Content uploaded by Hani S. Mahmassani
Author content
All content in this area was uploaded by Hani S. Mahmassani on Jun 09, 2016
Content may be subject to copyright.
Transpn. Res -,4,
Vol. 25A, No 5, pp 293-307, 1991 0191-2607/91 $3 00 +
00
Pnnted m Great Bream. © 1991
Pergamon
Press pie
SYSTEM PERFORMANCE AND USER RESPONSE
UNDER REAL-TIME INFORMATION IN A
CONGESTED TRAFFIC CORRIDOR
HANI S. MAHMASSANI and R. JAYAKRISHNAN
Department of Civil Engineering, The Umversity of Texas at Austin, Austin, TX 78712, U.S.A.
Abstract-A
modelhng framework is developed to analyze the effect of in-vehicle real time information
strategies on the performance of a congested traffic commuting corridor. The framework consists of a
special-purpose simulation component and a user decisions component that determines users' responses
to the supplied information. The user decisions component is microscopic and determines individual
commuters' route switching, at any node of the network, as a function of the supplied information. The
traffic simulation component moves vehicles in bundles or macroparticles at the prevailing local speeds,
as determined by macroscopic traffic relations. The framework allows the investigation of system perfor-
mance under alternative behavioral response mechanisms, as well as under different information strate-
gies. Results are presented for simulation experiments in a commuting corridor with a special network
structure that simplifies the network computations. The results illustrate the effect of the fraction of
users equipped with in-vehicle navigation systems on overall system performance. In addition, alternative
assumptions on user response reflecting varying degrees of optimizing behavior are explored. The model-
ling framework is shown to provide a useful approach for addressing key questions of interest in the
design of real time in-vehicle information systems.
1.
INTRODUCTION
The provision of information on a real-time basis to
individual vehicles in a traffic system is giving rise to
a new class of control strategies, and an expanded
realm of opportunities to influence traffic conditions
in increasingly congested urban and suburban areas.
This class of strategies also raises a number of funda-
mental questions that directly affect the ultimate ef-
fectiveness of any particular in-vehicle information
system configuration and information supply strat-
egy. These questions do not appear to have received
the level of attention accorded to the technological
aspects of these systems, or to the institutional as-
pects of their implementation. The hardware capa-
bilities exist for a wide range of in-vehicle guidance
schemes and system configurations (see French,
1986, 1989, for a review for some of the principal
technological options); however, little is known on
the relative effects of various information supply
strategies in terms of improving traffic conditions
for the individual drivers and for the system overall.
These impacts depend on (1) the existence of im-
provement opportunities in existing networks, (2)
user behavior and response to the supplied informa-
tion, and (3) the type, form, and extent of the infor-
mation available to different fractions of the user
population.
The performance of traffic networks under such
information strategies is the result of complex inter-
actions between and within several elements. User
decisions, made in real-time as well as from day-to-
day, determine the time-dependent distribution of
flows on the various components of the network.
Nonlinear interactions in the traffic stream on the
network links, and at the nodes, determine the asso-
ciated trip times, delays, and quality of traffic service
experienced by the tripmakers. Given readings of
these conditions on a real-time or quasi real-time
basis, a more or less centralized controller has the
ability to send various types of information to the
individual motorists, who in turn may change their
routing decisions. This dynamic interaction between
user decisions, traffic conditions, controller actions,
and information governs the overall performance of
the system.
In-vehicle information supply strategies differ in
terms of their characteristics along several important
dimensions, such as: (1) nature of the information
displayed (e.g. symbols, maps, text or a combina-
tion), (2) descriptive vs. prescriptive, (3) user-
optimized, so as to yield some "best" path for a par-
ticular driver, vs. system-optimized, reflecting some
central control logic, (4) whether based on actual
measured conditions (in preceding time step) or on
predicted values using some prediction logic to antic-
ipate conditions over the near-term given the mea-
sured values, and (5) frequency of update. To ad-
dress the relative merits of various combinations of
the above, it is necessary to predict how users might
respond to the particular information supplied, in
the context of a framework that captures the interac-
tions described earlier.
For analysis purposes, it is convenient to consider
the following four principal generic types of infor-
mation strategies:
1. Descriptive, stored information:
"static map" that
displays only stored information on time-
dependent trip times on the various links.
2. Descriptive, real-time information:
display of net-
work (or portions thereof) with indications of pre-
vailing congestion on the various links.
293
294 H. S. MAHMASSANI and R.
JAYAKRISHNAN
3. Descriptive, real-time information with individual
optimization:
the link-level information is pro-
cessed either on board or centrally to compute
the current shortest path from present position to
desired destination of given driver.
4. Controlled guidance:
the instructions given to us-
ers reflect a central controller's system level objec-
tives, subject to certain constraints to prevent un-
reasonable penalties to any individual tripmaker.
It is probable that user response will be governed
by different behavioral mechanisms and heuristics
under each type of information supply strategy. Un-
fortunately, little is available in the bodies of knowl-
edge on travel behavior, traffic theory, or network
assignment to provide an adequate basis for model-
ling user response and evaluating the above strate-
gies.
As noted above, the effectiveness of a particular
strategy must also consider the fraction of users with
access to information (of possibly different types).
Essentially, if every driver had access to the same
real-time information as the other drivers, and each
had a "myopic" self-optimizing capability (e.g. short-
est path routine), then severe congestion would likely
result if all drivers followed the prevailing "best"
paths. On the other hand, information accessible to
only a limited fraction of users would likely result in
benefits to these individuals, and possibly to the
other drivers as well (by diverting a sufficient num-
ber of informed drivers). Of course, equity issues are
associated with situations of unequal information
availability. The existence of a threshold level for
this fraction, which would need to be determined for
a given situation, ~s one of the hypotheses motivating
the work reported m this paper. The importance of
this parameter has recently been demonstrated in the
results of interactive experiments to study the day-
to-day dynamics of user decisions in commuting sys-
tems (Mahmassani and Stephan, 1988; Mahmassani
and Herman, 1990).
Existing methodologies for network assignment
and traffic systems analysis do not adequately cap-
ture the complex Interactions described earlier. Traf-
fic simulators typically require known time-varying
input flows on links and known movements from
one link to another. Virtually none of the models
used in practice consider the behavioral processes
that determine the formation of these flows. On the
other hand, network equilibrium assignment models
are aimed at static steady-state conditions, and as
such are used mostly for planning applications rather
than traffic operations analysis. Furthermore, the
prevailing equilibrium paradigm tends to be rather
limited in terms of the assumed behavior of tripmak-
ers, with shortest path selection being the rule used
almost exclusively in these models. Boyce (1988) dis-
cusses some of the difficulties of using static network
assignment models to analyze route guidance sys-
tems.
