Schedule Analysis and Optimization for Dinky Service

Schedule Analysis and Optimization for Dinky Service

Prof. Alain L. Kornhauser

Jueru Wang

Background

The Princeton Branch, also known as the Dinky line, starts its service since 1865, as downtown Princeton was taken off the mainline when the track was straightened. Dinky was opened to connect at Princeton Junction with the Northeast Corridor Line which serves between New York and Trenton Transit Center. The whole train trip of Dinky is between Princeton Station and Princeton Junction which is only around 2.7 miles and it only takes about 5 minutes for Dinky to travel one-way.

The line was once operated by the Pennsylvania Railroad and the Penn Central Transportation and it was transferred to the New Jersey Department of Transportation in 1976¹. The Princeton University bought the Princeton Station from New Jersey Transit in 1984 and get the right to move the station south at its own cost. Originally located at the south of Blair Hall on the Princeton University campus, the station was finally moved about 200 feet south to the current location².

The Princeton University is now considering about moving the station again in order to support the plan for a new art center in the area adjacent to McCarter Theatre³. The new station will be relocated approximately 460 feet south. The Dinky service will be maintained because from the long run, in this emerging age of train, the Dinky line still has the possibility to be incorporated into future plans for regional train service. If the station is moved, it is possible that the bus rapid transit will be introduced as a supplement to Dinky.

This report will first focus on the current performance of Dinky service on how the schedule of Dinky to meet the trains at Princeton Junction from New York or Trenton Transit Center. The performance is evaluated by the corresponding waiting time for passengers leaving from and coming to Princeton through Dinky. Then some daily operation data will be collected to get the reliability of both the travel time and the operation of the schedule. After the data collection, some recommendation based on the observation will be given to optimize the current schedule.

Introduction

Dinky provides rail service directly to the Princeton University campus from Princeton Junction, where New Jersey Transit and Amtrak trains that go to Newark, New York City, and Philadelphia can be boarded. Princeton Junction is the third station from Trenton. Two different schedules are followed for weekdays and weekends. For weekdays, there are 41 departures in each direction daily. The first train leaves the Princeton Junction Station at 4:50 am and the first one from Princeton leaves the station at 5:00 am. The last train to Princeton is 1:17 am and the last train to Princeton Junction is 1:27 am. For weekends and holidays, there are 29 departures in each direction. The early morning service is not provided during weekends and holidays. The first train leaves after 9 am and the train stops service before 1 am.

The number of the trains from New York to Trenton Transit Center which stop at the Princeton Junction is 49 every weekday, same as those from Trenton Transit Center to New York. The schedule of Dinky is designed to meet most of these trains in order not to let the passenger to wait. On weekends and holidays, the number will be reduced to 32 and, because there is no early morning service of Dinky during weekends, the number of the trains that meet the schedule is further reduced to 25 from New York to Trenton and 27 from Trenton to New York.

Most passengers from Princeton taking the Dinky need to take the eastbound trains from Trenton to the stations to the Newark airport and New York Penn Station. Bus services are provided at Princeton Junction station and there is a parking lot so part of the passengers who get off at Princeton Junction will drive their cars or take the bus services instead of Dinky. For the night services, there are not many passengers from Princeton to Princeton Junction through Dinky and most passengers take the night Dinky to get back to Princeton from both directions.

Analysis of the Schedules

1. 
   Description of Analysis
   

From the website of New Jersey Transit⁴, the general schedules of the Northeast Corridor Line and the daily operational schedules for Dinky are provided. The four schedules are combined together, including the schedules of the Northeast Corridor Line from and to New York and the schedules of Dinky from and to Princeton, to get the corresponding waiting time. The waiting time for passengers leaving from Princeton to New York or Trenton is calculated by the time difference between the arrival time of Dinky and the nearest next arrival time of Northeast Corridor Line at the Princeton Junction. Similarly, the waiting time for passengers coming to Princeton from New York or Trenton is calculated by the time difference between the arrival time of Northeast Corridor Line and the nearest next arrival time of Dinky at the Princeton Junction. The mean, median and variance of the waiting time are also calculated to analyze how well the schedules meet with each other. Although the trains also have a schedule for weekends which is different from the schedule for weekdays, the report will focus on the analysis for weekday trains because not only more trains are operating on weekdays, but also the number of passengers is larger than weekends. One more reason for not including weekend schedule analysis is that the data for weekends are similar to the one for weekdays.

