Fluid polling systems

We study N-queues single-server fluid polling systems, where a fluid is continuously flowing into the queues at queue-dependent rates. When visiting and serving a queue, the server reduces the amount of fluid in the queue at a queue-dependent rate. Switching from queue i to queue j requires two random-duration steps: (i) departing queue i, and (ii) reaching queue j. The length of time the server resides in a queue depends on the service regime. We consider three main regimes: Exhaustive, Gated, and Globally-Gated. Two polling procedures are analyzed: (i) cyclic and (ii) probabilistic. Under steady-state, we derive the Laplace-Stieltjes transform (LST), mean, and second moment of the amount of flow at each queue at polling instants, as well as at an arbitrary moment. We further calculate the LST and mean of the "waiting time" of a drop at each queue and derive expressions for the mean total load in the system for the various service regimes. Finally, we explore optimal switching procedures.