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The Transit Bottleneck Model
Available online at www.sciencedirect.com
Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
EWGT 2012 15th meeting of the EURO Working Group on Transportation
The transit bottleneck model Fabien Leurent*, Ektoras Chandakas Université Paris-Est, Laboratoire Ville Mobilité Transport, Ecole des Ponts ParisTech 6-8 avenue Blaise Pascal, 77455, Marne la Vallée, Cedex 2, France
Abstract The paper addresses the issue of passenger waiting and being stored at a station platform, from which point they plan to board transit services towards egress stations. Each transit service has a specific set of downstream egress stations and is operated at given frequency using homogeneous vehicles of limited available capacity. The model yields individual waiting time by egress station and the assignment of vehicle capacity to the flows by egress station. Two cases are distinguished, unsaturated versus saturated. The unsaturated case is addressed by standard line combination, where service frequency is added up among the routes that service a given egress station. The saturated case is addressed by making explicit the average number of passengers waiting on platform for a given egress station. From these passenger stocks is derived the individual probability to board a vehicle of limited capacity that service a given route hence a given subset of egress stations. Waiting passengers are assumed to be mingling on the origin platform. The subset of routes that service a given egress station, their vehicle capacities and the boarding probabilities induce a line capacity for that destination: to this is faced the passenger flow demanded during the assignment period, in a bottleneck model that yields an average waiting time per passenger. The vector of passenger stocks by egress station is shown to satisfy a fixed point problem. The existence and uniqueness of the solution are demonstrated on the basis of an equivalent, convex minimization program. A Newton-Raphson algorithm is recommended for computation and demonstrated in an instance of application. by Published Elsevier Ltd. and/or peer-review under responsibility the Program of Committee © 2012 Published The authors. by Selection Elsevier Ltd. Selection and/or peer-review underof responsibility the Program Committee Open access under CC BY-NC-ND license.
Keywords: mingled waiting; Transit assignment; Capacitated assignment; Bottleneck model.
* Corresponding author. Tel.: +33-164-152-111; fax: +33-164-152-140 E-mail address: [email protected]
1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Program Committee Open access under CC BY-NC-ND license. doi:10.1016/j.sbspro.2012.09.798
Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
1. Introduction In the dense core of metropolitan urban areas the transit network is frequently submitted to heavy passenger loads, especially at the peak hours on working days. Under such circumstances, various links of the network may provide insufficient capacity to passenger demand. At some station platforms, a traffic bottleneck may arise, in which the passengers experience an increased waiting time until boarding a vehicle of a convenient service. A number of models, static or dynamic, have been provided by the scientific community to deal with the total capacity in passengers of transit vehicles, hence of transit services by aggregating the vehicle runs throughout a given time period. On the static side, Gendreau [1] has modeled passenger queuing by an unbounded delay function that is increasing and convex with respect to boarding flow; however, the analytical solution is difficult in a practical setting. In [2], insufficient capacity is addressed by a fail-to-board probability: platform waiting is driven by a random process of successive Bernoulli trials to board, yielding an average wait time on the basis of the fail-to-board probability; excess flows of passengers are diverted from the network by assuming fictional links from station platform to destination nodes. On the dynamic side, [3] have modeled a traffic bottleneck to yield the queuing time per passenger at any instant. Similarly, in [4], each passenger joins a queue according to his arrival time at the node and experiences a travel time that stems from his position at the queue. Both the static and the dynamic models have been designed in the framework of traffic assignment to a transit network, in relation with passenger path choice. The paper is purported to present a novel model of passenger waiting on platform for transit services to their destination. The main assumptions are that (i) waiting passengers are mingled, (ii) those destined to a given station of egress make up a stock, of which the size is explicit, (iii) by transit service, the available capacity is faced to the stocks of the egress stations along the service route, yielding a probability of immediate boarding, (iv) by egress station, the average wait time on platform is same as in a traffic bottleneck. These assumptions are cast into a fixed point problem with respect to the destination stocks, and mathematical properties of solution existence and uniqueness are established. The rest of this paper is in four parts. To begin with, the platform model is developed in the form of necessary mathematical conditions which make up a fixed point problem. Then, a mathematical analysis of the fixed point problem is made, by formulating an equivalent problem of convex minimization, so as to yield properties of solution existence and uniqueness. Then, an application instance is carried on to demonstrate the main characteristics of the model, including the influence of passenger flow on the average journey time and the competition between destinations at a given station. Lastly, the conclusion points to the inclusion of the transit bottleneck model in the framework of traffic assignment to a transit network. 2. The queuing model 2.1. Definitions Let i be a given access station along line A , J i the set of egress stations j that are served by transit services z ∈ Zi . The subset of services that dwell at j coming from i is denoted as {z ∈ (i, j )} . During the period H , service z is operated at frequency f z′ by vehicles of residual capacity k ′z at station i (after the egress of the passengers destined to i ), yielding an available capacity of κ z′ = f z′k ′z in the period. It is assumed that passengers arrive at i under exogenous flow [q ij+ : j ∈ J i ] , yielding entry volumes H.q ij+ by egress station. Let f ij ≡ z∈(i , j ) f ij be the combined frequency between i and j of the services in Z i . It is assumed that
¦
the platform is shared by the services and that no vehicle can overtake another one, meaning that a timeminimizing passenger is eager to board a relevant vehicle as soon as it has some place available to him. Notation A designates this set of services where the routes and egress stations make up a connected component in the bipartite graph in Z i × J i that links the services to the stations that they serve.
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2.2. The unqueued case If capacity is available to each passenger as soon as he would like from his instant of arrival, then he can board the first relevant vehicle and the average wait time for destination j is:
wij = α / f ij .
(1)
In (1), parameter α models headway regularity: a value of 1 indicates memoryless vehicle arrivals, whereas ½ indicates perfect regularity (homogenous headways). Each service in A gets a share f z / f ij of volume Hqij+ . The resulting passenger flow, ( Hqij+ f z ) / f ij , yields a number ( Hq ij+ ) / f ij of passengers destined to j by vehicle of service z . There remains available capacity if and only if, by vehicle,
H
¦
j:(i , j )∋ z
qij+ fij
≤ k ′z .
