User Equilibrium in a Bottleneck under Multipeak Distribution of Preferred Arrival Time Fabien LEURENT
[email protected] Nicolas WAGNER
[email protected] Université Paris-Est, LVMT, Ecole des Ponts
CONTEXT Departure Time Choice And Dynamic Traffic Assignment Departure time is a microeconomic decision of each individual user on the basis of travel and schedule delay costs
The distribution of users subject to traffic loads and link capacities induces: • Queuing phenomena • Effect on route choice
Application To The French Trunk Road Network • LADTA, for Lumped Analytical DTA, a model and software developed at LVMT, is applied by motorway operator Cofiroute to the French trunk road network • Focus = trips on heavily trafficked summer days A54 Fig 1. The French trunk road network and a sample result
Flow of vehicles in pcu/h
A54 on 14/07/2007 Observed Flows
1600
Simulated Flows
1200 800 400 0 0
Flow in pcu/hour
2500
Preferred flow of arrival
2000
5
10
15
20
Time (in hour)
1500 1000 500 0 0,00
5,00
10,00
15,00
20,00
Two Distinctive Modeling Features
Time (in hour)
Fig 2. Origin-destination pair Paris – Lyons: distribution of preferred arrival times
Preferred arrival times: • are distributed among users • the distribution may exhibit several peaks in demand Schedule delay cost function: • is a modified V-shape function • applies to departure option in terms of day and time of day • was estimated on the basis of a Stated Preference survey (2007)
Fig 3. Schedule delay cost function.
TRB #09-2704
User Equilibrium in a Bottleneck under Multipeak Distribution of Preferred Arrival Time
THE MODEL A Supply Demand Framework Supply side : the flowing model • A single OD pair linked by a single arc • Congestion is modeled by a standard bottleneck model Demand side : the user model • The cost of departing at h is a function of the travel time and of the arithmetical lag between the preferred arrival time and the actual one • Users are heterogeneous w.r.t. their arrival time times • The set of users is represented by a distribution Xp along the preferred arrival times
Flowing model The travel time function X w is given by Q ( h) w(h) K in which Q is defined as follows:
x ( h ) K if Q ( h ) 0 or x ( h ) 0 Q (h) 0 otherwise where: • w is the waiting time • x+ is the flow rate of vehicles at the entrance of the bottleneck • Q is the volume of queued vehicles
User model • Given a travel time w , a user with preferred arrival time η has a cost function:
g
[ w]
(h, ) w(h) Dh w(h)
• The set of users is described by a distribution Xp over the set of preferred arrival time
Equilibrium statement Find an increasing function X () such that, 1 letting H p X p X , the following holds:
g Fig 4. – An example of distribution of preferred arrival times. Peak periods are when xp > K
[ w]
(h, H p (h)) g
[ w]
(h, H p (h))
w W( X )
Mathematical Analysis General properties of the equilibrium • Two differential equations in X+ : one for queued periods and one for unqueued • Each queued period can be divided in -Early sub-periods when users depart early, the entry flow is beyond capacity and the queue builds up -Late sub-periods when users depart late and the queue diminishes An existence result User equilibrium with general preferred arrival time distribution and V-shaped cost of schedule delay
Fig.5. -The equilibrium pattern of the distribution of departure time. Peak periods are wrapped into queued periods. The instants ħ, correspond to critical instants, dividing queued periods in early and late sub-periods
TRB #09-2704
User Equilibrium in a Bottleneck under Multipeak Distribution of Preferred Arrival Time
ALGORITHM Algorithm Philosophy • Under V-shape delay cost: only two departing flows are feasible, xe for early periods and xl for late periods, separated by critical times • The algorithm iteratively builds the queued periods by testing initial instants • Main variables: critical times and notional queue sizes, associated to a candidate initial queuing instant
Algorithm 1 maps a candidate initial instant for a queued period, into two sequences of critical times and notional queue sizes Algorithm 2 searches an interval [h1, h2] to identify a satisfactory initial queuing instant (dichotomy method) Algorithm 3 scans the overall period from beginning to end and identify each queued period in turn by use of Algorithm 2
Fig.6. –Iterative construction of the queued periods. The algorithm tests a candidate initial instant hˆ0 . The quantities W are “notional queue sizes”
Application Instance Two peak periods are progressively moved closer to each other. 1. Initially there are two distinct queued periods, with a single maximum in travel time. 2. The two queues are merged into a single one; there are two maxima in travel time. 3. Further, the two maxima collapse into a single one yielding the same pattern as in a single peak period.
Current Developments • Integration to a network assignment model: reference TRB #09-1832 • Model extension to non V-shape function of delay cost and non linear travel cost TRB #09-2704
