Occupation times of alternating renewal processes with Lévy applications

Abstract

This paper presents a set of results relating to the occupation time α(t) of a process X(·). The rst set of results concerns exact characterizations of α(t), e.g., in terms of its transform up to an exponentially distributed epoch. In addition we establish a central limit theorem (entailing that a centered and normalized version of α(t)/t converges to a zero-mean Normal random variable as t → ∞) and the tail asymptotics of P(α(t)/t ≥ q). We apply our ndings to spectrally positive Lévy processes re ected at the in mum and establish various new occupation time results for the corresponding model.We consider a stochastic process X(·) ≡ {X(t) : t ≥ 0} taking values on the state space E, and a partition of E into two disjoint subsets, denoted by A and B, i.e., E = A∪B and A∩B = ∅. Then, X(·) alternates between A and B. The successive sojourn times in A are (D i ) i∈N , and those in B are (U i ) i∈N . If (D i ) i∈N and (U i ) i∈N are independent sequences of i.i.d. random variables, the resulting process is an alternating renewal process,

Single-server queues under overdispersion in the heavy-traffic regime

“…The asymptotic (co-)variances v A , v S , and c A, S can be computed in an alternative way, using results from large deviations theory [17] ; a similar approach has been followed in e.g., [28,39] . With this approach, also higher (centered) moments of AðtÞ=t and SðtÞ=t can be calculated in closed form in the regime that t !…”

Single-server queues under overdispersion in the heavy-traffic regime

“…The asymptotic (co-)variances v A , v S , and c A,S can be computed in an alternative way, using results from large deviations theory [16]; a similar approach has been followed in e.g. [25,35]. With this approach, also higher (centered) moments of A(t)/t and S(t)/t can be calculated in closed form in the regime that t → ∞, as we point out below.…”