Modeling train delays in Urban Networks

Higgins and Kozan suggested in their article [37] a model to estimate expected positive delays. We report here their formulation of the model, writing in bold type between brackets the corresponding notation according to our formulation of the problem (see Section 2.1, in particular model [CTM-2]). We prefer not to change the notation since their formulation is mainly based on two index sets (one for the trains, the second for the blocks of the system) while our formulation considers only one index set (the set of events of the system).

In their model Higgins and Kozan consider the following specifications:

• delay: just positive delays are considered since early departures are not allowed for passenger trains(secondary delayy);

• source delay: the delay of a train that is not caused by another train (source delayd)

• minimum headway: the minimum length of time separating two trains traveling on the same edge, which is determined by the length of the blocks in the edge (headwayh);

• link: a section of track on which only one train is permitted at any time(block m);

• siding: a track link (a block) that can be used for the crossing and passing of trains.

Moreover they distinguish among three kinds of delays:

5.3 Modeling train delays in Urban Networks 91

• direct delays of trains, i.e. source delays;

• knock-on delays to other trains, i.e. forced delays due to driving/waiting activi-ties or to occupation of the scheduled block from a delayed train;

• delays due to late connections, where train connections include:

1. fixed connections;

2. fixed departure orders in a station;

3. commencements of a new service after arrival at the destination using the same physical train, i.e. turn-overs of trains.

They consider also the following assumptions:

• scheduled times contain some slack time;

• if a train arrives earlier, it will wait until the scheduled departure;

• trains may have different upper average velocity;

• waiting times at a station are included in the scheduled traveling times;

• trains can bypass a delayed one if there is a free siding or parallel track avaliable;

• conflicts are solved on a “first come first serve” basis (i.e. FIFO);

• the capacity at a station is determined by the number of track links in it.

The following parameters are considered in the model:

I =set of train services for one cycle of the schedule, usually daily,(set of trainst∈ T);

J =set of links for the whole suburban network(set of blocksm∈ M);

Qⁱ_j =set of remaining links from linkj∈Jalong which traini∈I is scheduled to travel;

Rⁱ_j =set of links prior to link j∈Jalong which traini∈I is scheduled to travel;

Qⁱ =set of links along which traini∈I is scheduled to travel from origin to destination, i.e.

Qⁱ= (Rⁱ_j,j,Qⁱ_j);

Vⁱ_j =the link immediately prior to linkj∈Jon which traini∈I is scheduled to travel,Vⁱ_j∈Rⁱ_j;

Yⁱ_j =scheduled departure time of traini∈I from linkj∈Qⁱ_j(scheduled timeπ);

T =duration of a source delay(value of the source delay variabled);

PRO(T,i,j) =probability that traini∈Iwill have a source delay of durationTon linkj∈J(in [37] neg. exponential);

E_{k j}ⁱ =amount of time traini∈Iis able to recover between linksk,j∈Qⁱ ( scheduled slack timec_{k j}).

The shortest possible travel time for traini∈Ibetween these two links isY_kⁱ−Yⁱ_j−E_{k j}ⁱ i.e.

scheduled travel time minus slack time (minimal traveling durationL_awherea= (k,j));

CONN =set of train connections. A train connection takes place between traini∈Iat linkj∈J and trainl∈Iat linkh∈J.Linksi,j∈J are located at the same station.(i,j,l,h)∈CONN (set of connectionsA^change).

The output of the model corresponds to the following information:

PRS(T L,i,j) =probability that traini∈Ihas a current delayT L at the departure of linkj∈J; 1T Cⁱ_j =source delay of traini∈Iat linkj∈Qⁱ (source delayd);

2T Cⁱ_j =delay of traini∈Iat linkj∈Qⁱ as a result of knock-on delays;

3T Cⁱ_j =delay of traini∈Iat linkj∈Qⁱ as a result of late connections;

T Cⁱ_j =delay of traini∈Iat linkj∈Qⁱ,T Cⁱ_j=¹T Cⁱ_j+²T Cⁱ_j+³T Cⁱ_j(secondary delayy).

Direct delays of trains

A traini∈ Iwith a current delayT Lhas a source delayT at linkk∈J, therefore its delay at link j∈Qⁱ_jis

max{T L−Eⁱ_{k j}+T, 0}

whereEⁱ_jkis the scheduled slack time from linkk∈Rⁱ_jtoj∈Qⁱ_j. Taking into account all possible source delaysT and current delaysT L, the expected train delay of train i∈Iat link j∈Qⁱdue to source delays is

We can distinguish two different cases: unidirectional and bidirectional track.

Unidirectional track

FurthermoreMj is defined as the minimum average time for a train to travel along link j∈J.

Considering the scheduled slack timeEⁱ_{k j}and just positive delays, the knock-on delay to traini∈Iat linkj∈Dⁱdue to trainl∈Iis

Extending the formulation in order to include all possible values of source delaysT and current delaysT LandT Mwe get

Under the assumption that in a delayed situation trains traveling in the same direction have priority over opposite trains, the knock-on delay on a bidirectional track can be defined as in (5.6) by adding the further condition that the set_{T L T M}^TS^l_kcontaints all

“same-direction” trains first (in ascending order with respect toY_kⁱ), followed by all

“opposite direction” trains.

Delay because of late connections

A scheduled connection between traini∈Iandl∈Ideparting from linksk∈Qⁱand h∈ Q^llocated in the same station is given, so thaticannot depart untillarrives, i.e.

Y_kⁱ ≥Y_h^l. Given the scheduled slack timeEⁱ_jkand supposing thatiandlhave current delaysT MandT Lrespectively and that there are just positive delays, the arrival delay of traini∈Iat link j∈Qⁱcan be defined as

g₃(i,k,l,h,j,T L,T M) =max{Y_h^l+T M−Y_kⁱ−T L−Eⁱ_jk, 0}

5.4 Some considerations about the normal distribution. 93 Taking into account all possible connections of trainiat link jwe get

E(³T Cⁱ_j) = X

Adding together the three delay components we obtain the overall expected delay of traini∈Iat link j∈Qⁱ

(5.8) E(T Cⁱ_j) =E(¹T Cⁱ_j) +E(²T Cⁱ_j) +E(³T Cⁱ_j)

This model to estimate expected delays (to trains or track links) has been tested on the suburban railway network in Queensland, Australia. The model estimations ofE(T C) were compared to stochastic simulations, results of which were used to reflect actual conditions.

The small model inaccuracy was reported to be due to the assessed distribution of current delay, being slightly different from the actual distribution for some trains.

Principal aim of this model is to evaluate the average delay of every train of the system. In future works, this average delay should be then added to the scheduled timetable to get a new timetable that at the current state of reseach is not guaranteed to be feasible, since no further checks are done concerning the capacity constraints of the model. This model seems more suitable for a program which purpose is to reduce the average delay of every train. Our formulation of the problem instead is based on the minimization of the total delay of the system. Moreover our formulation guarantees the feasibility of the “new timetable” for the problem.

5.4 Some considerations about the normal