We study the exact asymptotics for the distribution of the first time, τx , a Lévy process X t crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{X t ≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx >t} explicitly in both light- and heavy-tailed cases.

We analyse an ALOHA-type random multiple-access protocol where users have local interactions. We show that the fluid model of the system workload satisfies a certain differential equation. We obtain a sufficient condition for the stability of this differential equation and deduce from that a sufficient condition for the stability of the protocol. We discuss the necessary condition. Furthermore, for the underlying Markov chain, we estimate the rate of convergence to the stationary distribution. Then we establish an interesting and unexpected result showing that the main diagonal is locally unstable if the input rate is sufficiently small. Finally, we consider two generalisations of the model.

Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let S n =ξ1+⋯+ξn and M n =maxk≤n S k . Let τ=min{n≥1: S n ≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.