row totals. We put this point together with the earlier work with our matrix and the notion of most likely state to say that we want to subject to where: and this is the problem that we have solved above. Wilson adds another consideration; he constrains the system to the amount of energy available (i.e., money), and we have the additional constraint, where C is the quantity of resources available and is the travel cost from i to j. The discussion thus far contains the central ideas in Wilson’s work, but we are not yet to the place where the reader will recognize the model as it is formulated by Wilson. First, writing the function to be maximized using Lagrangian multipliers, we have: where and are the Lagrange multipliers, having an energy sense. Second, it is convenient to maximize the natural log (ln) rather than , for then we may use Stirling's approximation. so Third, evaluating the maximum, we have with solution Finally, substituting this value of back into our constraint equations, we have: and, taking the constant multiples outside of the summation sign Let we have which says that the most probable distribution of trips has a gravity model form, is proportional to trip origins and destinations. The constants , , and ensure that constraints are met. Turning now to computation, we have a large problem. First, we do not know the value of C, which earlier on we said had to do with the money available, it was a cost constraint. Consequently, we have to set to different values and then find the best set of values for and . We know what means – the greater the value of , the less the cost of average distance traveled. (Compare in Boltzmann's Law noted earlier.) Second, the values of and depend on each other. So for each value of , we must use an iterative solution. There are computer programs to do this. Wilson's method has been applied to the Lowry model. Issues Congestion One of the key drawbacks to the application of many early models was the inability to take account of congested travel time on the road network in determining the probability of making a trip between two locations. Although Wohl noted as early as 1963 research into the feedback mechanism or the “interdependencies among assigned or distributed volume, travel time (or travel ‘resistance’) and route