Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T05:58:55.312Z Has data issue: false hasContentIssue false
  • English
  • Français

POLLING SYSTEMS WITH TWO-PHASE GATED SERVICE

HEAVY TRAFFIC RESULTS FOR THE WAITING TIME DISTRIBUTION

Published online by Cambridge University Press:  25 September 2008

R. D. van der Mei  and
J. A. C. Resing
R. D. van der Mei
Affiliation:
Centre for Mathematics and Computer Science, Department of Probability and Stochastic Networks, Amsterdam, Netherlands and Faculty of Sciences, Vrije Universiteit, Department of Mathematics, Amsterdam, Netherlands E-mail: [email protected]
J. A. C. Resing
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 4 , October 2008 , pp. 623 - 651
Copyright
Copyright © Cambridge University Press 2008

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. & Fiems, D. (2007). Expected waiting time in symmetric polling systems with correlated vacations. Queueing Systems 56: 241–253.Google Scholar
2.Altman, E. & Kushner, H. (2002). Control of polling in presence of vacations in heavy traffic with applications to satellite and mobile radio systems. SIAM Journal of Control and Optimization 41: 217–252.Google Scholar
3.Athreya, K.B. & Ney, P.E. (1972). Branching processes. Berlin: Springer-Verlag.CrossRefGoogle Scholar
4.Blanc, J.P.C. (1992). An algorithmic solution of polling systems with limited service disciplines. IEEE Transactions on Communications 40: 1152–1155.CrossRefGoogle Scholar
5.Blanc, J.P.C. (1992). Performance evaluation of polling systems by means of the power-series algorithm. Annals of Operations Research 35: 155–186.Google Scholar
6.Brosh, E., Levy, H. & Avi-itzhak, B. (2007). SQF: A slowdown queueing fairness measure. Performance Evaluation 64: 1121–1136.Google Scholar
7.Coffman, E.G., Puhalskii, A.A. & Remain, M.I. (1995). Polling systems with zero switch-over times: A heavy-traffic principle. Annals of Applied Probability 5: 681–719.Google Scholar
8.Coffman, E.G., Puhalskii, A.A. & Remain, M.I. (1998). Polling systems in heavy-traffic: A Bessel process limit. Mathematics of Operation Research 23: 257–304.Google Scholar
9.Fricker, C. & Jaibi, M.R. (1994). Monotonicity and stability of periodic polling models. Queueing System 15: 211–238.Google Scholar
10.Groenevelt, R. & Altman, E. (2005). Analysis of alternating-priority queueing models with (cross) correlated switchover times. Queueing Systems 51: 199–247.CrossRefGoogle Scholar
11.Konheim, A.G., Levy, H. & Srinivasan, M.M. (1994). Descendant set: An efficient approach for the analysis of polling systems. IEEE Transactions on Communications 42: 1245–1253.Google Scholar
12.Kramer, G., Mukherjee, B. & Pesavento, G. (2001). Ethernet PON: Design and analysis of an optical access network. Photonic Network Communications. 3: 307–319.CrossRefGoogle Scholar
13.Kramer, G., Mukherjee, B. & Pesavento, G. (2002). Interleaved polling with adaptive cycle time (IPACT): A dynamic bandwidth allocation scheme in an optical access network. Photonic Network Communications. 4: 89–107.Google Scholar
14.Kramer, G., Mukherjee, B. & Pesavento, G. (2002). Supporting differentiated classes of services in Ethernet passive optical networks. OSA Journal of Optical Networking. 1: 280–298.Google Scholar
15.Kroese, D.P. (1997). Heavy traffic analysis for continuous polling models. Journal of Applied Probability 34: 720–732.Google Scholar
16.Kudoh, S., Takagi, H. & Hashida, O. (2000). Second moments of the waiting time in symmetric polling systems. Journal of The Operations Research Society of Japan 43: 306–316.CrossRefGoogle Scholar
17.Levy, H. & Sidi, M. (1991). Polling models: Applications, modeling and optimization. IEEE Transactions on Communications 38: 1750–1760.Google Scholar
18.Markowitz, D., Remain, M.I. & Wein, L.M. (2000). The stochastic economic lot scheduling problem: Heavy traffic analysis of dynamic cyclic policies. Operations Research 48: 136–154.CrossRefGoogle Scholar
19.Markowitz, D. & Wein, L.M. (2001). Heavy traffic analysis of dynamic cyclic policies: A unified treatment of the single machine scheduling problem. Operations Research 49: 246–270.Google Scholar
20.Olsen, T.L. (2001). Limit theorem for polling models with increasing setups. Probability in the Engineering, and Informational Sciences 15: 35–55.CrossRefGoogle Scholar
