Diffusion approximation for head-of-the-line processor sharing for two parallel queues
SIAM Journal on Applied Mathematics
Heavy Traffic Analysis of Two Coupled Processors
Queueing Systems: Theory and Applications
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Processor sharing for two queues with vastly different rates
Queueing Systems: Theory and Applications
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We consider a system of three parallel queues with Poisson arrivals and exponentially distributed service requirements. The service rate for the heavily loaded queue depends on which of the two underloaded queues are empty. We derive the lowest-order asymptotic approximation to the joint stationary distribution of the queue lengths, in terms of a small parameter measuring the closeness of the heavily loaded queue to instability. To this order the queue lengths are independent, and the underloaded queues and the heavily loaded queue have geometrically and, after suitable scaling, exponentially distributed lengths, respectively. The expression for the exponential decay rate for the heavily loaded queue involves the solution to an inhomogeneous linear functional equation. Explicit results are obtained for this decay rate when the two underloaded queues have vastly different arrival and service rates.