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## PowerPoint Slideshow about ' Competitive Queueing Policies for QoS Switches' - lisle

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Outline

- Motivation
- Model description
- Summary of Previous and new results
- Non-preemptive queue
- Two packet types
- Multiple packet types
- Preemptive queue lower bound
- Open Questions

Motivation

- Quality of Service
- Guaranteed performance
- Limited resources
- Premium Service

Motivation (cont.)

- Assured service
- Relative (not Guaranteed) Performance
- Different packet priorities (values)
- High Network Utilization

Our model

- Input: a stream of valued packets.
- Actions: either accept or reject a packet
- Send events: at integer times
- Benefit = Total value of the packets sent.
- Main Variations:
- Non-Preemptive FIFO Queue
- Preemptive FIFO Queue
- Delay-Bounded Queue
- Competitive Analysis: ρ = max {offline/online}

Previous Results

- Non-Preemptive Queue
- (2-1)/ lower bound for 2 values and Analyzes specific policies (AMRR00)
- Preemptive Queue
- 2-o(1) competitive greedy algorithm (KLMPSS01)
- 1.28 lower bound for 2 values (Sviridenko01)
- 1.30 competitive algorithm for 2 values (LP02)
- Delay-Bounded Queue (KLMPSS01)
- 2 competitive greedy algorithm
- 1.17 lower bound for -uniform bounded delay
- 1.414 ρ 1.618 for 2-variable bounded delay
- 1.25 ρ 1.434 for 2-uniform bounded delay

Summary of Our Results

- Non-preemptive queue
- Algorithm with ρ = (2-1)/
- optimal for 2 values
- tight(er) bounds for previous policies
- ρ = (ln()) for continuous values
- Preemptive queue
- General lower bound of 1.414
- Exact ρ =1.434 for queue size 2
- Delay-Bounded queue
- 1.366 ρ 1.414 for 2-uniform bounded delay
- ρ = 1.618 for 2-variable bounded delay

Non-Preemptive Lower bound - 2 values

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ON

OFF

[From AMRR 2000]

Online accepts xB packets.

Offline accepts B packets.

Ratio is x

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ON

OFF

Online accepts xB low and at most (1-x)B high.

Offline accepts B high value packets.

Ratio is [x+(1-x)]/

Ratio Partition (RP) Policy

- Always accept high value packets.
- Each high value packet marks /(-1) low value packets in the queue that arrived before it.
- Accept a low packet if you can mark it by filling the queue with high value packets.

RP Example (1)

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Let = 2,

Each high value marks 2 low values.

Lemma:

When the queue is full, all packets in it are marked.

RP Analysis

- Full queue:
- all low value packets are marked.
- Online marked packets bound:
- offline high value packets.
- Marking parameter balances:
- accepted low value packets
- slots for future high value packets.
- Optimizing the marking parameter givesρ=(2-1)/.
- Optimal competitive ratio.

Continuous Values

- Create n= ln() sub-queues
- Sub-queue k accepts values [k-1/n,k/n]
- Sub-queues take turns in sending
- Can be simulated by a FIFO queue.
- Competitive ratio of e ln()
- Lower bound: ln()+1

Preemptive Lowerbound

i

Z

i-1

B-1

- Stage i includes:
- A burst of B-1 i-1 packets followed by one i
- At the next Z times units, one i packet each unit
- End with B packets of value k
- Stop: if B-Z packets are preempted in a stage.
- Optimize i and Z=B/2
- the lower bound converges towards 1.414.
- For B=2 the bound is 1.434.

i-1

i-1

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i

Open Problems

- Non-Preemptive queue & continuous values
- Close the constant gap between the upper (e ln()) and lower (ln()+1) bounds
- Preemptive queue & continuous values
- Is there a policy which has ρ ≤ 2-ε
- Delay-Bounded queue:
- Better than Greedy for delay > 2

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