A risk of delay analysis is carried out subsequently to timetable-based assignment. Delays in integers are read out in seconds.
A vehicle journey item f is relevant for analysis when the actual likelihood of punctuality is smaller than one. Each relevant vehicle journey item f is considered separately. For the analysis, all connections Vi∈V that include a transfer or an alighting at destination are considered.
Determining delay situations
For each vehicle journey item f delay situations are determined. A delay situation is an interval d, for which all delays of f in d have the same impact on the subsequent course of the connections. The probability of occurrence P during a delay situation d is derived from the interval limits. The latter are calculated by applying exponential distribution with the parameter λ, multiplied by the probability of delay.
P(risk of delay d occurs) = P(arrival in d = [a,b]) = 
A search tree is created to calculate the delay situations of a vehicle journey item f. For this purpose, the transfers following f are considered. The transfers following f have a wait time. If f is less or as much delayed as the wait time, the transfer can still take place. If f is delayed more than the wait time, the transfer cannot be made. Let w1, w2,..,wj be wait times of the transfers uk from f to different vehicle journey items of all connections Vi that contain f in ascending order. Then interval d1= (0,w1] is the delay situation of f, where no transfer is missed. For a delay w1+1, at least one of the transfers is missed. For all affected connections from f onwards, alternative connections are searched for via the Branch and Bound search, so that the passenger either remains seated in f or switches to a later departing journey or walks to the destination zone. In any case, the arrival time of the connection at the destination zone then changes by a time period of ∆t. This change in arrival time depends on the arrival times of the alternative journeys found. There is also a minimum wait time w‘2 for the alternative connections, by which f can be delayed without missing one of the new transfers. This results in delay situation d2 = (w1,min(w2,w‘2)] for the vehicle journey item under consideration. Thus, by searching for alternative connections in case of missed connections, all relevant delay situations D can be calculated successively. If the maximum delay tmax is exceeded, the search is terminated.
Illustration 168: Delay situations with wait time
Illustration 168 is described below: The wait times w i for the transfers at the stop station are shown together with the resulting delay situations di for the vehicle journey item f under consideration. All connections containing f start at A Village and end at X City.
- d1=(0,w1] In this interval, the connecting journey Vehicle journey 2 Train can still be reached.
-
If the first connecting journey is missed, the alternative connection is found by continuing the journey to B-Village and switching connections there. This can be done up until the delay 
of f. -
Once a delay time of 
is reached, it is worthwhile switching at the Station again and taking Vehicle journey 3 Train. This connecting journey is reached up to a delay of w1+w2. -
In this delay situation, it is worthwhile switching at the Station again to Vehicle journey 4 Train in order to arrive at X-City the earliest possible.
Calculating the risk of delay
a) At connections and their transfers
A choice is calculated for the alternative connections of a delay situation, so they can be assigned a volume. For each connection and delay situation, the change in travel time ∆t is calculated as a weighted mean from the volumes of the alternatives and their change in travel time. If no alternative connections were found for a delay situation d, the input parameter of the assumed travel time extension 
̅ is set for ∆t if no alternative connection is found.
The risk of a transfer u after vehicle journey item f of a connection V then results from the product of "probability of the delay situation" and the resulting "travel time extension".
The risk also includes a term ε which represents the delay of delay situations no longer examined. Here the maximum across all possible delays and the expected value is multiplied by the remaining probability of occurrence. As this calculation only serves a rough estimate, the termination conditions should not be set too strictly. Therefore, among other things, the maximum analyzed delay tmax should not be chosen too small.
b) At the alighting destination
The arrival time of the connections does not only change for the passengers transferring after vehicle journey item f, but also for those passengers alighting after f, who from there will walk to the destination zone. In this case, there is no need to search for alternative connections. Their change in travel time always corresponds exactly to the delay of f. The risk for an alighting destination a after f of a connection V therefore results from:
The total of risks of all connections using this alighting destination, multiplied by the volume of the respective connection, equals the total delay risk of an alighting destination Ra.
To calculate the risk of transfer or alighting destination per person, the sum of the total volumes of all affected connections is divided.
c) Risk of delay of connections
The risk of a delayed connection per person is calculated as the total of all transfers occurring in the connection and the risk of delay at the alighting destination.
The total risk of delay is calculated by multiplying this by the total volume of the connection.
d) Risk of delay of transfers
The total risk of delay for a transfer u from vehicle journey item f to vehicle journey item e can be calculated by summing up all connections containing the transfer u, multiplied by the volume of the respective connection.
Share of OD trips with relevant delay
a) Relevant delay of a connection
The proportion of passengers AVfu with a relevant delay of a connection V after vehicle journey item f at transfer u is calculated as follows:
The proportion of passenger journeys with a relevant delay AVfa, with an alighting destination after f, is inverse to the probability that f arrives later than the relevant delay time:
A connection V is relevant if it contains at least one vehicle journey item whose delay at a transfer or alighting destination leads to a relevant delay of the connection. The proportion of delayed passenger trips AV is therefore inverse to the proportion where there is no relevant delay for any of the transfers or alighting destinations.
b) Relevant delay of a transfer
The share of passenger journeys with a relevant delay for a transfer u is calculated as a weighted mean over the individual values of the connections that include this transfer.
Passenger journeys with a relevant delay
To calculate absolutely delayed passenger journeys, multiply the proportions of the delayed passenger journeys by the volume of the respective item (connection, transfer, alighting).
Influence of planned connections on delay situations
During the operation of a timetable, generally more connections are realized than are calculated via delay analysis with Visum. This is due to the fact that connections often wait for a previous connection. This behavior can be modeled by creating planned connections (Using Visum: Managing planned connecting journeys). Planned connections can be created between two subsequent vehicle journey items. A planned connection requires a maximum wait time for the previous vehicle journey item or a connection probability. These parameters are then used to adjust the probabilities of the delay situations.
If delay situations d1,…,dn have been determined for vehicle journey item f and connection probabilities or maximum wait times of the connections under consideration are given, these are taken into account in a second step. In these cases, the delay situations must be adjusted, as connections can still be reached after their planned departure.
For a connection 
between vehicle journey items f and g let there be a connection probability Preach. In this case, the probability of arising delay situations must be checked and adjusted if required. All delay situations d1,…,dk whose probabilities together are less than or equal to connection probability Preach are set to zero and 〖P(d〗_1) is increased to P_reach. This procedure reflects the fact that passengers are more likely to reach the planned connection by waiting in order to arrive without delay. Delay situations d2,…,dk, together with P(d1), are less likely to occur than the connection probability and therefore cannot occur. If there is a delay situation dk for which P(d1)+P(d2)+⋯+P(dk-1) < Preach and P(d1)+P(d2)+⋯+P(dk) > Preach applies, the probability of occurrence is reduced according to P(dk). An example of how the probabilities are adjusted is given in Table 195. It is based on the connection in Illustration 168. If a maximum wait time for the planned connection is given instead of a connection probability, the calculation is based on the wait time. In this case, the exponential distribution of the arriving vehicle journey item is used.
If a connection probability and a maximum wait time are given for a planned connection, only the connection probability is used for the calculation.
|
Delay situation d1 |
d1 |
d2 |
d3 |
d4 |
d5 |
|
Probability of occurrence P(di) |
0.3 |
0.1 |
0.1 |
0.05 |
0.05 |
|
New calculated probabilities of occurrence with connection probability Preach = 0.45 |
0.45 |
0 |
0.05 |
0.05 |
0.05 |
Table 195: Adjustment of connection probabilities of delay situations of the planned connection from f to vehicle journey item 3 of Illustration 168
