### What is a Distribution Model?

A distribution model produces a new origin-destination trip matrix to reflect new trips in the future made by population, employment and other demographic changes so as to reflect changes in people's choice of destination. They are used to forecast the origin-destination pattern of travel into the future and produce a trip matrix which can be assigned in an assignment model of put into a mode choice model. The trip matrix can change as a result of improvements in the transport system or as a result of new developments, shops, offices etc and the distribution model seeks to model these effects so as to produce a new trip matrix for the future travel situation.

### Input Future Year Trip Ends

To reflect new levels of trip making in the future or to reflect a changes to the disposition of shops, offices, new developments etc, the distribution model needs data about the number pf trips generated by each zone and number of trips attracted to each zone. These are input as a set of 'trip ends'. A set of trip ends is obtained from a trip matrix by summing the row totals to give the number of trips coming out of each zone (called the origin trip ends for that zone) and by summing the column totals for each zone to give the number of trips going to each zone (called the destination trip ends for that zone). Each zone therefore has an origin and a destination trip end. If these were obtained from the base year trip matrix then they are the base year trip ends

To reflect future levels of trip making, the base year trip ends have to be changed to be future year trip ends and the future year trip ends have to be input into the distribution model so as to give the future year trip matrix. Making the future year trip ends from the base year trip ends is the job of the trip end model but in simple terms if you know that travel is growing at 7% per annum compound growth then travel levels will double every 10 years. So if our base year is the year 2000, we can take our base year trip ends and forecast the trip ends for the year 2010 by multiplying each trip end by a growth factor of 2.

These future year trips ends can be put into the distribution model to give a trip matrix for the year 2010. In practice you will find that some areas will grow more than others and that some zones will be developed with new houses shops, offices etc while other zones won't be. You can adjust the growth factor for each zone (or set of zones) to reflect the differential growth and this differential growth is reflected in the new origin-destination pattern given by the trip matrix output from the distribution model.

### The Furness Distribution Model

The future year trip matrix can be derived from the base year trip matrix so that its row and column totals match the future trip ends. The simplest model for doing this is the Furness distribution model. This operates as shown below where a cell of the matrix is the number of trips from one origin zone to one destination zone as follows:

1. The base year matrix cells for one row are multiplied by the growth for that zone and all rows are done in turn. The matrix so obtained will have its origin trip ends matching the future year origin trip ends which is what we wanted to achieve, however the column totals will not in general match the future year destination trip end so:

2. The matrix cells for each column are multiplied by the ratio of the future year destination trip end to the column total achieved in 1 above so that the resulting matrix will have its column total matching its future year destination trip end.

3. However its row total will not generally match its future year origin trip end so steps 1 and 2 are repeated successively until the row and column total are both close to the future year origin and destination trip ends. The process stops when they are close enough (eg to within a few trips).

The Furness model generally converges quite quickly The Furness mathematics has been explored in research and this has shown that if the matrix cells are all greater than zero then it converges to a unique solution. It does generally work if some cells are zero and can work where some are negative too. It is frequently used in transport modelling and even if more sophisticated distribution models are used. quite often Furness is used for the external to external movements or for forecasting goods vehicles and freight.

There are two primary weaknesses of Furness. The first weakness is that if a cell in the matrix is zero, no matter how much it is factored it always remains zero. Quite often a zone would have few trips in the base year because it does not have many people living there, nor jobs nor shops etc so not many trips come from or go to there. Whereas in the future the zone may be fully developed with houses, shops, factories etc. No matter how many trips are forecast to originate or be destined for it, if most of the zone's cells are zero in the base year, with Furness, they will remain zero in the future. One method for getting round this is to 'seed' all the zero cells with a value (eg one trip, or to assume a distribution of trips from the zone in question to every other zone and from every other zone to the zone in question ). The resulting matrix for this zone is therefore dependent on the input assumptions and it puts the onus on the modeller to get the seeding right.

The second weakness of the Furness is that it is not sensitive to changes in the transport system. It is known that if the transport system is improved people will adjust their choice of destination to make the most of additional destination opportunities which have suddenly become much more accessible. For example if a new motorway is built which now connects you to a superstore, you are more likely to use the new superstore. The same goes for job or education opportunities or you may move house so as to have better job opportunities made available by the new transport infrastructure. The same goes for bus and rail. In Victorian times the (then new) railway opened up holiday and travel destinations for people like never before. This second weakness is more insurmountable and one generally adopts a more sophisticated model such as the gravity model to overcome this weakness. It also overcomes the first weakness. (Although a variant of Furness was developed called Time Function Iteration which took the distance or generalised cost between the origin and destination zone as a starting point, from which to apply Furness's row and column balancing - which is a step towards the gravity model.)

