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

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


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


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.


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]


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


Our contribution (once again)

Lower bound of e/(e-1) on the competitive ratiofor the fractional model

This talk: we assume that B = 1


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 :


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 =>


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!


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




Uniform strategy is better so far (in terms of the total load).But by deferring injection, the adversary gained a better configuration!



Deferred injections help reducing the total load!


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?


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


Improving uniform strategy

Set of n full buffers



Design rationale: inside and outside of the average load decrease (approximately) at the same pace.


This strategy is - reducing

Competitive ratio: for



How good is strategy S1?

n+j rounds

reducing properties of uniform strategies + simple counting




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)!


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:


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!

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

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