Distributed Selfish Replication under Node Churn

1. Distributed Selfish Replication under Node Churn Eva Jaho, Ioannis Koukoutsidis, Ioannis Stavrakakis, Ina Jaho Advanced Networking Research Group National and Kapodistrian University of Athens Οctober 2007

2. Overview Setting of a distributed replication group: N nodes, M objects rij: request rate of node j for object i Cj: capacity of node j tl: local access cost, tr: remote access cost, ts: access cost from an origin server Presence of node churn each node is “active” or “available” with a certain probability (ON probability) πj origin server tl <tr<ts ts tr vj tl

3. Access cost of a node under a given placement • Pj = {set of objects replicated by node j} • global placement P = {P1, P2, …, PN} • P-j= P - Pj • mean access cost per unit time for node j : (the cost for an unsuccessful query is negligible)

4. Game formulation • At the beginning of the game, each node has stored Cj objects in decreasing order of rij values • During the game, nodes play sequentially and make changes to their placements so as to decrease their access cost at the end of the game • Each node knows the global placement P prior to making its move (some kind of communication exists) • The game is studied as a dynamic noncooperative game

5. Strategies • Greedy local strategy: nodes locally replicate their most requested objects • Greedy churn-unaware strategy: nodes change their initial placements to minimize their imminent access cost. However, they falsely consider other nodes to be always ON • Greedy churn-aware strategy: nodes change their initial placements to minimize their imminent access cost, considering the probabilities with which other nodes are ON

6. Greedy churn-aware strategy • Each node changes its initial placement to minimize its average access cost immediately after its move • For an object e replicated at node j, define the average eviction cost as: • For an object i not replicated at node j, define the average insertion gain as:

7. Greedy churn-aware strategy (contd.) • Set of “eviction candidates” of node j, Єj = {e1j, e2j, …, e|Єj|j} • Eviction candidates indexed by increasing costs: Le1,j ≤ Le2,j ≤ … ≤ Le|Єj|,j • Set of “insertion candidates” of node j, Ij = {i1j, i2j, …, i|Ij|j} • Insertion candidates indexed by decreasing costs: Gi1,j ≥ Gi2,j ≥ … ≥ Gi|Ij|,j • Node j makes a maximum number mjof changes ekj <- ikj, k=1,…,mj s.t (mj ≤ min(|Єj| ,| Ij|))

8. Greedy churn-aware strategy with multiple rounds • Each node applies the greedy churn-aware strategy in each round of the game • The same order of the play is maintained in each round • Theorem: the algorithm ends in a finite number of rounds irrespective of the order of play in each round • Proof: At each step, each player may evict an object owned by a number of nodes to insert an object owned by: • a) a smaller number of nodes (or none) • b) a larger number of nodes with smaller probability that at least one of them is ON • Hence, at a certain epoch in the future either all nodes have no objects in common, or no further replacements are possible

9. Equilibrium properties • The strategy may not arrive in a Nash equilibrium • Proof: . . … . . . 1 2 N-2 N-1 N We show that the greedy churn-aware strategy is not always sequentially rational for player N-1. Suppose both N-1, N evict the same object e. That is, the following conditions hold: • Gi,N-1 > Le,N-1 • Gi’,N > Le,N (i’ may be equal to i) If Gi,N-1 < Le,N, then the move e <- i is not sequentially rational for node N-1 (Le,N > Le,N-1)

10. Mistreatment under the greedy churn-aware strategy Given that the churn-aware strategy is followed by all nodes, we say a node is mistreated when its incurred access cost is higher than its greedy-local cost • for N=2 nodes, mistreatment never occurs (the 2nd node only evicts objects belonging to the 1st node, so the access cost of node 1 is not decreased) • for N≥3, mistreatment may occur

11. Mistreatment under the greedy churn-aware strategy (contd.) • In the homogeneous case (rij ri for all i, j, Cj= C), where less reliable nodes play first (π1 ≤ π2 ≤ … ≤ πN) • If the set of objects evicted by node j+1 are also evicted by node j, for all j = 1,…, N-1, the greedy churn-aware strategy is mistreatment-free. • The proof follows by showing that subsequent nodes have decreasing gain when making the kth feasible replacement, k = 1,2, …

12. Numerical evaluation • We study cases where nodes have similar request rates for objects, so that mutual benefits emerge by cooperation • Request rates drawn from Zipf distribution s≈0.8-0.9 • tl=1, tr=10, ts=100 • N=10, M=50 • C=10

13. Case studies • Case I • Nodes have the same request rates for each object • Case II • Nodes have different request rates and different priorities for objects

14. Access costs (case I) • Under an LRF order, the greedy churn-aware strategy significantly improves performance • When all nodes follow the greedy churn-unaware strategy, MRF better than LRF order • Repeating the greedy churn-aware strategy for multiple rounds only yields a small benefit to some nodes

15. Access costs (case I-cntd.)

16. Potential gains of nodes by playing again after 1 round (case I)

17. Participation gain (case I) • Gain of a node if it follows the common churn-aware strategy, vs. keeping the greedy local placement

18. Access costs (case II)

19. Mistreatment example • Set of objects: {1, 2, 3, 4, 5}, set of nodes: {1, 2} • C1=4, C2=1 • r1={0.5, 0.4, 0.3, 0.2, 0.1}, r2={0.4, 0.3, 0.5, 0.2, 0.1} • π1=0,9, π2:variable • tl=1, tr=10, ts=100 • Placements • greedy local: P1={1, 2, 3, 4}, P2={3} • greedy churn-unaware when node 1 plays first: P1={1, 2, 4, 5}, P2={3} • Greedy churn-aware when node 1 plays first: P1={1, 2, 3, 4}, P2={5} when π2 ≤ 0.74 P1={1, 2, 4, 5}, P2={3} when π2 > 0.74

20. Mistreatment example (cntd.) • The greedy churn-unaware strategy causes mistreatment • to node 1 when π2≤0.74 • The greedy churn-aware strategy is always better than • the greedy local

21. Conclusions • In the majority of test cases, the greedy churn-aware strategy • reduces access cost over the greedy local and greedy churn-unaware strategy in most of the nodes • alleviates mistreatment problems • the LRF order is fair and incites nodes to participate in the game