(2)
This condition must be checked to ensure that there is no queuing – at least no remaining stock of passengers that keep waiting just after a vehicle relevant to them has left. 2.3. The queued case If some passengers cannot board the first vehicle relevant to them, then they have to wait for other vehicles to arrive and to supply them with some capacity. Let us assume that there is a fictive, stationary state of passenger traffic on the platform, with a number v j of passengers waiting for destination j . When a vehicle of service z arrives with available capacity k ′z , there are candidate passengers in number of
n z = ¦ j∈z v j .
(3)
Their individual probability to board, assuming mingled waiting hence equity among them, is
π z = min{1, k z′ / nz } (with π z = 1 if n z = 0 ).
(4)
Then the number of passengers boarding a vehicle of service z to exit at j is π z v j , and the passenger flow during H to j via z is − = f ′π v . qijz z z j
The total flow rate to j from
(5)
i during H is the throughput rate as follows:
− =( qij− = ¦ j∈z qijz ¦ j∈z f z′π z )v j .
(6)
Let us denote ( fπ) j ≡ ¦ j∈z f z′π z . Queuing may eventually occur when at least one service has π z < 1 , meaning saturation of that service. Let us derive necessary conditions on the π z and v j variables that characterize service availability and waiting. A bottleneck model is assumed by egress station j , with arrival rate q ij+ on [0, H ] and discharge rate q ij− on
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w j 0 + [0, H j ] , in which w j 0 is the mean wait time between arriving at the queue front end in eligible place and the arrival of a relevant vehicle. A reasonable value could be w j 0 = α / f ij ; for the moment, let us keep w j 0 as an independent variable so as to reserve one degree of freedom. H
y = q+ x N
w0
Hj
w0 qFig. 1. Cumulative flows at bottleneck.
Figure 1 depicts the dynamics of the cumulated flows and the waiting queue: if q − < q + then there is a ‘triangle of waiting’ that lies between the cumulated curve of passenger arrival, y = q ij+ .x , shifted to the right due to the minimum waiting w j 0 , and the cumulated curve of passenger departure, y = q ij− ( x − w j 0 ) for x − w j 0 ∈ [0, H j ] . Flow conservation implies that:
N j = H .q ij+ = H j .q ij− .
(7)
The average number of passengers waiting for destination j , v j , is equal to the total time spent at waiting,
WijA , divided by the time interval, H ′j = w j 0 + H j . Letting N ′j = N j + w j 0 qij− and ( fπ) j ≡ ¦ j∈z f z′ π z , the total wait amounts to the area between the two large triangles minus the area of the small triangle:
WijA =
2 N 2j 1 N ′j ( − − + − w 2j 0 q ij− ) . 2 q ij q ij
On replacing WijA with v j H ′j = v j ( w j 0 + H j ) and N ′j with N j + w j 0 q ij− , we get that
v j (w j 0 + vj
w j0 Nj
+
Nj q ij−
)=
2 N 2j 1 Nj ( − + 2 w j 0 N j + w 2j 0 q ij− − + − w 2j 0 q ij− ) 2 q ij q ij
Nj Nj 1 1 − w j0 = ( − ) after dividing by N j and replacing q ij− by v j ( fπ) j + ( fπ) j 2 v j ( fπ) j qij
(8)
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2v j
( fπ) j w j 0 HN j
+2
qij+ Nj
[1 − ( fπ) j w j 0 ] =
qij+ vj
− ( fπ) j after multiplying by 2qij+ ( fπ) j / N j
At this stage, let us set w j 0 = 1 /( fπ) j so as to simplify into
qij+ 2 vj = − ( fπ) j . HN j vj
(9)
This model is taken for all traffic regimes, including the unqueued state in which the left hand side is negligible, yielding qij+ ≈ ( fπ) j v j = qij− . It is expected that
q ij+ 2 v j << , i.e. 2v 2j << q ij+ HN j = N 2j , or HN j vj
v j << N j / 2 .
(10)
Referring to (3) and (7), the π z depend on the v j through the nz . Thus there is a circular dependency between our variables, in other words a Fixed Point Problem (FPP) with respect to vector v = [v j : j ∈ J i ] . In the next section, it is shown that the FPP (3), (7) and (9) has a solution, which is unique. Also included is a computation scheme. Lastly, let us mention a related problem definition that distinguishes the queuing regimes and would be more rigorous, but less tractable: qij+ − ( fπ) j ∈ {0, 2 v j } Definition of Rigorous Fixed Point Problem. Search for v = [v j : j ∈ J i ] such that HN j vj
∀j ∈ J i , or equivalently [
qij+ vj
− ( fπ) j ].[
qij+ vj
2 v ]=0 . − ( fπ) j − HN j j
3. Mathematical analysis of the Fixed Point Problem Here the Fixed Point Problem is shown to be equivalent to the first order optimality conditions of a minimization program which is strictly convex. As a such program has a solution that is unique, this is also the unique solution to the FPP. 3.1. Admissible set Denote v = [v j : j ∈ J i ] a vector of passenger stocks by egress station (from entry station i ) . Define the feasible set as V = {v ≥ 0 : ∀j ∈ J i , v j ≤ N j and ( fπ) j v j ≤ qij+ } .
Lemma 1. Set V is (a) closed and (b) compact. Proof. (a) Let G j : v 6 G j ( v) = ( fπ) j v j . This is a continuous function as v 6 nz = ¦ z∈(i, j ) v j is continuous, as is n z 6 π z = min{1, k ′z /n z } for n z > 0 and 1 at n z = 0 . Thus v 6 ( fπ) j is continuous as a linear combination of continuous functions, and its product by v j , yielding G j , is continuous too. As G j is
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continuous and the sets [0, qij+ ] are closed, the reciprocal sets G −j 1 ([0, q ij+ ]) are closed. Therefore V is closed as the intersection of a finite number of closed sets G −j 1 ([0, q ij+ ]) and [0, N j ] . (b) As V ⊂ [0, N ] with N = [ N j : j ∈ J i ] , the multidimensional interval [0, N ] is bounded hence compact. So is V , a closed subset of [0, N ] .
