This paper provides a direct approach to obtaining formulas for derivatives of functionals of point processes in rare perturbation analysis ([2], [6]). Results are obtained for arbitrary (not necessarily stationary) point processes in and d, d 2, under transparent conditions, close to minimal. Formulas for higher-order derivatives allow one to construct asymptotical expansions. The results can be useful in sensitivity analysis, in light traffic theory for queues and for computation by simulation of derivatives at positive intensity, while the computation of the derivatives via statistical estimation of the functional itself and its increments usually gives poor results.

An approximate model for the study of platoon formation on two-lane highways is discussed in detail. The model assumes that the two-lane highway is divided in each traffic direction into alternating road sections of fixed lengths. The passing in one type of section is unrestricted and the passing in the other one is prohibited. It is assumed that there are slow and fast vehicles on the highway and that inputs follow independent Poisson processes. The results include the distribution of the number of vehicles in a platoon and the average speed of a typical fast vehicle.

A general procedure is outlined for obtaining lower bounds and approximations to the amount of congestion in queues with low traffic. Some detailed formulae are given for a number of single-server systems and compared with exact solutions where available. Results are also given for a discrete time system in which departures clash with new arrivals.