We consider a two-node tandem queueing network in which the upstream queue has renewal arrivals with generally distributed service times, and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. The reuse of service times creates strong dependence at the second queue, making its workload difficult to analyze. To investigate the evolution of workload in the second queue, we introduce and study a process M, called the plateau process, which encodes most of the information in the workload process. We focus on the case of infinite-variance service times and show that under appropriate scaling, workload in the first queue converges, and although the workload in the second queue does not converge, the plateau process does converge to a limit M that is a certain function of two independent Lévy processes. Using excursion theory, we derive some useful properties of M and compare a time changed version of it to a limit process derived in previous work.
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Stochastic Processes and their Applications
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Gromoll, C., Terwilliger, B., & Zwart, B. (2019). Heavy traffic limit for the workload plateau process in a tandem queue with identical service times. Stochastic Processes and their Applications, 130(3), 1435–1460. doi:10.1016/j.spa.2019.05.007