For a future transfer, the delays of the feeder- and connecting train determine the classification of a transfer. To illustrate the dynamical change in delays and classification, Fig. 5.1 gives an example where the status of some specific interchange is monitored for two hours.

**Fig. 5.1:** The figure sketches exemplarily the development of the delay in minutes of a single feeder vehicle,
two hours before the departure of a connecting vehicle of a planned interchange. Depending
on the size of the delay and the potential to regain, PANDA classifies the transfer as**SAFE,**
**UNCERTAIN,** **CRITICAL**or**BREAK**[Lem+14].

In this case, 40 minutes before the departure of the connecting train, an additional delay
cause the transfer to be classified as **BREAK. After this delay waiting is no longer an option;**

dispatchers should communicate new routes for affected passengers. The next step for working with the defined classifications for transfers is their specific implementation. We created three classification schemes [LMHR16] which will be explained in the next section.

### 5.3.1 Classification Schemes

The EAN modeling the real-world network receives information about realized arrival and departure times every minute. Whenever new information about the realized event times is available, we propagate it through the network. Received information about event times arrive as one of three types.

**delay messages** messages about realized past events

**automated predictions** messages about future events generated by algorithms of the Deutsche
Bahn

**manual predictions** messages about future departure times from human dispatchers in case of
decisions, this type has priority over automated predictions

Not every event of the route of a train will receive an external prediction. Information about events during a trip have to be consistent for later usage in the simulation. Generating missing

**5.3** Terms and Classification **51**

timestamps as well as the timestamps used in delay propagation during simulation is difficult.

For this task, data about the acceleration, and driving speeds is often useful. When there is no information about infrastructural data, speeds, and acceleration of trains, two types of approaches for predictions are common. The first one utilizes a statistical model using large sets of data [KL17]. We apply a second type of prediction utilizing heuristical propagation rules.

We propagate the current delay along the activity edges of a train. If there are defined buffer timess along an activity edge, we subtract those from the propagated delay. In the simulations made in the following experiments, we used the latter method.

Included in this heuristic propagation is the application of waiting time rules. Given a number
*wt** _{a}*∈N0 for a transfer activity

*a*∈

*A*

*, it shall be maintained if the connecting train has to wait at most*

_{transf er}*wt*

*a*minutes compared to its original schedule. We also consider the duration of the interchange itself, which we denote with

*l*

*. By*

_{a}*w.t*

*:=*

^{maxwait}*max{w.t*

*+*

^{sched}*wt*

*|a= (v, w)∈*

_{a}*A*

*} the maximum waiting time induced by any delayed feeder train is denoted.*

_{transf er}Unfortunately, this heuristic can be too pessimistic on tracks where the driver can make up
lost time. The classification needs to assess the situation correctly, even if there is no information
on minimum traveling times. Therefore, every event is assigned a lower bound representing
the theoretically earliest arrival using a general assumption on minimum traveling times. In
practice, it is typically assumed that every train can make up 7% on every driving arc. Using
this assumption during propagation for each arrival event *arr* ∈*V* we obtain a lower bound
*arr.t** ^{lb}* on its realization time.

The definitions up to this point suffice to create a simple classification scheme using bounds.

### Classification by Lower Bounds

We formalize the first classifier in the following fashion. An interchange arc is denoted as *a*=
(arr, dep)∈*A**transf er*. We classify it as BREAK if*arr.t** ^{lb}*+

*l*

*a*

*> max(dep.t*

^{pred}*, dep.t*

*+*

^{sched}*wt*

*a*+

*δ). We look at the earliest possible arrival of the passenger at the departure platformarr.t** ^{lb}*+l

*. If this is greater than*

_{a}*dep.t*

*+*

^{sched}*wt*

*a*+

*δ, it is considered as BREAK. The parameterδ >*0 specifies a “safety margin”. An increase in this parameter makes the classification more conservative. In this method, we also check the predicted departure time of the connecting train

*dep.t*

*and consider if an existing delay of the departure could make the transfer valid. An arc is classified as SAFE, if*

^{pred}*arr.t*

*+*

^{pred}*l*

*≤*

_{a}*dep.t*

*. A transfer is classified as UNCERTAIN if it is not SAFE but*

^{pred}*arr.t*

*+*

^{pred}*l*

*≤*

_{a}*dep.t*

*+*

^{sched}*wt*

*. If a transfer could not fit in any of those distinctions, it is CRITICAL. This classification scheme will later be referred to as “STANDARD”.*

_{a}### Classification by Transfer Probabilities

The second classification scheme makes use of the massive amount of data about past transfers. It is based on probabilities and has been suggested by the Deutsche Bahn. Dispatchers of Deutsche Bahn had access to a similar version of this scheme giving an indication about future transfers using only three classification states. The type of train has an impact on the probabilities if

**52** **Chapter 5** Timing of Train Disposition

connections are maintained. Here dispatchers distinguish between InterCity Express (ICE), InterCity (IC), and regional trains.

