Market Equilibrium for Load Balancing in Wireless Networks

1. An Application of Market Equilibrium in Distributed Load Balancing in Wireless Networking Title Page Algorithms and Economics of Networks UW CSE-599m

2. Reference • Cell-Breathing in Wireless Networks, by Victor Bahl, MohammadTaghi Hajiaghayi, Kamal Jain, Vahab Mirrokni, Lili Qiu, Amin Saberi

3. Wireless Devices • Wireless Devices • Cell-phones, laptops with WiFi cards • Referred as clients or users interchangeably • Demand Connections • Uniform for cell-phones (voice connection) • Non-uniform for laptops (application dependent)

4. Access Points (APs) • Access Points • Cell-towers, Wireless routers • Capacities • Total traffic they can serve • Integer for Cell-towers • Variable Transmission Power • Capable of operating at various power levels • Assume levels are continuous real numbers

5. Clients to APs assignment • Assign clients to APs in an efficient way • No over-loading of APs • Assigning the maximum number of clients • Satisfying the maximum demand

6. One Heuristic Solution • A client connects to the AP with the best signal and the lightest load • Requires support both from AP and Clients • APs have to communicate their current load • Clients have WiFi cards from various vendors running legacy software • Limited benefit in practice

7. We would like … • A Client connects to the AP with the best received signal strength • An AP j transmitting at power level Pj then a client i at distance dij receives signal with strength Pij = a.Pj.dij-α where a and α are constants • Captures various models of power attenuation

8. Cell Breathing Heuristic • An overloaded AP decreases its communication radius by decreasing power • A lightly loaded AP increases its communication radius by increasing power • Hopefully an equilibrium would be reached • Will show that an equilibrium exist • Can be computed in polynomial time • Can be reached by a tatonement process

9. Market Equilibrium – A distributed load balancing mechanism. • Demand = Supply • No Production • Static Supply • Analogous to Capacities of APs • Prices • Analogous to Powers at APs • Utilities • Analogous to Received Signal Strength function

10. Analogousness is Inspirational • Our situation is analogous to Fisher setting with Linear Utilities

11. Fisher Setting Linear Utilities Goods Buyers

12. Clients assignment to APs APs Clients

13. Analogousness is Inspirational • Our situation is analogous to Fisher setting with Linear Utilities • Get inspiration from various algorithms for the Fisher setting and develop algorithms for our setting • We do not know any reduction – in fact there are some key differences

14. Differences from the Market Equilibrium setting • Demand • Price dependent in Market equilibrium setting • Power independent in our setting • Is demand splittable? • Yes for the Market equilibrium setting • No for our setting • Under mild assumptions, market equilibrium clears both sides but our solution requires clearance on either one side • Either all clients are served • Or all APs are saturated • This also means two separate linear programs for these two separate cases

15. Three Approaches for Market Equilibrium • Convex Programming Based • Eisenberg, Gale 1957 • Primal-Dual Based • Devanur, Papadimitriou, Saberi, Vazirani 2004 • Auction Based • Garg, Kapoor 2003

16. Three Approaches for Load Balancing • Linear Programming • Minimum weight complete matching • Primal-Dual • Uses properties of bipartite graph matching • No loop invariant! • Auction • Useful in dynamically changing situation

17. Another Application of Market Equilibria in Networking • Fleisher, Jain, Mahdian 2004 used market equilibrium inspiration to obtain Toll-Taxes in Multi-commodity Selfish Routing Problem • This is essentially a distributed load balancing i.e., distributed congestion control problem

18. Linear Programming Based Solution • Create a complete bipartite graph • One side is the set of all clients • The other side is the set of all APs, conceptually each AP is repeated as many times as its capacity • The weight between client i and AP j is wij = α.ln(dij) – ln(a) • Find the minimum weight complete matching

19. Theorem • Minimum weight matching is supported by a power assignment to APs • Power assignment are the dual variables • Two cases for the primal program • Solution can satisfy all clients • Solution can saturate all APs

20. Case 1 – Complete matching covers all clients

21. Case 1 – Pick Dual Variables

22. Write Dual Program

23. Optimize the dual program • Choose Pj = eπj • Using the complementary slackness condition we will show that the minimum weight complete matching is supported by these power levels

24. Proof • Dual feasibility gives: -λi ≥ πj – wij= ln(Pj) – α.ln(dij) + ln(a) = ln(a.Pj.dij-α) • Complementary slackness gives: xij=1 implies -λi = ln(a.Pj.dij-α) • Together they imply that i is connected to the AP with the strongest received signal strength

25. Case 2 – Complete matching saturates all APs

26. Case 2 – The rest of the proof is similar

27. Optimizing Dual Program • Once the primal is optimized the dual can be optimized with the Dijkstra algorithm for the shortest path

28. Primal-Dual-Type Algorithm • Previous algorithm needs the input upfront • In practice, we need a tatonement process • The received signal strength formula does not work in case there are obstructions • A weaker assumption is that the received signal strength is directly proportional to the transmitted power – true even in the presence of obstructions

29. Cell-towers Cell-phones

30. 10 40 10 30 Start with arbitrary non-zero powers

31. 10 40 10 30 RSS Powers and Received Signal Strength 8 8 4 7

32. 10 40 10 30 Max RSS Equality Edges 8 8

33. 10 40 10 30 Equality Graph Desirable APs for each Client

34. 10 40 10 30 Maximum Matching Maximum Matching, Deficiency = 1

35. 10 40 10 30 Neighborhood Set T S Neighborhood Set

36. 10 40 10 30 Deficiency of a Set T S Deficiency of S = Capacities on T - |S|

37. Simple Observation Deficiency of a Set S≤ Deficiency of the Maximum Matching Maximum Deficiency over Sets ≤ Minimum Deficiency over Matching

38. Generalization of Hall’s Theorem Maximum Deficiency over Sets = Minimum Deficiency over Matching Maximum Deficiency over Sets = Deficiency of the Maximum Matching

39. 10 40 10 30 Maximum Matching Maximum Matching, Deficiency = 1

40. 10 40 10 30 Most Deficient Sets Two Most Deficient Sets

41. 10 40 10 30 Smallest Most Deficient Set S Pick the smallest. Use Super-modularity!

42. 10 40 10 30 Neighborhood Set T S Neighborhood Set

43. 10 40 10 30 Complement of the Neighborhood Set S Tc Complement of the Neighborhood Set

44. 10 40 10r 30r Initialize r. S Tc Initialize r = 1

45. 10 40 10r 30r About to start raising r. S Tc Start Raising r

46. 10 40 10r 30r Equality edges about to be lost. S Tc Equality edge which will be lost

47. 10 40 10r 30r Useless equality edges. S Tc Did not belong to any maximum matching

48. 10 40 10r 30r Equality edges deleted. S Tc Let it go

49. 10 40 10r 30r All other equality edges remain. S Tc All other equality edges are preserved!

50. 10 40 20 60 A new equality edge added S Tc At some point a new equality appears. r =2