Abstract
A time-invariant fluid model of a polling system is considered. It consists of finitely many servers and buffers with unlimited sizes. The buffers receive inflows of work from the outside, work leaves the system after processing by a server. Every server works only with buffers from an associated zone of service, which may overlap for various servers, is able to serve at most one buffer at a time and so has to switch, from time to time, among buffers, the switch-over times are nonzero. We present a criterion for existence of a scheduling and service protocol that makes the system stable in the sense that the total amount of work in the buffers remains bounded as time progresses. The necessity part of this result is concerned with the widest class of protocols, including dynamic ones that are centralized and have access to the full information about the events in the system. Meanwhile, we show that every stabilizable system can be stabilized in a fully decentralized fashion via a simple static protocol, e.g., by a protocol that is based on independent round robin scheduling of the servers and for every server, employs only time measurement.
Highlights
1 Introduction We consider polling systems [Kleinrock, 1976] where finitely many and mostly independent queues share common sources of service; every server is able to serve at most one queue at a time and has to switch among queues from time to time
Control of polling systems is a twofold discipline: for every server, a scheduling protocol gives the queue to visit whereas a service protocol regulates the amount and rate of service given to the current queue
Stability of a polling system in this sense can be linked to stability of an associated deterministic fluid model, see, e.g., [Rybko and Stolyar, 1992; Chen, 1995; Dai, 1995; Fricker and Jaıbi, 1998; Down, 1998; Foss and Kovalevskii, 1999; Bramson, 2008]
Summary
We consider polling systems [Kleinrock, 1976] where finitely many and mostly independent queues share common sources of service (finite capacity servers); every server is able to serve at most one queue at a time and has to switch among queues from time to time. In [Dai, 1995], an unified approach to stability study is laid down for a general class of stochastic multiserver multi-class open queueing networks with a given queueing discipline by showing that such a network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. By using this result, it is shown that the usual traffic condition ensures positive Harris recurrence of the network under a number of specific standard disciplines. One more inevitable minimal requirement is that the system closed by the control algorithm at hands should be solvable and give rise to a specific process from any feasible initial state (X, Q)
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