Value of travel time reliability: A review of current evidence
Carlos Carrion
⇑
, David Levinson
Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive SE, Minneapolis, MN 55455, USA
article info
Article history:
Received 25 February 2011
Received in revised form 8 December 2011
Accepted 5 January 2012
Keywords:
Variability
Reliability
Travel time
Scheduling
Meta-analysis
abstract
Travel time reliability is a fundamental factor in travel behavior. It represents the temporal
uncertainty experienced by travelers in their movement between any two nodes in a net-
work. The importance of the time reliability depends on the penalties incurred by the trav-
elers. In road networks, travelers consider the existence of a trip travel time uncertainty in
different choice situations (departure time, route, mode, and others). In this paper, a sys-
tematic review of the current state of research in travel time reliability, and more explicitly
in the value of travel time reliability is presented. Moreover, a meta-analysis is performed
in order to determine the reasons behind the discrepancy among the reliability estimates.
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Two of the most important values obtained from travel demand studies are the value of travel time (VOT), and the value of
travel time reliability (VOR). The former links the monetary values travelers (or consumers) place on reducing their travel
time (i.e. savings). The latter connects the monetary values travelers place on improving the predictability (i.e reducing
the variability) of their travel time.
The concept of value of travel time has a long established history through the formulation of time allocation models from a
consumer theory background. These models are reviewed thoroughly in Jara-Diaz (2007); a full chapter is dedicated to them.
Also, the reader may refer to Small and Verhoef (2007) for a (succinct) review of the theory. In addition, more than a hundred of
empirical estimates of the value of travel time has been carried out by both researchers and practitioners. Reviews such as
Abrantes and Wardman (2011), Shires and de Jong (2009), Wardman (1998, 2001, 2004), Zamparini and Reggiani (2007b,a)
serve as compilations of VOT estimates, and summaries of what has become mainstream knowledge in the field of travel
demand. In contrast, the value of travel time reliability is a ‘‘newcomer’’ to this field, and although it has received increased
attention, the procedures for quantifying it are still a topic of debate. The differences among studies span almost every aspect
such as: experimental design (e.g. presentation of reliability to the public in stated preference [SP] investigations); theoretical
framework (e.g. scheduling vs. centrality-dispersion); variability (unreliability) measures (e.g. interquartile range, standard
deviation; a requirement in the centrality-dispersion framework); setting (or estimating) the preferred arrival time (e.g. assum-
ing work start time as preferred arrival time in the scheduling approach); data source (e.g. revealed preference [RP] vs. state
preference [SP]); and others. As a consequence, value of reliability estimates also exhibit a significant variation across studies.
In this paper, a systematic review of the theoretical and empirical research on travel time reliability is presented. Firstly,
the concept of travel time reliability is discussed as it has become ‘‘defined’’ in the literature. Secondly, the most common
theoretical models (scheduling and mean-variance [centrality-dispersion]) are described; others (e.g. mean-lateness) are
0965-8564/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tra.2012.01.003
⇑
Corresponding author.
E-mail addresses: carri149@umn.edu (C. Carrion), dlevinson@umn.edu (D. Levinson).
URL: http://nexus.umn.edu (D. Levinson).
Transportation Research Part A 46 (2012) 720–741
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also briefly covered. Thirdly, the empirical evidence is compiled, and surveyed; the similarities and discrepancies of results
across studies are discussed. Fourthly, a meta-analysis is performed to identify the sources of variations in travel time reli-
ability estimates, and to provide an even more objective comparison of the reasons behind the estimates variability across
studies. Lastly, the article is concluded.
2. Travel time reliability: concepts
The concept of travel time can be defined as the time elapsed when a traveler displaces between two (distinct) spatial
positions. Certainly, this definition is applicable to any transportation mode (or combinations of them) regardless of the
inherent differences across them. This is expected as travel time is typically understood as a one dimensional quantity (var-
iable). Furthermore, travel time can be divided into several components depending on the analyst. For example, travel time
of public transit modes tends to be split into waiting time, in-vehicle time, transfer time, and others.
