for the second leg of the mixed mode trip. The total park&ride demand for O-D pair pq is 
.
Note that the above formulation does not include an explicit impedance term for the parking lots (e.g. to model the effects of parking cost). Dropping this term does not cause any loss of generality of the following development, as it can be thought already included in one of the leg impedances.
See Spiess (1993) for a more ``practical'' discussion of this uncapacitated mixed mode model (as a special case of an activity chain model), including details on its implementation using the convolution module of the EMME/2 software (Spiess 1984; INRO 1996). A practical application of this model for the Puget Sound Regional Council has been reported by Blain (1994).
From a mathematical programming point of view, it is important to note that the above parking lot choice model is equivalent to the following convex minimization problem:
subject to
By applying the Kuhn-Tucker optimality conditions (see e.g. Luenberger 1984), it can be shown that (1) corresponds indeed to the optimal solution of the above optimization problem.
Parking lot choice model with explicit capacities
We now turn our attention to the parking lot choice model in which an explicit capacity 