There is increasing interest in the assignment of
time-dependent network flows, as reflected in the
published literature. However, most studies have as-
sumed the time-varying demand pattern to be given,
and have solved for the link flows, as a function of
time, given the O-D trips rates (as a function of time)
and the physical and operational characteristics of
the transportation network. Early contributions in
this area include the work of Yagar (1976) on numer-
ical techniques for the dynamic traffic assignment
problem; a proposal to extend this approach to in-
clude some sensitivity to route guidance strategies
has recently been presented by Van Aerde and Yagar
(1988), though the behavioral aspects of user re-
sponse to information appear to be limited. A special
case of the dynamic traffic assignment problem has
been addressed by Sheffi, Mahmassani, and Powell
(1981) in the context of estimating network clearing
times during emergency evacuations. A mathemati-
cal programming formulation and solution proce-
dure was presented by Merchant and Newhauser
(1978) for system-optimal assignment of a discretized
time-varying demand pattern for multiple origins to
a single destination. This formulation was recently
expanded by Carey (1987). A pure network formula-
tion of the problem of determining the system-
optimal joint scheduling and routing of trips in an
urban corridor with a single destination was recently
given by Chang, Mahmassani, and Engquist (1988).
A related problem is that of finding the time-
dependent departure pattern that satisfied dynamic
user equilibrium (DUE) conditions, which extend the
usual static UE to include the trip timing decision.
Unfortunately, the analytical complexity of the
problem in a network context has limited these con-
tributions to highly idealized situations of a single
origin-destination pair connected by one or more
routes (Hendrickson and Kocur, 1981; de Palma et
al.,
1983; Mahmassani and Herman, 1984). In gen-
eral, the state of the art of dynamic assignment in a
network context remains in its early stages of devel-
opment, both in terms of incorporating user deci-
sions and modelling congestion in the network.
A dynamic simulation and assignment framework
has recently been developed by Mahmassani, Chang,
and Herman (1986) to investigate the day-to-day dy-
namics of traffic patterns in a single-destination
commuting corridor with parallel routes (Mahmas-
sani and Chang, 1986; Chang eta/., 1985). It consists
of (1) a special-purpose macroparticle traffic simula-
tor (MPSM), which simulates vehicular movement
on freeways and arterials given time-dependent input
functions, and (2) a user decisions component, which
determines the time-dependent departure functions
resulting from individual departure time and route
choice decisions of commuters in response to experi-
enced congestion in the system and available exoge-
nous information. However, these decisions are up-
dated on a day-to-day basis, rather than on a
real-time basis, as would be the case under in-vehicle
System performance
information availability. The behavioral basis for the
models is derived from laboratory experiments (Mah-
massani, Chang, and Herman, 1986), so they are not
directly applicable to the problem of evaluating the
effects of route-guidance systems. Furthermore, the
network representation is limited to parallel routes,
which is not adequate for this application, where
opportunities for switching routes should be avail-
able for the problem to be meaningful.
In this paper, we develop a simulation framework
for investigating the effect of different parameters
on the performance of a congested urban traffic sys-
tem under real-time in-vehicle information. In addi-
tion to representing vehicular movement, the simula-
tion explicitly models the path selection decisions of
individual motorists along their journey in response
to supplied information. A simulation approach is
adopted in order to deal with some of the complexity
of the interactions taking place in the system, which
precludes analytic approaches for all but the most
highly idealized situations, and to provide the degree
of experimental control needed to gain the kind of
fundamental insights motivating this study. The sim-
ulation is limited to the particular network structure
of a commuting corridor with alternate major paral-
lel facilities, with connecting links at regular intervals
that allow for path switching. A description of the
commuting context and of the modelling framework
is presented in the next section. The model is illus-
trated through a set of simulation experiments that
focus on the effect of two parameters: the fraction
of users with access to information and the mean
indifference band across the user population, which
controls the propensity to switch. The details of these
experiments are given in Section 3. The experimental
results are analyzed in Section 4, followed by con-
cluding comments in Section 5.
2. MODELLING FRAMEWORK
2.1. The commuting context and general
assumptions
Consider the commuting corridor, shown in Fig.
l, consisting of three major parallel facilities (ex-
pressways or major arterials), used by adjoining resi-
dents for the work commute to points downstream
and user response 295
(typically a CBD). The facilities are connected by
crossover links that allow switching between facilities
at given intervals. The catchment area of the corridor
is conveniently subdivided into S residential sectors.
In the morning commute, tripmakers depart accord-
ing to a time-dependent cumulative departure func-
tion
Q,(O,
for sector i, i = 1 .... S. They load the
three alternative facilities according to
Q,,(t), r =
3
1,2,3, such that ~
Q,,(t) = Q,(t).
Suppose that some users are equipped with a re-
ceiver that allows them to obtain information on pre-
vailing traffic conditions on all links of the network
(or suitably preselected subset thereof). An on-board
microprocessor further analyzes this information
from the standpoint of the driver's remaining trip to
the desired work destination. Without specifying all
the details of the in-vehicle screen display, it is suffi-
cient to assume for the purposes of this analysis that
the driver can see and become aware of the best path
(least travel time, for now; possibly other attributes
or generalized cost in subsequent implementations),
and associated travel time, from present location to
the destination. In addition, the driver also becomes
aware of the remaining travel time to the destination
along his/her current path. In the commuting con-
text of Fig. 1, "current path" carries the particular
connotation of staying on the same major facilityt.
A further simplification in our context is that an
alternate path from one's present location consists
of a crossover to another major facility, followed by
the remaining portion of that facility to the destina-
tion.
Effectively, the driver can select which link to
follow among those emanating from the upcoming
node. One of these links corresponds to the driver's
current path. Let
TTCj(k)
denote the travel time on
the
current
path from node k to driver fs destina-
tion, and
TTB~(k)
the travel time along the
best
path.
Most discussions of how drivers respond to real-time
information have assumed that drivers will follow
a local strategy of switching to the best path if
?In a more general network, "current path" would refer
to the path (to the destination) evoked by the driver at the
time it was selected.
HWY-1
Fig. 1. The commuting corridor network.
r N I
I
C
,I
DI
296 H. S. MAHMASSANI and R.
JAYAKRISHNAN
TTBI(k) < TTC~(R).
