A Transit graph for static macro assignment

This page is a description of a graph structure for a transit network, used for static, link-based, frequency-based assignment. For the sake of clarity and conciseness, we won't go over all the details of the transit graph features.

Let's start by giving a few short definitions:

• transit definition from Wikipedia:

system of transport for passengers by group travel systems available for use by the general public unlike private transport, typically managed on a schedule, operated on established routes, and that charge a posted fee for each trip.

• transit network: a set of transit lines and stops, where passengers can board, alight or change vehicles.

• assignment: distribution of the passengers (demand) on the network (supply), knowing that transit users attempt to minimize total travel time, time or distance walking, time waiting, number of transfers, fares, etc...

• static assignment : assignment without time evolution. Dynamic properties of the flows, such as congestion, are not well described, unlike with dynamic assignment models.

• frequency-based (or headway-based) as opposed to schedule-based : schedules are averaged in order to get line frequencies. In the schedule-based approach, distinct vehicle trips are represented by distinct links. We can see the associated network as a time-expanded network, where the third dimension would be time.

• link-based: the assignment algorithm is not evaluating paths, or any aggregated information besides attributes stored by nodes and links. In the present case, each link has an associated cost (travel time) c [s] and frequency f [1/s].

We are going at first to describe the input transit network, which is mostly composed of stops, lines and zones.

Transit stops and stations

Transit stops are points where passenger can board, alight or change vehicles. Also, they can be part of larger stations, where stops are connected by transfer links.

In this figure, we have two stops : A and B, which belong to the same station (in red).

Transit lines

A transit line is a set of services that may use different routes.

Transit routes

A routes is described by a sequence of stop nodes. We assume here the routes to be directed. For example, we can take a simple case with 3 stops:

In this case, the L1 line is made of two distinct routes:

• ABC
• CBA.

But we can have many different configurations:

• a partial route at a given moment of the day: AB,
• a route with an additional stop : ABDC
• a route that does not stop at a given stop: AC

So lines can be decomposed into multiple "sub-lines" depending on the distinct routes, with distinct elements being part of the same commercial line. In the above case, we would have for example:

line id	commercial name	stop sequence	headway (s)
L1_a1	L1	ABC	600
L1_a2	L1	ABDC	3600
L1_a3	L1	AB	3600
L1_a4	L1	AC	3600
L1_b1	L1	CBA	600

The headway is associated to each sub-line and corresponds to the mean time range between consecutive vehicles. It is related to the inverse of the line frequency. The frequency is what is used as link attribute in the assignment algorithm.

Line segments

A line segment is a portion of a transit line between two consecutive stops. With the previous example line L1_a1, we would get two distinct line segments:

line id	segment index	origin stop	destination stop	travel_time (s)
L1_a1	1	A	B	300
L1_a1	2	B	C	600

Note that we included a travel time for each line segment. This is another link attribute used by the assignment algorithm.

Transit assignment zones and connectors

In order to assign the passengers on the network, we also need to express the demand in the different regions of the network. This is why the network area is decomposed into a partition of transit assignment zones, for example into 4 non-overlapping zones:

Then the demand is express as a number of trips from each zone to each zone: a 4 by 4 Origin/Destination (OD) matrix in this case.

Also, each zone centroid is connected to some network nodes, in order to connect the supply and demand. These are the connectors.

We now have all the elements required to describe the assignment graph.

The Assignment graph

Link and node types

The transit network is used to generate a graph with specific nodes and links used to model the transit process. Links can be of different types:

• on-board
• boarding
• alighting
• dwell
• transfer
• connector
• walking

Nodes can be of the following types:

• stop
• boarding
• alighting
• od
• walking

Here is a figure showing how a simple stop is described:

The waiting links are the boarding and transfer links. Basically, each line segment is associated with a boarding, an on-board and an alighting link.

Transfer links appear between distinct lines at the same stop:

They can also be added between all the lines of a station if increasing the number of links is not an issue.

walking links connect stop nodes, while connector links connect the zone centroids (od nodes) to stop nodes:

Connectors that connect od to stop nodes allow passengers to access the network, while connectors in the opposite direction allow them to egress. Walking nodes/links may be used to connect stops from distant stations.

Link attributes

Here is a table that summarize the link characteristics/attributes depending on the link types:

link type	from node type	to node type	cost	frequency
on-board	boarding	alighting	trav. time	$\infty$
boarding	stop	boarding	const.	line freq.
alighting	alighting	stop	const.	$\infty$
dwell	alighting	boarding	const.	$\infty$
transfer	alighting	boarding	const. + trav. time	dest. line freq.
connector	od or stop	od or stop	trav. time	$\infty$
walking	stop or walking	stop or walking	trav. time	$\infty$

The travel time is specific to each line segment or walking time. For example, there can be 10 minutes connection between stops in a large transit station. A constant boarding and alighting time is used all over the network. The dwell links have constant cost equal to the sum of the alighting and boarding constants.

We can use more attributes for specific link types, e.g.:

• line_id: for on-board, boarding, alighting and dwell links.
• line_seg_idx: the line segment index for boarding, on-board and alighting links.
• stop_id: for alighting, dwell and boarding links. This can also apply to transfer links for inner stop transfers.
• o_line_id: origin line id for transfer links
• d_line_id: destination line id for transfer links

Next, we are going see a classic transit network example with only four stops and four lines.

A Small example : Spiess and Florian

This example is taken from Spiess and Florian [1]:

Travel time is indicated on the figure. We have the following four line characteristics:

line id	route	headway (min)	frequency (1/s)
L1	AB	12	0.001388889
L2	AXY	12	0.001388889
L3	XYB	30	0.000555556
L4	YB	6	0.002777778

Passengers want to go from A to B, so we can divide the network area into two distinct zones: TAZ 1 and TAZ 2. The assignment graph associated to this network has 26 links:

Here is a table listing all links :

link id	link type	line id	cost	frequency
1	connector		0	$\infty$
2	boading	L1	0	0.001388889
3	boading	L2	0	0.001388889
4	on-board	L1	1500	$\infty$
5	on-board	L2	420	$\infty$
6	alighting	L2	0	$\infty$
7	dwell	L2	0	$\infty$
8	transfer		0	0.000555556
9	boarding	L2	0	0.001388889
10	boarding	L3	0	0.000555556
11	on-board	L2	360	$\infty$
12	on-board	L3	240	$\infty$
13	alighting	L3	0	$\infty$
14	alighting	L2	0	$\infty$
15	transfer	L3	0	0.000555556
16	transfer		0	0.002777778
17	dwell	L3	0	$\infty$
18	transfer		0	0.002777778
19	boarding	L3	0	0.000555556
20	boarding	L4	0	0.002777778
21	on-board	L3	240	$\infty$
22	on-board	L4	600	$\infty$
23	alighting	L4	0	$\infty$
24	alighting	L3	0	$\infty$
25	alighting	L1	0	$\infty$
26	connector		0	$\infty$

References

[1] Heinz Spiess, Michael Florian, Optimal strategies: A new assignment model for transit networks, Transportation Research Part B: Methodological, Volume 23, Issue 2, 1989, Pages 83-102, ISSN 0191-2615, https://doi.org/10.1016/0191-2615(89)90034-9.