Abstract
We study the limiting behavior of gated polling systems, as their dimension (the number of queues) tends to infinity, while the system's total incoming workflow and total switchover time (per cycle) remain unchanged. The polling systems are assumed asymmetric, with incoming workflow obeying general Lévy statistics, and with general inter-dependent switchover times. We prove convergence, in law, to a limiting polling system on the circle. The derivation is based on an asymptotic analysis of the stochastic Poincaré maps of the polling systems. The obtained polling limit is identified as a snowplowing system on the circle—whose evolution, steady-state equilibrium, and statistics have been recently investigated and are known.
Similar content being viewed by others
References
J. Bertoin, Lévy Processes, (Cambridge University Press, 1996); J. Bertoin, Subordinators: examples and applications, Lecture notes in mathematics 1717, Springer, 1999.
E.G. Coffman Jr. and E.N. Gilbert, A continuous polling system with constant service times, IEEE Trans. Inf. The. IT-32 (1986) 584–591.
E.G. Coffman Jr. and E.N. Gilbert, Polling and greedy servers on the line, Que. Sys. 2 (1987) 115–145.
I. Eliazar, Gated polling systems with Lévy inflow and inter-dependent switchover times: A dynamical-systems approach, Queueing Systems 49 (2005) 49–72.
I. Eliazar, The snowblower problem, Queueing Systems 45 (2003) 357–380.
S.W. Fuhrmann and R.B. Cooper, Applications of the decomposition principle in M/G/1 vacation models to two continuum cyclic queueing models, AT&T Tech. Jour. 64 (1985) 1091–1098.
O. Kallenberg, Random Measures 3rd edn, (Academic Press, 1997).
D.E. Knuth, The Art of Computer Programing (Addison Wesley) 1973, (see Vol. III, pp. 254–255 and 259–264).
D.P. Kroese and V. Schmidt, A continuous polling system with general service times, Ann. Appl. Prob. 2(4) (1992) 906–927.
D.P. Kroese and V. Schmidt, Queueing systems on the circle, Z. Oper. Res. (1993) 37(3) 303–331.
D.P. Kroese and V. Schmidt, Single-server queues with spatially distributed arrivals, Que. Sys. 17 (1994) 317–345.
D.P. Kroese and V. Schmidt, Light-traffic analysis for queues with spatially distributed arrivals, Math. Oper. Res. 21 (1996) 135–157.
H. Takagi, Analysis of Polling Systems, (MIT Press, Cambridge, MA, 1986).
H. Takagi, Queueing analysis of polling systems: an update, in: Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) 267–318.
H. Takagi, Queueing analysis of polling models: Progress in 1990–1994, in: Frontiers in Queueing: Models and Applications in Science and Engineering, ed. J.H. Dshalalow (CRC Press Boca Raton, 1997) pp. 119–146.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eliazar, I. From Polling to Snowplowing. Queueing Syst 51, 115–133 (2005). https://doi.org/10.1007/s11134-005-2401-5
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11134-005-2401-5