distributed selfish replication under node churn
DESCRIPTION
Distributed Selfish Replication under Node Churn. Eva Jaho, Ioannis Koukoutsidis, Ioannis Stavrakakis, Ina Jaho. Advanced Networking Research Group National and Kapodistrian University of Athens Ο ctober 2007. Overview. Setting of a distributed replication group: N nodes, M objects - PowerPoint PPT PresentationTRANSCRIPT
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Distributed Selfish Replication under Node Churn
Eva Jaho, Ioannis Koukoutsidis, Ioannis Stavrakakis, Ina Jaho
Advanced Networking Research GroupNational 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
vj
tr
ts
tl
tl <tr< ts
origin server
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)
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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:
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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 mj of changes ekj <- ikj, k=1,…,mj s.t
(mj ≤ min(|Єj| ,| Ij|))
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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
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
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
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
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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
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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
Access costs (case I-cntd.)
Potential gains of nodes by playing again after 1 round (case I)
Node 1 2 3 4 5 6 7 8 9 10
LRF 1.52 1.61 0.83 1.93 0 0 0 0 0 0
MRF 0 0 0 0 0 0 0 0 0 0
Random order
2.81 2.73 0 2.94 2.69 0 0 0 0 0
πj=0.5 j=1…N
0.35 0.40 0 0 0 0 0 0 0 0
Participation gain (case I)
Gain of a node if it follows the common churn-aware strategy, vs. keeping the greedy local placement
Access costs (case II)
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