1. Online Packet Switching Techniques and algorithms Yossi Azar Tel Aviv University

2. Motivation Current networks are mostly packet-based (Internet) QoS guarantees essential to most network applications Steady traffic increase + constant fluctuation lead to packet loss Objective: transmit “valuable” packets

3. Single queue switch 4 7 9 B FIFO queue with bounded capacity ( B ) Packets marked with values One packet transmitted each time step Objective: maximize total transmitted value

4. Greedy single queue admission control (preemptive) Algorithm G Accept packets greedily. Packet accepted if: Queue not full -or- Packet with smallest value discarded from queue

5. Online Greedy is not optimal …… t = 1 B ε ε ε ε ε ε ε B ε ε ε ε ε ε ε εεεε ε ε εεε ε 1 1 t = 2 1 1 ε εε1 1 t = 3 1 1 1 1 ε ε11 1 Same goes on… t=B+2 … … ε 1 1111 B 1 1 1 1 B 1 1 1 1 1111 No more packets arrive 1 1 1 1 … … B 2B G Opt

6. Single queue – results Upper bound: –[KLMPSS '01] – Greedy is 2-competitive –[KMvS '03] – 1.98-competitive –[BFKMSS '04] – 1.75-competitive Lower bound: –[AMZ ’03] – 1.41

7. Multi-Queue QoS switch m bounded capacity FIFO queues Single output port, one packet transmitted each time step Objective: maximize total transmitted value … … … … … B m 3 1 4 9 7

8. Multi-queue switch - results Arbitrary values: –[AR '03] – 4-competitive algorithm –[AR '04] – 3-competitive algorithm Unit value: –[AR '03] – deterministic 2-competitive randomized 1.58-competitive –[AS '04] – deterministic 1.89 –[AL '04] - deterministic 1.58 ( )

9. Special case – unit packets Model remains the same All packets have equal (unit) value Goal: maximize number of transmitted packets Motivation: IP networks Better algorithms for this case B m 1 1 1 11

10. Lower bound for unit-value B=1 Packet arrives to each queue As long as ON has at least two full queues: –ON empties some queue –Adversary empties queue not used by ON –New packet arrives to this queue

11. Lower bound - construction t=1 X 2 t=2t=3 X 3 t=4 X 4 t=7 X 7 No more packets arrive ON OPT

12. Getting below 2-competitive “Any” algorithm is 2-competitive Randomized 1.58-compeititve (AR ‘03) (Albers+Schmidt ’04): –Any Greedy is at least 2-competitive –First deterministic 1.89 Deterministic 1.58 (large buffers) (AL ’04) AS 0. Partition into busy periods 1.If load(max_queue) > B/2 – use max_queue 2.Otherwise, if there are queues that were never full – use max_queue among them 3.Otherwise, use max_queue

13. Multi-Queue QoS switch m bounded capacity FIFO queues Single output port, one packet transmitted each time step Objective: maximize total transmitted value … … … … … B m 3 1 4 9 7

14. 4-competitive upper bound Based on reduction to single-queue Generic Scheme: (A+Richter ’03) Single queue admission control Multi-queue scheduling + admission control C-competitive2C-competitive

15. Model Relaxation Relaxation: –packets can be transmitted in any order, not only FIFO –preemption allowed Optimal solution remains unchanged Relaxation adds considerable strength to online algorithms

16. Relaxed model – algorithm Relax Algorithm Relax Admission control: Greedy algorithm (G) in each queue (optimal non-fifo) Scheduling: Transmit packet with largest value in all queues

17. t = 4t = 3 Relax demonstration 19 7 24 24 91 7 9 7 9 t = 1t = 2 3 9 71 17 9 3 9 9 7 7 9 9 9 9 t = 5 7 7 t = 8 431

18. Generic Scheme Algorithm M(A) (A – admission control for single queue) Maintain online simulation of Relax Admission control: according to A Scheduling: according to Relax

19. M(G) - demonstration Relax simulation 9 4 3 4 39 t = 1 3 39 9 4 4 9 3 t = 2 7 97 9 79 79 9 9 9 t = 3 8 8 8 8 8 4 8 t = 4 7 9 t = 5 4 8

20. Algorithm Relax - analysis Theorem 1: Relax is 2-competitive in relaxed model. Proof: Relies on potential function Based on minimum weighted perfect matching in a graph that measures the distance between the values on Relax and OPT

21. Algorithm M(A) - analysis Theorem 2: C M(A) ≤ C Relax ∙C A = 2∙C A Proof: Relax is 2-competitive In each queue we lose a factor of C A compared to non-fifo, by transforming the input sequence

22. Compact σ i : 4 2 1 7 1 9 7 7 31 2 9 1 4 7 time 1 3 1 4 3 7 1 3 2 4 7 9 7 3 4 3 4 33 7 2 91 σ i : σ* i : time

23. Corollaries Preemptive: 4-competitive (using KLMPSS ’01) Unit-value: 2-competitive 2-values, preemptive: 2.6-comp. (using LP ’02) Non-preemptive: 2e∙ln(max / min)-competitive (using AMZ ’03)

24. Zero-One Principle (A+Richter ’04) Analysis of packets with arbitrary values is complicated Goal: reduce to “simpler” sequences Zero-one principle: Comparison-based algorithm (given a network) Sufficient to analyze 0/1 sequences (with arbitrary tie breaking)

25. Comparison-based algorithms Informally, A is comparison-based if decisions made based on relative order between values Notation: A(σ) – possible output sequences, ties broken in every possible way V(σ) – total value of sequence

26. Zero-one principle Theorem: Let A be comparison-based (deterministic or randomized). A achieves c-approximation if and only if A achieves c-approximation with respect to all 0/1 sequences, for all possible tie breaking

27. Zero-one principle - proof 1 x ≥ t Define: f t (x) = 0 otherwise 159433 011111 001000 σ:σ: f 3 (σ): f 6 (σ):

28. Proof – continued Claim1: Sequence can be broken into sum of 0/1 sequences using f t : Claim 2: For comparison-based A, sequence σ, and t ≥ 0 :

29. Putting it all together: - 0/1 sequence Claim 2Claim 1

30. Application 1 B m Algorithm TLH Admission control: greedy, independently in each queue Scheduling: Transmit packet with largest value among all packets at head of queues 0/1 principle -> TLH is 3-competitive

31. CIOQ switch............. 11 NN N×N switch Virtual output queues at input ports Speedup S Objective: maximize total transmitted value

32. Results General CIOQ switch: –arbitrary packet values –Any speedup (Kesselman+Rosen ’03): –Linear in speedup -or- –Logarithmic in value range (A+Richter ’04): –constant-competitive algorithm

33. Dynamic Routing All models can be generalized to networks, with switches at the nodes –Line topology –Cycles –Trees –General networks With / without routing decisions

34. Example (line) Dynamic Routing on a line of length k 0/1 principle -> simple alg. is (k+1)-comp. Greedy is at least k 0.5 -competitive (AKOR ’03) Example (tree) Merging trees (KLMP ’03)

35. Summary Single queue Multiple queues Multiple queues – unit packets Zero-One principle CIOQ switch Networks (e.g. line, tree)

36. Open problems Explore connections between different models General theorems to facilitate analysis Improve upper bounds of specific problems: –Single-queue switches –Multi-queue switches –Dynamic routing on a line –General networks