6.3 Reliability of Transfers
6.3.1 Reliability Measure
Consider two otherwise equivalent connectionsaandbwith 2 interchanges each. Connec-tionahas 5 minutes buffer time for each interchange, connectionbhas 9 minutes buffer time for the first and 1 minute for the second interchange. Although the sum is identical, connectionaseems somehow more secure. If any one of its trains is delayed by at most 5 minutes, there is no problem. But if the second train for connectionbis only 2 minutes late, the connection breaks. It gets even more complicated if we compare connections with different numbers of interchanges. Connection a from above might be considered safer than a connection with just one train change and 0 minutes buffer.
ivWe will formally define the buffer time later.
6.3 Reliability of Transfers 59 6.3.1.1 Definition of the Buffer Time for a Single Interchange
To obtain a confidence measure for the reliability of transfers there are a number of factors to consider: the excess time when changing trains, empirical values for delays at the station, the number of passengers entering the connecting train / leaving the feeding train, the current situation at neighboring stations, condition of the tracks, . . .
As most of these are not easy to observe, impossible to estimate, and/or simply not available beforehand, we opted to use the availablebuffer timeto base our calculation on.
Apart from the arrivalarrs(f) and departure timedeps(t) of the feeding trainf and the connecting train t at stationswe need the change time cts(f, t) to define the buffer time. Time cts(f, t) is the time needed to walk from train f to train t. It depends on the station layout (number of platforms, distances in the station) and the platforms the trains stop at (for more details see Section 1.1). In the basic case we can determine the buffer time
buf :=deps(t)−arrs(f)−cts(f, t).
Scenario with delays In a scenario with delay information additional values determine the buffer time. We have the minimum change time minct(f, t) that is applicable if a train arrives late. In this case announcements in the train or even railway personnel guiding passengers can make it possible to change faster. The change time can often be decreased by up to two minutes. Furthermore, connecting trains may wait for the arrival of feeding trains according to a set of waiting rules (cf. Chapter 7.2.2). We can increase the buffer time by the number of additional minutes available, e.g. if t would wait for trainf at stationsup towts(t, f) minutes and is alreadyxminutes late we have wait+s(t, f) = max{wts(t, f)−x,0} additional minutes. If train t waits for train f, it departs not later thandepscheds (t) +wts(t, f). The additional waiting time is zero if train t has a higher delay thanwts(t, f) on its own.
For an interchange i from train t to f with ∆ := dep(t)−arr(f) for the (possibly delayed) event times, we now get the buffer time:
buf(i) :=
∆−cts(f, t) +wait+s(t, f) ifcts(f, t)≥∆
0 +wait+s(t, f) ifmincts(f, t)≤∆< cts(f, t)
⊥ if ∆< mincts(f, t)
For the first case we simply apply the rules from above. When a train arrives too late to use change timects(f, t), we switch to using the minimum change time. In that case, we do not want to gain additional buffer time in our model by defining buf(i) as
∆−mincts(f, t) > 0. Therefore, the buffer time is set to zero in the second interval.
In the last case the interchange will break as the train would have to wait more than wts(t, f) minutes to make the interchange possible (as train t arrived later than time depsched(t) +wts(t, f)−mincts(f, t)).
6.3.1.2 Reliability Rating
We assume we have a buffer time for each of the interchanges of a connection and want to determine the reliability of transfers for the whole connection.
In his Bachelor thesis Kai Mehrungsk¨otter [Meh07] investigated evaluation functions to determine the reliability of transfersrel(c) for the whole connectioncwith interchanges i1, . . . , ik:
Min Obtaining the reliability measure from the minimum over the buffer times of all interchanges rel(c) := minj(buf(ij)) is not sufficient, e.g. a connection with two interchanges, with 1 and 20 minutes buffer, is regarded as equally reliable as a connection with 1 and 2 minutes buffer time.
Sum Defining the reliabilityrel(c) :=P
j(buf(ij)) as the sum of the buffer times is not a good measure, e.g. two interchanges with 3 minutes of buffer each are considered better than only one interchange with 5 minutes buffer.
Arith If we define the reliabilityrel(c) :=
P
j(buf(ij))
k as the average of the buffer times, it fixes the previous example, but as we disregard the number of interchanges and the distribution of buffer times, this is still not enough. For example, two interchanges with 1 and 9 or 4 and 6 as well as one interchange with 5 minutes buffer, all evaluate to the same result.
To overcome all these drawbacks we decided to calculate a probability factorsec(ij) for each interchangeij and to determine the reliability as
rel(c) :=
k
Y
j=1
(sec(ij)).
6.3.1.3 Reliability-Functions for an Interchange
Now we want to determine the security factor sec(ij) for a single interchange ij with a buffer of x = buf(ij). We need some preliminary considerations. According to the schedule construction a railway company does not consider a buffer time of zero minutes as an endangered or impossible interchange. So we cannot set the security factor for zero minutes to 0% but have to choose a much higher value instead, e.g.η= 60%. From the introductory example it is clear that a connection without interchanges should obtain the highest possible reliability measure of 100%. Connections with at least one interchange can obtain a reliability of at mostµ <1, e.g.µ= 99%. On the other hand we do not want to impose a buffer of say at least 100 minutes to reach the highest possible reliabilityµ.
Here we could chooseθ = 45 minutes (as within 60 minutes another train operates the same route for most frequencies).
With this consideration we can discuss the following examples:
linear With a linear function
sec(x) :=
½ µ−η
θ x+η : 0≤x≤θ
µ :otherwise
the increase of the buffer by δminutes increases the security factor by µ−ηθ δ. This is clearly not intended, as the increase in reliability from 0 to 5 minutes should be much higher than that from 30 to 35 minutes.
discrete For interval boundsx0= 0, . . . , xn andy0=η < y1 < . . . < yn=µa discrete function
sec(x) :=
½ yj :xj≤x < xj+1 for some j µ :x > xn
requires too many intervals and guessing or fine-tuning too manyyj. Additionally, it is not strictly monotonically increasing, so a small change in buffer time either changes nothing or jumps fromyj toyj+1.
6.3 Reliability of Transfers 61
Parameters
η 60%
µ 99%
θ 45min
α 8
x sec(x)
0 60%
1 65%
4 75%
6 80%
8 85%
x sec(x)
12 90%
13 91%
14 92%
15 93%
17 94%
x sec(x) 19 95%
21 96%
24 97%
30 98%
45 99%
Table 6.1: The parameters for the reliability functionsec(x) and sample values for some buffer timesx.
piecewise linear For a piecewise linear function we need interval bounds and constant slopesaj, with a0 > ai > . . . an in the intervals we again have to determine and adjust too many values for intervals and slopes.
exponential Addressing all the concerns from above we use an exponential function sec(x) =µ−eln(µ−η)−1αx.
The formula evaluates to values in the range [η, µ], as we have
x→∞lim sec(x) = lim
x→∞µ−eln(µ−η)−α1x=µ−0 =µ for largex, and forx= 0:
sec(0) =µ−e(lnµ−η)−α10=µ−eln(µ−η)=µ−(µ−η) =η.
We only have to fix the parameter αdefining the steepness. A nice choice seems to beα= 8, which results in the values given in Table6.1, e.g. it gives us for our
“long waiting time”θa reliability ofsec(θ) =µ−0.0014(≈µ) which rounds toµ.
Thus, we selected the exponential function to calculate the reliability for a single inter-change. Different sets of parameters have been evaluated in [Meh07] and we finally agreed on the parameters in Table 6.1(left).