Decay

Alongside OD-matrices, the model can compute an accessibility indicator for each origin: a single value that summarises how many destinations of interest are reachable, weighted by how far away they are. This is commonly called a gravity-based or distance-decay accessibility measure.

What is distance decay?

The further away a destination, the less likely it is that someone will actually travel there. Distance decay formalises this by applying a weight to each destination that decreases with travel time or distance. The result, often called D_i, is the total weighted count of destinations reachable from origin i:

$$D_i = \sum_j w_j \cdot f(t_{ij})$$

where $w_j$ is the "mass" of destination $j$ (e.g. number of jobs), $t_{ij}$ is the travel time from $i$ to $j$, and $f$ is the decay function.

The decay function

The model uses a log-logistic (dist_logit) decay function with three parameters:

$$f(t) = \frac{1}{1 + e^{a + b \cdot \ln(t) + c \cdot t}}$$

The parameters a, b, and c control the shape of the curve: where it peaks, how steeply it falls off, and whether it has a long tail. They are estimated empirically from observed travel behaviour and differ by mode:

Mode	a	b	c
Car	−8.658	2.492	0.01164
Bike	−7.957	2.675	0.01198
Public transport	−12.330	2.908	0.01282

These parameters are defined in ModelParameters/Advanced/Distance_Decay and were calibrated by PBL based on travel survey data. They can be adjusted if a different decay shape is needed for a specific analysis.

How it is computed

The decay calculation is integrated into the routing step itself, using the dijkstra_m64 function with the interaction(v_i, w_j, dist_logit(alpha, beta, gamma)) option. This means the weighted sum is computed on-the-fly during routing, without first materialising the full OD-matrix. The output is D_i per origin: the gravity-weighted accessibility score.

For the doubly-constrained variant (used in the legacy congestion model), a balancing factor C_j per destination is also computed, ensuring that the total weighted demand adds up consistently. In the standard accessibility output, only the singly-constrained D_i is exported.

OD-matrix vs. accessibility

The model offers two complementary output types:

Output	What it contains	When to use
OD-matrix	Travel time (and optionally distance) for each origin–destination pair	When you need pair-level data for further modelling, e.g. modal split or assignment
Accessibility (D_i)	One value per origin: weighted sum of reachable destinations	When you want a spatial indicator of how well a location is served

Both can be exported simultaneously. The OD-matrix is controlled by the ExportTable_Traveltimes container; the accessibility indicator by ExportTable_Decayed. Which is included in the output depends on the configured export parameters.

Parameters

Parameter	Location	Description
car_a, car_b, car_c	ModelParameters/Advanced/Distance_Decay	Decay shape parameters for car
bike_a, bike_b, bike_c	ModelParameters/Advanced/Distance_Decay	Decay shape parameters for bike
ov_a, ov_b, ov_c	ModelParameters/Advanced/Distance_Decay	Decay shape parameters for public transport
items_of_interest	ModelParameters/Advanced	The destination weight attribute (e.g. nr_banen for jobs). Determined automatically from the chosen DestSet.
MaxCarTime / MaxCyclingTime_Org2Dest	ModelParameters	Maximum travel time cutoff. Destinations beyond this are excluded entirely.