Xia Zhou*, Stratis Ioannidis ♯, and Laurent Massoulié + * University of California, Santa Barbara ♯ Technicolor Research Lab, Palo Alto + Technicolor Research Lab, Paris

Swarm: set of users interested in the same file 1 Seed

[Hajek and Zhu 10] Unstable when λ > s! Missing-piece syndrome: Each peer waiting for only one piece 2 Seed λ peers per sec s chunks per sec Online P2P Networks

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

4 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?

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 5

Motivation A model for universal swarms Main results Stability of universal swarms Content exchange designs in universal swarms Conclusion and future works 6

Peer requests one chunk i K Peers requesting the same chunk form a peer swarm 7 ? ??

Peer has cache size of C Peer may use cache to store chunks it is not interested in 8 Cache ? Request C Stored chunks f K

Peers arrive with full caches Peers requesting i and caching f arrive according to a Poisson process with rate λ i, f 9 Time ??? CacheRequest C ?

Online P2P: random sampling Mobile P2P: contact when within transmission range 10 Time ? × ?? ×

One peer contacts other peers according to a Poisson process with rate N(t): number of peers in the system at time t 11 N(t)N(t) Contact rate 0 ≤ β < 1 Constant-bandwidth N(t)N(t) Contact rate μ β = 1 N(t)N(t) Contact rate 1 < β ≤ 2 Contact-constrained Interference- constrained

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) 12 ??? Conversion probabilityA, A’  B, B’

Motivation Model for universal swarms Main results Stability of universal swarms Content exchange designs in universal swarms Conclusion and future works 13

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 14 For all β For all content exchange policies

Question 1: How does inter-swarm exchange affect the system stability? 15

16 Let > 0 be the arrival rate of peers requesting i and storing j. 16 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!

17 At most one swarm can blow up! ???

Motivation Model for universal swarms Main results Stability of universal swarms Content exchange designs in universal swarms Conclusion and future works 18

Question 2: How should chunks be exchanged across swarms? 19

20 -- 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!

Centralized tracker maintains valuation v i for each chunk i Positive v i : chunk i needs more replicas Negative v i : chunk i needs fewer replicas Replace the chunks with negative valuation with that with positive valuations … … ??

-- the number of peers requesting chunk i in the optimal state -- the number of peers storing chunk i in the optimal state 22 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

Evaluations based on fluid trajectories in MATLAB Numerically solving ODEs 23 Valuation-guided content exchange improves the system stability

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 24

27 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!