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 ?ctober 2007
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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) pj

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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

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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

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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,
• ?j e1j, e2j, , e?jj
• Eviction candidates indexed by increasing costs
  Le1,j Le2,j Le?j,j
• Set of insertion candidates of node j,
• Ij i1j, i2j, , iIjj
• Insertion candidates indexed by decreasing costs
  Gi1,j Gi2,j GiIj,j
• Node j makes a maximum number mj of changes ekj
  lt- ikj, k1,,mj s.t
• (mj min(?j , 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
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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 gt Le,N-1
• Gi,N gt Le,N (i may be equal to i)
• If Gi,N-1 lt Le,N, then the move e lt- i is not
  sequentially rational for node N-1 (Le,N gt
  Le,N-1)

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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 N2 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 N3, mistreatment may occur

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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 (p1
  p2 pN)
• If the set of objects evicted by node j1 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,

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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
• s0.8-0.9
• tl1, tr10, ts100
• N10, M50
• C10

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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

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Access costs (case I-cntd.)
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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
pj0.5 ?j1N 0.35 0.40 0 0 0 0 0 0 0 0
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Participation gain (case I)

• Gain of a node if it follows the common
  churn-aware strategy, vs. keeping the greedy
  local placement

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Access costs (case II)
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Mistreatment example

• Set of objects 1, 2, 3, 4, 5, set of nodes
  1, 2
• C14, C21
• r10.5, 0.4, 0.3, 0.2, 0.1, r20.4, 0.3, 0.5,
  0.2, 0.1
• p10,9, p2 variable
• tl1, tr10, ts100
• Placements
• greedy local P11, 2, 3, 4, P23
• greedy churn-unaware when node 1 plays first
  P11, 2, 4, 5, P23
• Greedy churn-aware when node 1 plays first
• P11, 2, 3, 4, P25 when p2 0.74
• P11, 2, 4, 5, P23 when p2 gt 0.74

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Mistreatment example (cntd.)

• The greedy churn-unaware strategy causes
  mistreatment
• to node 1 when p20.74
• The greedy churn-aware strategy is always better
  than
• the greedy local

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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