21.Olsen, T.L. & Van der Mei, R.D. (2003). Periodic polling systems in heavy-traffic: Distribution of the delay. Journal of Applied Probability 40: 305–326.Google Scholar
22.Olsen, T.L. & Van der Mei, R.D. (2005). Periodic polling systems in heavy-traffic: Renewal arrivals. Operations Research Letters 33: 17–25.Google Scholar
23.Park, C.G., Han, D.H., Kim, B. & Jun, H.-S. (2005). Queueing analysis of symmetric polling algorithm for DBA scheme in an EPON. In Choi, B.D., (ed.) Proceedings of the Korea–Netherlands Joint Conference on Queueing Theory and its Applications to Telecommunication Systems, Seoul, Korea. pp. 147–154.Google Scholar
24.Quine, M.P. (1972). The multitype Galton–Watson process with ρ near 1. Advances in Applied Probability 4: 429–452.CrossRefGoogle Scholar
25.Raz, D., Levy, H. & Avi-Ithzak, B. (2003). A resource allocation queueing fairness measure. In Proceedings of the ACM Sigmetrics. pp. 130–141.Google Scholar
26.Reiman, M.I., Rubio, R. & Wein, L.M. (1999). Heavy traffic analysis of the dynamic stochastic inventory-routing problem. Transportation Science 33: 361–380.Google Scholar
27.Reiman, M.I. & Wein, L.M. (1998). Dynamic scheduling of a two-class queue with setups. Operations Research 46: 532–547.Google Scholar
28.Resing, J.A.C. (1993). Polling systems and mulitype branching processes. Queueing Systems 13: 409–426.Google Scholar
29.Takagi, H. (1986). Analysis of polling systems. Cambridge, MA: MIT Press.Google Scholar
30.Takagi, H. (1990). Queueing analysis of polling models: an update. In Takagi, H. (ed.) Stochastic analysis of computer and communication systems. Amsterdam: North-Holland, 267–318.Google Scholar
31.Takagi, H. (1991). Application of polling models to computer networks. Computer Networks and ISDN Systems. 22, 193–211.Google Scholar
32.Takagi, H. (1997). Queneing analysis of polling models: progress in 1990–1994. In Dshalalow, J.H., (ed.) Frontiers in queueing: Models and applications in science and technology. Boca Raton, FL: CRC Press, pp. 119–146.Google Scholar
33.Van der Mei, R.D. (1999). Distribution of the delay in polling systems in heavy traffic. Performance Evoluation 31: 163–182.Google Scholar
34.Van der Mei, R.D. (1999). Delay in polling systems with large switch-over times. Journal of Applied Probability 36: 232–243.Google Scholar
35.Van der Mei, R.D. (2002). Waiting-time distributions in polling systems with simultaneous batch arrivals. Annals of Operations Research 113: 157–173.CrossRefGoogle Scholar
36.Van der Mei, R.D. (2007). Towards a unifying theory on branching-type polling models in heavy traffic. Queueing Systems 57: 29–49.Google Scholar
37.Van der Mei, R.D. & Levy, H. (1997). Polling systems in heavy traffic: Exhaustiveness of the serivce disciplines. Queueing Systems 27: 227–250.Google Scholar
38.Van der Mei, R.D. & Resing, J.A.C. (2007). Analysis of polling systems with two-stage gated serivce: fairness versus efficiency. In Mason, L., Drwiega, T. and Yan, J. (eds.) Managing traffic performance in converged networks: the interplay between convergent and divergent forces. Berlin: Springer-Verlag, pp. 544–555.Google Scholar
39.Van der Mei, R.D. & Winands, E.M.M. (2007). Polling models with renewal arrivals: A new method to derive heavy-traffic asymptotics. Performance Evalution 64: 1029–1040.Google Scholar
40.Van der Mei, R.D. & Winands, E.M.M. (2008). Mean value analysis for polling models in heavy traffic. Performance Evaluation 65: 400–416.CrossRefGoogle Scholar
41.Vatutin, V.A. & Dyakonova, E.E. (2002). Multitype branching processes and some queueing systems. Journal of Mathematical Sciences 111: 3901–3909.Google Scholar
42.Vishnevskii, V.M. & Semenova, O.V. (2006). Mathematical methods to study the polling systems. Automation and Remote Control 67: 173–220.Google Scholar
43.Wierman, A. & Harchol-Balter, M. (2003). Classifying scheduling policies with respect to unfairness in an M/G/1. In Proceedings of the ACM Sigmetrices, pp. 238–249.Google Scholar
44.Winands, E.M.M., Adan, I.J.B.F. & Van Houtum, G.J. (2006). Mean value analysis for polling systems. Queueing Systems 54: 35–44.Google Scholar
45.Winands, E.M.M. (2006). Branching-type polling systems with large setups. Techincal report, Technische Universiteit Eindhoven.Google Scholar