However, this may be an ex-
treme assumption. Experimental evidence presented
by Mahmassani and Stephan (1988) suggests that
commuter route choice behavior, much like that of
departure time, exhibits a boundedly-rational char-
acter (originally proposed by Simon (1955) as a
model of decision making in complex environments,
and discussed by Mahmassani and Chang (1985,
1987) for commuting decisions). Such behavior can
be operationalized with a simple satisficing decision
rule, whereby the motorist switches from his current
path only if the improvement in remaining travel
time exceeds some threshold level (expressed either
in relative terms or absolute terms). This can be
stated as:
6j(k) = I I if TTC~(k) - TTBJk)
> max(~j -
TTCj(k), Tj)
(1)
(0 otherwise
where
61(k) is a binary indicator variable equal to 1 when
user j switches from the current path to the best
alternate, and 0 if the current path is maintained,
~1 is the threshold level for user j, as a fraction of
the remaining trip time on the current path, with, of
course ~ _> 0, v~, and
• ~ is an absolute minimum travel time improve-
ment below which userj will not switch routes.
Plausibility is the main justification for the above
rule. The threshold level may reflect perceptual fac-
tors, preferential indifference, or persistence and
aversion to switching (for example, May
et al.
(1989)
report a "freeway bias" in commuters' path selection
in the L.A. area). Switching involves a cost in mental
stress and more demanding maneuvering. Further-
more, even though the driver is receiving current in-
formation, traffic conditions may still change by the
time the driver actually travels the downstream links
along an alternate path. The threshold level, hereaf-
ter referred to as an indifference band, provides a
simple mechanism to reflect these factors. In our
model, this band is expressed in relative terms, and
is conveniently thought of as a percent improvement
in remaining trip time over the current path. How-
ever, a minimum absolute level ~j is also provided,
in order to retain a meaningful threshold effect and
avoid unintended switching when
TTCj(k)
becomes
small as the driver nears the destination.
It is natural to question the above model of path
switching in response to real-time information. In
the absence of extensive observational evidence,
which is rather difficult to obtain when the technol-
ogy being evaluated is not quite in place yet, no claim
can be made that any one model of behavior pro-
vides the only possible representation of reality. For
this reason, sensitivity analyses are conducted with
respect to the parameters of the above switching
mechanism, via the simulation experiments described
in the next section. After all, one would not want to
predicate major investment decisions on results that
cannot be shown to be robust with respect to the
underlying behavioral assumptions. In these experi-
ments, the relative indifference band )b is treated as
a random variable distributed across the user popula-
tion, whereas the minimum threshold Tj is assumed
for convenience to be identical for all users, as de-
scribed in the next section, following the presenta-
tion of the simulation logic.
2.2. The simulator
The simulator is comprised of two principal com-
ponents: a traffic simulator and a user behavior com-
ponent. Given the network representation and link
characteristics, the first component will take a time-
dependent loading pattern (i.e. the functions
Q,,(t),
vt, r) and handle the movement of vehicles on links,
as well as the transfers between links. These transfers
require instructions that direct vehicles approaching
the downstream node of a link to the desired outgo-
ing link. The user behavior component is the source
of these instructions. In this study, it consists of the
rules, described earlier, which specify how a particu-
lar driver (with a given origin and destination) ap-
proaching a given node selects the next link to take.
It receives information on prevailing link conditions
from the traffic simulator, via a network processing
subcomponent which calculates the pertinent mea-
sures of effectiveness at the path level, namely
TTCj(k)
and
TTBj(k).
The traffic simulator per se is a general-purpose
tool that is not exclusively tied to the particular net-
work structure considered here. The user behavior
component essentially implements the particular
rules described earlier, though its modularity allows
easy testing of alternative behavioral rules. However,
the network path processor is at present specific to
the particular type of commuting corridor consid-
ered here with major parallel facilities connected by
crossover links. This special structure, with its regu-
lar node numbering scheme, allows rapid calculation
of the path trip times from any node, through simple
recursive formulae, and simplifies path tracking for
individual vehicles.
The traffic simulator is an extension of the pre-
viously mentioned MPSM code (Chang
et al.,
1985),
previously limited to a single facility or multiple in-
dependent parallel facilmes with no crossover links.
It can now handle a network of nodes and links with
general typology and link characteristics. The
MPSM logic, adapted from plasma physics, is re-
tained for traffic movement on individual links.
Traffic is not treated as a compressible fluid; vehicles
are moved in bunches, or macroparticles, at the pre-
vailing local speeds, consistent with a relation be-
tween the average speed and the prevailing concen-
tration. The particular relation used here is a simple
modified Greenshields model of the form:
v = Vo + ( v/- Vo)(l - K/K)" (2)
System performance and user response 297
where V and K are the average speed and density,
respectively, in the given highway segment; V/and
V0 are the free mean speed and minimum speed, re-
spectively; K~ is the maximum or jam concentration;
and a is a parameter that captures the sensitivity of
speed to changes in concentration. Note that other
functional forms may be used for this relation; we
have found the present one adequate for the purpose
of capturing the character of the congestion in the
system and estimating the principal traffic perfor-
mance measures. Of course, different parameter val-
ues can be specified for different highway sections.
In previous work, we have used macroparticles of
5 to 20 vehicles (Chang et ai., 1985; Mahmassani and
Jayakrishnan, 1988), and found values in this range
adequate for estimating trip times and similar quan-
tities. In the present study, however, we are moving
vehicles individually (i.e. macroparticle of size one),
primarily because the route switching decision rules
in the behavior component are applied at the individ-
ual user level. Nevertheless, the traffic simulation is
still not microscopic because the interactions in the
traffic stream are modelled macroscopically using
the average speed-concentration relation. Micro-
scopic details of car following and lane changing are
not modelled.
Traffic simulation follows a deterministic, fixed
time-step approach. At each time step (of the order
of 0.1 minute in the present experiments), vehicles
are moved at the prevailing local speed on the same
link or transferred to another link. The latter is de-
termined by the user decisions component, according
to the switching rules described earlier. The concen-
trations are updated and the corresponding average
speeds are calculated for the next time step. Of
course, a transfer from one link to another is com-
pleted only if sufficient capacity is available on the
receiving link; otherwise, queues will develop. The
travel times for the three alternate paths are calcu-
lated from each node based on the current link
speeds (including the crossover links) and, when ap-
plicable, an expected link-end queue waiting time,
which in turn is based on the queue discharge rate in
the previous time steps (up to 30 time steps in the
previous experiments). The model does not presently
model the details of signalized operation at junc-
tions; the mechanism to do so is relatively straight-
forward, by suitably constraining the interlink trans-
fers.
