Placement of Web-Server Proxies with Consideration of Read and Update Operations on the Internet

1. Placement of Web-Server Proxies with Consideration of Read and Update Operations on the Internet Parallel Programming Team 6 B92902039莊謹譽 B92902054李苡嬋 B92902092張又仁

2. Outline • Introduction • Problem Formulation • Optimal Placement of k Proxies • Optimal Number of Proxies

3. Outline • Introduction • Problem Formulation • Optimal Placement of k Proxies • Optimal Number of Proxies

4. Introduction (1/3) • Caching – alleviate traffic congestion & improve the response time of Web servers. • Client-based: • cache the file recently accessed by the clients • Server-based: • server proxy stores replicated data, acts as a “front end” of this Web server to its client.

5. Introduction (2/3) • Server-based caching: • advantages: • improve client response times • Distribute the workload of the Web server • reduce the network traffic • disadvantages: • need to update data at the proxies. cost up.

6. Introduction (3/3) • Two subproblems discussed: • Given k proxies, find the optimal placement of the proxies in the network • For an unconstrained number, find the optimal number of proxies and their placement. • such that the overall access cost (read & update) is minimized. • using dynamic programming.

7. Outline • Introduction • Problem Formulation • Optimal Placement of k Proxies • Optimal Number of Proxies

8. Notation (1/3) • s: server. • w: the update frequency of server. • r(v): the access frequency to s of node v. • SPT: the shortest path tree. • the leaf nodes of the SPT are only proxies. • only one copy of the data traverses over it. • Multicast vs.Unicast

9. SPT for updating proxies SPT Ts

10. Notation (2/3) • : the path connecting u & v in Ts. • : distance function = • : the first proxy that is met while going from client v to s along tree Ts. • : the hit ratio of a proxy.

11. Equation (for read) • The cost for client v to access s • Total cost for all clients in Ts to access s ---- equation for reads

12. Notation (3/3) • P: a set of proxies in the network • SPT(s,P): the SPT rooted from s

13. Equation (for update) • The overall cost to update the proxies: ---- equation for update

14. Equation (total cost) • Total cost of all clients in Ts to access s with a set of proxies P: C(Ts,P) = (read) (update)

15. Outline • Introduction • Problem Formulation • Optimal Placement of k Proxies • Optimal Number of Proxies

16. Optimal Placement of k Proxies • Notation: • Tv • u is to the left of v

17. Optimal Placement of k Proxies • Notation: • Lu,v = {x: x∈ Tv and x is to the left of u } • Ru,v = {x: x ∈ Tvand x Tu∪Lu,v} • Lu,v,x = {y: y∈ Rv,u and y is to the left of x }

18. Optimal Placement of k Proxies

19. Optimal Placement of k Proxies This term is irrelevant to the proxy placement P.

20. Optimal Placement of k Proxies Optimal Placement of k Proxies • Define the placement of proxies recursively. • Server is regarded as one of the proxy • C(Tv,k) (or C*(Rv,u,k)) is the minimal access cost by placing k proxies in Tv (or Rv,u)

21. Optimal Placement of k Proxies • When t = 1

22. Optimal Placement of k Proxies • When t > 1, we can always find a node u, u ∈ Tv and uv, which satisfies: • That is: C( Tv, t ) = C( Lv,u ) + C( Tu, t’ ) + C*( Rv,u, t - t’ ) + w*d( v,u ) • Where constant ! when u,v given

23. Optimal Placement of k Proxies • The recursive equation:

24. Optimal Placement of k Proxies • Theorem 1. Equation (4) and (5) are correct. • Proof. • Prove equation (5). • When t = 1, trivially correct. • When t > 1, define C*’(Rv,u,t). C*’( Rv,u, t ) = the right-hand side of formula (5), i.e., To prove: C*( Rv,u, t ) = C*’( Rv,u,t )

25. Optimal Placement of k Proxies • C*(Rv,u, t) C*’( Rv,u, t) (1/2) • Let Popt: optimal placement in Rv,u → C*( Rv,u, t ) = C*( Rv,u, Popt ) • Popt is known find a proxy node x in Rv,u, such that:

26. Optimal Placement of k Proxies • C*(Rv,u, t) C*’( Rv,u, t) (2/2) • C*( Rv,u, Popt ) = C( Lv,u,x ) + C( Tx, Popt Tx ) + C*( Rv,x, Popt Ru,x ) + w*d(x, (u,v) ) • Since the sub-placement may not be optimal • Substituing these into (7)

27. Optimal Placement of k Proxies • C*(Rv,u, t) C*’( Rv,u, t) (1/2) • Find a x, such that: • Lv,u,x has no proxy. • Tx has a placement with t’ proxies. • Rv,x has a placement with t-t’ proxies. • See their union • : a t proxies placement in Rv,u but may not be optimal

28. Optimal Placement of k Proxies • C*(Rv,u, t) C*’( Rv,u, t) (2/2) • x and t’ are arbitrary value

29. Optimal Placement of k Proxies

30. Algorithm 1 Case 0 • Case 0: return (computed) Initialize Case 1 Case 2 • Case 1: only one proxy • Case 2: recursion of eq. (4)

31. Algorithm 1 – complexity • C[ Tv, t ] : 2-d array ( n*k ) C*[ Rv,u, t ] : 3-d array ( n*n*k ) n times k times constant

32. Outline • Introduction • Problem Formulation • Optimal Placement of k Proxies • Optimal Number of Proxies

33. Optimal Number of Proxies • Let P be a set of proxies placed in Tv ( the size of P is unknown ) note: v is always a proxy.

34. Optimal Number of Proxies • : no poxy placed in Tv except v • : some proxies placed in Tv • Obviously,

35. Optimal Number of Proxies • Consider : • when there are proxies other than v in Tv, we can always find a node u, uTv, uv, which satisfies: • a proxy is placed at u; • no proxy is placed in Lv,u ( Lv,u could be empty ) • No proxy is placed in • The cost for updating proxy node u is d(u,v) • Tv is partitioned into Lv,u, Tu, and Rv,u

36. Optimal Number of Proxies • Thus we have • Where • For all possible dividing point u:

37. Optimal Number of Proxies (11)

38. Optimal Placement of k Proxies • Theorem 3.Equation (10) and (11) are correct. • Proof. • Prove equation (11). • From definition in (11), (12)

39. Optimal Number of Proxies • Comparing (11) with (12) • We need to prove: • Prove and

40. Optimal Number of Proxies • Prove (1/2) • Let Poptbe the optimal placement in Rv,u, that is: • Popt is known, find the proxy node x satisfies: • The cost for updating proxy x is • Rv,u is partitioned into three parts by x: • Lv,u,x, Tx, and Rv,x

41. Optimal Number of Proxies • Prove (2/2)

42. Optimal Number of Proxies • Prove (1/2) • For any x Rv,u, • : optimal placement in Tv • : optimal placement in Rv,x • Consider their union • : a placement in Rv,u, but may not be optimal

43. Optimal Number of Proxies • Prove (2/2) • x is an arbitray value

44. Optimal Number of Proxies

45. Optimal Number of Proxies

46. Optimal Number of Proxies n entries n*n entries n times

47. A numerical example • Update frequency : 12

48. Simulation Setup • Inet topology generator, • http://topology.eecs.umich.edu/inet/ • Default n = 3037 • r(v) randomly generated in [0,100] • w : number of update operation • α : read-write ratio

49. Performance comparisons • Traffic reduction ratio • 3 algorithm: • Greedy • Optimal • Random

50. Performance comparisons (R vs. k) • α = 0 • Difference between opt and greedy usually within 10% • R rises sharply at a small number of proxies • R does not change much as the access frequency changes.