In road networks, travel time may be split into two components: free flow time, and additional time. The former refers to
the amount of time it takes a driver to arrive at his/hers destination without encountering any (or very little) traffic. The lat-
ter refers to each increase of travel time due to variations in the traffic conditions. These variations may be predictable (e.g.
peak-hour congestion), or unpredictable (e.g. vehicular crashes).
The predictable variations are events (i.e. traffic congestion) expected by travelers, and thus travelers (in principle) per-
form the necessary adjustments to offset the added costs (e.g. departing earlier to avoid arriving late at work). Such events
(i.e. traffic congestion) are by itself a topic of interest to many researchers focusing on traffic flow theory (see Daganzo (2007)
for an introduction to traffic flow theory). In transportation research, the morning peak-hour congestion is considered as a
classic problem of trip scheduling under deterministic traffic conditions. Vickrey (1969) presented a solution to the problem
with a single deterministic bottleneck model between an origin and destination, fixed and homogeneous travel demand, and
endogenous departure time (i.e. trip scheduling choices). This model was further extended by Arnott et al. (1990), Laih
(1994), Arnott et al. (1993), Arnott et al. (1994), Garcia (1999), Newell (1987), Daganzo and Garcia (2000), Daganzo
(1985), Daganzo (1995) and others. The interested reader on bottleneck theory, and its pricing applications may consult Yang
and Huang (2005) and Small and Verhoef (2007).
The unpredictable variations are directly linked to the uncertainty of travel time. This uncertainty has been divided in
three elements by Wong and Sussman (1973): variation between seasons and days of the week; variation by changes in tra-
vel conditions because of weather and crashes or incidents; and variations attributed to each traveler’s perception. Nicholson
and Du (1997) lists also the components of uncertainty as variations in the link flows and variations in the capacity. There-
fore, the unpredictable variations trace their source at both the demand side (e.g. traveler’s heterogeneous behavior) and
supply side (e.g. traffic signal failure) of a transportation system.
Travel time reliability is closely linked to the unpredictable variations. This suggests that travelers choose under an uncer-
tain environment as they may fail to predict their exact travel time before scheduling their trips (i.e. choosing a departure
time). In the case of predictable variations, the travelers may adjust their departure time choice, and still be certain of arriv-
ing on time at their destinations. This is true even in a transportation system with high congestion. Notice that travelers are
choosing under a certain environment. Therefore, it’ll be incorrect to consider predictable variations as examples of travel
time (un) reliabiility (Bates et al., 2001). It should be noted that this travel time uncertainty may also extend to other choice
dimensions (e.g. mode, route). Furthermore, the concept of travel time (un)reliability is defined as interchangeable with tra-
vel time variability (or unpredictable variation) in the transportation research literature; high variability means high unre-
liability, and vice versa. Consequently, it is natural to think of travel time in two dimensions: frequency, and magnitude. In
other words, travel time defined as a distribution in the probability theory sense. In this way, travel time (un)reliability can
be associated as a measure of spread to the travel time distribution. Distinct approaches have been proposed to model travel
time reliability, and they are reviewed in the subsequent section. Moreover, similarities (i.e. travel time composed of deter-
ministic and random elements) may be drawn to other transportation modes despite the fact that the concepts were mostly
explained with a focus on road transportation.
