Stationary queues with one autonomous server

Abstract

Given a stationary stochastic continuous demand of service σ(θ t ω) d t with ∫ σ(ω) P(dω) < 1, we construct real stationary point processes (T n, n ∈ Z)[T n < T n+1, lim ±∞ T n = ±∞] such that T n+1-T n=D + ∫ T n T n-1 σΘ t Dt (n ∈ Z) for a given constant D \\2>0. These point processes correspond to a service discipline for which a single server services during the time intervals [ T n , T n+1 [ the demand of service accumulated during the proceding intervals [ T n−1 , T n [ and take a rest of fixed duration D.