Exponents for the tails of distributions in some polling models

The tail asymptotics of the distribution of the waiting-time W in some polling models is investigated. When this is of the form P[W > x] ∼ αx^β e^−ηx for some α, β, η, we show how to calculate the exponents β and η, and we establish the extent and form of their dependence on the distributions of the service-time and switchover-time. The exponents are expressed in terms of the fixed points and Lyapunov exponents of a dynamical system which we associate with the recursion which is used to calculate the moment generating functions of the waiting time.