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Coupon Replication Systems

Coupon Replication Systems. Laurent Massouli é & Milan Vojnović Microsoft Research Cambridge, UK. System we look at. file dissemination by file swarming file sliced into chunks (we say coupons) user granted initial coupon from server other coupons collected by replication between users

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Coupon Replication Systems

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  1. Coupon Replication Systems Laurent Massoulié & Milan Vojnović Microsoft Research Cambridge, UK

  2. System we look at • file dissemination by file swarming • file sliced into chunks (we say coupons) • user granted initial coupon from server • other coupons collected by replication between users • scalability • slicing into K chunks reduces server load by factor 1/K • ex. BitTorrent • partial view • greedy: top 4 uploaders • random search: optimistic unchoke • approx direct reciprocity of exchanged bits

  3. Scale • large number of distinct coupons • ex. movie files • 2 GB file length & ¼ MB coupon length= 8000 coupons • same for some software binaries (ex. Linux Redhat) • large num. of concurrent users for popular files • ~ 1000 [The Lord of the Rings, Pouwelse et al 2005]

  4. Related work • a few theoretical studies • Yang & de Veciana (Infocom’04) • service capacity & scaling • Qiu & Srikant (Sigcomm’04) • macroscopic population dynamics • some empirical work • Izal et al (PAM’04) • Pouwelse et al (IPTPS’05)

  5. Our work • model of probabilistic replication • we call: coupon replication system • population dynamics model • system performance captured by • mean file download time • closed system: leftover users with incomplete collections • sheds light how critical is: • replication strategy (who to peer with? which coupon to replicate?) • user altruism (users offer coupons after collected all distinct coupons)

  6. Outline • open system • two peering strategies • How long it takes to download a file? • by-products: stability results • closed system • no new user arrivals (flashcrowd end-phase) • How many users are left with incomplete collection? • conclusion

  7. Open system: assumptions • peering strategy LAYER • peer with a random user having same number of coupons • peering strategy FLAT • peer with a random user • RANDOM PULL • instigator user copies a random coupon of interest from the encounter user • throughout users assumed non-altruistic • after completing coupon collection offer no coupons

  8. Model • Xc(t) = number of users with coupon collection c at time t • each user initiates encounters at instants Poi(1) • X is Markov process: fixed arrival rate of users with collection c user arrives with coupon collection c Prob (c encounters s) defined to capture either LAYER or FLAT user with collection c enlarges its collection with coupon i

  9. Large population limit • scaled process XN: • Kurtz: XN/N converges uniformly on finite intervals to x • object of our study arrival rate departure rate

  10. LAYER • number of users in layer i = • sojourn time in layer i = • general convergence results (not on slides; see paper) • Result: mean file download time = K+O(1) • asymptotically optimal as K tends to  (Little’s law)

  11. FLAT • analysis more difficult than for LAYER • results under symmetric arrival rates and initial value • Result: (i) (ii) mean file download time = • asymptotically optimal as K tends to  • same as for LAYER Ti = sojourn time in layer i

  12. Mean file download time for LAYER & FLAT mean file download time / (K-1) flat layer optimum K

  13. Sojourn times per layer for LAYER & FLAT flat sojourn time in layer k layer k

  14. Closed system • Problem • a closed population of users • given initial coupon collections over users • How many users are left with incomplete collection? • models flashcrowd end-phase • leftover users impose workload on server • partial result: “last-missing coupon” • each user has initially all but 1 coupon

  15. Phase transition • Result: for initial point x1x2…xK, the limit point is • Result: assume • X1 = N, X2 +…+ XK =(K-1)N • K ~ a log(N) then if leftover users =

  16. Conclusion • good news on file swarming • open system • for both LAYER & FLAT, mean file download time asymptotically optimal for large coupon collection • last-missing coupon • for K / log(N) sufficiently large, number of leftover users ~ log(log(N))

  17. Outlook • results suggest • replication strategy & user altruism not critical • open: • stability of FLAT • beyond last-missing coupon • heterogeneous download/upload capacities • topology effects – beyond random mixing

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