Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T19:54:32.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Queueing in space

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Eitan Altman  and
Hanoch Levy
Eitan Altman*
Affiliation:
INRIA
Hanoch Levy*
Affiliation:
Rutcor
*
* Postal address: INRIA Centre Sophia Antipolis, 2004 Route des Lucioles, 06560 Valbonne, France.
** Postal address: RUTCOR, Rutgers University, POB 5062, New Brunswick, NJ 08903–5062, USA. On leave from Tel-Aviv University.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a problem in which a single server must serve a stream of customers whose arrivals are distributed over a finite-size convex space. Under the assumption that the server has full information on the customer location, obvious service policies are the FCFS and the greedy (serve-the-closest-customer) approaches. These algorithms are, however, either inefficient (FCFS) or ‘unfair' (greedy).

We propose and study two alternative algorithms, the gated-greedy policy and the gated-scan policy, which are more ‘fair' than the pure greedy method. We show that the stability conditions of the gated-greedy are p < 1 (where p is the expected rate at which work arrives at the system), implying that the method is at least as efficient (in terms of system stability) as any other discipline, in particular the greedy one. For the gated-scan policy we show that for any p < 1 one can design a stable gated-scan policy; however, for any fixed gated-scan policy there exists p < 1 for which the policy is unstable. We evaluate the performance of the gated-scan policy, and present bounds for the performance of the gated-greedy policy.

These results are derived for systems in which the arrivals occur on a two-dimensional space (a square) but they are not limited to this configuration; rather they hold for more complex N-dimensional spaces, in particular for serving customers in (three-dimensional) convex space and serving customers on a line.

Keywords

ERGODICITY CONDITIONSFAIRNESSOPTIMAL STABILITY REGIONPOLLING ON THEPLANEGATED-GREEDY REGIMEGATED-SCAN REGIMEPERFORMANCE EVALUATION;BOUNDS

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 26 , Issue 4 , December 1994 , pp. 1095 - 1116
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Altman, E. and Foss, S. (1993) Polling on a graph with general arrival and service time. INRI A report No. 1992.Google Scholar
[2] Altman, E. and Spieksma, F. (1994) Geometric ergodicity and moment stability of station times in polling systems. Stoch. Models. Google Scholar
[3] Altman, E., Konstantopoulos, P. and Liu, Z. (1992) Stability, monotonicity and invariant quantities in general polling systems. Queueing Systems, Special Issue on Polling Models, ed. Takagi, H. and Boxma, O., 11, 35–57.CrossRefGoogle Scholar
[4] Altman, E. Foss, S. Riehl, E. and Stidham, S. (1994) Sample path analysis of token rings. In The Fundamental Role of Teletraffic in the Evolution of Telecommunication Networks, Proc. 14th Internat. Teletraffic Congr. Antibes Juan-les-Pins, pp. 811–820.CrossRefGoogle Scholar
[5] Baccelli, F. and BréMaud, P. (1980). Palm Probabilities and Stationary Queues. Lecture Notes in Statistics, Springer-Verlag, Berlin.Google Scholar
[6] Bartholdi, J. J. and Platzman, L. K. (1988) Heuristic based on spacefilling curves for combinatorial problems in euclidean space. Management Sci. 34, 291–305.CrossRefGoogle Scholar
[7] Bearwood, J., Halton, J. and Hammersley, J. (1959) The shortest path through many points. Proc. Camb. Phil. Soc. 55, 299–327.CrossRefGoogle Scholar
[8] Bertsimas, D. J. and Van Ryzin, G. (1991) A stochastic and dynamic vehicle routing problem in the euclidean plane. Operat Res. 39, 601–615.CrossRefGoogle Scholar
[9] Boxma, O. J., Levy, H. and Yechiali, U. (1992) Cyclic reservation schemes for efficient operation of multiple-queue single-server systems. Ann. Operat. Res. 35, 187–208.CrossRefGoogle Scholar
[10] Cohen, J. W. (1982) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[11] Coffman, E. G. Jr and Gilbert, E. N. (1987) Polling and greedy servers on the line. Queueing Systems 2, 115–145.CrossRefGoogle Scholar
[12] Coffman, E. G. Jr and Stolyar, A. (1993) Continuous polling on graphs. Prob. Eng. Inf. Sci. 7, 209–226.CrossRefGoogle Scholar
[13] Harel, A. and Stulman, A. (1994) Polling, greedy, and horizon servers on a circle. Operat Res. CrossRefGoogle Scholar
[14] Fricker, C. and Jaibi, M. R. (1992) Monotonicity and stability of periodic polling models. Report FEW 559, Department of Economics, Tilburg University.Google Scholar
[15] Georgiadis, L. and Szpankowski, W. (1992) Stability of token passing rings. Queuing Systems, Special Issue on Polling Models ed. Takagi, H. and Boxma, O., 11, 7–33.CrossRefGoogle Scholar
[16] Khamisy, A., Altman, E. and Sidi, M. (1992) Polling systems with synchronization constraints. Ann. Operat. Res. Special Issue on Stochastic Modeling of Telecommunication Systems. ed. Nain, P. and Ross, K. W., pp. 231–267.CrossRefGoogle Scholar
[17] Kroese, D. P. and Schmidt, V. (1994) Light-traffic analysis for queues with spatially distributed arrivals.CrossRefGoogle Scholar
[18] Levy, H., Sidi, M. and Boxma, O. J. (1990) Dominance relations in polling systems. Queueing Systems 6, 155–172.CrossRefGoogle Scholar
[19] Meyn, S. P. and Tweedie, R. L. (1992) Stability of Markovian processes I: criteria for discrete time chains. Adv. Appl. Prob. 24, 542–574.CrossRefGoogle Scholar
[20] Resing, J. A. C. (1991) Polling systems and multi-type branching processes. Report BS-R9128, C.W.I., Amsterdam.Google Scholar
[21] Tweedie, R. L. (1983) Criteria for rates of convergence of Markov chains, with application to queueing and storage theory. In Probability, Statistics and Analysis, ed. Kingman, J. F. C. and Reuter, G. E. H., London Math. Society Lecture Notes Series 79, pp. 260–276. Cambridge University Press.CrossRefGoogle Scholar
[22] Tweedie, R. L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar
[23] Zhdanov, V. S. and Saksonov, E. A. (1979) Conditions of existence of steady-state modes in cyclic queuing systems. Autom. Remote Control 40, 176–184.Google Scholar