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Online Packet Switching Techniques and algorithms

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Online Packet Switching Techniques and algorithms Yossi Azar Tel Aviv University – PowerPoint PPT presentation

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Title: Online Packet Switching Techniques and algorithms


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
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

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
t 2
t 3
Same goes on
tB2
No more packets arrive
G
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B
Opt
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2B
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

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

10
Lower bound for unit-value
  • B1
  • 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
t1
t2
t3
t4
t7
ON
No more packets arrive
X 2
X 3
X 4
OPT
X 2
X 3
X 4
X 7
12
Getting below 2-competitive
  • Any algorithm is 2-competitive
  • Randomized 1.58-compeititve (AR 03)
  • (AlbersSchmidt 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
  • If load(max_queue) gt B/2 use max_queue
  • Otherwise, if there are queues that were never
    full use max_queue among them
  • 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

14
4-competitive upper bound
  • Based on reduction to single-queue
  • Generic Scheme (ARichter 03)

C-competitive
2C-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
Relax demonstration
t 4
t 3
t 1
t 2
t 5
t 8
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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
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t 1
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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 CM(A) CRelaxCA 2CA
  • Proof
  • Relax is 2-competitive
  • In each queue we lose a factor of CA
  • compared to non-fifo, by transforming the input
    sequence

22
  • Compact si

si
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time
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23
Corollaries
  • Preemptive 4-competitive (using KLMPSS 01)
  • Unit-value 2-competitive
  • 2-values, preemptive 2.6-comp. (using LP 02)
  • Non-preemptive 2eln(max / min)-competitive
  • (using AMZ 03)

24
Zero-One Principle (ARichter 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(s) possible output sequences, ties broken in
    every possible way
  • V(s) 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 ft(x)
  • 0 otherwise

28
Proof continued
  • Claim1 Sequence can be broken into sum
  • of 0/1 sequences using ft
  • Claim 2 For comparison-based A, sequence
  • s, and t 0

29
  • Putting it all together

30
Application 1
  • 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 -gt TLH is
  • 3-competitive

31
CIOQ switch
  • NN switch
  • Virtual output queues at input ports
  • Speedup S
  • Objective maximize total transmitted value

32
Results
  • General CIOQ switch
  • arbitrary packet values
  • Any speedup
  • (KesselmanRosen 03)
  • Linear in speedup
  • -or-
  • Logarithmic in value range
  • (ARichter 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 -gt simple alg. is (k1)-comp.
  • Greedy is at least k0.5-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
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