On Mutually Interfering Parallel Servers Subject to External Disturbances

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Abstract
This paper considers a continuous-time non-Markovian parallel queueing system subject to external disturbances. The servers are mutually interfering in that their service rates are nonlinearly interdependent functions of the controls applied by the servers, and external discrete-valued continuous-time random disturbances. At certain time epochs, namely, every Δ time units, the servers may adjust their service rates by changing the values of their controls; however, the system may change its state several times between successive decision epochs. The stability region of the system is established and a service rate control policy π* is provided, where an arrival rate vector in the interior of the region is sufficient for stability under π*, and a vector in the closure is necessary for stability under any policy. The stability region depends on Δ and the variations of the disturbances between decision epochs, and π* does not require knowledge of the arrival rates. The stability region is not in general monotonic in Δ, but under perfect continuous control (Δ = 0) the stability region is a superset of that under Δ > 0. This queueing model captures essential features of resource allocation and stochastic control problems encountered in a number of telecommunication, transportation, and manufacturing systems.

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