3. Theoretical frameworks
3.1. Centrality-dispersion
The approach is mostly known in the context of risk-return models in finance. A decision-maker looks to maximize the
option’s return while minimizing its associated risk. The option’s return is represented by the expected value, and the risk by
the variance (see Markowitz (1999) for an overview). In a transportation context, the framework is based on the notion that
not only travel time is a source of disutility, but also travel time variability (or unreliability). Thus, the formulation (with a
linear-additive form) of the model, in a consumer theory background, is as follows:
U ¼
c
1
l
T
þ
c
2
r
T
ð1Þ
The traveler is minimizing the sum of the two terms (objective function for an unspecified choice dimension): the
‘‘expected’’ travel time of the trip, and the travel time variability of the trip. The ‘‘expected’’ travel time (
l
T
) is included
C. Carrion, D. Levinson / Transportation Research Part A 46 (2012) 720–741
721
as a centrality measure (e.g. mean) of the travel time distribution. The travel time variability (
r
T
) is included as dispersion
measure (e.g. standard deviation) of the travel time distribution. The
c
coefficients are exogenous parameters. Typically, the
choice dimension is route choice, and the centrality (dispersion) measure is mean (variance or standard deviation) among
studies using this approach. Mean-variance is also usual name the approach is known in the transportation literature,
despite the fact that the centrality and dispersion measures vary among studies.
In the transportation literature, the framework was introduced by Jackson and Jucker (1982). Their original formulation is:
Minimize EðT
p
Þþk
k
VarðT
p
Þ
p 2 P
AB
k
k
> 0 ð2Þ
A traveler k has a priori information of the mean (E(T
p
)), and variance (Var(T
p
)) of the travel time distribution for each
route in their choice set (P) between an origin-destination pair (AB). k
k
indicates the degree of risk aversion of the traveler
k. The choice dimension is the route. Succinctly, a traveler k, with a degree of risk aversion k
k
, chooses the route that min-
imizes the objective function Eq. (2) given the expected and variance of the travel time distribution. The model proposed by
Jackson and Jucker (1982) is usually estimated using discrete choice methods with the linear-additive specification given in
Eq. (1). In this utility form plus a travel cost variable (
c
3
C), marginals rate of substitution may be computed to obtain impor-
tant quantities such as the value of travel time (VOT), value of travel time reliability (VOR), and the reliability ratio (RR).
These are defined formally in the previous order as,
VOT ¼
@U=@
l
T
@U=@C
ð3Þ
VOR ¼
@U=@
r
T
@U=@C
ð4Þ
RR ¼
@U=@
r
T
@U=@
l
T
¼
VOR
VOT
ð5Þ
In essence, this framework is based on expected utility theory developed by Von Neuman and Morgenstern (1944). The
theory prescribes a set of axioms about how decision-makers deal with risky prospects (set of alternatives where a choice
is selected) based on distinct states of nature (or states of the world). In simple words, there are several alternatives with
several possible states of natures (the distribution of outcomes for each alternative), and associated to each combination
of alternative and state of nature there is an outcome. In the transportation context, the set of alternatives could be routes,
modes, schedules. The states of nature could be traffic signal failure, crashes, and others. The outcomes are likely to be the
distribution of travel times for each alternative. In addition, the decision-maker ranks the risky prospects through the
assumption of the existence of an ordinal utility function (i.e. U = f(outcome); the utility function associates a single real
number to each outcome), and prefers the alternative with the highest expected utility (E(U)). Furthermore, an important
feature of the expected utility framework is based on decision-making under risk. In other words, there’s a different between
risk, where probabilities are known or at least knowable, and uncertainty, where probabilities are unknown. This difference
may not be particularly useful for most practical purposes, or it may be irrelevant by considering subjective probability, and
the axiomatic approach of expected utility theory (Takayama (1993) Chapter 5). Readers may also refer to Mas-Colell et al.
(1995)Varian (1978), for treatments of expected utility theory.
The functional form of the utility function is not restricted by the axioms. In fact, the functional form chosen should be
based on its close description of a decision-maker’s behavior. The functional form determines the risk preferences of the deci-
sion-maker. Functional forms may be selected based on regression analysis of experiments (e.g. gambling games that provide
observations revealing the utility function) or computationally convenient forms (Hazell and Norton (1986) Chapter 5).
In the transportation literature, several functional forms have been considered to understand the risk behavior of travel-
ers. Polak (1987) considered an alternative formulation to Jackson and Jucker (1982), where he defined the utility function of
the traveler as a polynomial of second degree with respect to the travel time variable (T). Formally,
U ¼
c
1
T þ
c
2
T
2
ð6Þ
This functional form Eq. (6) is known in the microeconomics literature (see Varian, 1978, Chapter 11, pp. 189) as equiv-
alent to the mean-variance model under expected utility theory. This can be seen by applying the expectation operator to Eq.
