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Critical queues and reflected stochastic processes

Final Report Summary - CRITIQUEUE (Critical queues and reflected stochastic processes)

Reflected stochastic processes are stochastic processes that are restricted to stay positive. Scalings limits arise by considering a sequence of stochastic processes, and appropriately scaling space and time. This project was centered around scaling limits for reflected stochastic processes. These scaling limits provide simple approximations for complicated stochastic processes, usual in their most critical and most relevant regime. The scaling limits also provide insight into the macroscopic behavior of the processes, and the different phenomena occurring at different space-time scales. Our primary motivation stemmed from queueing theory, the branch of applied probability that deals with congestion phenomena. Congestion levels are typically nonnegative, which is why reflected stochastic processes arise naturally in queueing theory. Other applications of reflected stochastic processes are in the fields of branching processes and random graphs. We have studied critically-loaded queueing systems (close to 100% utilization), also referred to as queues in heavy traffic. Heavy-traffic analysis typically reduces complicated queueing processes to much simpler (reflected) limit processes or scaling limits. This makes the analysis of complex systems tractable, and from a mathematical point of view, these results are appealing since they can be made rigorous. Within the large body of literature on heavy-traffic theory and critical stochastic processes, we launched two new research lines:
(i) Time-dependent analysis through scaling limits and (ii) Dimensioning stochastic systems via refined scaling limits and optimization.
Both research lines involved mathematical techniques that combine stochastic theory with asymptotic theory, complex analysis, functional analysis, and modern probabilistic methods.