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
Walking	as bike	as bike	as bike
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. PBL has not provided parameters for walking; walk_a, walk_b and walk_c are set to the bike values until it does.

Since 25 September 2026 each mode uses its own parameter set: the travel time template (Calc_Traveltimes_T) receives the mode ('walk' or 'bike', the e-bike uses the bike set). Before, walking, cycling and e-bike accessibility were all computed with the car parameters (audit 3.2), so decayed accessibility indicators of those modes from earlier runs are not comparable with new ones.

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 and e-bike
walk_a, walk_b, walk_c	ModelParameters/Advanced/Distance_Decay	Decay shape parameters for walking (for now the bike values)
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.