The details of the traffic simulation are not of
primary importance to the central concerns of the
present paper. Additional information on this aspect
can be found in Chang et al., (1985) and Mahmas-
sani and Jayakrishnan (1988). In the next section,
we describe the application of the modelling frame-
work to investigate the effect of in-vehicle informa-
tion in the commuting context of Fig. 1. Also dis-
cussed are several aspects of the simulation that
pertain to the treatment of users with different infor-
mation availability characteristics, and to the user
decisions component.
3. THE SIMULATION EXPERIMENTS
3.1. The corridor network
The network studied is that of Fig. 1, with three
major highway facilities, each consisting of 9 links of
one-mile length each, and connecting the adjoining
residential sectors to the same destination (CBD).
For simplicity, and with no loss of generality, equal
access time to the three facilities was assumed,
though queueing at the entry points of the facility is
possible. Crossover links that interconnect the three
major highways are provided at the third, fourth,
fifth, and sixth miles from the CBD. All three facili-
ties as well as the crossover links are assumed to have
two lanes in each direction. Of the major facilities,
Hwy-I is the fastest with a 55 mph free mean speed,
followed by Hwy-2 with 45 mph, and Hwy-3 with 35
mph. All crossover links have a free mean speed of
45 mph. The values of the parameter ~ (in eqn 2)
were set so as to yield a speed of Vf/2 at a concentra-
tion of 2K/3, resulting in 0.86, 0.94, and 1.1 (for
facilities 1, 2, and 3, respectively); a value of 0.94
was used for all the crossover links.
It can be noted that this network exhibits similar
features to an actual corridor network, in the Austin
area, that was the site of a recent related observa-
tional study of the variability of trip times during
the evening peak-period (Jones et al., 1989).
3.2. The loading patterns
The area surrounding the three facilities is subdi-
vided into one-mile sectors, which contain the corre-
sponding parallel links of the three facilities. Only
sectors 1 through 6 (with Sector 1 denoting the most
distant from the CBD) are assumed to be residential
sectors which generate commuting traffic. As ex-
plained earlier, it is necessary to specify the time-
dependent departure functions Q,,(t), for each facil-
ity r = 1,2,3 in each sector i = 1 .... 6. No vehicle
generation is assumed onto the crossover links.
Two different loading patterns are considered in
these experiments. The first is simply referred to as
loading pattern one; the second satisfies the afore-
mentioned dynamic user equilibrium conditions, for
a particular user utility function. In both cases, a
total of 9,600 commuters, split equally among the
six residential sectors, share the use of the facilities
in the corridor during the morning peak.
Under the first pattern, commuters in each sector
split equally among the three facilities, and depart
uniformly over a 20-minute period, at a rate of 26.67
vehicles per minute for each facility. The loading
periods for each sector are staggered with a time lag
of five minutes between adjacent sectors, with Sector
1 starting first.
The second loading pattern satisfies the stochastic
version of dynamic user equilibrium (DUE) condi-
tions, in that no user can improve his/her random
utility by unilaterally switching either departure time
or route. Note that the utility function here includes
not only the travel time but also the schedule delay
298
H. S. MAHMASSANI and R. JAYAKRISHNAN
experienced by the user (i.e. the difference between
one's actual and desired arrival times). The particular
utility function used for illustrative purposes in these
experiments is based on empirical results presented
by Hendrickson and Plank (1984), and reported for
completeness in the Appendix. The DUE pattern was
solved for using an iterative procedure, the details of
which are outside the scope of the present paper,
and which will be included in a forthcoming paper
focusing on DUE issues. To calculate the schedule
delays, it was assumed, with no loss of generality,
that all users have identical work start times.
3.3. Experimental factors and simulation cases
The simulation experiments were conducted to in-
vestigate the sensitivity of the system's performance,
under the information strategy described in the pre-
vious section, with respect to two principal factors:
(1) the fraction of users with access to information,
and (2) the mean relative indifference band, which
captures the propensity of users to switch in response
to information.
Fraction of users with information.
To examine
the effect of this fundamental parameter in the
large-scale deployment of any in-vehicle information
system, five levels were considered, spanning the
spectrum from luxury gadget to universal availabil-
ity: 0.10, 0.25, 0.50, 0.75, and 1.00. Information
availability status is assigned randomly and indepen-
dently to each vehicle as it is generated, according to
the specified fraction. Of course, in the universal
access case, this assignment is automatic.
Mean relative indifference band.
The quantity ~j
in eqn (1) governs users' response to the supplied
information and their propensity to switch. As noted
earlier, we treat it as a random variable; when gener-
ated, a user is assigned randomly and independently
a value for ~. For convenience, ~j is assumed to
follow a triangular distribution, with mean ~ and
range of ~/2. To examine the robustness of the re-
suits vis ~t vis the underlying switching behavior
rules, we consider five different levels of ~: 0.0, 0.1,
0.2, 0.3, and 0.5. In the no band (~ = 0.0) case, all
users are assumed to have a zero band, and thus to
always switch to an alternate path if it offers an
improvement in travel time, no matter how small its
magnitude. The minimum improvement rj (in eqn 1)
is taken to be identical across users, and equal to one
minute, though not in the zero band case, where no
minimum improvement restriction is imposed.
For each level of ~, simulations were conducted
for each of the five fractions of users with informa-
tion, resulting in 25 different cases for each of the
two loading patterns described above. In addition,
the cases with no information (i.e. switching) were
simulated for each loading pattern so as to provide a
reference for comparison. All the simulation experi-
ments were performed on a CRAY X-MP supercom-
puter; this computing environment was needed pri-
marily to meet the extensive memory requirements
associated with applying the behavioral rules at the
individual vehicle level, and tracking individual vehi-
cle paths. The results are discussed in the next section.
4. RESULTS
4.1. Loading pattern 1
Table 1 reports summary statistics on the switch-
ing activity taking place in connection with the first
loading pattern. The trend reflected by the column
listing the fraction of those drivers with access to
information who make at least one switch conforms
to intuition, in that switching activity decreases as
drivers require some minimum threshold of trip time
improvement in order to deviate from their initially
selected paths. Furthermore, for a given mean band,
this fraction decreases as the availability of the infor-
mation spreads across the user population. Thus the
relative usefulness of the real-time information to
individual drivers (in terms of uncovering opportuni-
ties that warrant a switch from the current path)
decreases as the information becomes more widely
available, again conforming to intuition.
