Stationary queues with one autonomous server

Given a stationary stochastic continuous demand of service σ(θ_tω) dt with ∫ σ(ω)P(dω) < 1, we construct real stationary point processes $\text{(T}{}_{\mathrm{n}}\text{, n ∈}\text{Z}\text{)T}{}_{\mathrm{n}}\text{< T}{}_{\mathrm{n}}\text{+1, lim}{}_{\mathrm{\pm \infty }}\text{T}{}_{\mathrm{n}}\text{= ±∞]}$ such that

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.