Exact Asymptotics for a Multitimescale Model with Applications in Modeling Overdispersed Customer Streams

Abstract

In this paper we study the probability ξn(u) := P (Cn un), with Cn := A(ψnB(ϕn)) for Lévy processes A(·) and B(·), and ϕn and ψn non-negative sequences such that ϕnψn = n and ϕn → ∞ as n → ∞. Two timescale regimes are distinguished: a 'fast' regime in which ϕn is superlinear and a 'slow' regime in which ϕn is sublinear. We provide the exact asymptotics of ξn(u) (as n → ∞) for both regimes, relying on change-of-measure arguments in combination with Edgeworth-type estimates. The asymptotics have an unconventional form: the exponent contains the commonly observed linear term, but may also contain sublinear terms (the number of which depends on the precise form of ϕn and ψn). To showcase the power of our results we include two examples, covering both the case where Cn is lattice and non-lattice. Finally we present numerical experiments that demonstrate the importance of taking into account the doubly stochastic nature of Cn in a practical application related to customer streams in service systems; they show that the asymptotic results obtained yield highly accurate approximations, also in scenarios in which there is no pronounced timescale separation.We proceed by explaining that there are two timescale regimes. (i) If ϕ n is superlinear, it is anticipated that B(ϕ n )/ϕ n is close to b, such that ξ n (u) resembles P(A(bn) un). We refer to this AMS 2000 subject classifications: 60F10, 60G51, 60K37

Exact asymptotics in an infinite-server system with overdispersed input

“…Following the analysis presented in [8,Section 3], √ ϕ n ∆ n converges to (τ ⋆ σ Q − √ 2π) −1 as n → ∞. We thus arrive at the following result, which is the counterpart of [8, Thm.…”

“…This aligns with the expansion featuring in [8, Section 3], but with different coefficients w k (that take into account the effect of the leaving jobs). • Again, as a next step the probability ξ n (u) is rewritten using the ϑ n -twisted version of N n ; from that point on, the proof precisely follows the one in [8,Section 3].…”

“…evidently, the coefficients v k are different from those in [8,Section 2], as there only the arrival process was considered (i.e., without jobs potentially leaving the system). • The next step is to express the probability ξ n (u) using the ϑ n -twisted version of N n .…”

“…Then it is a matter of applying uniform (in x, that is) bounds on Q n (M n x) − Φ(x), with Φ(·) denoting the cumulative distribution function of a standard Normal random variable; such an Edgeworth expansion is derived in precisely the same way as in [8,Appendix A]. Notice that the lattice version, as in [8,Remark 2], needs to be applied, due to the fact that A(·) attains integer values. Combining the above, the following counterpart of [8, Thm.…”

“…Here we focus on the case that ϕ n is sublinear, such that ψ n → ∞ as n → ∞. We follow the line of reasoning used in [8,Section 3]; observe that (1) remains valid. The argumentation is as follows:…”