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Online Packet Switching Techniques and algorithms Yossi Azar Tel Aviv University 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 Single queue switch B 9 7 4 • FIFO queue with bounded capacity (B) • Packets marked with values • One packet transmitted each time step • Objective: maximize total transmitted value 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 Online Greedy is not optimal Same goespackets on… arrive t=B+2 tNo =3 1more 2 B B B B 1ε 1ε 1ε 1ε 1ε ε 1ε ε 1ε ε … … ε G 1ε … 1 1 B 1ε 1ε 1 1ε 1ε ε 1 1 1 1 … … 1 Opt 1ε 1 2B … 1 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 Multi-Queue QoS switch … 3 9 … 7 … m 1 … 4 … B • m bounded capacity FIFO queues • Single output port, one packet transmitted each time step • Objective: maximize total transmitted value 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 ( B m ) 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 1 1 1 m 1 1 B 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 t=3 t=2 t=4 t=7 t=1 Lower bound - construction ON X4 2 3 No more packets arrive OPT X X7 2 3 4 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 Multi-Queue QoS switch … 3 9 … 7 … m 1 … 4 … B • m bounded capacity FIFO queues • Single output port, one packet transmitted each time step • Objective: maximize total transmitted value 4-competitive upper bound • Based on reduction to single-queue • Generic Scheme: (A+Richter ’03) Single queue admission control C-competitive Multi-queue scheduling + admission control 2C-competitive 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 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 Relax demonstration t = 521348 9 74 912 1 9 17 4 92 9 1 9 37 3 7 9 7 7 4 79 7 3 1 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 M(G) - demonstration Relax simulation 9 t = 54312 9 73 79 93 84 8 4 73 79 93 84 8 84 9 9 8 7 4 3 9 4 9 8 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 Algorithm M(A) - analysis Theorem 2: CM(A) ≤ CRelax∙CA = 2∙CA Proof: • Relax is 2-competitive • In each queue we lose a factor of CA compared to non-fifo, by transforming the input sequence Compact σi: σi : 2 4 7 1 3 1 4 3 2 9 1 3 1 4 7 9 7 7 3 time σ*i : 2 4 1 7 1 3 2 9 1 4 3 7 3 1 3 4 7 9 7 time 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) 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) 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 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 Zero-one principle - proof 1 Define: ft(x) = x≥t 0 otherwise σ: 3 1 5 9 4 3 f3(σ): 1 0 1 1 1 1 f6(σ): 0 0 0 1 0 0 Proof – continued Claim1: Sequence can be broken into sum of 0/1 sequences using ft: V ( ) V ( ft ( ))dt t 0 Claim 2: For comparison-based A, sequence σ, and t ≥ 0 : V ( ft ( A( ))) V ( A( ft ( ))) Putting it all together: Claim 1 V ( A( )) Claim 2 V ( f ( A( ))) dt t t 0 V ( A( f ( )))dt t t 0 t 0 1 V (Opt( ft ( ))) dt c Claim 1 ft ( ) - 0/1 sequence 1 V ( ft (Opt ( ))) dt c t 0 1 V (Opt ( )) c Application 1 Algorithm TLH • Admission control: greedy, independently in each queue m B • Scheduling: Transmit packet with largest value among all packets at head of queues 0/1 principle -> TLH is 3-competitive CIOQ switch 1 . . . 1 . . . . . . • • • • . . . N N N×N switch Virtual output queues at input ports Speedup S Objective: maximize total transmitted value 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 Dynamic Routing • All models can be generalized to networks, with switches at the nodes – – – – Line topology Cycles Trees General networks • With / without routing decisions Example (line) • Dynamic Routing on a line of length k 0/1 principle -> simple alg. is (k+1)-comp. • Greedy is at least k0.5-competitive (AKOR ’03) Example (tree) • Merging trees (KLMP ’03) Summary • Single queue • Multiple queues • Multiple queues – unit packets • Zero-One principle • CIOQ switch • Networks (e.g. line, tree) 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