2. 
   Results of Analysis
   

The results of the analysis contain four parts. The overall performance will be shown first to give a general idea. Then, as mentioned before, there are 4 types of connections in our analysis: from New York to Princeton, from Princeton to Trenton, from Trenton to Princeton and from Princeton to New York. So the analysis will be made according to each type. Then, the trains will be further divided into express trains (those that only make limited stops between Princeton Junction and New York) and local trains (all the others).

1. 
   For the all 49 trains during weekdays.
   

mean	Median	variance
12.53	10	92.05

Table 1: Statistical result of the overall performance for all weekday trains

Fig 1: The box plot of the overall performance for all weekday trains

As can be seen from the plot, most waiting times are around 10 minutes and there are some extreme cases that have very long waiting time. The duration between two trains of Northeast Corridor Line departing from the same terminal is around 25 minutes for both ends. There are more trains leaving in the morning from Trenton than those from New York while more trains are leaving from New York during the evening. The possible reason for the extreme cases in the schedule is that not every train from origin has a train matching with it to the destination. For example, there will be one Dinky departing from Princeton Junction at 10:04 am and the next one leaving Princeton Junction will be one hour later. During that 1-hour, there will be 3 trains arriving at Princeton Junction from New York. So for the first train arriving at the station will have the longer waiting time than the following trains since all the passengers from the three trains need to take the same train at 11:04 am.

There is one more situation in the data. That is the zero waiting time cases. In this report, zero waiting time is considered to be possible for passengers. However, it is not desirable to have zero waiting time. Passengers need some time to get off the Dinky and get to the platforms, and therefore, usually zero waiting time means missing the corresponding train unless the corresponding train is delayed.

2. 
   The waiting time are divided according to the type
   

See the appendix 1 for the detailed wait time of different types

	from New York to Princeton	from Princeton to Trenton	from Trenton to Princeton	from Princeton to New York
Mean(unit: min)	10.22	12.75	14.97	12.16
Median(unit:min)	7	11	15	10
var(unit:min^2)	47.63	132.98	113.81	67.29

Table 2: Statistical results of each type

Fig 2: Box plot of Wait time for trains from NY to Princeton	Fig 3: Scatter plot of Wait time for trains from NY to Princeton
Fig 4: Box plot of Wait time for trains from Princeton to Trenton	Fig 5: Scatter plot of Wait time for trains from Princeton to Trenton
Fig 6: Box plot of Wait time for trains from Trenton to Princeton	Fig 7:Scatter plot of Wait time for trains from Trenton to Princeton
Fig 8: Box plot of Wait time for trains from Princeton to NY	Fig 9:Scatter plot of Wait time for trains from Princeton to NY

After the waiting times are divided into 4 types, the distributions of the waiting time become clearer. Generally speaking, the waiting time for passengers from Princeton to New York and the trains from New York to Princeton are shorter and more concentrated than the waiting time for passengers leaving for and from Trenton. The results are mainly based on the design of the schedules for the Northeast Corridor Line. As the table has shown, there is always a 10 – 20 minutes difference for the arrival time for NEC trains at the Princeton Junction from different directions (The time difference is around 10 minutes for an eastbound train after a westbound train while this number increases to around 20 minutes for a westbound train after an eastbound train). And the difference for NEC trains from the same direction is around 20 – 40 minutes which is the same as the frequency of Dinky trains during most time of the day. So the schedule for Dinky trains can always fit one direction better than the other. In this case, the schedule is designed to fit the train from and to New York better because these lines have more passengers.

3. 
   Express trains:
   

mean	median	variance
11.13	7	75.91

Table 3: Statistical result of overall performance of express train

	from New York to Princeton	from Princeton to Trenton	from Trenton to Princeton	from Princeton to New York
Mean(unit: min)	8.56	14.26	10.86	10.80
Median(unit:min)	7	13	6.5	8
var(unit:min^2)	20.16	135.92	97.45	40.96

Table 4: Statistical results of each type

Fig 10: Box plot of Wait time for Express trains from NY to Princeton	Fig 11: Scatter plot of Wait time for Express trains from NY to Princeton
Fig 12: Box plot of Wait time for Express trains from Princeton to Trenton	Fig 13: Scatter plot of Wait time for Express trains from Princeton to Trenton
Fig 14: Box plot of Wait time for Express trains from Trenton to Princeton	Fig 15: Scatter plot of Wait time for Express trains from Trenton to Princeton
Fig 16: Box plot of Wait time for Express trains from Princeton to NY	Fig 17: Scatter plot of Wait time for Express trains from Princeton to NY

The wait time for the express trains from Princeton to Trenton is much longer than the time for other three cases. The results show that most of the waiting times for other three cases are around 10 minutes and there are only a few special cases of long waiting time. Express trains are not designed to stop at every station between New York and New Brunswick so it is reasonable that they have shorter waiting time.