For a train of type*trainT ype* which is currently delayed by*d*minutes, and a time horizon of
*h* minutes we have an empirical density function *f**trainT ype,h,d*

∆ :Z7→ [0,1] for the probability
that the current delay will change by *x* minutes in*h*minutes from now. With the help of this
function we can derive the probability*f**trainT ype,h,d*that future event*v*for this train will occur at
time*t*=*v.t** ^{sched}*+

*d*+

*x, whered*+

*x*≥0 holds. In the following notation ∆ is not a parameter.

It only marks, that the function is applied to a duration and not a timestamp. We define
*f**trainT ype,h,d*(t) :Z7→[0,1], with *f**trainT ype,h,d*(t) =*f**trainT ype,h,d*

∆ (x), x∈Z*.*

Using this function, we can compute the distribution of departure and arrival times for all future
events with respect to the current delay scenario. For a given transfer, we can calculate the
joined probability *p* of cases where the time is sufficient to comply with the minimum transfer
times. The underlying assumption is that the probability of delays is independent.

Next we explain the formula used for calculating the probability of a future transfer arc
*a*= (arr, dep)∈*A. We derive the probability distribution that the transfer will be maintained*
as follows. As before denote by *l** _{a}*the minimum transfer time and by

*dep.t*

*the maximum waiting time of the departing train. Thus, unless the departing train itself is delayed, it will wait at most until*

^{maxwait}*t*

*=*

_{max}*dep.t*

*. The transfer activity requires at least*

^{maxwait}*l*

*minutes. The probability*

_{a}*p*that a transfer will be maintained if the feeder train arrives not later than

*t*

*−*

_{max}*l*

*is*

_{a}*p*=

*t**max*−l*a*

X

*t=0*

*f**arr.trainT ype,arr.h,arr.d*(t) (5.1)

The probability that a transfer will be maintained after *t** _{max}* depends on the distribution of the
departing train as well. In this case

*p*=

∞

X

*t*1=t*max*−l*a*+1

∞

X

*t*2=t1+l*a*

*f**arr.trainT ype,arr.h,arr.d*(t1)·*f**dep.trainT ype,dep.h,dep.d*(t2) (5.2)
where*arr.h*and*dep.h*denote the current time horizon for the arrival event*arr*and the departure
event *dep, and where* *arr.d*and *dep.d* denote the current delays of the arriving and departing
train, respectively. This classification scheme will late be referred to as STOCHASTIC. The last
step needed for classification is a partition of the final probabilities into intervals for each state.

These intervals are shown in Table 5.1.

### Classification by Fuzzy Logic

This third classification scheme proposed in [LMHR16] utilizes fuzzy logic for classification. The concept of fuzzy logic differs from classical logic, where variables are either true or false. It is a form of many-valued logic, where the truth value of variables may be any real number between

**5.3** Terms and Classification **53**

**class** **rule**

**SAVE** *p*≥0.96

**UNCERTAIN** 0.6≤*p*≤0.96
**CRITICAL** 0.05≤*p*≤0.6

**BREAK** *p*≤0.05

**Tab. 5.1:** Translation of probability values of a transfer into classification classes.

**Fig. 5.2:** Fuzzification of the two variables *current delay* and *catch*−*up potential for a transfer arc*
*a*= (arr, dep) [Lem+14].

zero and one. The reason for choosing this concept is the representation of the vagueness of information present in real-world scenarios. The basic attributes of this scheme are the same as those used by the STANDARD scheme. We will refer to this classification scheme as FUZZY. To classify a transfer concerning uncertainty, we use a classifier based on fuzzy logic. We consider three linguistic variables for the feeder’s arrival event:

• the*current delay* with possible values *on time, small delay* and *strong delay,*

• the*catch-up* potential with values*possible* and *impossible,*

• the*state* of a transfer with values SAFE, UNCERTAIN, CRITICAL, and BREAK.

Figure 5.2 shows for an arrival event*arr* how the variables *arr.t** ^{lb}* are fuzzified into linguistic
variables

*current delay*and

*catch-up potential*with certainty

*p*

*and*

_{d}*p*

*, respectively. We use the interference rules shown in Table 5.2 to determine the state of a transfer. The lines of the table are to be read as IF current delay is*

_{r}*current delay*AND regain potential is

*regain potential*THEN the transfer is classified as

*class. We use the maximum of*

*p*

*and*

_{d}*p*

*to compute the certainty*

_{r}*p*

*t*of a transfer.

We now introduced three classification schemes producing different outcomes. In the next section, we measure their performance against one another and calculate the potential benefit using the best of the three schemes.

**54** **Chapter 5** Timing of Train Disposition

**current delay** **catch-up potential** **class**

on time SAFE

small delay possible SAFE

small delay impossible UNCERTAIN

strong delay possible CRITICAL

strong delay impossible BREAK

**Tab. 5.2:** Fuzzy inference rules.