Car following route choice, crashes-and the Lambert W- function.

The Lambert W -function is the multi-valued inverse of the complex function w-we. This paper shows how this function arises naturally in three different areas of road traffic research. Following a description of the main properties of W(z), the first application is to delay differential equations of the car following model. It is shown that the use of W leads to a simpler derivation of oscillatory solutions of the linear model. The Lambert W-function arises in combinatorics too, and there is a short discussion of potential applicatons to road networks and route choice. The third and major application is to empirical Bayes methods and the Poisson-lognormal distribution. Neither the distribution nor the resulting posterior expectation have closed-form expressions, but it is shown that the posterior mode can be expressed directly in terms of W, and that this leads to a good approximation to the posterior expectation. Implications of these results are discussed. (A) Reprinted with permission from Elsevier. For the covering abstract see ITRD E134766.