2021/12/03 by Remigijus Leipus, Leipus, Remigijus, Vytautė Pilipauskaitė +3 · 1 citation
Business, Management and Accounting · Decision Sciences · Mathematics · #60F05 #60G18 (Primary) #60G22 #60G51 #60G52 #60G60 #60K25 (Secondary) #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2112.01893
openalex publication_date 2021/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss joint spatial-temporal scaling limits of sums Aλ,γ (indexed by (x,y) ∈ ℝ2+) of large number O(λγ) of independent copies of integrated input process X = \X(t), t ∈ ℝ\ at time scale λ, for any given γ>0. We consider two classes of inputs X: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that normalized random fields Aλ,γ tend to an α-stable Lévy sheet (1< α<2) if γ< γ0, and to a fractional Brownian sheet if γ> γ0, for some γ0>0. We also prove an `intermediate' limit for γ= γ0. Our results extend previous works Mikosch et al. (2002), Gaigalas, Kaj (2003) and other papers to more general and new input processes.