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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. Bit-Torrent Swarms.
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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
Bit-Torrent Swarms • Swarm: set of users interested in the same file • Seed
Bandwidth Under-Utilization • Online P2P Networks • [Hajek and Zhu 10] • Unstable when λ> s! • Missing-piece syndrome: Each peer waiting for only onepiece • Seed • s chunks per sec • λ peers per sec
Bandwidth Under-Utilization • Mobile P2P Networks • Cached content is shared in a P2P fashion (eg. bluetooth) • Opportunistic communication • May not encounter the content they are interested in • ?
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?
Our Contributions • A versatile modelfor 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
Outline • Motivation • A model for universal swarms • Main results • Stability of universal swarms • Content exchange designs in universal swarms • Conclusion and future works
Peer Swarms • Peer requests one chunk iK • Peers requesting the same chunk form a peer swarm • ? • ? • ?
Peer Caches • Peer has cache size of C • Peer may use cache to store chunks it is not interested in • Cache • Request • C • ? • Stored chunks fK
Peer Arrivals • Peers arrive with full caches • Peers requesting iandcaching f arrive according to a Poisson process with rate λi, f • Cache • Request • C • ? • ? • ? • ? • Time
Peer Contact Process • Online P2P: random sampling • Mobile P2P: contact when within transmission range • × • × • Time • ? • ? • ?
Peer Contact Process (Cont.) • One peer contacts other peers according to a Poisson process with rate • N(t): number of peers in the system at time t • 0 ≤ β < 1 • β = 1 • 1< β ≤ 2 • Contact rate • Contact rate • Contact rate • μ • Contact-constrained • Constant-bandwidth • Interference-constrained • N(t) • N(t) • N(t)
Content Exchange Policy • 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) • A, A’ B, B’ • Conversion probability • ? • ? • ?
Outline • Motivation • Model for universal swarms • Main results • Stability of universal swarms • Content exchange designs in universal swarms • Conclusion and future works
Methodology: Fluid Limit • 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 • For allβ • For all content exchange policies
Universal Swarms • Question 1: How does inter-swarm exchange affect the system stability?
Stability of Static-Cache Policy • Let > 0 be the arrival rate of peers requesting i and storing j. 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! 16
Only One Swarm Can Become Unstable! • ? • ? • ? • At most one swarm can blow up!
Outline • Motivation • Model for universal swarms • Main results • Stability of universal swarms • Content exchange designs in universal swarms • Conclusion and future works
Universal Swarms • Question 2: How should chunks be exchanged across swarms?
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!
BARON: Valuation-Guided Replication • Centralized tracker maintains valuation vi for each chunk i • Positivevi: chunk i needs more replicas • Negativevi: chunk i needs fewer replicas • Replace the chunks with negative valuation with that with positive valuations 2 1 0 -1 … • ? • ?
BARON: Valuation Design • -- the number of peers requesting chunk i in the optimal state • -- the number of peers storing chunk iin the optimal state 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
BARON: Numerical Results • Evaluations based on fluid trajectories in MATLAB • Numerically solving ODEs • Valuation-guided content exchange improves the system stability
Conclusion and Future Works • Universal swarms achieve better stabilityeven 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
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 ifor which Moreover, for β in [0,1], the number of peers requesting item igrows to infinity, while the number of peers requesting other items remains bounded. • At most one swarm can blow up!