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An Optimal Lower Bound for Buffer Management in Multi-Queue Switches Marcin Bieńkowski

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 2/19 Problem definition Discrete time divided into rounds. Any number of packets arrive (at the beginning of a round) Algorithm may transmit one packet (during a round) Buffers have limited capacity (each equal B) Packet overflow => packets get lost network switchoutput m input queues (buffers) Round 1Round 3Round 2Round 4

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 3/19 Online problem, algorithm does not know the future Adversary: adds packets to buffers = creates input Algorithm: decides from which buffer to transmit Competitive ratio: Competitive analysis throughput of the optimal offline algorithm on throughput of online algorithm on Goal: maximize throughput = number of transmitted packets

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 4/19 Fractional vs. randomized vs. deterministic deterministic algorithms in standard model deterministic algorithms in fractional model randomized algorithms in standard model Best competitive ratios: May send fractions of packets. The total load transmitted in one round is at most 1 harder for the algorithm, easier for the adversary This talk: A lower bound on the competitive ratio for the fractional model.

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 5/19 Previous landscape of results (competitive ratios) [1] Azar, Richter (STOC 03): work conserving alg. [2] Albers, Schmidt (STOC 04): lower bounds [3] Azar, Litichevskey (ESA 04): fractional (by online matching) + transformation from fractional to deterministic [4] Random Permutation algorithm (STACS 05) fractional: randomized: deterministic: 1 1 1 2 [1] for m >> B 1.4659 [2] 1.5 [4] Upper bounds: any B and m For any B and large m 1.4659 [2] 1.5 [4] [3] 2 [1] [3] [2]

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 6/19 This paper: the new landscape [1] Azar, Richter (STOC 03): work conserving alg. [2] Albers, Schmidt (STOC 04): lower bounds [3] Azar, Litichevskey (ESA 04): fractional (by online matching) + transformation from fractional to deterministic [4] Random Permutation algorithm (STACS 05) fractional: randomized : deterministic: 1 1 1 2 [1] for m >> B 1.4659 [2] 1.5 [4] Upper bounds: any B and m For any B and large m 1.4659 [2] 1.5 [4] [3] 2 [1] [3] [2] NEW IMPLIED For any B and large m

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 7/19 Our contribution (once again) Lower bound of e/(e-1) on the competitive ratio for the fractional model This talk: we assume that B = 1

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 8/19 Old lower bound by Albers and Schmidt (1) Uniform strategy for the adversary: Fill all buffers at the beginning Repeat: wait a round and inject a packet to the most loaded buffer Total load of ALG At the beginning: After round 1: After round 2: … After round :

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 9/19 Old lower bound by Albers and Schmidt (2) We call a strategy (T,L)-reducing if it takes T rounds it reduces the total load (even applied to full buffers) to at most L OPT can serve the input losslessly. Uniform strategies: are -reducing Best competitive ratio of ¼ 1.4659 achieved for (T,L)-reducing strategy =>

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 10/19 Deferring injections (1) In other words: Adversary tries to inject as many packets and as soon as possible, while still being able to serve the sequence losslessly. Can the adversary win anything by deferring the injection, e.g., waiting for 2 rounds and then injecting 2 packets at once? In the analysis of the Random Permutation algorithm, it is argued that it is not the case. Let’s check!

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 11/19 Deferring injections: example & comparison Uniform strategy: 8 rounds of uniform strategy. Inject a packet after round 9 Inject a packet after round 10 Inject a packet after round 11 Strategy with deferred injection: 8 rounds of uniform strategy. Do not inject a packet after round 9 Inject two packets after round 10 Inject a packet after round 11 After round 8: total load = 4.874 After round 8: total load = 4.874 After round 10: total load = 4.138 After round 10: total load = 4.299 Uniform strategy is better so far (in terms of the total load). But by deferring injection, the adversary gained a better configuration! After round 11: total load = 3.824 After round 11: total load = 3.799 Deferred injections help reducing the total load!

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 12/19 What did we learn from the last example? Uniform strategies reduce the load roughly by (m-1)/m in each step. This becomes less effective when buffers are less populated. Remedy: At that time fill simultaneously a subset of buffers and then apply uniform strategy only to this subset. How to generalize this idea?

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 13/19 Improving uniform strategy Set of n full buffers Adversarial strategy for buffers: Fill all buffers For in Attack n buffers and denote them Execute uniform strategy on for rounds Wlog., in these rounds ALG transmits the load only from

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 14/19 Improving uniform strategy Set of n full buffers Adversarial strategy for buffers: Fill all buffers For in Attack n buffers and denote them Execute uniform strategy on for rounds Design rationale: inside and outside of the average load decrease (approximately) at the same pace.

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 15/19 This strategy is - reducing Competitive ratio: for Adversarial strategy for buffers: Fill all buffers For in Attack n buffers and denote them Execute uniform strategy on for rounds How good is strategy S 1 ? n+j rounds reducing properties of uniform strategies + simple counting

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 16/19 Adversarial strategy for buffers: Fill all buffers For in Attack n buffers and denote them Execute uniform strategy on for rounds This is a transformation! What we did: On the basis of a strategy for n buffers… … treating it as a black box … … we created a more efficient strategy for buffers. We may apply this transformation again (and again)!

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 17/19 Series of strategies (Neglecting rounding issues, problems with lower-order terms, and other gory details) Uniform on buffers: -reducing on buffers: -reducing on buffers: -red. … … …. In the limit: strategy for M buffers that is -reducing Competitive ratio:

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M. BieńkowskiAn Optimal Lower Bound for Buffer Management in Multi-Queue Switches 18/19 Final remarks When B > 1, all arguments remains intact. We showed a lower bound of Open problem: what is the exact competitive ratio for small m? The approach of Albers and Schmidt yields a lower bound 16/13 for m = 2 which is matched [B., Mądry 08], [Kobayashi, Miyazaki, Okabe 08] The approach of Albers and Schmidt stops to be optimal for m > 8 (deferring injections are better).

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