4. 
   Local Trains 
   

mean	median	variance
13.71	10.5	103.41

Table 5: Statistical result of overall performance of local train

	from New York to Princeton	from Princeton to Trenton	from Trenton to Princeton	from Princeton to New York
Mean(unit: min)	11.69	11.42	18.33	13.22
Median(unit:min)	8.5	8	18	10
var(unit:min^2)	68.94	131.77	105.38	87.49

Table 6: Statistical result of each type

Fig 18: Box plot of Wait time for Local trains from NY to Princeton	Fig 19: Scatter plot of Wait time for Local trains from NY to Princeton
Fig 20: Box plot of Wait time for Local trains from Princeton to Trenton	Fig 21: Scatter plot of Wait time for Local trains from Princeton to Trenton
Fig 22: Box plot of Wait time for Local trains from Trenton to Princeton	Fig 23: Scatter plot of Wait time for Local trains from Trenton to Princeton
Fig 24: Box plot of Wait time for Local trains from Princeton to NY	Fig 25: Scatter plot of Wait time for Local trains from Princeton to NY

Compared with the express trains, the local trains have longer waiting times. The schedules are designed to have more waiting time than the express trains because the local trains have to stops at more stations than express trains, and thus, the local trains have more variation in their travel time. Schedule for more waiting time can help the local trains to avoid the cases that passengers may miss the Dinky at Princeton Junction if the local trains arrive later than the schedule time.
Reliability of Arrival time

1. 
   Description of Analysis
   

After the analysis of the current schedule, we need to find out how well the trains follow the schedule, which is the reliability of arrival time. Here delay is defined as the length of time between the scheduled time and the arrival time. If trains arrive earlier than the scheduled time, the delay for such trains will be negative. Only the delays at the Princeton Junction for both westbound and eastbound trains are needed. To get the data, two weeks have been spent on the trains and time is recorded both by hands and a GPS device. However, it is hard for a GPS device to define the stop so there might be a source of error.

2. 
   Results
   

See the appendix 4 and 5 for the detailed arrival time of trains.

	Delay:	W(For next Dinky):	W(Fr previous Dinky):
Mean :(min)	4.95	14.42	17.35
Variance:(min^2)	24.53	56.86	154.74

Table 7: Statistical result of delay for trains from NB
Fig 26: Scatter plot of delay for trains from NB

	Delay:	W(For next Dinky):	W(Fr previous Dinky):
Mean :(min)	1.73	22.12	16.40
Variance:(min^2)	16.54	84.80	111.41

Table 8: Statistical result of delay for trains from Trenton

Fig 27: Scatter plot of delay for trains from Trenton

The overall performance of trains on following the schedules is acceptable. As can be seen from the figures, the delays of trains from Trenton, most of which are shorter than 5 minutes, are much shorter and more stable than those of trains from New York, which are around 5 minutes. The results are reasonable since Trenton is one end of the Northeast Corridor Line and there is only one stop between Trenton and Princeton Junction. But for trains from New York, there are many stops before the trains reach Princeton Junction and therefore the delay is accumulated through the previous trip. There are also some extreme cases of long delay which is the result of the cancellation of trains due to bad weather.

Reliability of Travel time

1. 
   Description of Analysis
   

Dinky is now operating under a fixed time schedule. If a demand responsive schedule is desired, the travel time of trains from other stations to Princeton Junction is needed because in a demand responsive schedule, Dinky’s operation time depends on the situation of Northeast Corridor Line trains at next station instead of a fixed time. The travel time between New Brunswick station and Princeton Junction has been recorded instead of the travel time between New York Penn Station and Princeton Junction for two reasons. The first reason is the travel time from New York is too long, which is not convenient for data collection. Another reason is that the travel time from New York varies more since trains have to stop at more stations. A relatively stable travel time is needed to develop a demand responsive schedule. The travel time is calculated by the time difference between the arrival time and the departure time from last stop. The time is recorded both by hand and a GPS device. So there still exists the problem of how a GPS device defines the tart point and stop point.

2.