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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• 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!
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