Lemma 2. Any solution to the FPP must belong to the admissible set. Proof. From (9) and v ≥ 0 ,
qij+
− ( fπ*) j v*j
q ij+ 2 * ≥ 0 . Furthermore, v j ≤ * hence v *j ≤ N j / 2 which HN j vj
meets the second requirement for v* to belong to V . 3.2. Objective function Let us define two mappings as follows: F j ( v) ≡ ( fπ) j − qij+ / v j , yielding mapping F = [ F j : j ∈ J i ] , and
~ F j ( v) ≡
qij+ 2 ~ ~ vj − + ( fπ) j , yielding mapping F = [ F j : j ∈ J i ] . HN j vj
These mappings have been designed so as to yield cross derivatives that are symmetrical in (v j , v r ) , meaning ~ that there are potential functions f and f from which they are derived respectively: ~ ~ ∇f ( v) = [ F j ( v) : j ∈ J i ] and ∇f ( v) = [ F j ( v) : j ∈ J i ] .
Lemma 3. Let δ jr = 1 if j = r or δ jr = 0 otherwise, and denote π~ z ( v) = π z ( v).1{n z ( v )< k ′z } . Then:
∂F j ( v) ∂vr ~ ∂F j ( v) ∂v r
= =
qij+ v 2j
δ jr −
¦
z∈ j ∩ r
~ πz , nz ( v)
∂F j ( v) 2 . δ jr + HN j ∂v r
∂π z ∂ k z′ = min{1, } ∂vr ∂vr nz ′ πz k z ∂n z =− = − 1{ z∈r} nz nz2 ∂vr
Proof.
∂ k z′ ∂vr nz
f z′
is
equal
to
0
if
k z′ ≥ n z ( v )
meaning
π z ( v) = 1
~ ∂π z πz = − 1{ z∈r} whatever the case. Consequently, ∂vr nz ~ ~ ∂ ( fπ) j ∂π z πz π = ¦ f z′ = − ¦ f z′ 1{z∈r} = − ¦ f z′ z . ∂vr ∂vr nz nz z∈ j z∈ j z∈ j ∩ r
On combining, it holds that
~ ∂F j ∂Fr ∂F j ∂F~r ~ = = and . Thus both F j and F j have derivatives that are symmetrical, i.e. ∂v r ∂v j ∂v r ∂v j
or
else
to
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The associated potential functions may be defined in the following way, from a reference point v 0 : ~ v v ~ ~ ∂f ∂f ~ f ( v) = F (v) dv j and f ( v) = F (v) dv j . By construct, = Fr and = Fr . j v0 j j v0 j ∂v r ∂v r
¦³
¦³
Theorem 1. (i) The potential function f is convex on V . ~ (ii) The potential function f is strictly convex on V .
~ Proof. (ii) stems from (i) in a straightforward way, since f = f +
¦r 2(vr − v0r ) 2 / HN r
in which the first
term is convex and the second is strictly convex. (i) Let us show that the Hessian of f is a positive matrix: ∀v ′ = [v r′ : r ∈ J i ] ,
∂2Fj ∂2 f vr′ v′j = ¦ vr′ v′j r , j ∂vr ∂v j r , j ∂vr
¦
~ q+ π f z′ z = ¦ ir2 vr′2 − ¦ vr′ v′j ¦ n z ( v) r, j z∈ j ∩ r r vr ~ q+ π = ¦ ir2 vr′2 − ¦ f z′ z (n′z ) 2 in which n′z = ¦ vr′ = ¦ v′j nz z∈J i r j r vr As v belongs to V , q ij+ ≥ ( fπ ) j v j and v j ≤ n z for z ∈ (i, j ) , hence
¦
qir+
2 r vr
vr′2 ≥ ¦ r
vr′2 ¦ z∈r f z′π z v′2 = . ¦ f z′π z ¦ r vr
z∈Z i
r∈z
vr
On the other hand, n′z2 v′r 2 vr v′r 2 =[ ¦ ] =[ ¦ ] to which we apply the Cauchy - Schwarz inequality nz vr r ∈z n z r ∈z n z
vr vr′2 vr′2 vr ].[ ¦ ]= ¦ since ¦ =1 n v v n r ∈z z r ∈z r r∈ z r r∈ z z
≤[ ¦ Thus :
∂2 f vr′ v′j ≥ ¦ r , j ∂vr ∂v j z∈ J i
¦
vr′2 2 )n′z r ∈z v r
f z′ (π z − ~ π z )( ¦
≥ 0 since π z ≥ ~ πz This ensures that the Hessian matrix of f is positive, hence that f is a convex function. 3.3. Existence and uniqueness of solution
~ Theorem 2. (i) Function f has a unique minimum on V . (ii) The FPP has a solution on V , which is unique. Proof. (i) V being a closed and compact set, a strictly convex function on it admits a minimum value at some unique point v* . There, the first order optimality conditions are that ~ ∇f v* .( v − v*) ≥ 0 ∀v ∈ V* , V* being a restriction of V in the vicinity of v* .
Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
∀j ∈ J i , v *j cannot be zero since at zero the j-th component of the gradient would be very much negative, ~ indicating that an increase in v j would enable one to decrease f . ~ Similarly, the condition qij+ = ( fπ*) j .v*j cannot be achieved because in that case F j ( v*) > 0 , meaning that a ~ decrease in v j would enable one to decrease f . Lastly, condition v *j = N j cannot hold because at such a point ~ ~ F j = 2 − 1 + ( fπ) j ≥ 0 , meaning that a decrease in v j would enable one to decrease f . H H ~ ~ Thus any minimum point of f must be interior to V , which implies that ∇f = 0 at v* , hence that v* ~ satisfies the FPP problem. The uniqueness of solution stems from the strict convexity of f . 3.4. Solution algorithm A Newton – Raphson algorithm is appropriate to solve the minimization program, since the first and second order derivatives are easy to evaluate at each current point. The convergence criterion only involves the norm of ~ ~ the gradient function, ∇f = [ F j : j ∈ J i ] . From step n to n + 1 , the state vector is updated by 2 ~ −1
v n +1 := v n + [∇ f ]
~ ∂F j ~ 2~ : r , v ∈ J i ] the Hessian matrix. v n .(∇f ) v n , denoting by [∇ f ] = [ ∂vr
3.5. Solution revision If the basic solution to the FPP involves some components v *j that are quite small with respect to N j , then an ~ adapted FPP with F j replaced by F j on these components will yield a revised solution with the corresponding egress stations in an unqueued state. 4. An application instance Let us use a classroom instance to demonstrate the behaviour of the transit bottleneck model. The instance is adapted from [5], where it has been designed to compare various capacitated assignment models. Three parallel services connect an origin A with a destination D. The two services ML2 and ML3 also serve node B, which can serve both for an origin and a destination, as shown in figure 2. Table 1 summarizes the operational characteristics. Table 1. Operational characteristics of the service routes Route
Frequency (veh/h)
Vehicle Capacity (pass/veh)
Service Capacity (pass/h)
ML1
6
50
300
ML2
12
100
1,200
ML3
6
150
900
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Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
18’
ML1
$
$·
ML2
10’
10’
'
% ML3
14’
'·
14’
Fig. 2. The service network of the application instance
4.1. Passenger traffic flow states The expected travel time by access – egress station pair was computed on the basis of the passenger stock (for the waiting time) and the weighted average of the flows assigned to the transit services. Let us focus on the simpler origin-destination pair from B to D to distinguish three flow regimes. Maintaining flow q +AD = 0 , the + is increased progressively in order to saturate service ML2 then service ML3. Figures 3(a) and 3(b) flow q BD
illustrate the three flow regimes, by relating the in-vehicle time ( t BD , in dashed line) and the journey time ( g BD = t BD + w BD , in continuous line), with the passenger stock waiting at station B’, v D , and the arriving + . passenger flow, q BD Therefore three flow regimes can be distinguished on the transit line, as depicted in Figures 3 and 4: I. Uncongested regime; passenger flow is weak and does not saturate any capacity constraint. II. Semi-congested regime: at least one service is saturated, while others have still some capacity available to accommodate the additional flow. The access – egress flow is assigned to the alternative services within the reference period H . Both the expected waiting time and the in-vehicle travel time are increased. III. Congested regime: all the services are saturated. The access – egress flow q ij+ cannot be served within the
reference period, H . Insufficient capacity results in a significant increase in the expected waiting time by access – egress pair. •
Fig. 3. Average Travel Time and its In-vehicle part w.r.t. (a) stock of passengers waiting and (b) exogenous passenger flow.
Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
Fig. 4. Relation between Passenger Stock waiting to board and arriving passenger flow
+ q BD .
Figure 4 illustrates the relationship between the arriving passenger flow and the passenger stock. Observe that the congested state begins at a flow slightly higher than the combined residual capacity of the services ( z κ z′ = 1,500 pass / h ). This stems from the simplifying assumptions in the bottleneck model. However, the
¦
difference is insignificant in the calculation of expected waiting time and flow share by transit service. 4.2. Destination competition When multiple destination flows share the same transit services, their respective partial passenger stocks compete for boarding. In the unsaturated regime, the competition does not alter the respective service assignment that is proportional to the service frequencies. When at least one of the routes is saturated, the passenger flows are assigned in a more complex way, due to the joint application of the partial passenger stocks per egress station. In the instance, given a moderate flow q +AD = 800 from A to D, the flow q +AB from A to B is increased progressively until all routes are saturated. Figure 5 depicts the service shares for the flow from A to D. In the unsaturated regime, services ML1, ML2, ML3 are assigned 25%, 50% and 25% respectively of the flow, according to the frequencies. Once a transit service is at capacity, the waiting time is increased and the flow shares are changed. In the instance, ML2 is the + ML 2 is reduced down to p AD =0.40 at first service to be saturated, at q +AB =1,209, and its proportion of flow q AD
q +AB =1,558. The saturation of ML3 creates a competition between the B and D partial stocks at A for the services ML2 and ML3. Therefore, ML1 receives the flow unable to board the other services and its flow share increases ML1 ML1 from p AD =0.30 (at q +AB =1,558) to p AD =0.377 (at q +AB =1,638). When all services are at capacity, the + using boarding competition reduces the probability to board ML2 and ML3 and therefore the proportion of q AD ML1 =0.441 at q +AB =2,500. ML1 increases to p AD
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Fig. 5. The cumulative passenger flow share of each transit service.
5. Conclusion A set of physical assumptions have been provided to model passenger waiting on a transit platform: mingled passengers, explicit stock, fair probability to board under insufficient capacity. By making two further assumptions about, respectively, minimum wait time and treatment of an un-queued stock similar to that of a queued stock, a mathematical formulation has been obtained that is easy to solve and possesses nice properties of solution existence and uniqueness. The transit bottleneck model is designed as a platform model to be used in the framework of passenger traffic assignment to a transit network, as in [6], in which passenger flows are assigned to service legs between pairs of access and egress stations along a given transit line, on a lower network layer dedicated to the detailed operations of vehicles and services – whereas passenger path choice is addressed at the upper layer of network which is comprised of pedestrian and line leg links only.
Acknowledgements This research was supported by the Research and Education Chair on “The socio-economics and modeling of urban public transport”, operated by Ecole des Ponts ParisTech in partnership with the Transport Organizing Authority in the Paris Region (STIF), to which we are grateful.
References [1] Gendreau M. (1984), Etude Approfondie d’un modèle d’équilibre pour l’affectation des Passagers dans les Réseaux de Transport en Commun, Pub. 384, Centre de Recherche Sur les Transports, Université de Montréal [2] Kurauchi F., Bell M.G.H., Schmocker J.D. (2003), Capacity Constrained Transit Assignment with Common Lines, Journal of Mathematical Modelling and Algorithms 2: 309 – 327 [3] Poon M.H., Wong S.C., Tong C.O. (2004). A dynamic schedule-based model for congested transit networks, Transportation Research Part B: Methodological, 38/ 4: 343-368. [4] Hamdouch Y., Lawphongpanich S. (2008), Schedule-based transit assignment model with travel strategies and capacity constraints, Transportation Research Part B 42: 663-684
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[5] Leurent F., Askoura Y. (2010) The person capacity of a transit route: a review, assessment and benchmark of static models for network traffic assignment, Paper Presented at TRB Meeting 2010\ [6] Leurent F., Chandakas E., Poulhes A. (2011) User and service equilibrium in a structural model of traffic assignment to a transit network, in Zak J (ed) The State of the Art in the European Quantitative Oriented Transportation and Logistics Research – 14th Euro Working Group on Transportation & 26th Mini Euro Conference & 1st European Scientific Conference on Air Transport. Elsevier Procedia Social and Behavioral Sciences, 20: 495-505.