### The Concept of Generalised Cost

To be sensitive to the separation between the origin and the destination, we need a measure of this 'separation'. It needs to take account of the transport system, where it is not just distance that is important but other variables too are important and the common ones being: travel time (usually termed in-vehicle time or ivt) fare, waiting time, walking time, whether the traveller has to interchange, petrol cost, parking charge and toll.

The idea of generalised cost was developed (or adopted from economics) to encompass all these variables. Generalised cost is simply the weighted sum of all these variables so the generalised cost of going from origin to destination would be the weighted sum of these variables for the origin to destination zone pair in question. These variables can be taken from the networks used in the assignment process. They can be 'skimmed' from the networks in the form of a matrix - not of trips this time but where each cell represents the value of a variable. So the in-vehicle time skim would contain the time spend in going from every zone to every other zone, the fare skim would contain the cost of travel from every origin zone to every destination zone and so on for all variables. These skim matrices can be combined into one overall measure of the 'separation' between every pair of zone by weighting each matrix and adding them all together.

### The Concept of a Deterrence Function

When researchers started to look at the effect of distance (or generalised cost) on the number of trips it became clear that people's behaviour is such that the further away the destination the less frequently they travel there. They looked at different functions to find the best relationships with varying degrees of success. The general consensus and now normal practice is to assume a negative exponential relationship (or deterrence function) with generalised cost, so that the measure of separation is given as e to the power of (a calibration constant multiplied by generalised cost). The calibration constant is negative which means that the higher the generalised cost the lower the number of trips. This calibration constant needs to be estimated for the deterrence function but first we must look at a new form for our distribution model.

(Some practitioners also use a term generalised cost raised to the power of another calibration constant, which multiplies the former (negative exponential) function. This combined function has the name: 'Tanner Function' after John Tanner who suggested it. However in some cases the additional term does not add much 'explanation' to the model and can be omitted. If the Tanner Function is used, the deterrence function has two parameters to be estimated rather than just one.)

### The Gravity Distribution Model

The gravity model gets its name from the idea of gravity where the 'pull' between two stars is proportional to the size of the stars and inversely proportional to (some function of) the distance between them. This is similar to travel between areas where the amount of travel between two areas can be considered as being proportional to their population, numbers of jobs, schools, factories, offices etc but inversely proportional to the distance (or some measure of the separation or deterrence) between them. When researchers started looking at this they found that generally this relationship holds up quite well - the bigger the towns the more travel there was between them and the further apart towns were, the less travel there was between them. The amount of pull between the origin zone and the destination zone is given as the origin and destination trip ends respectively.

As far as the function of the distance between them is concerned, we have our concept of the deterrence function outlined above. The deterrence function has a calibration constant (or two if you use a Tanner Function) which must be fitted while the trip ends have row and column factors which must also be fitted. This fitting procedure is called calibration and is undertaken as part of the process of building the transport model. Calibration can give you quite a different matrix from that which went into the process (the base year observed trip matrix) and the accuracy of the model is dependent on getting a 'good' calibration.

In some cases the deterrence function is different for different sorts of people, for different types of trip (eg journey to work, education, shopping, leisure, holiday etc), for travel during different times of the day or for those with or without a car. To incorporate these differences it is necessary to have different matrices for each different type of trip and/or traveller. This can produce a 'better' calibration. In addition different area to area movements may have different deterrence functions and it is sometimes necessary to have different calibration constants for each different area to area movement. With all these variables calibration can be a long time consuming process. It is also risky because it may be that there is no deterrence effect which means that a gravity model cannot be calibrated. Modelling is not without risk - the risk of not being able to build a model at all!

When the gravity distribution model has been calibrated, it can be used. We can use the same calibration constant for the deterrence function and assume that it holds for the future year. We also need the future year generalised costs and future year trip ends and with these and the deterrence function, we can forecast a future year trip matrix.

This future year trip matrix will be sensitive to the transport networks and to the future levels of trip making from and to each zone. It will also overcome our other problem (encountered with Furness that of having zero cells in the base year) because we use the generalised cost matrix as our starting point and this comes from the transport networks and can be calculated for every cell.

The gravity model is there a more robust distribution model. It contains sensitivity to the levels of trip making from and to each zone and is sensitive to the transport networks. However it can be quite a volatile model and even when calibrated the resulting trip matrix may be quite different from the observed trip matrix from which it was derived. It is therefore necessary to check the matrix produced against the observed matrix at a sector to sector level to check that the model is working correctly.