(6), and using a simple identity (Var(X) E(X
2
) (E(X))
2
),
EðUÞ¼
c
1
EðTÞþ
c
2
ðEðTÞÞ
2
þ
c
2
VarðTÞ ð7Þ
An important consideration is that the omission of the additional term [(E(T))
2
] in Eq. (7) might bias the estimates of
c
2
,
especially when the formulation in Eq. (6) is accurate. In addition, the
c
2
indicates whether the traveler prefers alternatives
(e.g. routes) with high variance of travel time (risk prone), low variance of travel time (risk averse) or only cares about the
expected travel time (risk neutral). Furthermore, higher degrees of polynomials may be specified, and consequently in ex-
pected utility forms will lead to higher moments to be included. Another formulation proposed by Polak (1987) is
722 C. Carrion, D. Levinson / Transportation Research Part A 46 (2012) 720–741
U ¼e
c
1
T
ð8Þ
This functional form Eq. (8) is also known in the microeconomics literature (see Varian, 1978, Chapter 11, pp. 189–190); it
describes a traveler with absolute risk aversion.
Senna (1994) introduced a more general form based on the previous mentioned work, where the utility function is given
by a algebraic term of degree b.
U ¼
c
1
T
b
ð9Þ
The utility function can be written in terms of expected utility, by applying the expectation operator, and by considering
some simple identities such as the definition of covariance, in the following form:
EðUÞ¼
c
1
ET
b
2
2
þ
c
1
Var T
b
2
ð10Þ
The Eq. (10) exhibits certain properties. The b parameter estimates the degree of risk aversion/proneness by the travelers.
Another important property is that the value of time and the value of variability (reliability) depend directly on the travel
time distribution. The reader should refer to appendix 2 in Senna (1994) for the mathematical proofs.
It should be noted that all the previous
c
coefficients are parameters to be estimated, and expected to be negative.
3.2. Scheduling delays
Historically, this approach has been linked to the departure time choice (or trip scheduling) studies. The basis for the ap-
proach rests on the time constraints (e.g. work start time) a traveler may face, and thus it associated costs due to early or late
arrival. This leads to the idea of a traveler intrinsic choice of a preferred arrival time (PAT); the point of reference that delim-
its whether an arrival is early or late. Gaver (1968) is one of its earliest proponents. He introduced a theoretical framework
for describing variability in trip-scheduling decisions. He considered distinct head start strategies for given delay distribu-
tions along with the costs of arriving early or late. In addition, statistical estimation procedures (non-parametric and para-
metric) are provided to estimate the probability density distribution of the trip delay, when it is unknown to the researcher.
Vickrey (1969), as described previously, also considered the trade off travelers face between queue delay, and schedule delay
of arriving early or late at work. Furthermore, a similar hypothesis is the existence of a ‘‘safety margin’’ advocated by Knight
(1974) and Pells (1987). Knight (1974) suggested that travelers consider a slack time (i.e. safety margin) between their (aver-
age) arrival time and their work start time. This safety margin allows the reduction of the probability of late arrivals, and
implies that travelers have a preference of arriving early to work (i.e. existence of positive utility for the time spent at work
before work start time). In essence, Knight (1974) hypothesizes that the departure time chosen happens when the marginal
utility of time spent at home is equal to the marginal utility of arriving early to work plus the marginal utility of arriving late
to work. Pells (1987) further argued that two opposite existing factors are at play: the need to minimize the frequency of late
arrivals, and maximize the time spent at home relative to the early time spent at work. Travelers meet the first factor by
allocating a safety margin, and they meet the second factor by maintaining the safety margin at required levels (i.e. safety
margins are acceptable when there’s more time spent at home relative to early time spent at work).