Of course, as the mean band increases, the num-
ber of users who switch more than once decreases
drastically. The statistics for switching from each of
the three main highway facilities, reported in the last
three columns of Table 1, follow expectations in that
more switching takes place from the slowest (under
uncongested conditions) facility, Hwy-3, followed by
Hwy-2, and then Hwy-l, which has the highest free
mean speed. The vast majority of the switches in the
no-band case are triggered by improvements of less
than 10070 of the remaining travel time. This reflects
the very large number of only marginally productive
switches that might be available in a network, and
that the availability of information may induce users
to make, with a possibly detrimental systemwide ef-
fect, as discussed hereafter.
Figure 2 depicts the variation of the systemwide
trip time with the fraction of the population with
access to information, under each of the five as-
sumed indifference band levels. Note that the trip is
expressed as a percent of the total trip time under
the no information base case (i.e. no switching); thus
values in excess of 100070 correspond to a worsening
of systemwide performance compared to the do-
nothing case. Such worsening occurs under the zero-
band assumption, when users switch any time that
an alternate path offers some improvement in trip
time, no matter how small, over the current path.
After reaching a minimum when the fraction of the
population with information is around 25070, the to-
tal trip time exhibits an increasing trend, resulting in
some cases in a worsening relative to the do-nothing
situation.
On the other hand, tempering switching propen-
sity through an indifference band leads to an im-
provement relative to the no-information ease. The
best systemwide improvement takes place under a
mean relative indifference band of 0.2 to 0.3 (i.e.
when individual drivers on average reject opportuni-
System performance and user response
Table 1. Switching stausttcs for loading pattern 1
299
Fraction
w~th lnfo
0.1
0 25
0 50
0.75
100
Mean Relative
Indifference
Band
0.0
0.I
0.2
0.3
0.5
O0
O1
02
0.3
0.5
0.0
0.1
0.2
0.3
0.5
0.0
0.I
0.2
0.3
0.5
0.0
0.I
0.2
0.3
0.5
Fracuon
of
Drivers
with Info
that SwRch
0.773
0.622
0.553
0 397
0.239
0668
0 594
0
458
0.326
0.240
0.665
0.483
0.358
0.295
0.207
0.627
0.388
0.299
0.278
0.186
0.532
0.324
0.279
0.261
0.168
Number of Total Number of
Driven wRh Swt~hes from
at Last a
Second Swl~h Hwy- 1 Hwy-2 Hwy-3
468 166 483 556
333 192 240 494
337 323 217 324
257 246 185 204
180 170 107 131
647 332 961 935
638 339 539 1165
308 261 293 836
310 291 243 547
378 346 281 318
1782 801 1945 2224
1109 615 1002 1808
857 626 577 1369
252 413 475 778
596 526 363 698
3309 2366 2728 2729
1109 719 I149 2029
878 837 631 1563
629 770 787 1077
387 448 453 827
3312 2156 3148 3120
1770 1370 1365 2147
1227 1183 785 1942
921 1020 887 1520
338 454 586 914
ties that offer less than a 20% improvement in the
remaining trip time). At the other extreme, a band
of 0.5 misses too many opportunities, resulting in a
systemwide improvement of no more than 2~/0. In
general, the marginal systemwide effectiveness of ad-
ditional information diffusion decreases signifi-
cantly, especially after it reaches 25% of the trip-
maker population. The maximum systemwide im-
provement obtained here is just over 7%.
To examine the incidence of the impacts of infor-
mation, Fig. 3 depicts the average trip time (ex-
pressed as a percent of the corresponding average in
the do-nothing, no information situation) experi-
enced by those who have access to information, in
the various cases considered. Figure 4 shows similar
information but for those tripmakers with no access
to information (which explains why there are no val-
ues corresponding to the l.O fraction of users with
~
100
90
~, oo
0~- 01
-- 02
0.3
Jh 0.5
0 00 0 25 0 SO 0 75 1.00
b'I:V~TI~ Of ~S:I$ wrrH INFO
Fig. 2. Variation of total travel time, as a percent of no information case with fraction of users with information, for each
mean relative indifference band: Loading pattern !.
300 H.S. MAHMASSANI and R. JAYAKRISHNAN
110
100,
90,
80 O0
| i | !
0 25 0 50 0.75 1 00
FRACTION OF USERS WiTH INFO
~- 00
01
---- 02
03
,I, 05
F]g. 3. Average trip time for users with information, as percent of value for no info base case: Loading pattern 1.
information). Several important phenomena are il-
lustrated by these graphs. First, users with informa-
tion could do worse than they would have in the
do-nothing situation, when at the same time those
without information experience a reduction in their
average trip time relative to the base case (thereby
still resulting in the systemwide improvement seen
earlier). Second, this worsening is experienced by us-
ers when they switch too readily, as seen in the zero
band for higher fractions with information. Third,
benefits are incurred by those without information
in most of the cases considered in these simulations.
Fourth, the relative performance of the two groups
is strongly dependent on the underlying switching
propensity, and/or the fraction of information. The
group with information shows a trend of decreasing
benefits as the fraction with information increases.
On the other hand, the group without information
shows increasing benefits as the fraction with infor-
mation increases. The results highlight that the rela-
tive effectiveness of information to those who have
access to it depends not only on the relative scarcity
of this information, but also on the response strategy
followed by the user. In these experiments, a relative
indifference band between 0.2 and 0.3 provides an
adequate cushion against the volatility that may be
generated by massive flows of other users responding
to the same information. Similar phenomena are ex-
mined next for the dynamic user equilibrium load-
ing pattern.
4.2. Dynamic user equilibrium loading pattern
Table 2 reports similar statistics on switching ac-
tivity as Table 1, but for the dynamic user equilib-
rium departure pattern. While we find similar trends
regarding the effect of the fraction with information
0 00 0 25 0.50 0 75
FRACTION OF Uf, ERS wrlvI INFO
~. 00
O--" 01
---- 02
03
& 05
Fig. 4. Average trip time for users with no information, as percent of value for no info base case: Loading pattern i.
