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Queueing and scheduling in random environments

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Nicholas Bambos  and
George Michailidis
Nicholas Bambos*
Affiliation:
Stanford University
George Michailidis*
Affiliation:
University of Michigan
*
∗ Postal address: Department of Management Science and Engineering and Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA.
∗∗ Postal address: Department of Statistics, University of Michigan, Ann Arbor, MI 48109-1092, USA. Email address: gmichail@umich.edu
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Abstract

We consider a processing system, composed of several parallel queues and a processor, which operates in a time-varying environment that fluctuates between various states or modes. The service rate at each queue depends on the processor bandwidth allocated to it, as well as the environment mode. Each queue is driven by a job traffic flow, which may also depend on the environment mode. Dynamic processor scheduling policies are investigated for maximizing the system throughput, by adapting to queue backlogs and the environment mode. We show that allocating the processor bandwidth to the queues, so as to maximize the projection of the service rate vector onto a linear function of the workload vector, can keep the system stable under the maximum possible traffic load. The analysis of the system dynamics is first done under very general assumptions, addressing rate stability and flow conservation on individual traffic and environment evolution traces. The connection with stochastic stability is later discussed for stationary and ergodic traffic and environment processes. Various extensions to feed-forward networks of such nodes, the multi-processor case, etc., are also discussed. The approach advances the methodology of trace-based modelling of queueing structures. Applications of the model include bandwidth allocation in wireless channels with fluctuating interference and allocation of switching bandwidth to traffic flows in communication networks with fluctuating congestion levels.

Keywords

Stability of queuesprocessing networkdynamic schedulingbandwidth allocationcomputer networkwireless network

MSC classification

Primary: 60K25: Queueing theory 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60K37: Processes in random environments
Secondary: 60G55: Point processes

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 293 - 317
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Armony, M. and Bambos, N. (2003). Queueing dynamics and maximal throughput scheduling in switched processing systems. Queueing Systems 44, 209–252.CrossRefGoogle Scholar
[2] Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, New York.Google Scholar
[3] Bambos, N. and Michailidis, G. (1995). On the stationary dynamics of parallel queues with random server connectivities. In Proc. 34th Conf. Decision Control (New Orleans, LA), IEEE Control Systems Society, pp. 3638–3643.Google Scholar
[4] Bambos, N. and Michailidis, G. (2001). Processor scheduling in fluctuating environments. Adaptive bandwidth allocation for throughput maximization. Tech. Rep. NetLab-2001-11/01, Engineering Library, Stanford University.Google Scholar
[5] Bambos, N. and Michailidis, G. (2001). Queueing networks of random link topology; stationary dynamics of maximal throughput schedules. Tech. Rep. NetLab-2001-10/02, Stanford University.Google Scholar
[6] Bambos, N. and Michailidis, G. (2002). On parallel queueing with random server connectivity and routing constraints. Prob. Eng. Inf. Sci. 16, 185–203.CrossRefGoogle Scholar
[7] Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. John Wiley, Chichester.Google Scholar
[8] El-Taha, M. and Stidham, S. (1999). Sample-Path Analysis of Queueing Systems. Kluwer, Boston, MA.Google Scholar
[9] Loomis, L. H. and Sternberg, S. (1990). Advanced Calculus. Revised Edition. Jones and Bartlett, Boston, MA.Google Scholar
[10] Lott, C. and Teneketzis, D. (2000). On the optimality of an index rule in multichannel allocation for single-hop mobile networks with multiple service rates. Prob. Eng. Inf. Sci. 14, 259–297.Google Scholar
[11] Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
[12] Petersen, K. (1983). Ergodic Theory. Cambridge University Press.Google Scholar
[13] Shakkottai, S. and Stolyar, A. L. (2002). Scheduling for multiple flows sharing a time-varying channel: the exponential rule. In Analytic Methods in Applied Probability. In memory of Fridrikh Karpelevich (Amer. Math. Soc. Trans. Ser. 2 207), ed. Suhov, Yu. M., American Mathematical Society, Providence, RI, pp. 185–201.Google Scholar
[14] Stolyar, A. L. (2001). MaxWeight scheduling in a generalized switch: state space collapse and equivalent workload minimization under complete resource pooling. Preprint.Google Scholar
[15] Tassiulas, L. (1997). Scheduling and performance limits of networks with constantly changing topology. IEEE Trans. Inf. Theory 43, 1067–1073.Google Scholar
[16] Tassiulas, L. and Ephremides, A. (1993). Dynamic server allocation to parallel queues with randomly varying connectivity. IEEE Trans. Inf. Theory 39, 466–478.CrossRefGoogle Scholar
[17] Walters, P. (1982). Introduction to Ergodic Theory (Graduate Texts Math. 79). Springer, New York.Google Scholar
[18] Wasserman, K. and Olsen, T. L. (2001). On mutually interfering parallel servers subject to external disturbances. Operat. Res. 49, 700–709.Google Scholar