# zbMATH — the first resource for mathematics

Busy period analysis, rare events and transient behavior in fluid flow models. (English) Zbl 0826.60086
A fluid process $$(V_t : t \geq 0)$$ with piecewise continuous paths and reflection at 0 is considered where the slopes of the paths are governed by a finite state space Markov process $$(J_t : t \geq 0)$$. A busy period of type $$i$$ starts form $$V_0 = 0$$, $$J_0 = i$$ and ends when $$V$$ returns to 0 the first time thereafter. The first result is a set of linear equations for the mean value of a busy period of type $$i$$ in terms of the steady state quantities. Its distribution is given via Laplace transform. The main topic of the paper is to compute approximations and bounds for transient probabilities of the process. E.g., the cycle maximum is shown to have exponential tails, a central limit estimate for large deviation probabilities is given, the rate of convergence to the steady-state is computed, and an approximation for the time until reaching equilibrium. Applications to quick simulations of rare events are shown.