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Market Equilibrium for Load Balancing in Wireless Networks

This paper explores the application of market equilibrium theory in achieving load balancing in wireless networks, specifically in the assignment of clients to access points. Three different approaches, including convex programming, primal-dual, and auction based methods, are discussed. The paper also highlights another application of market equilibrium in toll taxes for multi-commodity selfish routing problems. The linear programming based solution is presented with a minimum weight complete matching approach.

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Market Equilibrium for Load Balancing in Wireless Networks

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  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

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