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

2. An Optimal Lower Bound for Buffer Management in Multi-Queue Switches
Problem definition
Round 3
Round 1
Round 4
Round 2
network
network
output
switch
m input queues (buffers)
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
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

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

4. Fractional vs. randomized vs. 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 ratio for the fractional model.
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

5. Previous landscape of results (competitive ratios)1
[2]
1.5 [4]
2 [1]
[3]
fractional: 1
[2]
1.5 [4]
2 [1]
[3]
randomized:
[2]
[3] 1
2 [1]
deterministic:
For any B and large m
for m >> B
Upper bounds: any B and m
[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)
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

6. This paper: the new landscape1
[2]
1.5 [4]
2 [1]
[3]
fractional:
NEW 1
[2]
1.5 [4]
2 [1]
[3]
randomized:
IMPLIED
[2]
[3] 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
[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)
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

7. 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

8. 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. 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 achieved for
(T,L)-reducing strategy =>
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

10. 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. 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 10:
total load = 4.138
After round 8:
total load = 4.874
After round 11:
total load = 3.824
After round 11:
total load = 3.799
After round 8:
total load = 4.874
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!
Deferred injections help reducing the total load!
An Optimal Lower Bound for Buffer Management in Multi-Queue Switches

12. 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. 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. 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. 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

16. 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

17. 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

18. 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

19. Thank you for your attention!