Two-parameter process limits for infinite-server queues with dependent service times via chaining bounds

We prove two-parameter process limits for infinite-server queues with weakly dependent service times satisfying the ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-mixing condition. The two-parameter processes keep track of the elapsed or residual service times of customers in the system. We use the new methodology developed in Pang and Zhou (Stoch Process Appl 127(5):1375–1416, 2017) to prove weak convergence of two-parameter stochastic processes. Specifically, we employ the maximal inequalities for two-parameter queueing processes resulting from the method of chaining. This new methodology requires a weaker mixing condition on the service times than the ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-mixing condition in Pang and Whitt (Queueing Syst 73(2):119–146, 2013), as well as fewer regularity conditions on the service time distribution function.