Diffusion Approximation for Fair Resource Control—Interchange of Limits Under a Moment Condition

Heng-Qing Ye () and David D. Yao ()
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Heng-Qing Ye: Department of Logistics and Maritime Studies, Hong Kong Polytechnic University, Hong Kong
David D. Yao: Department of Industrial Engineering and Operations Research, Columbia University, New York 10027

Mathematics of Operations Research, 2021, vol. 46, issue 3, 869-894

Abstract: In a prior study [Ye HQ, Yao DD (2016) Diffusion limit of fair resource control–Stationary and interchange of limits. Math. Oper. Res . 41(4):1161–1207.] focusing on a class of stochastic processing network with fair resource control, we justified the diffusion approximation (in the context of the interchange of limits) provided that the p th moment of the workloads are bounded. To this end, we introduced the so-called bounded workload condition, which requires the workload process be bounded by a free process plus the initial workload. This condition is for a derived process, the workload, as opposed to primitives such as arrival processes and service requirements; as such, it could be difficult to verify. In this paper, we establish the interchange of limits under a moment condition of suitable order on the primitives directly: the required order is p ∗ > 2 ( p + 2 ) on the moments of the primitive processes so as to bound the p th moment of the workload. This moment condition is trivial to verify, and indeed automatically holds in networks where the primitives have moments of all orders, for instance, renewal arrivals with phase-type interarrival times and independent and identically distributed phase-type service times.

Keywords: Primary: 60K25; 90B15; secondary: 60F17; 60J20; Primary: queueing network/limit theorem; secondary: Markov process/diffusion model; resource-sharing network; diffusion limit; stationary distribution; interchange of limits; uniform stability (search for similar items in EconPapers)
Date: 2021
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