Tighter bounds for the two-level arrivals system without intermittent overload

Develop tighter bounds for the mean queue length E[Q] in the two-level arrivals single-server queue with exponential service rate μ, where the arrival rate alternates between λ_H and λ_L according to a two-state continuous-time Markov chain with switching rates α_H and α_L, specifically in the non-intermittent-overload regime λ_H < μ. The goal is to improve upon existing basic bounds for this case by deriving sharper, explicit bounds within this model.

Background

The paper presents tight bounds for the two-level arrivals system under intermittent overload (λ_H > μ) and in regimes of fast or slow switching, by leveraging a drift-based framework with relative arrivals. These bounds are complemented by general heavy-traffic results.

However, for the non-intermittently-overloaded regime (λ_H < μ), the authors note that only basic bounds from prior work are available and that extending their framework to obtain tighter results in this case remains challenging. They explicitly state that deriving sharper bounds for this regime is future work.