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Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
EWGT 2012 15th meeting of the EURO Working Group on Transportation
The transit bottleneck model Fabien Leurent*, Ektoras Chandakas Université Paris-Est, Laboratoire Ville Mobilité Transport, Ecole des Ponts ParisTech 6-8 avenue Blaise Pascal, 77455, Marne la Vallée, Cedex 2, France
Abstract The paper addresses the issue of passenger waiting and being stored at a station platform, from which point they plan to board transit services towards egress stations. Each transit service has a specific set of downstream egress stations and is operated at given frequency using homogeneous vehicles of limited available capacity. The model yields individual waiting time by egress station and the assignment of vehicle capacity to the flows by egress station. Two cases are distinguished, unsaturated versus saturated. The unsaturated case is addressed by standard line combination, where service frequency is added up among the routes that service a given egress station. The saturated case is addressed by making explicit the average number of passengers waiting on platform for a given egress station. From these passenger stocks is derived the individual probability to board a vehicle of limited capacity that service a given route hence a given subset of egress stations. Waiting passengers are assumed to be mingling on the origin platform. The subset of routes that service a given egress station, their vehicle capacities and the boarding probabilities induce a line capacity for that destination: to this is faced the passenger flow demanded during the assignment period, in a bottleneck model that yields an average waiting time per passenger. The vector of passenger stocks by egress station is shown to satisfy a fixed point problem. The existence and uniqueness of the solution are demonstrated on the basis of an equivalent, convex minimization program. A Newton-Raphson algorithm is recommended for computation and demonstrated in an instance of application. by Published Elsevier Ltd. and/or peer-review under responsibility the Program of Committee © 2012 Published The authors. by Selection Elsevier Ltd. Selection and/or peer-review underof responsibility the Program Committee Open access under CC BY-NC-ND license.
Keywords: mingled waiting; Transit assignment; Capacitated assignment; Bottleneck model.
* Corresponding author. Tel.: +33-164-152-111; fax: +33-164-152-140 E-mail address: [email protected]
1877-0428 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Program Committee Open access under CC BY-NC-ND license. doi:10.1016/j.sbspro.2012.09.798
Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
1. Introduction In the dense core of metropolitan urban areas the transit network is frequently submitted to heavy passenger loads, especially at the peak hours on working days. Under such circumstances, various links of the network may provide insufficient capacity to passenger demand. At some station platforms, a traffic bottleneck may arise, in which the passengers experience an increased waiting time until boarding a vehicle of a convenient service. A number of models, static or dynamic, have been provided by the scientific community to deal with the total capacity in passengers of transit vehicles, hence of transit services by aggregating the vehicle runs throughout a given time period. On the static side, Gendreau [1] has modeled passenger queuing by an unbounded delay function that is increasing and convex with respect to boarding flow; however, the analytical solution is difficult in a practical setting. In [2], insufficient capacity is addressed by a fail-to-board probability: platform waiting is driven by a random process of successive Bernoulli trials to board, yielding an average wait time on the basis of the fail-to-board probability; excess flows of passengers are diverted from the network by assuming fictional links from station platform to destination nodes. On the dynamic side, [3] have modeled a traffic bottleneck to yield the queuing time per passenger at any instant. Similarly, in [4], each passenger joins a queue according to his arrival time at the node and experiences a travel time that stems from his position at the queue. Both the static and the dynamic models have been designed in the framework of traffic assignment to a transit network, in relation with passenger path choice. The paper is purported to present a novel model of passenger waiting on platform for transit services to their destination. The main assumptions are that (i) waiting passengers are mingled, (ii) those destined to a given station of egress make up a stock, of which the size is explicit, (iii) by transit service, the available capacity is faced to the stocks of the egress stations along the service route, yielding a probability of immediate boarding, (iv) by egress station, the average wait time on platform is same as in a traffic bottleneck. These assumptions are cast into a fixed point problem with respect to the destination stocks, and mathematical properties of solution existence and uniqueness are established. The rest of this paper is in four parts. To begin with, the platform model is developed in the form of necessary mathematical conditions which make up a fixed point problem. Then, a mathematical analysis of the fixed point problem is made, by formulating an equivalent problem of convex minimization, so as to yield properties of solution existence and uniqueness. Then, an application instance is carried on to demonstrate the main characteristics of the model, including the influence of passenger flow on the average journey time and the competition between destinations at a given station. Lastly, the conclusion points to the inclusion of the transit bottleneck model in the framework of traffic assignment to a transit network. 2. The queuing model 2.1. Definitions Let i be a given access station along line A , J i the set of egress stations j that are served by transit services z ∈ Zi . The subset of services that dwell at j coming from i is denoted as {z ∈ (i, j )} . During the period H , service z is operated at frequency f z′ by vehicles of residual capacity k ′z at station i (after the egress of the passengers destined to i ), yielding an available capacity of κ z′ = f z′k ′z in the period. It is assumed that passengers arrive at i under exogenous flow [q ij+ : j ∈ J i ] , yielding entry volumes H.q ij+ by egress station. Let f ij ≡ z∈(i , j ) f ij be the combined frequency between i and j of the services in Z i . It is assumed that
¦
the platform is shared by the services and that no vehicle can overtake another one, meaning that a timeminimizing passenger is eager to board a relevant vehicle as soon as it has some place available to him. Notation A designates this set of services where the routes and egress stations make up a connected component in the bipartite graph in Z i × J i that links the services to the stations that they serve.
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2.2. The unqueued case If capacity is available to each passenger as soon as he would like from his instant of arrival, then he can board the first relevant vehicle and the average wait time for destination j is:
wij = α / f ij .
(1)
In (1), parameter α models headway regularity: a value of 1 indicates memoryless vehicle arrivals, whereas ½ indicates perfect regularity (homogenous headways). Each service in A gets a share f z / f ij of volume Hqij+ . The resulting passenger flow, ( Hqij+ f z ) / f ij , yields a number ( Hq ij+ ) / f ij of passengers destined to j by vehicle of service z . There remains available capacity if and only if, by vehicle,
H
¦
j:(i , j )∋ z
qij+ fij
≤ k ′z .