Another important contribution is by Small (1982), based on some of the previous articles (mostly Gaver (1968) and
Vickrey (1969)). He formulates a theoretical model based on the traditional utility maximization framework (i.e. consumer
behavior; see (Varian, 1978)) with insights from time allocation models (e.g. Becker (1965), DeSerpa (1971), Bruzelius
(1979)). Small (1982)’s model consists of tying explicitly the departure time choice, and also adding a workplace constraint
(i.e. an equation linking departure time, and working hours with merits or penalties to the wage rate; workplace policies
where pay is docked by tardiness or bonuses are given for arrival on time) to the utility function of a traveler. In this
way, the traveler’s utility is influenced by the departure time, and also the value of time is influenced by the workplace con-
straint. Properties of this formulation may be reviewed in Jara-Diaz (2007, Chapter 2, pp. 67–69), and Carrion (2010, Chapter
2, pp. 11–15). Furthermore, he specifies a functional form for the (indirect) utility of scheduling:
Uðt
d
; PATÞ¼
c
1
T þ
c
2
SDE þ
c
3
SDL þ
c
4
DL ð11Þ
This is a linear-additive form, where the
c
coefficients are parameters to be estimated, and expected to be negative. In this
equation, the travel time (T) is not only included but also the scheduling delays which are divided by early (SDE; defined as
Max(0,PAT [T + t
d
])) and late (SDL; defined as Max(0,[T + t
d
]) PAT) arrivals according to a preferred arrival time (PAT), and
a binary term DL to indicate whether it is a late arrival or not (SDL > 0). The SDE, SDL and DL terms represent scheduling
considerations for the workplace constraint. t
d
is the decision variable (usually a continuous real variable for mathematical
models), and it represents the traveler’s departure time choice. Up until this point, the scheduling delay framework describes
travelers’ choices under certainty. Moreover, Bates et al. (2001) points out that capacity restrictions (i.e. t
d
is no longer inde-
pendent of T; travelers cannot choose the same t
d
as queueing is now present) of the transportation facility readily translates
this framework to one extensively studied using bottleneck models (e.g. Arnott et al. (1990), Laih (1994), Arnott et al. (1993),
Arnott et al. (1994)). This implies (as discussed in Section 2) the decomposition of travel time into: free flow travel time, and
additional travel time due to recurrent congestion.
C. Carrion, D. Levinson / Transportation Research Part A 46 (2012) 720–741
723
The model proposed by Small (1982) is usually estimated using discrete choice methods (i.e. the departure times are dis-
crete intervals if scheduling is the choice situation) with the linear-additive specification given in Eq. (11). In this utility form
plus a travel cost variable (
c
5
C), marginals rate of substitution may be computed to obtain important quantities such as the
value of travel time (VOT), value of scheduling delay early (VSDE), and the value of scheduling delay late (VSDL). Researchers
often discard the lateness penalty variable (DL), because it adds a discontinuity that is inconvenient to mathematical opti-
mization models (gradient-based), and a missing lateness penalty may translate into a higher lateness scheduling delay in
econometric models. These are defined formally in the previous order as,
VOT ¼
@U=@T
@U=@C
ð12Þ
VSDE ¼
@U=@SDE
@U=@C
ð13Þ
VSDL ¼
@U=@SDL
@U=@C
ð14Þ
3.2.1. Scheduling delays + dispersion
In Noland and Small (1995), the previous scheduling approach is extended to include explicitly the uncertainty of travel
time (i.e. unpredictable variation; see Section 2). This uncertainty is expressed in the form of a random variable (T
r
; preserv-
ing Noland and Small (1995) notation) with a given probability density function, and with the restriction of being greater or
equal to zero.