System performance and user response
Table 2. Switching statistics for the dynamic user equilibrmm departure pattern
301
Fraction
wlth Info
0
1
0 25
0 50
075
100
Mean Relanve
Indifference
Band
0.0
0.1
0.2
0.3
0.5
0.0
0.1
02
03
0.5
O0
0.1
0.2
0.3
0.5
0.0
0.1
0.2
0.3
0.5
0.0
0.1
0.2
0.3
0.5
Fracoon
of
Drivers
wlth Info
that Swuch
0.659
0.422
0 272
0 133
0.000
0 572
0
298
0.230
0.091
0000
0 475
0.231
0 163
0.060
0.000
0.401
0.185
0.124
0.046
0.000
0.334
0.154
0.099
0.037
0.000
Number of Total Number of
Drivers with Switches from
at least a
Second Switch Hwy- 1 Hwy-2 Hwy-3
0 0 345 283
0 0 131 271
0 0 0 259
0 0 0 127
0 0 0 0
1 0 624 728
0 0 72 623
0 0 0 543
0 0 0 215
0 0 0 0
0 0
889 1403
0 0 0 1107
0 0 0
784
0 0 0 290
0 0 0 0
47 22 1104 1811
0 0 0 1335
0 0 0
895
0 0 0 331
0 0 0 0
43 8 1198 2045
0 0 0 1479
0 0 0 952
0 0 0 360
0 0 0 0
and the mean indifference band on switching activ-
ity, the most notable result is the markedly lower
switching activity associated with the DUE departure
pattern, reflecting fewer opportunities for individual
drivers to reduce their respective trip times by chang-
ing routes. Most of these opportunities appear to be
out of the slower Hwy-3.
To appreciate why traffic conditions could appar-
ently be so different in the same commuting context,
compare the plots in Fig. 5, of the cumulative depar-
ture functions onto each of the three highway facili-
ties for sectors 2 and 5 (i.e. of the functions Q,~(t)
and Qs~(0, r = 1,2,3) under loading pattern 1, to
those in Figs. 6 and 7 (one for each sector) under the
DUE patternt. Essentially, the DUE pattern associ-
ated with the particular utility function shown in the
tThe ordinate on these graphs is the ratio of cumulative
number of departures onto the particular facihty divided
by the total number of departures from the given sector.
Note that the three plots (corresponding to each of the three
facilities) are identical for loading pattern 1, since users
split equally among the three routes.
CUMULATIVE DEPARTURE PATrERNS IN SECTORS 2 & 5
°']
o
§.,t
u 4,
• ~ a M
/
e m
Tin((~NUTES)
Sectoq S
Fig. 5. Cumulauve department functions on each highway facility for sectors 2 and 5 under loading pattern 1.
302
H.S. MAHMASSANI and
R.
JAYAKRISHNAN
CUMULATIVE DEPARTURE PATTERNS IN SECTOR 2
e.+ 1
Hwy-3..~.{~- ~ "'--~
,L, "I
~'~f~ Hwy - 2
TIME (MINUTES)
Fig. 6. Cumulative departure functions on each highway facility under DUE loading pattern: Sector 2. (fraction with
info = 0.5, Indifference band = 0.2).
Appendix is considerably more spread out than load-
ing pattern I, starting up to 80 minutes before the
work start time. Therefore, less congestion is associ-
ated with it than with the steeper pattern I. This is
confirmed by the plots of the vehicular concentra-
tions, as a function of time, on the fifth section of
each of the three facilities, for loading pattern l (Fig.
8) and DUE (Fig. 9). Similar information is shown
for the eighth section in Figs. 10 and l I.
Under the considerably less congested DUE pat-
tern, one would expect that the potential of informa-
tion to reduce systemwide trip times will diminish.
Figure 12 presents similar plots as Fig. 2 (namely the
systemwide trip time as a percent of the no informa-
tion base case value) for the DUE departure pattern.
The results differ from those of Fig. 2 in that while
no cases are encountered where the supply of infor-
mation increases the total time, the reductions at-
tained are generally smaller in relative magnitude
than those under the more congested first loading
pattern. The maximum improvement is about 5%.
The widest band leads to almost no improvement,
CUMULATIVE DEP~,RTURE PATTERNS IN SECTOR 5
1.4 "1
~'l Hwy-3
• , ,,.~/~Z.-- Hwy - 2
Fig. 7. Cumulative departure fimctions on each highway facility under DUE loading pattern: Sector 5. (fraction with
info = 0.5, Indifference band = 0.2).
System
performance
and user response 303
!
aL•j
e, o
o
,
.!,,i
! el .J II. I
"i
;ii
o_
i, j'l
-r .'ipd
*/I
,..i i
, ;IF"
0,~ ,,'/
o m m
CONCENTRATIONS IN SECTOR 5
I I I
.ii
I
I
'i i
TIME (MINUTES)
Fig. 8. Time-varying concentration profile, as a fraction of maximum concentration, in segment 5 of each major highway
facility under loading pattern I. (fraction with info = 0.5, Indifference band = 0.2).
since it generates no switching. The same general
trends as in the first loading pattern are present here,
in that the marginal effectiveness of information dra-
matically decreases, and in some cases actually re-
verses (as in the no band situation) after a certain
fraction of the population, between 25 to 50070, is
equipped with in-vehicle information devices.
Figures 13 and 14 present similar results as Figs.
3 and 4 for the DUE pattern, reflecting the relative
improvement of average trip time for users with in-
formation (Fig. 13) and those without it (Fig. 14).
The results differ from those under the first loading
pattern in that users with information appear to out-
perform those without information in virtually all
cases. Thus, the incidence of benefits depends on the
initial loading pattern and associated overall conges-
tion levels. In a system with greater spreading of the
peak and thus less congestion, information availabil-
ity allows users to fine tune their path selection, and
their locally optimal moves contribute to global im-
provements, unlike the more congested situations.
Perhaps one of the more interesting results of
these simulations, and one that is not explicitly visi-
ble in the previous figures, is how the trip times com-
pare under the two loading patterns in the absence
of information. This comparison provides an indica-
1o-I
CONCENTRATIONS IN SECTOR 5
oe~
u
I" ~" J"~ Hwy- 1
• hue (uluuTEs)
Fig. 9. Time-varying concentration profile, as a fraction of maximum concentration, in segment 5 of each major highway
facility under DUE loading pattern. (fraction with info = 0.5, Indifference band = 0.2).
304
£
"5
ee
X
,T
o* ; ,~ ;o
H. S. MAHMASSANI
and R.
JAYAKRISHNAN
CONCENTRATIONS IN SECTOR 8
/ ~ ..y-3!
~\ ' ,~ ~ ; J
j~ Jr,!,
H.y-1 ,1. ~ l,lh,,,,tii,l~Ji.L d ~i
• /,
',,,!"
;t,lt~llt..'f. ~. I~
z.
-,',;
'~
I'
:: JU',!! ILJ ~i~
!'