(2)
This condition must be checked to ensure that there is no queuing – at least no remaining stock of passengers that keep waiting just after a vehicle relevant to them has left. 2.3. The queued case If some passengers cannot board the first vehicle relevant to them, then they have to wait for other vehicles to arrive and to supply them with some capacity. Let us assume that there is a fictive, stationary state of passenger traffic on the platform, with a number v j of passengers waiting for destination j . When a vehicle of service z arrives with available capacity k ′z , there are candidate passengers in number of
n z = ¦ j∈z v j .
(3)
Their individual probability to board, assuming mingled waiting hence equity among them, is
π z = min{1, k z′ / nz } (with π z = 1 if n z = 0 ).
(4)
Then the number of passengers boarding a vehicle of service z to exit at j is π z v j , and the passenger flow during H to j via z is − = f ′π v . qijz z z j
The total flow rate to j from
(5)
i during H is the throughput rate as follows:
− =( qij− = ¦ j∈z qijz ¦ j∈z f z′π z )v j .
(6)
Let us denote ( fπ) j ≡ ¦ j∈z f z′π z . Queuing may eventually occur when at least one service has π z < 1 , meaning saturation of that service. Let us derive necessary conditions on the π z and v j variables that characterize service availability and waiting. A bottleneck model is assumed by egress station j , with arrival rate q ij+ on [0, H ] and discharge rate q ij− on
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w j 0 + [0, H j ] , in which w j 0 is the mean wait time between arriving at the queue front end in eligible place and the arrival of a relevant vehicle. A reasonable value could be w j 0 = α / f ij ; for the moment, let us keep w j 0 as an independent variable so as to reserve one degree of freedom. H
y = q+ x N
w0
Hj
w0 qFig. 1. Cumulative flows at bottleneck.
Figure 1 depicts the dynamics of the cumulated flows and the waiting queue: if q − < q + then there is a ‘triangle of waiting’ that lies between the cumulated curve of passenger arrival, y = q ij+ .x , shifted to the right due to the minimum waiting w j 0 , and the cumulated curve of passenger departure, y = q ij− ( x − w j 0 ) for x − w j 0 ∈ [0, H j ] . Flow conservation implies that:
N j = H .q ij+ = H j .q ij− .
(7)
The average number of passengers waiting for destination j , v j , is equal to the total time spent at waiting,
WijA , divided by the time interval, H ′j = w j 0 + H j . Letting N ′j = N j + w j 0 qij− and ( fπ) j ≡ ¦ j∈z f z′ π z , the total wait amounts to the area between the two large triangles minus the area of the small triangle:
WijA =
2 N 2j 1 N ′j ( − − + − w 2j 0 q ij− ) . 2 q ij q ij
On replacing WijA with v j H ′j = v j ( w j 0 + H j ) and N ′j with N j + w j 0 q ij− , we get that
v j (w j 0 + vj
w j0 Nj
+
Nj q ij−
)=
2 N 2j 1 Nj ( − + 2 w j 0 N j + w 2j 0 q ij− − + − w 2j 0 q ij− ) 2 q ij q ij
Nj Nj 1 1 − w j0 = ( − ) after dividing by N j and replacing q ij− by v j ( fπ) j + ( fπ) j 2 v j ( fπ) j qij
(8)
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2v j
( fπ) j w j 0 HN j
+2
qij+ Nj
[1 − ( fπ) j w j 0 ] =
qij+ vj
− ( fπ) j after multiplying by 2qij+ ( fπ) j / N j
At this stage, let us set w j 0 = 1 /( fπ) j so as to simplify into
qij+ 2 vj = − ( fπ) j . HN j vj
(9)
This model is taken for all traffic regimes, including the unqueued state in which the left hand side is negligible, yielding qij+ ≈ ( fπ) j v j = qij− . It is expected that
q ij+ 2 v j << , i.e. 2v 2j << q ij+ HN j = N 2j , or HN j vj
v j << N j / 2 .
(10)
Referring to (3) and (7), the π z depend on the v j through the nz . Thus there is a circular dependency between our variables, in other words a Fixed Point Problem (FPP) with respect to vector v = [v j : j ∈ J i ] . In the next section, it is shown that the FPP (3), (7) and (9) has a solution, which is unique. Also included is a computation scheme. Lastly, let us mention a related problem definition that distinguishes the queuing regimes and would be more rigorous, but less tractable: qij+ − ( fπ) j ∈ {0, 2 v j } Definition of Rigorous Fixed Point Problem. Search for v = [v j : j ∈ J i ] such that HN j vj
∀j ∈ J i , or equivalently [
qij+ vj
− ( fπ) j ].[
qij+ vj
2 v ]=0 . − ( fπ) j − HN j j
3. Mathematical analysis of the Fixed Point Problem Here the Fixed Point Problem is shown to be equivalent to the first order optimality conditions of a minimization program which is strictly convex. As a such program has a solution that is unique, this is also the unique solution to the FPP. 3.1. Admissible set Denote v = [v j : j ∈ J i ] a vector of passenger stocks by egress station (from entry station i ) . Define the feasible set as V = {v ≥ 0 : ∀j ∈ J i , v j ≤ N j and ( fπ) j v j ≤ qij+ } .
Lemma 1. Set V is (a) closed and (b) compact. Proof. (a) Let G j : v 6 G j ( v) = ( fπ) j v j . This is a continuous function as v 6 nz = ¦ z∈(i, j ) v j is continuous, as is n z 6 π z = min{1, k ′z /n z } for n z > 0 and 1 at n z = 0 . Thus v 6 ( fπ) j is continuous as a linear combination of continuous functions, and its product by v j , yielding G j , is continuous too. As G j is
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continuous and the sets [0, qij+ ] are closed, the reciprocal sets G −j 1 ([0, q ij+ ]) are closed. Therefore V is closed as the intersection of a finite number of closed sets G −j 1 ([0, q ij+ ]) and [0, N j ] . (b) As V ⊂ [0, N ] with N = [ N j : j ∈ J i ] , the multidimensional interval [0, N ] is bounded hence compact. So is V , a closed subset of [0, N ] .
