Distributed Selfish Replication under Node Churn

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## Distributed Selfish Replication under Node Churn

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**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**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**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)**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**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**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:**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|))**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**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)**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**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, …**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**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**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**Potential gains of nodes by playing again after 1 round**(case I)**Participation gain (case I)**• Gain of a node if it follows the common churn-aware strategy, vs. keeping the greedy local placement**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**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**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