Heavy Traffic Limit with Discontinuous Coefficients via a Nonstandard Semimartingale Decomposition

This paper studies a single server queue in heavy traffic, with general interarrival and service time distributions, where arrival and service rates vary discontinuously as a function of the (diffusively scaled) queue length. It is proved that the weak limit is given by the unique-in-law solution to a stochastic differential equation in $[0,\infty )$ with discontinuous drift and diffusion coefficients. The main tool is a semimartingale decomposition for point processes introduced by Daley and Miyazawa in 2019, which is distinct from the Doob-Meyer decomposition of a counting process. Although the use of this tool is demonstrated here for a particular model, we believe it may be useful for investigating the scaling limits of queueing models very broadly.

Funding: R. Atar received financial support from the Israel Science Foundation [Grants 1035/20 and 3240/25].