Diffusion Approximation for an Overloaded X Model Via a Stochastic Averaging Principle

In previous papers we developed a deterministic fluid approximation for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. The control aims to keep the two queues at a pre-specified fixed ratio. We supported the fluid approximation by establishing a many-server heavy-traffic functional weak law of large numbers (FWLLN) involving an averaging principle. In this paper we develop a refined diffusion approximation for the same model based on a many-server heavy-traffic functional central limit theorem (FCLT).

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