risk averse.

Note that only the ratio N/s matters, the nominal values of N and s having no importance.

β and γ are not exactly in the ratio 0.8:3.0 in order to get round values for the fractions β/(β+ γ) and γ /(β+ γ).

There is also the special case β = γ. In sum, when β = γ, travel time variability leads to peak spreading and peak reduction, but no peak shifting occurs. The MSCTTV behaves as when β < γ or β > γ.

Published in: Transportation Research Part C (2014), Vol.47, pp.47-60 DOI: http://dx.doi.org/10.1016/j.trc.2013.12.004

Let us note that the VTTV is not the same for the uniform and normal distributions.

A comparison of the MSCTTV for the uniform and normal distributions, not presented here for the sake of conciseness, shows that they are indeed very close whatever the value of σ.

Acknowledgments

The authors would like to thank first the organizers of the EWGT 2012 conference, and the referees and the audience for useful comments on this paper. They would also like to thank Neila Bhouri and Nadir Farhi for bringing forward this special issue, the three anonymous referees, again for very perceptive comments and suggestions, and the editor Matthew Karlaftis.