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

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### An Optimal Lower Bound for Buffer Management in Multi-Queue SwitchesMarcin 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

Round 1

Round 3

Round 2

Round 4

network

network

m input queues (buffers)

output

switch

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

Competitive analysis

Online problem, algorithm does not know the future

- Adversary: adds packets to buffers = creates input
- Algorithm: decides from which buffer to transmit

Competitive ratio:

Goal: maximize throughput = number of transmitted packets

throughput of the optimal offline algorithm on

throughput of online algorithm on

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

Fractional vs. randomized vs. deterministic

deterministic

algorithms in fractional model

randomized

algorithms in standard model

deterministic

algorithms

in standard model

harder for the algorithm, easier for the adversary

May send fractions of packets. The total load transmitted in one round is at most 1

Best competitive ratios:

This talk: A lower bound on the competitive ratiofor the fractional model.

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

Previous landscape of results (competitive ratios)

2 [1]

[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)

1

1.4659 [2]

1.5 [4]

[3]

fractional:

1

2 [1]

1.4659 [2]

1.5 [4]

[3]

randomized:

[3]

[2]

1

2 [1]

deterministic:

For any B and large m

for m >> B

Upper bounds: any B and m

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

This paper: the new landscape

2 [1]

[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)

1

1.4659 [2]

1.5 [4]

[3]

fractional:

NEW

1

2 [1]

1.4659 [2]

1.5 [4]

[3]

randomized:

IMPLIED

[3]

[2]

1

2 [1]

deterministic:

For any B and large m

For any B and large m

for m >> B

Upper bounds: any B and m

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

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

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

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

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

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

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
- Injecttwo 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 11:

total load = 3.824

After round 10:

total load = 4.299

After round 11:

total load = 3.799

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!

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

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

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

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

How good is strategy S1?

Adversarial strategy for buffers:

- Fill all buffers
- For in
- Attack n buffers and denote them
- Execute uniform strategy on for rounds

n+j rounds

This strategy is - reducing

Competitive ratio: for

reducing properties of uniform strategies + simple counting

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

This is a transformation!

Adversarial strategy for buffers:

- Fill all buffers
- For in
- Attack n buffers and denote them
- Execute uniform strategy on for rounds

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

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

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).

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

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