On the Stability and Optimality of Universal Swarms

1. 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

2. Bit-Torrent Swarms • Swarm: set of users interested in the same file • Seed

3. 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

4. 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 • ?

5. 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?

6. 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

7. Outline • Motivation • A model for universal swarms • Main results • Stability of universal swarms • Content exchange designs in universal swarms • Conclusion and future works

8. Peer Swarms • Peer requests one chunk iK • Peers requesting the same chunk form a peer swarm • ? • ? • ?

9. 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

10. 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

11. Peer Contact Process • Online P2P: random sampling • Mobile P2P: contact when within transmission range • × • × • Time • ? • ? • ?

12. 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)

13. 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 • ? • ? • ?

14. Outline • Motivation • Model for universal swarms • Main results • Stability of universal swarms • Content exchange designs in universal swarms • Conclusion and future works

15. 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

16. Universal Swarms • Question 1: How does inter-swarm exchange affect the system stability?

17. 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

18. Only One Swarm Can Become Unstable! • ? • ? • ? • At most one swarm can blow up!

19. Outline • Motivation • Model for universal swarms • Main results • Stability of universal swarms • Content exchange designs in universal swarms • Conclusion and future works

20. Universal Swarms • Question 2: How should chunks be exchanged across swarms?

21. 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!

22. 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 … • ? • ?

23. 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

24. BARON: Numerical Results • Evaluations based on fluid trajectories in MATLAB • Numerically solving ODEs • Valuation-guided content exchange improves the system stability

25. 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

26. Thank you!

27. Backup

28. 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!