L
'i~ i
'" ' V '~" ",It,' "~,
,,-j
.~.,/
!" ~ ,~
v
,:
:
CJ.
'. ""/
i I~ /
Hwy-2 -/'-' / I]., :
/r, vl/
.;/
•
/" / Hwy 3
"--'
. ' ,L if/
TIME (MINUTES)
:' i~Hwy-2
Hwy I~. i i
]
'1
w
Fig. 10. Time-varying concentration profile, as a fraction of maximum concentration, in segment 8 of each major highway
facility under loading pattern 1. (fraction with info = 0.5, Indifference band = 0.2).
tion of the relative effectiveness of peak spreading
compared to route control. Remarkably, the total
trip time under the DUE pattern is 48.65% of the
value obtained in the first loading pattern. When
compared to the best improvement of about 7%
achieved with self-directed route switching, peak
spreading may hold considerably more potential to
reduce trip times in some situations. The determining
ingredient in this regard is obviously the kind of
loading pattern under present conditions, and the
opportunities for improvement that it offers. The
importance of characterizing traffic conditions in ac-
tual networks and the manner in which these are
utilized by tripmakers has been recognized in related
observational work (Jones
et al.,
1989). However,
considerably more needs to be done in terms of de-
veloping the observational basis for assessing the po-
tential benefits of information in any given situation.
These issues are discussed further in the next section.
$. DISCUSSION AND CONCLUDING COMMENTS
The modelling framework developed in this paper
and its application to the simulation experiments
provides many insights into the effect of certain
types of in-vehicle real-time information on overall
CONCENTRATIONS IN SECTOR 8
Hwy-1
~, Hwy-2
~- ' z,'~-~ ~',
/
.-.,>-.tiC_....--" \ ',:;
_:.:;;.:; ,/-~J~ Hwy - 3 ,. \
TIME (MINUTES)
Fig. 1 I. Time-varying concentration profile, as a fraction of maximum concentration, in segment 8 of each major highway
facility under DUE loading pattern. (fraction with info = 0.5, Indifference band = 0.2).
System performance and user response 305
101 t
100, --" --"
-" "
99 ' [ ~, 0 0
[
Ol
~. 02
98, '--"-'O--- 03
•
05
07
95
04
0 00 0.25 0,50 0.75 1 00
FRACTION OF USERS WIT14 INFO
Fig. 12. Variation of total travel time, as a percent of no information case, with fraction of population with information
for each mean relative indifference band: Dynamic user equilibrium loading pattern.
traffic system performance, and thus on the poten-
tial of in-vehicle information systems to improve
conditions in congested networks. As such, the
model allows the identification of the key underlying
parameters and the exploration of their effect. The
experiments reported in this paper are only an illus-
tration of some of the insights that can be obtained,
and more importantly of the fundamental questions
that should be addressed in connection with the ap-
plication of in-vehicle information technologies.
The results highlight the complexity and subtlety
of the effect of information on system performance,
thereby precluding general conclusio,is and clarify-
ing possible misconceptions that information will au-
tomattcally lead to improvements in traffic condi-
tions. Several factors determine the impact of
information, as illustrated in the results shown in
this paper. Not least of these factors is the manner
in which users respond to the information. The re-
sults have illustrated that myopic local actions by
individual drivers may actually result in worse out-
comes for them as well as systemwide, as happened
under the no band situation. In this regard, a strat-
egy of switching paths only when the improvement,
under prevailing conditions, exceeds 20% of the re-
maining trip time seemed to work best for individual
users receiving information as well as for the overall
system. Users may come to such a conclusion with
repeated experience with the technology, or they may
possibly need to be instructed in proper response
strategies.
The issue of information availability is another
critical one in the debate regarding the implementa-
tion of the technology. Our results suggest little addi-
tional systemwide benefits beyond a certain fraction
of users who have access to information, with this
85
i I i I
0 00 0.25 0.50 0.75 1 00
FRACTt~ OF USERS WITH Id:O
~.
oo
• ---0"-- 01
-- 02
---'O-- 03
• 05
F~g 13. Average trip time for users with information, as percent of value for no info base case: DUE loading pattern.
101
10o
90
07
0.00
H. S. MAHMASSANI and R. JAYAKRISHNAN
306
0.25 0.50 0 75
FRACTION OF USERS WITH INFO
~. O0
--'-0"-- 01
---- 0.2
03
&
05
Fig. 14. Average trip time for users with no information, as percent of value for no info base case: DUE loading pattern.
fraction being in the range of 0.25 to 0.5 in these
experiments. Of course, one should keep in mind
that our experiments considered only one informa-
tion supply strategy, namely one that coveys real-
time descriptive information that is then self-
optimized for the user.
The dependence of the benefits on existing condi-
tions is clearly borne out by the cases considered
here. The question is whether actual systems are
closer to the more congested loading pattern or to
the particular DUE pattern considered. The results
of related laboratory experiments have suggested
greater peaking than that of the DUE pattern, more
like loading pattern 1 (Mahmassani
et al.,
1986). It
is nevertheless worthy to note the relative potential
of time shifting, and the role of in-vehicle and out-
of-vehicle information to influence this aspect. Of
course, there is a limit to how much one can spread
the peak, and the Los Angeles area provides a vivid
example of these limits. Travel behavior research has
been concerned with the constraints within which in-
dividual tripmakers operate; the recently introduced
concept of "operating envelopes" may provide a use-
ful framework for further discussion of these issues
(Lee-Gosselin
et al.,
1989).
Naturally, it is not wise to generalize from the
experiments presented here. Nevertheless, they pro-
vide a start and a useful framework to examine many
of the fundamental questions mentioned earlier.
Much work remains to be done on several aspects of
this problem. We have only considered the role of
information to improve routine incident-free com-
muting patterns. Another very important situation
to consider is the occurrence of traffic disrupting
incidents, where real-time information may poten-
tially be of greater value. The traffic simulation
model already has the capability to introduce ran-
domly generated incidents. A set of simulation ex-
periments will be performed to further investigate
this aspect. The simulation model will be extended
to more general networks; the task is conceptually
simple but computationally demanding. The behav-
ioral framework remains to be refined and extended
to address different information supply strategies.
The indifference band mechanism presented here is
a simple tool to parameterize users' propensity to
switch routes in response to information. The behav-
ioral processes are undoubtedly more complex, and
the human factors aspects of screen display reading,
mental processing, and driving should not be under-
estimated. Laboratory experiments can be very use-
ful in this regard, and some are presently contem-
plated by the authors. Finally, the observational
study of how users actually use actual networks re-
mains an important task, as explained earlier.
