Dynamic Internet Congestion with Bursts
Stefan Schmid
Roger Wattenhofer
Distributed Computing Group, ETH Zurich
13th International Conference On High Performance Computing (HiPC)
Bangalore, India, December 2006
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TCP Congestion Control
• The available bandwidth changes dynamically over time depending on the demands of other computers.
• In order to prevent collapses, hosts in the
Internet collaboratively reduce load in busy
times of high congestion!
• Successful strategy: TCP congestion control - Additive Increase, Muliplicative Decrease (AIMD)
- Indications for congestion: e.g., packet loss
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Selfish Behavior (2)
• Some participants may not care about stability of Internet, but selfishly aim at maximizing own throughput!
• Given the dynamics of the available bandwidth, selfish throughput maximization constitutes an optimization problem!
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In this Paper…
• Introduction of models for dynamic changes of congestion.
• Study of selfish (online) algorithms which maximize throughput.
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Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 11
Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 12
Model (1)
• We divide time into rounds t, for t = 1, 2, ….!
• The available bandwidth at time t is ut
• The selfish sender uses a sending rate xt at time t
• Selfish player does not know ut: All a sender knows is whether her sending in the last round was larger than the available bandwidth (i.e., xt>ut, hence congestion!), or not (binary feedback).
- If xt>ut packets are dropped by routers.
- Consequently, a selfish transfer protocol has to decide xt without knowing the present or future available bandwidth: framework for online algorithms!
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Model (2)
• The optimization problem can be formalized as follows!
• Gain of optimal (offline algorithm) OPT:
• Gain of online algorithm ALG:
Maybe harsh, but retransmissions, timeouts, etc. is overhead!
t
rate
ut
xt
Packets come through,
but opportunity costs!
Sending rate too large,
no transmission at all!
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Model (3)
• Goal of the online algorithm is to send always at the rate of the available bandwidth, or slightly lower!
• We are interested in minimizing the strict competitive ratio (worst-case!):
That is, the gain of ALG should be almost as large as the one of the optimal offline algorithm OPT!
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Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 16
Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 17
Multiplicative Dynamics (1)
• If ut can change arbitrarily over time, there is no competitive algorithm: ut can always be chosen slightly smaller than xt!
• However, assuming arbitrary changes may also be too pessimistic!
• Consequently, we want to restrict the dynamics.
• Model 1: Multiplicative dynamics changes max by a constant factor μ, i.e., an adversary (worst-case!) can choose the available bandwidth from the interval
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Multiplicative Dynamics (2)
• Online Algorithm: After a round with sending rate lower or equal the available bandwidth, increase rate by a factor of μ, otherwise reduce sending rate by a factor μ3
• Analysis: - After a „bad“ round, there will always be a „good“ round due to the sharp cut of the sending rate.
- Good rounds are at most μ4-competitive.
- The gain of OPT in bad round is at most a factor μ larger than the gain of ALG in the preceding good round.
- Consequently,
Stefan Schmid, ETH Zurich @ HiPC 2006 19
Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 20
Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 21
Bursty Dynamics (1)
• So far: Adversary can change congestion by at most a constant factor in each round.
• There are many additional models for congestion dynamics, waiting for efficient online algorithms!
• One dynamics model studied on the network layer is network calculus!
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Bursty Dynamics (2)
• Network Calculus is used to analyse queuing strategies in networks from a worst-case perspective (worst-case queuing)!
• Network Caculus are not only interesting on the network layer, but may serve as a good dynamics model on the transport layer as well!
• In our paper, we propose to study Network Calculus models for congestion control!
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Network Calculus (1)
• Traditional Network Calculus- Defines arrival curves (e.g., leaky-bucket arrival curve)- Traffic coming out of a router is assumed to adhere to arrival curve.- If this is the case, bounds for queue lengths and delays can be computed (with min-plus algebra).
Arrival curve:
max burst b and rate r
Total number of bits coming out of
router should never exceed arrival
curve attached at all points!
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Network Calculus (2)
• Leaky-bucket arrival curve allows for bursts in the traffic, as long as they are only temporal.
• After quite times with low rates, power can be accumulated for another traffic burst.
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Dynamic Network Calculus Congestion
• We adopt these properties and allow our congestion adversary to change the available bandwidth with bursts!
• The adversary can choose the new bandwidth as follows:
• Thereby,
Arrival curve: accumulate
during quiet times with few changes,
but at most factor B
Change in round t
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Results
• Upper Bound: Online algorithm which cuts sending rate by half after bad rounds, and increases the rate by μ B1/3 yields a competitive ratio of
• Lower Bound: No online algorithm can achieve a competitive ratio better than
against a Network Calculus adversary.
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Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 28
Talk Overview
• Model
• Multiplicative Dynamics
• „Bursty Dynamics“
• Open Research Questions and Conclusion
Stefan Schmid, ETH Zurich @ HiPC 2006 29
Open Research Questions
• Selfish TCP: A real threat?
• Verification of model in practice!
• Fill gap between our upper and lower bound!
• Randomized algorithms (also for multiplicative adversary)
• Other arrival curves, study of different dynamics
• More generally: Adaption and analysis of network calculus for other dynamic models! Limitations?
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Discussion
• Selfishness in congestion control
- Devise throughput maximizing protocols
• Network Calculus: An interesting model for dynamics! - Lots of future research! - However, challenging analysis!
• Transport layer: Algorithmically less understood than other layers!
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Questions and Comments?
Stefan SchmidDistributed Computing Group
http://dcg.ethz.ch/members/stefan.html
Thank you for your attention!
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