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

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

2. 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

3. 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

4. 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

5. 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

6. 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

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 1.4659 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 • 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

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. 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. 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!