Acknowledgements-This
paper ~s based on research con-
ducted as part of an ongoing research program at the Uni-
versity of Texas at Austin on advanced information techno-
logies for network traffic control, which is funded from
three sources: a grant from the Advanced Technology Pro-
gram of the Texas Higher Education Coordinating Board,
the U.S. Department of Transportation through the South-
western Regional Research Center, and the General Motors
Research Laboratories. The authors are indebted to Robert
Herman and Richard Rothery for many useful suggestions
regarding various aspects of this problem. Of course, the
authors remain solely responsible for the contents and con-
clusions of this paper.
REFERENCES
Boyce D. E. (1988) Route guidance systems for improving
urban travel and location choices.
Transpn. Res.o
22A,
275-282.
Carey M. (1987) Opumai time-varying flows on congested
networks.
Operattons Research,
35, 58-69.
Chang G.-L., Mahmassani H. S. and Herman R. (1985)
A macroparticle traffic simulation model to investigate
peak-period commuter decision dynamics.
Transpn.
Res. Rec.,
1005, 107-120.
Chang G.-L., Mahmassani
H. S.
and Engquist
M. L. (1988)
System optimal trip scheduling and routing in commut-
ing networks.
Transpn. Res. Rec.,
1251, 54-65.
French R. L. (1986) Historical overvzew of automobile nav-
igation technology. IEEE Vehicular Technology Con-
ference, Dallas, Texas.
System performance and user response 307
French R. L. (1989) The roles of cooperative programs in
developing vehicular navigation and route gmdance sys-
tems. In C. Hendrickson and K. Sinha (Eds.),
Proceed-
mgs of the First International Conference on Apphca-
ttons of Advanced Technologies in Transportataon
Engineering,
American Society of Civil Engineers.
Hendrickson C. and Kocur G. (1981) Schedule delay and
departure time decisions in a determinisuc model.
Transpn. Sci.,
15, 62-77.
Hendrickson C. and Plank E. (1984) The flexibility of depar-
ture times for work trips.
Transpn. Res.,
18A, 25-36.
Jones E. G., Mahmassani H. S., Herman R. and Walton
C. M. (1989) Travel time variability in a commuting
corridor: implications for electronic route guidance. In
C. Hendnckson and K. Sinha (Eds.),
Proceedings of
the First Internanonal Conference on Apphcations of
Advanced Technologies m Transportation Engineermg,
American Society of Civil Engineers.
Lee-Goselin M., Hanson S. and Mahmassani H. (1989) Op-
eratmns envelopes as a heuristic for understanding
travel choice. 68th Annual Meeting of the Transporta-
tion Research Board, Washington, D.C.
Maher M. J. and Akcelik R. (1977) Route control-
simulation experiments.
Transpn. Res.,
11, 25-31.
Mahmassani H. S. and Chang G.-L. (1985) Dynamic as-
pects of departure time choice behavior in a commuting
system: theoretical framework and experimental analy-
sis.
Transpn. Res. Rec.,
1037, 88-101.
Mahmassani H. S. and Chang G.-L. (1986) Experiments
with departure t~me choice dynamics of urban commut-
ers.
Transpn. Res.,
20B(4), 297-320.
Mahmassani H. S. and Chang G.-L. (1987) On boundedly-
rational user equilibrium in transportation systems.
Transpn. Sct.,
21(2), 89-99.
Mahmassani H. S., Chang G.-L. and Herman R. (1986)
Individual decismns and collective effects in a simulated
traffic system.
Transpn. Sct.,
20(4), 258-271.
Mahmassani H. and Herman R. (1984) Dynamic user equi-
librium departure time and route choice on idealized
traffic arterials.
Transpn, Sci.,
18(4). 362-384.
Mahmassani H. and Herman R. (1990) Interactive experi-
ments for the study of tripmaker behaviour dynam-
ics in congested commuting systems. In Jones. P.
(Ed.),
Developments in Dynamic and Activity-Based
Approaches to Travel Analysts,
Gower Aldershot (272-
298).
Mahmassani H. S. and Jayakrishnan R. (1988) Dynamic
analysis of lane closure strategies.
ASCE Journal of
Transportatton Engineering,
114(4), 476--496.
Mahmassani H. S. and Stephan D. G. (1988) Experimental
investigation of route and departure time dynamics of
urban commuters.
Transpn. Res. Rec.
(in press).
May A., Martello M., A1-Deek H. and Sanders W. (1989)
Potential benefits of in-vehicle information systems
in a real life corridor under recurring and incident-in-
duced congestion. In C, Hendrickson and K. Sinha
(Eds.),
Proceedings of the First International Con-
ference on Applications of"Advanced Technologies in
Transportation Engineering,
American Society of Civil
Engineers.
Merchant D. K. and Newhauser G. L. (1978) A model and
an algorithm for the dynamic traffic assignment prob-
lem.
Transpn. Sci,,
12, 183-199.
Sheffi Y., Mahmassam H. and Powell W. (1981) A trans-
portation network evaluation model.
Transpn. Res.,
16A, 209-218.
Simon H. (1955) A behavioral model of rational choice.
Quarterly Journal of Economics,
69, 99-118.
Van Aerde M. and Yagar S. (1988) Dynamic integrated
freeway/traffic signal networks: a routing-based model-
ling approach.
Transpn. Res.,
22A(6), 445--454.
Yagar S. (1976) Emulation of dynamic equilibrium in traf-
fic networks. In Florian, M. (Ed.),
Traffic Equilibrium
Methods,
Spnnger-Verlng, Berhn, 240-264.
APPENDIX
The particular utihty function and parameters used to
derive the dynamic user equilibrium loading pattern is mod-
ified from an empirically calibrated function reported by
Hendrlckson and Plank (1984) for Pittsburgh area com-
muters. The funcuon was developed in a random utility
framework, using a muitinomial logit model form. The
modificauon consisted of the deletion of terms that are not
pertinent to the present study.
The systematic component of the function used is as
follows:
U,, = -0.008TT, -
O.021CTtr - O.O0042(SDE,,) 2
- O.148(SDLt,) + O.OI4(SDL,,) 2
where
/1',, is the utility of departure time t on route r
TT r is the free flow travel time on route r
CT,,
is the additional trip thne (i.e. in excess of TTr) on
route r for departure time t
SDE,,
is the schedule delay corresponding to an early arrival
SDL,,
is the schedule delay corresponding to a late arrival.
TR(A)
25:5-F
