On the Stability and Optimality of Universal Swarms

Xia Zhou*, Stratis Ioannidis♯, and Laurent Massoulié+

*University of California, Santa Barbara♯Technicolor Research Lab, Palo Alto

+Technicolor Research Lab, Paris

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• Swarm: set of users interested in the same file

Bit-Torrent Swarms

Seed

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[Hajek and Zhu 10]• Unstable when λ > s!• Missing-piece syndrome: Each peer waiting for only one

piece

Bandwidth Under-Utilization

Seed

λ peers per sec

s chunks per sec

Online P2P Networks

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Bandwidth Under-Utilization

• Cached content is shared in a P2P fashion (eg. bluetooth)• Opportunistic communication• May not encounter the content they are interested in

Mobile P2P Networks

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Universal Swarms

Key idea: Exchange chunks across swarms upon bandwidth under-utilization

Question 1: How does such inter-swarm exchange affect stability?

Question 2: How should items be exchanged among swarms?

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• A versatile model for universal swarms

• Universal swarms achieve better stability compared to autonomous swarms

• Only one swarm can become unstable!

• Optimal replication ratios that minimize the time for peers to retrieve interested content

Our Contributions

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• Motivation

• A model for universal swarms

• Main results• Stability of universal swarms• Content exchange designs in universal swarms

• Conclusion and future works

Outline

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• Peer requests one chunk i K• Peers requesting the same chunk form a peer

swarm

Peer Swarms

??

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• Peer has cache size of C• Peer may use cache to store chunks it is not

interested in

Peer Caches

Cache

?

Request

C

Stored chunks f K

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• Peers arrive with full caches• Peers requesting i and caching f arrive

according to a Poisson process with rate λi, f

Peer Arrivals

Time

? ? ?Cache Reques

t

C ?

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• Online P2P: random sampling• Mobile P2P: contact when within transmission range

Peer Contact Process

Time

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×

? ?

×

• One peer contacts other peers according to a Poisson process with rate

• N(t): number of peers in the system at time t

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Peer Contact Process (Cont.)

N(t)

Contact rate

0 ≤ β < 1

Constant-bandwidth

N(t)

Contact rate

μ

β = 1

N(t)

Contact rate

1 < β ≤ 2

Contact-constrained

Interference-constrained

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• If encountering requested chunk: Grab-and-Go• Otherwise:• Static-cache policy: no change on cached chunks• Alternatives: updating cached contents

• Requested chunk and cached chunks define a peer class• N(t): system state at time t (# of peers in each peer class)

Content Exchange Policy

? ??

Conversion probability

A, A’ B, B’

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• Motivation

• Model for universal swarms

• Main results• Stability of universal swarms• Content exchange designs in universal swarms

• Conclusion and future works

Outline

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The evolution of the universal swarm system can be approximated arbitrarily well by the solution of a system of ODEs that depend on the conversion probabilities

Methodology: Fluid Limit

• For all β

• For all content exchange policies

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Question 1:

How does inter-swarm exchange affect the system stability?

Universal Swarms

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Stability of Static-Cache PolicyLet > 0 be the arrival rate of peers requesting i and storing j.

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Theorem: The system is stable under the static cache policy if and only if:

• Independent of β and cache size C

• The system is stable even if arrivals of peers requesting i exceed arrivals of peers storing i!

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Only One Swarm Can Become Unstable!

• At most one swarm can blow up!

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?

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• Motivation

• Model for universal swarms

• Main results• Stability of universal swarms• Content exchange designs in universal

swarms

• Conclusion and future works

Outline

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Question 2:

How should chunks be exchanged across swarms?

Universal Swarms

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Optimal Demand and Supply

-- the number of peers requesting chunk i (demand)

-- the number of peers storing chunk i (supply)

Theorem: Under the grab-and-go principle, the average sojourn time of a peer in the system is minimized when

where .

• The optimal supply is C times the demand!

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• Centralized tracker maintains valuation vi for each chunk i

• Positive vi : chunk i needs more replicas

• Negative vi : chunk i needs fewer replicas

• Replace the chunks with negative valuation with that with positive valuations

BARON: Valuation-Guided Replication

2 1 0 -1

…

??

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-- the number of peers requesting chunk i in the optimal state

-- the number of peers storing chunk i in the optimal state

BARON: Valuation Design

Optimal:

Valuation:

No need to know arrival rates and contact rates, but only the cache size C

Need to track the demand and supply dynamically

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• Evaluations based on fluid trajectories in MATLAB

• Numerically solving ODEs

BARON: Numerical Results

Valuation-guided content exchange improves the system stability

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• Universal swarms achieve better stability even with the simplest replication strategy

• At most one swarm can blow up!

• Optimal supply linearly proportional to the demand

• BARON extends the stability region using valuations

• Better understanding of the dynamics under more sophisticated content exchange mechanisms

• Peer incentives• Removing the assumption of one-chunk request

Conclusion and Future Works

Thank you!

Backup

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Only One Swarm Can Become Unstable!

Let > 0 be the arrival rate of peers requesting i and storing j.

Theorem: There exists at most one item i for which

Moreover, for β in [0,1], the number of peers requesting item i grows to infinity, while the number of peers requesting other items remains bounded.

• At most one swarm can blow up!