Competitive Buffer Management with Packet Dependencies

Alex Kesselman, GoogleBoaz Patt-Shamir, Tel Aviv University

Gabriel Scalosub, University of Toronto

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

2

Motivation: Video Streaming

• Smart encoding:– Suffices to recover many

• Every video frame is fragmented into packets• Restoration depends on recovering all packets• If packets are lost:– Affects other packets as well (become redundant)– Streaming: retransmission is not an option

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

3

Buffering Schematics

buffer management

incoming packets

outgoingstream

dropped packets

finite linkspeed (“drain rate”)

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

4

Buffer Model

• Single FIFO queue of size • Discrete time:– Delivery substep

• One packet delivered from head of queue

– Arrival substep• Packets arrive• Some packets may be dropped• Packets accommodated in the buffer

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

5

Traffic and Goal

• Traffic: frames consist of packets

– E.g.

• Goal: Maximize number of whole frames delivered

• Methodology: Competitive analysis– Algorithm is -competitive if for every arrival

sequence

frame ‘s j-packet

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

6

Previous Work

• Buffer management for QoS– Multi-valued packets– Constant competitive ratio for

finite values

• Some results for multiple buffers

• No results for co-dependent packets!

[Kesselman et al., 2001],[Englert & Westerman, 2006],

and many more

[Mansour et al., 2000],

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

7

Our Results

• Frames consisting of k>1 parts

• Bad news: No finite competitive ratio in general

• Good news: If frame parts are consistently ordered– There exists an algorithm with c.r. – All deterministic algorithms have c.r.

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

8

Preliminaries

• Offline– Closely related to k-DM (as hard)– Simple greedy algorithm is a (k+1)-approximation

• Online (arbitrary traffic)– Not much you can do

…

time

OPT

ALG

…

packetsALG

OPT

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

9

Restricted Traffic

• Problem:– Selective unbounded delay/burstiness

• Model requirement (solution):– Both ALG and OPT have to deal with same delay/burstiness

• Order-respecting traffic:– Frame order induced by j-packets is the same for every j

• OK: 2.1 (frame 2, part 1), 3.1, 2.2, 3.2• Not OK: 3.1, 2.1, 2.2, 3.2

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

10

All Is Well if Order-Respecting?

• Answer: Yes and No

• No:– Any deterministic algorithm has competitive ratio at least

• Yes:– A natural preemptive greedy approach– Conservative non-preemptive approaches

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

11

Greedy Algorithm (GA)

• Consider k=2

At every time t where overflow occurs, keep packets according to the following priority:

1. 2-packets whose 1-packets were delivered2. Complete frames (both 1-packets and 2-packets)3. Remaining 1-packets

• Theorem: GA is constant-competitive for k=2

• What if k>3?

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

12

Greedy Algorithm for k=3

GA

OPT

time

OPT

ALG

… ……

…

1-packets

2-packets

3-packets

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

13

Static-Partitioning Algorithm (SPA)

• Intuition– Think ahead: focus on admission control– Virtually partition the buffer into k levels of size

• Buffer is still FIFO!!

– Level j only holds j-packets– Level j accepts j-packets that are “evenly” spaced in time

• Alternating accept/reject periods

– Levels synchronize on frame index• Ensures delivered packets correspond to the same frame

• Extra perk: non-preemptive

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

14

Example: SPA for k=2

• Consider level 1, i.e., 1-packets

• 1-sync frame indices:• Accepts first 1-packets after every 1-sync– Specifically, has sufficient buffer space

time

Accept packetsWait time unitsAccept packetsWait time units

is the first 1-packet arriving after reject period is the first 1-packet arriving after reject period

A R RA

is the first 1-packet

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

15

Example: SPA for k=2

• Consider level 2, i.e., 2-packets

• 2-sync indices 1-sync indices• Accepts first 2-packets after every 2-sync– Specifically, has sufficient buffer space

time

Accept packetsWait time units

A R

is the first 2-packet

Accept packetsWait time units

A R

is the first 2-packet of a 1-sync arriving after reject period is the first 2-packet of a 1-sync arriving after reject period

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

16

Analysis of SPA for arbitrary k

• Consider two consecutive k-syncs :

– SPA delivers out of frames

– OPT delivers at most out of frames

• Intuition: at most out of every level

• Theorem: SPA is -competitive

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

17

Summary

• A new model in buffer management– Traffic has inter-packet dependencies

• Highly applicable to, e.g., video streaming

• First analytic results (still a lot to discover…)– Competitive algorithms (and lower bounds)– Complexity

Dagstuhl 2009 Competitive Buffer Management with Packet Dependencies

18

Still Open

• Gap: vs.

• Randomization– Useful in the packet-weights models

• How does it work for real traffic?– Is greedy still an option?

• Using forward-error-correction (FEC)– Suffices to deliver m-of-k– Some preliminary results, but still a lot to discover

• High-order dependencies

Thank You!