Lemma 2. Any solution to the FPP must belong to the admissible set. Proof. From (9) and v ≥ 0 ,
qij+
− ( fπ*) j v*j
q ij+ 2 * ≥ 0 . Furthermore, v j ≤ * hence v *j ≤ N j / 2 which HN j vj
meets the second requirement for v* to belong to V . 3.2. Objective function Let us define two mappings as follows: F j ( v) ≡ ( fπ) j − qij+ / v j , yielding mapping F = [ F j : j ∈ J i ] , and
~ F j ( v) ≡
qij+ 2 ~ ~ vj − + ( fπ) j , yielding mapping F = [ F j : j ∈ J i ] . HN j vj
These mappings have been designed so as to yield cross derivatives that are symmetrical in (v j , v r ) , meaning ~ that there are potential functions f and f from which they are derived respectively: ~ ~ ∇f ( v) = [ F j ( v) : j ∈ J i ] and ∇f ( v) = [ F j ( v) : j ∈ J i ] .
Lemma 3. Let δ jr = 1 if j = r or δ jr = 0 otherwise, and denote π~ z ( v) = π z ( v).1{n z ( v )< k ′z } . Then:
∂F j ( v) ∂vr ~ ∂F j ( v) ∂v r
= =
qij+ v 2j
δ jr −
¦
z∈ j ∩ r
~ πz , nz ( v)
∂F j ( v) 2 . δ jr + HN j ∂v r
∂π z ∂ k z′ = min{1, } ∂vr ∂vr nz ′ πz k z ∂n z =− = − 1{ z∈r} nz nz2 ∂vr
Proof.
∂ k z′ ∂vr nz
f z′
is
equal
to
0
if
k z′ ≥ n z ( v )
meaning
π z ( v) = 1
~ ∂π z πz = − 1{ z∈r} whatever the case. Consequently, ∂vr nz ~ ~ ∂ ( fπ) j ∂π z πz π = ¦ f z′ = − ¦ f z′ 1{z∈r} = − ¦ f z′ z . ∂vr ∂vr nz nz z∈ j z∈ j z∈ j ∩ r
On combining, it holds that
~ ∂F j ∂Fr ∂F j ∂F~r ~ = = and . Thus both F j and F j have derivatives that are symmetrical, i.e. ∂v r ∂v j ∂v r ∂v j
or
else
to
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The associated potential functions may be defined in the following way, from a reference point v 0 : ~ v v ~ ~ ∂f ∂f ~ f ( v) = F (v) dv j and f ( v) = F (v) dv j . By construct, = Fr and = Fr . j v0 j j v0 j ∂v r ∂v r
¦³
¦³
Theorem 1. (i) The potential function f is convex on V . ~ (ii) The potential function f is strictly convex on V .
~ Proof. (ii) stems from (i) in a straightforward way, since f = f +
¦r 2(vr − v0r ) 2 / HN r
in which the first
term is convex and the second is strictly convex. (i) Let us show that the Hessian of f is a positive matrix: ∀v ′ = [v r′ : r ∈ J i ] ,
∂2Fj ∂2 f vr′ v′j = ¦ vr′ v′j r , j ∂vr ∂v j r , j ∂vr
¦
~ q+ π f z′ z = ¦ ir2 vr′2 − ¦ vr′ v′j ¦ n z ( v) r, j z∈ j ∩ r r vr ~ q+ π = ¦ ir2 vr′2 − ¦ f z′ z (n′z ) 2 in which n′z = ¦ vr′ = ¦ v′j nz z∈J i r j r vr As v belongs to V , q ij+ ≥ ( fπ ) j v j and v j ≤ n z for z ∈ (i, j ) , hence
¦
qir+
2 r vr
vr′2 ≥ ¦ r
vr′2 ¦ z∈r f z′π z v′2 = . ¦ f z′π z ¦ r vr
z∈Z i
r∈z
vr
On the other hand, n′z2 v′r 2 vr v′r 2 =[ ¦ ] =[ ¦ ] to which we apply the Cauchy - Schwarz inequality nz vr r ∈z n z r ∈z n z
vr vr′2 vr′2 vr ].[ ¦ ]= ¦ since ¦ =1 n v v n r ∈z z r ∈z r r∈ z r r∈ z z
≤[ ¦ Thus :
∂2 f vr′ v′j ≥ ¦ r , j ∂vr ∂v j z∈ J i
¦
vr′2 2 )n′z r ∈z v r
f z′ (π z − ~ π z )( ¦
≥ 0 since π z ≥ ~ πz This ensures that the Hessian matrix of f is positive, hence that f is a convex function. 3.3. Existence and uniqueness of solution
~ Theorem 2. (i) Function f has a unique minimum on V . (ii) The FPP has a solution on V , which is unique. Proof. (i) V being a closed and compact set, a strictly convex function on it admits a minimum value at some unique point v* . There, the first order optimality conditions are that ~ ∇f v* .( v − v*) ≥ 0 ∀v ∈ V* , V* being a restriction of V in the vicinity of v* .
Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
∀j ∈ J i , v *j cannot be zero since at zero the j-th component of the gradient would be very much negative, ~ indicating that an increase in v j would enable one to decrease f . ~ Similarly, the condition qij+ = ( fπ*) j .v*j cannot be achieved because in that case F j ( v*) > 0 , meaning that a ~ decrease in v j would enable one to decrease f . Lastly, condition v *j = N j cannot hold because at such a point ~ ~ F j = 2 − 1 + ( fπ) j ≥ 0 , meaning that a decrease in v j would enable one to decrease f . H H ~ ~ Thus any minimum point of f must be interior to V , which implies that ∇f = 0 at v* , hence that v* ~ satisfies the FPP problem. The uniqueness of solution stems from the strict convexity of f . 3.4. Solution algorithm A Newton – Raphson algorithm is appropriate to solve the minimization program, since the first and second order derivatives are easy to evaluate at each current point. The convergence criterion only involves the norm of ~ ~ the gradient function, ∇f = [ F j : j ∈ J i ] . From step n to n + 1 , the state vector is updated by 2 ~ −1
v n +1 := v n + [∇ f ]
~ ∂F j ~ 2~ : r , v ∈ J i ] the Hessian matrix. v n .(∇f ) v n , denoting by [∇ f ] = [ ∂vr
3.5. Solution revision If the basic solution to the FPP involves some components v *j that are quite small with respect to N j , then an ~ adapted FPP with F j replaced by F j on these components will yield a revised solution with the corresponding egress stations in an unqueued state. 4. An application instance Let us use a classroom instance to demonstrate the behaviour of the transit bottleneck model. The instance is adapted from [5], where it has been designed to compare various capacitated assignment models. Three parallel services connect an origin A with a destination D. The two services ML2 and ML3 also serve node B, which can serve both for an origin and a destination, as shown in figure 2. Table 1 summarizes the operational characteristics. Table 1. Operational characteristics of the service routes Route
Frequency (veh/h)
Vehicle Capacity (pass/veh)
Service Capacity (pass/h)
ML1
6
50
300
ML2
12
100
1,200
ML3
6
150
900
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18’
ML1
$
$·
ML2
10’
10’
'
% ML3
14’
'·
14’
Fig. 2. The service network of the application instance
4.1. Passenger traffic flow states The expected travel time by access – egress station pair was computed on the basis of the passenger stock (for the waiting time) and the weighted average of the flows assigned to the transit services. Let us focus on the simpler origin-destination pair from B to D to distinguish three flow regimes. Maintaining flow q +AD = 0 , the + is increased progressively in order to saturate service ML2 then service ML3. Figures 3(a) and 3(b) flow q BD
illustrate the three flow regimes, by relating the in-vehicle time ( t BD , in dashed line) and the journey time ( g BD = t BD + w BD , in continuous line), with the passenger stock waiting at station B’, v D , and the arriving + . passenger flow, q BD Therefore three flow regimes can be distinguished on the transit line, as depicted in Figures 3 and 4: I. Uncongested regime; passenger flow is weak and does not saturate any capacity constraint. II. Semi-congested regime: at least one service is saturated, while others have still some capacity available to accommodate the additional flow. The access – egress flow is assigned to the alternative services within the reference period H . Both the expected waiting time and the in-vehicle travel time are increased. III. Congested regime: all the services are saturated. The access – egress flow q ij+ cannot be served within the
reference period, H . Insufficient capacity results in a significant increase in the expected waiting time by access – egress pair. •
Fig. 3. Average Travel Time and its In-vehicle part w.r.t. (a) stock of passengers waiting and (b) exogenous passenger flow.
Fabien Leurent and Ektoras Chandakas / Procedia - Social and Behavioral Sciences 54 (2012) 822 – 833
Fig. 4. Relation between Passenger Stock waiting to board and arriving passenger flow
+ q BD .
Figure 4 illustrates the relationship between the arriving passenger flow and the passenger stock. Observe that the congested state begins at a flow slightly higher than the combined residual capacity of the services ( z κ z′ = 1,500 pass / h ). This stems from the simplifying assumptions in the bottleneck model. However, the
¦
difference is insignificant in the calculation of expected waiting time and flow share by transit service. 4.2. Destination competition When multiple destination flows share the same transit services, their respective partial passenger stocks compete for boarding. In the unsaturated regime, the competition does not alter the respective service assignment that is proportional to the service frequencies. When at least one of the routes is saturated, the passenger flows are assigned in a more complex way, due to the joint application of the partial passenger stocks per egress station. In the instance, given a moderate flow q +AD = 800 from A to D, the flow q +AB from A to B is increased progressively until all routes are saturated. Figure 5 depicts the service shares for the flow from A to D. In the unsaturated regime, services ML1, ML2, ML3 are assigned 25%, 50% and 25% respectively of the flow, according to the frequencies. Once a transit service is at capacity, the waiting time is increased and the flow shares are changed. In the instance, ML2 is the + ML 2 is reduced down to p AD =0.40 at first service to be saturated, at q +AB =1,209, and its proportion of flow q AD
q +AB =1,558. The saturation of ML3 creates a competition between the B and D partial stocks at A for the services ML2 and ML3. Therefore, ML1 receives the flow unable to board the other services and its flow share increases ML1 ML1 from p AD =0.30 (at q +AB =1,558) to p AD =0.377 (at q +AB =1,638). When all services are at capacity, the + using boarding competition reduces the probability to board ML2 and ML3 and therefore the proportion of q AD ML1 =0.441 at q +AB =2,500. ML1 increases to p AD
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Fig. 5. The cumulative passenger flow share of each transit service.
5. Conclusion A set of physical assumptions have been provided to model passenger waiting on a transit platform: mingled passengers, explicit stock, fair probability to board under insufficient capacity. By making two further assumptions about, respectively, minimum wait time and treatment of an un-queued stock similar to that of a queued stock, a mathematical formulation has been obtained that is easy to solve and possesses nice properties of solution existence and uniqueness. The transit bottleneck model is designed as a platform model to be used in the framework of passenger traffic assignment to a transit network, as in [6], in which passenger flows are assigned to service legs between pairs of access and egress stations along a given transit line, on a lower network layer dedicated to the detailed operations of vehicles and services – whereas passenger path choice is addressed at the upper layer of network which is comprised of pedestrian and line leg links only.
Acknowledgements This research was supported by the Research and Education Chair on “The socio-economics and modeling of urban public transport”, operated by Ecole des Ponts ParisTech in partnership with the Transport Organizing Authority in the Paris Region (STIF), to which we are grateful.
References [1] Gendreau M. (1984), Etude Approfondie d’un modèle d’équilibre pour l’affectation des Passagers dans les Réseaux de Transport en Commun, Pub. 384, Centre de Recherche Sur les Transports, Université de Montréal [2] Kurauchi F., Bell M.G.H., Schmocker J.D. (2003), Capacity Constrained Transit Assignment with Common Lines, Journal of Mathematical Modelling and Algorithms 2: 309 – 327 [3] Poon M.H., Wong S.C., Tong C.O. (2004). A dynamic schedule-based model for congested transit networks, Transportation Research Part B: Methodological, 38/ 4: 343-368. [4] Hamdouch Y., Lawphongpanich S. (2008), Schedule-based transit assignment model with travel strategies and capacity constraints, Transportation Research Part B 42: 663-684
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[5] Leurent F., Askoura Y. (2010) The person capacity of a transit route: a review, assessment and benchmark of static models for network traffic assignment, Paper Presented at TRB Meeting 2010\ [6] Leurent F., Chandakas E., Poulhes A. (2011) User and service equilibrium in a structural model of traffic assignment to a transit network, in Zak J (ed) The State of the Art in the European Quantitative Oriented Transportation and Logistics Research – 14th Euro Working Group on Transportation & 26th Mini Euro Conference & 1st European Scientific Conference on Air Transport. Elsevier Procedia Social and Behavioral Sciences, 20: 495-505.
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