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

4

7

9B

• 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

…

…

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

ε ε1

1

t = 3

1

1

1

1

ε

ε1 1

1

Same goes on…t=B+2

…

…

ε

1

1 1 1 1

B1

1

1

1

B1

1

1

1

1111

No more packets arrive

1

1

1

1 …

…

B

2B

G

Opt

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

• 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

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 ( )

mB

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

1 1

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

Lower bound - construction t=1

X 2

X 2

t=2t=3

X 3

X 3

t=4

X 4

X 4

t=7

X 7

No more packetsarrive

ON

OPT

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)

AS0. Partition into busy periods1. If load(max_queue) > B/2 – use

max_queue2. Otherwise, if there are queues that were

never full – use max_queue among them3. Otherwise, use max_queue

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

4-competitive upper bound

• Based on reduction to single-queue

• Generic Scheme: (A+Richter ’03)

Single queue admission control

Multi-queuescheduling

+ admission control

C-competitive 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

t = 4t = 3

Relax demonstration

1 9

7

24 24

91

7

9

7

9

t = 1t = 2

3

9

7 1 17

9

3

9

9

7

7

9 9

9

9

t = 5

7

7

t = 8

4 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

4

3

4

39

t = 1

3

39

9

4

4

93

t = 27

9 7

9

7 9

79

99

9

t = 3

8

8

8

88

48

t = 4

7

9

t = 5

4

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:

42

17

197

731

29

1

47

time

13

143

713

24

797

34

343 3

7291

σi :

σ*i :

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 (deterministicor randomized).A achieves c-approximation if and only if A achieves c-approximation with respectto all 0/1 sequences, for all possible tie

breaking

Zero-one principle - proof

1 x ≥ t Define: ft(x) =

0 otherwise

1 5 9 4 33

0 1 1 1 11

0 0 1 0 00

σ:

f3(σ):

f6(σ):

Proof – continued

Claim1: Sequence can be broken into sumof 0/1 sequences using ft:

Claim 2: For comparison-based A, sequenceσ, and t ≥ 0 :

)))((()))((( tt fAVAfV

0

))(()(t

t dtfVV

Putting it all together:

)))((( tfAV )))((( tfOptVc

1

0t

0t

dt dt

0

)))(((t

t dtAfV ))(( AV

0

)))(((1

t

t dtOptfVc

))((1 OptVc

)(tf - 0/1 sequence

Claim 2Claim 1

Claim 1

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

CIOQ switch

.

.

.

.

.

.

. .

. .

. .

1 1

NN

• 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