Useful stochastic bounds in time-varying queues with service and patience times having general joint distribution

Consider a first-come, first-served single server queue with an initial workload $x>0$ and customers who arrive according to an inhomogeneous Poisson process with rate function λ:[0,∞)→[0,λₕ ] for some λₕ>0. For each i, let Sᵢ (resp., Yᵢ) be the service (resp., patience) time of the i'th customer and assume that (S₁,Y₁),(S₂,Y₂),… is an iid sequence of bivariate random vectors with non-negative coordinates. A customer joins if and only if his patience time is not less than his prospective waiting time (i.e., the left-limit of the workload process at his arrival epoch). Let τ(x) be the first time when the system becomes empty and let N^*_λ(·) be the arrival process of those who join the queue. In the present work we suggest a novel coupling technique which is applied to derive stochastic upper bounds for the functionals: {equation*} ∫_0τ(x)g∘ W_x(t){ d}t\ \ and\ \ ∫_0τ(x)g∘ W_x(t){ d}N^*_λ(t)\,, {equation*} where Wₓ(·) is the workload process in the queue and g(·) is any lower semi-continuous function. We also demonstrate how to utilise these bounds via some examples under the additional assumption that λ(·) is periodic.