EðUðt
d
ÞÞ ¼
Z
1
0
Uðt
d
Þf ðT
r
ÞdT
r
¼
c
1
EðTÞþ
c
2
EðSDEÞþ
c
3
EðSDLÞþ
c
4
P
L
ð15Þ
The objective function of the traveler changes (also the utility function is traded for a trip cost form in Noland and Small
(1995), but we choose to keep it for coherency), and now the consumer maximizes the expected utility E(U(t
d
)) by choosing
the optimal t
d
(see Eq. (15)) for a given probability density function of T
r
. The elements of Eq. (15) include the scheduling
costs for early (SDE) vs. late (SDL) arrival at work presented earlier (see Eq. (11)), but also the last term employs the distri-
bution of the random variable (T
r
) in order to compute the probability of being late. P
L
is simply E(DL) (note DL is an indicator
function) conditional on t
d
. Therefore, the last term P
L
also contains the costs of travel time unreliability as the dispersion (or
variability) of the travel time distribution affects the calculated probabilities. In addition, travel time dispersion (or variabil-
ity) may increase the propensity of early arrivals, and thus high earliness costs can be incurred. This implies variability and
scheduling costs are related. In fact, Bates et al. (2001) argues that
c
2
E(SDE)+
c
3
E(SDL) may approximate the
c
0
2
r
T
in the
centrality-dispersion model (see Section 3.1) under certain conditions: travel time distribution is independent of departure
time;
c
4
= 0 in Eq. (15) or no lateness penalty; departure time is continuous; congestion dynamics are neglected as in travel
time is independent of departure time. Such mathematical properties and others are discussed in detail in Bates et al. (2001).
Recent work by Fosgereau and Karlstrom (2010) proved mathematically the previous statement by Bates et al. (2001)
(scheduling models approximated by mean-variance models). They indicate this can be achieved with knowledge not only
of the estimated parameters (
c
1
and
c
2
in Eq. (1)) for the expected travel time and variance, but also the travel time distribu-
tion, and the optimal probability of being late. This proof follows the assumptions presented earlier by Bates et al. (2001), and
also the obvious assumptions the mean of random variable (T
r
; they actually use a standardized form with mean 0 and var-
iance 1) is defined (i.e exists), and that it has an invertible distribution. These assumptions are more general than the previous
ones of assuming the density function of the random variable (T
r
) follows an uniform or exponential distribution (see Bates et
al., 2001; Polak, 1996; Noland and Small, 1995; Small and Verhoef, 2007). The interested reader should refer directly to Fos-
gereau and Karlstrom (2010), especially appendix A for more details. An empirical verification is also included in the paper.
It should be noted that all the previous
c
coefficients are parameters to be estimated, and expected to be negative.
Other recent work has followed different paths: inclusion of risk attitudes in scheduling models (Senbil and Kitamura,
2004; Michea and Polak, 2006; Schwanen and Ettema, 2009; Li et al., 2010); alternative formulation of schedule early
(SDE), and schedule late (SDL) (Tilahun and Levinson, 2010); and scheduling preferences with non-constant marginal util-
ities or time-varying parameters (Tseng and Verhoef (2008) and Fosgereau and Engelson (2011) and Jenelius et al. (2011)).
In Li et al. (2010), a non-linear utility specification (they assume a utility function of the form U ¼
c
1
x
1
a
1
a
, where x is any
variable in the model and
a
represents risk attitude) is used like in the other mentioned studies (e.g. Michea and Polak
(2006)), but the parameter indicating risk attitude (
a
) was assumed random, and thus the parameters of its population den-
sity function can be estimated using a mixed logit formulation. The idea of risk attitudes has been considered before in
microeconomics, and discussed implicitly in Section 3.1.
Tilahun and Levinson (2010) introduces a new approach for measuring SDE and SDL in Eq. (11) consisting of two mo-
ments: the first representing on average how early the traveler has arrived by using that route; and the second representing
on average how late that individual arrived by using that particular route. They assume that the deviation of the two mo-
ments (average late or average early) from the most frequent experience is a representative way of getting together the pos-
sible range and frequencies experienced by the travelers. Thus, this measure may considers scheduling constraints as well,
albeit not separately from (un)reliability of travel time.
724 C. Carrion, D. Levinson / Transportation Research Part A 46 (2012) 720–741
