In this paper we consider a ring of N >1 queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a polling model.
Each of the queues is fed by a non-decreasing L´evy process, which can be different during each of the consecutive periods within the server’s cycle. The N-dimensional L´evy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams.
For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch.
Our analysis heavily relies on establishing a link between fluid (L´evy input) polling systems and multitype Jirina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the branching property for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.
|Place of Publication||Eindhoven|
|Number of pages||16|
|Publication status||Published - 2009|