Ergodicity and analysis of the process describing the system state in polling systems with two queues |
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| Institution: | 1. College of Computer Science and Technology, Zhejiang University, Hangzhou 310058, China;2. Department of Computer Science and Technology, Hefei University of Technology, Hefei 230009, China;3. School of Mathematical Science, Peking University, Beijing 100871, China;4. Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;5. Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, USA;6. School of Humanities, Zhejiang University, Hangzhou 310058, China;7. University of California Berkeley, Berkeley, CA 94720, USA;8. Guanghua School of Management, Peking University, Beijing 100871, China;9. Department of Government & Department of Statistics, Harvard University, Cambridge, MA 02138, USA;1. VU University Amsterdam, Amsterdam, Netherlands;2. Centre for Mathematics and Computer Science, Amsterdam, Netherlands;1. Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;2. Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands;3. Alcatel-Lucent Bell Labs, P.O. Box 636, Murray Hill, NJ 07974-0636, USA |
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| Abstract: | Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition. |
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