Overland flow as a queueing process: The B/D/1 queue with an arbitrary service time

We study the single server B/D/1 FIFO queue with Bernoulli inter-arrival times t (t∈{0,1}) and arbitrary deterministic service time s (s<1). This queue is introduced in hydrology to model overland flow generation during a rainfall event on a sloping flat soil surface with infiltration rate distributed randomly in space. The service time, the inter-arrival time and the waiting time stand respectively for the rainfall rate, the infiltration rate and the overland flow-rate. The main performance measure of interest in hydrology is the mean overland flow-rate. Its variation with the rainfall rate is of primary importance for peak-flow forecasting. For this purpose the queue must be solved for any real value of s. Two solutions, one exact and one approximate, are presented. The exact one is developed for rational values of s. The queue is transformed into a discrete time queue with a slot equal to 1/q, where q is the denominator of the irreducible fraction s=p/q (p and q integers). The waiting time and the queue length distributions are derived, and for each value of p and q there is a characteristic equation to solve inside the unit disk. The approximate, and simpler, solution is an extrapolation of the special case where s is a unit fraction (s=1/q) to an arbitrary value of s. Simulations show that this extrapolation seems to be acceptable only for the mean and standard deviation of the waiting time.