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An Optimal Lower Bound for Buffer Management in Multi-Queue Switches Marcin Bieńkowski. 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) - PowerPoint PPT Presentation
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An Optimal Lower Bound for Buffer Management in Multi-Queue Switches
Marcin Bieńkowski
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 2/19M. Bieńkowski
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 network
switchoutput m input queues (buffers)
Round 1 Round 3Round 2 Round 4
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 3/19M. Bieńkowski
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
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 4/19M. Bieńkowski
Fractional vs. randomized vs. deterministic
deterministicalgorithms
in standard model
deterministicalgorithms
in fractional model
randomizedalgorithms
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 ratiofor the fractional model.
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 5/19M. Bieńkowski
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 mFor any B and large m
1.4659 [2] 1.5 [4] [3]
[3]
2 [1]
2 [1]
[3]
[2]
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 6/19M. Bieńkowski
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 mFor any B and large m
1.4659 [2] 1.5 [4] [3]
[3]
2 [1]
2 [1]
[3]
[2]
NEW
IMPLIED
For any B and large m
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 7/19M. Bieńkowski
Our contribution (once again)
Lower bound of e/(e-1) on the competitive ratiofor the fractional model
This talk: we assume that B = 1
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 8/19M. Bieńkowski
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 :
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 9/19M. Bieńkowski
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 =>
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 10/19M. Bieńkowski
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!
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 11/19M. Bieńkowski
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!
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 12/19M. Bieńkowski
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?
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 13/19M. Bieńkowski
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
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 14/19M. Bieńkowski
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.
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 15/19M. Bieńkowski
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 S1?
n+j rounds
reducing properties of uniform strategies + simple counting
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 16/19M. Bieńkowski
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)!
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 17/19M. Bieńkowski
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:
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches 18/19M. Bieńkowski
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).
Thank you for your attention!