Research questions

In this simulation study, we will consider the Uithoflijn, the new tram line between the central station of Utrecht and de Uithof which will start operating in the future. (see also We will study the operational performance of this line. You have to answer the following questions:

Performance measures

Important performance measures are:

On weekdays, trams will be driving between 6 AM and 9:30 PM at a constant frequency. To obtain stable measurements, you have to use the performance between 7 AM and 7 PM. You also have to report the performance measures during the morning peak from 7:30 AM until 9:30 AM.

Problem description

Passenger model

You may assume that the passengers arrive at the stops according to a Poisson Process from which the rate varies over the day. This rate is determined per 15 minutes.

For the Uithoflijn, we do not have passenger counts, but we can predict them using the data for bus route 12 and the passenger prognosis from province of Utrecht. This information is enclosed in the set of data files.

When a tram arrives at a stop, first passengers exit the tram and after that passenger enter the tram.You may assume that passengers enter the tram in a FIFO order, i.e. passengers enter the tram in order of arrival time at the stop.

The capacity of the tram is 420 passengers. If the tram is full, passenger that want to enter wait for the next tram. Passengers that came to a stop to take the tram will never leave because of a long waiting time. The stops do not have capacity limits.

Driving times

The Uithoflijn consists of 7.9 km of track, with 9 stops. In the following tables, a list of all stops are given. In these tables, some more information is given. Because the tram follows more or less the same route as bus route 12, the corresponding stops are also given. In the last column of each table, the average driving time between each pair of stops is given. Note that this driving time is pure driving time, without the dwell time, where the dwell time is the time that a tram is standing still at the stop.

Driving times suffer from variation and should therefore be modelled as stochastic variables. We assume that the probability distribution can be based on the driving times from the SUNIj line (line to Nieuwegeid and Ijsselstein). These driving are available is the set of data files.

We assume that all variation is included in the stochastic model and do not model other disturbances during driving.

Stop Corr. stop route 12 Distance(km) Avg. Driving time(sec)
Centraal Station CS Centrumzijde
Vaartsche Rijn Bleekstraat 1.4 134
Galgenwaard Galgenwaard 3.1 243
Kromme Rijn De Kromme Rijn 0.6 59
Padualaan Padualaan 0.8 101
Heidelberglaan Heidelberglaan 0.4 60
UMC AZU 0.4 86
WKZ 0.6 78
P+R De Uithof 0.6 113
Table 1: Route information Uithoftram Centraal Station to Uithof

Stop Corr. stop route 12 Distance(km) Avg. Driving time(sec)
P+R De Uithof
WKZ 0.6 110
UMC AZU 0.6 78
Heidelberglaan Heidelberglaan 0.4 82
Padualaan Padualaan 0.4 60
Kromme Rijn De Kromme Rijn 0.8 100
Galgenwaard Galgenwaard 0.6 59
Vaartsche Rijn Bleekstraat 3.1 243
Centraal Station CS Centrumzijde 1.4 135
Table 2: Route information Uithoftram Uithof to Centraal Station


Qbuzz (U-OV) works with a timetable for the departure at the end stations. At an end station, a tram will not depart before the departure time given in the timetable. This timetable is defined as follows. We first fix our frequency, suppose for example that this is 15 (trams per hour). We assume that the frequency is fixed during the complete day, i.e. from 6.00 to 21.30. Than we fix the departure times at P+R de Uithof as

In the timetable we use a one-way driving time of 17 minutes. Each timetable is based on a specific planned turn-around time of q minutes at the endstations. Suppose that we choose q=5 minutes. Then a tram that leaves P+R at t, leaves CS at t+17+5=t+22 and then P+R again at t+44, etc. Consequently the departure times at CS will be: We assume that there are always enough trams to execute the timetable.

Stations: intermediate stops

At the intermediate stops there is a platform at each side of the tracks. A tram can not switch track but stops and leaves after the exit and entry of passengers is finished.

A tram can only enter an intermediate stop, when the previous tram in the same direction has left at least 40 second ago.

The time that a tram is standing still at the stop is called dwell time. The dwell time is modelled as a stochastic variable and depends on the number of passengers leaving and entering the tram. The average dwell time d is given by d = 12.5 + 0.22 p(in) + 0.13 p(out), where p(in) and p(out) are the number of passengers entering and leaving the tram, respectively. For the distribution of the dwell time, we use a gamma distribution with shape parameter k=2 and a minimum value of 0.8d. The tram dwells at every stop.

Stations: end points

The end point CS and P+R De Uithof have a special layout. At each end point there are two parallel tracks with a platform in between. At the end point, all passengers in the tram get out and at the same platform new passengers get in. In the mean time, to switch driving direction, the driver has to walk to the driver's cab at the other side of the tram. This is all included in the turn-around time. The timetable includes a planned turn-around time q. Technically, the minimum turn-around time is 3 minutes. When the tram arrives at the end station, it tries to depart at the time of the next `unexecuted' trip in the timetable. However, if the next `unexecuted' trip in the time table is within 3 minutes (so e.g. in 2 minutes), this would imply that the actual time for turn-around becomes shorter than 3 minutes. Since this is not allowed, the tram leaves after exactly 3 minutes.

Before the platform section there is a so-called `cross-switch' through which the trams can move to another track. We model the switch as a server with a processing time of one minute, a switch can process one tram at a time. There is one exception, two trams that do not cross can use the switch at the same time. Arriving trams have priority over departing trams. A schematic layout is given in Figure 1. Note that here multiple interpretations are possible: 1) What we originally meant and happens in practice, is that a tram can only start crossing the switch after it has been empty for 1 minute, but crossing the switch itself takes hardly any time. 2) Another explanation that (understandably) has been chosen is that the tram physically occupies the switch for 1 minute. So crossing the switch takes 1 minute. (In the grading both alternatives will be considered to be correct)

Figure 1: Layout end station.

Tram depot

During the night, trams are located at a depot which is connected to the P+R end station. The trams leave in the morning to start up the timetable and return in the evening after the timetable is finished. Trams are allowed to drive without passengers at the start or end of the days. Since the frequency is fixed over the day, trams do not move to and from the depot during the day. If there are delays during the day, this cannot be solved by taking additional trams from the depot.

Optional features

Additional research questions:

Data for the Simulation

You find files that contains data that you will need for the tram simulation through this link (you will get the password separately). These files contain: