Utility Maximizing Routing to Data Centers

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1 0-0 Utility Maximizing Routing to Data Centers M. Sarwat, J. Shin and S. Kapoor (Presented by J. Shin) Sep 26, 2011

2 Sep 26, Outline 1. Problem Definition - Data Center Allocation 2. How to construct Data Center Allocation 3. Mathematical Model 4. Greedy Approach - single server (GA-I) 5. Greedy Approach - multiple server (GA-II) 6. Auction Algorithm (Auction) 7. Simulation Results : utility, congestion, exe. time 8. Conclusion

3 Sep 26, Problem Definition Objective : Maximize the utility of users. Constraint : 1. Supply Constraint : Each source (or data/service center) i has a supply of goods, bounded by a capacity function a i. 2. Budget Constraint : Each sink (or user) j has a bound on the incoming set of data, specified by a budget b j.

4 Sep 26, Problem Definition - Example Figure 1: One instance of Data Center Allocation Problem

5 Sep 26, Network Topology for Simulation Figure 2: AT&T Network Topology

6 Sep 26, To Data Center Allocation Problem Convert the original problem into Data Center Allocation Problem Defined on a Weighted Bipartite Graph. Given a pair of data center i and user j, add edge e ij between i and j with p ij (p ij represents the price of utilizing the path). One such measure is the number of hops between i and j. u ij represents the benefit of using the path. One such benefit is the minimum capacity amongst of edges on the path.

7 Sep 26, Mathematical Model - Primal P: Maximize u ij f ij subject to i S,j T j T f ij a i i S (1) i S p ijf ij b j j T (2) f ij 0 i S, j T (3) u ij, p ij : utility and price using i by j f ij : amount of data from i to j a i, b j : capacity function of i and budget function of j

8 Sep 26, Mathematical Model - Dual The dual of (P) is as follows denoted by (D). D: Minimize i S α i a i + j T β j b j subject to α i u ij p ij β j i S, j T (4) α i 0 i S (5) β j 0 j T (6) α i and β j represent dual variables.

9 Sep 26, GA-I - High Level Idea Objective : Selects a data center which maximizes their utility. Constraint: Each user uses at most a single path. 1: Let us start with j = 1 in a round robin way. 2: Pick i s.t. utility is maximized. 3: Calculate current amount of data from i. 4: Calculate budget of user j. 5: Assign i to j under those constraints. The algorithm iterates at most n u times.

10 Sep 26, GA-I - Pseudocode 1: for j = 1 n u do 2: i := max k u kj /p kj 3: t f := k f ik 4: t p := k p kjf kj 5: x := min(a i t f, (b j t p )/p ij ) 6: end for

11 Sep 26, GA-II - High Level Idea Objective : Selects a data center maximizing their utility. Constraint: Each user is able to access multiple servers. 1: Find (i, j) s.t. maximizing utility. 2: Calculate current amount of data from i. 3: Calculate budget of user j. 4: Assign i to j under those constraints. The algorithm may iterate for all pairs.

12 Sep 26, : i, j : visited ij := false 2: for i = 1 n d do 3: for j = 1 n u do GA-II - Pseudocode 4: (s i, t j ) := arg max (i,j) visitedij =false u ij /p ij 5: visited si t j := true 6: f s := k f s i k and p t := k p kt j f ktj 7: f si,t j := min(a si f s, (b tj p t )/p si t j ) 8: end for 9: end for

13 Sep 26, Greedy Approach is Not Enough! Figure 3: A counter example of Auction vs. GA-I Let a 1 = a 2 = 1, b 1 = b 2 = 1, p 11 = p 12 = p 21 = p 22 = 1. Let u 11 = u 22 = 9, u 12 = 10 and u 21 = i=1 j=1 u ijf ij = 11 vs. 2 2 i=1 j=1 u ijf ij = 18

14 Sep 26, What is Auction? An auction algorithm is an intuitive method for solving the classical assignment problem. Suppose there are n agents with budget e i and m goods. At each iteration, an agent will increment his bid to acquire his preferred good, and it terminates when each agent are satisfied. Users bid up to their own budget allowed to maximize their utility. Data centers assigns a user an edge with available supply (capacity).

15 Sep 26, Auction - 1 1: while i s.t. α i > 0 and j f ij < a i do 2: Pick j : max j (u ij p ij β j ) 3: if i p ijf ij = b j then 4: Find a source i : y i j = β j 5: Shift flow from i to i 6: else 7: Sink is unsaturated, push max. flow possible 8: Under supply and demand constraint 9: end if 10: end while

16 Sep 26, : if β j = 0 then Auction - 2 2: β j ϵ max i u ij /p ij 3: else if i : f ij > 0, y ij = β j then 4: β j β j 5: β j β j (1 + ϵ) 6: else 7: //no update required 8: end if

17 Sep 26, Simulation Results - Environment Environment : A Intel Core 2 Duo(2 GHZ) processor, 2 GB memory and Windows 32-bit operating system. Compiler : Visual Studio 2008 on a Windows operating system. Parameters : Given a i, b j, p ij, u ij and ϵ = Output : average 10 different instances of the same parameter(utility,congestion,exetime)

18 Sep 26, Simulation Results - E-I Figure 4: Auction vs. GA-II vs. GA-I -Utility

19 Sep 26, Simulation Results - E-I Figure 5: Auction vs. GA-II - Congestion

20 Sep 26, Simulation Results - E-I Figure 6: Execution Time of Auction in Seconds

21 Sep 26, Simulation Results - E-II Figure 7: Auction vs. GA-II vs. GA-I - Utility

22 Sep 26, Simulation Results - E-II Figure 8: Auction vs. GA-II - Congestion

23 Sep 26, Simulation Results - E-II Figure 9: Execution Time of Auction in Seconds

24 Sep 26, Conclusion 1. Greedy approach is easy to implement and fast, but as expected the method does not match up to the optimal. 2. Since the auction approach is quite efficient, the usage of the method for optimally routing data center traffic is also practical. 3. Further improvements in the efficiency of the algorithm are possible. 4. Future work would extend the algorithm to non-linear objective functions.

Market Equilibrium using Auctions for a Class of Gross-Substitute Utilities

Market Equilibrium using Auctions for a Class of Gross-Substitute Utilities Rahul Garg 1 and Sanjiv Kapoor 2 1 IBM T.J. Watson Research Center,USA. 2 Illinois Institute of Technology, Chicago, USA Abstract.