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**An Application of Market Equilibrium in Distributed Load**Balancing in Wireless Networking Title Page Algorithms and Economics of Networks UW CSE-599m**Reference**• Cell-Breathing in Wireless Networks, by Victor Bahl, MohammadTaghi Hajiaghayi, Kamal Jain, Vahab Mirrokni, Lili Qiu, Amin Saberi**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)**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**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**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**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**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**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**Analogousness is Inspirational**• Our situation is analogous to Fisher setting with Linear Utilities**Fisher Setting Linear Utilities**Goods Buyers**Clients assignment to APs**APs Clients**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**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**Three Approaches for Market Equilibrium**• Convex Programming Based • Eisenberg, Gale 1957 • Primal-Dual Based • Devanur, Papadimitriou, Saberi, Vazirani 2004 • Auction Based • Garg, Kapoor 2003**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**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**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**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**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**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**Optimizing Dual Program**• Once the primal is optimized the dual can be optimized with the Dijkstra algorithm for the shortest path**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**Cell-towers**Cell-phones**10**40 10 30 Start with arbitrary non-zero powers**10**40 10 30 RSS Powers and Received Signal Strength 8 8 4 7**10**40 10 30 Max RSS Equality Edges 8 8**10**40 10 30 Equality Graph Desirable APs for each Client**10**40 10 30 Maximum Matching Maximum Matching, Deficiency = 1**10**40 10 30 Neighborhood Set T S Neighborhood Set**10**40 10 30 Deficiency of a Set T S Deficiency of S = Capacities on T - |S|**Simple Observation**Deficiency of a Set S≤ Deficiency of the Maximum Matching Maximum Deficiency over Sets ≤ Minimum Deficiency over Matching**Generalization of Hall’s Theorem**Maximum Deficiency over Sets = Minimum Deficiency over Matching Maximum Deficiency over Sets = Deficiency of the Maximum Matching**10**40 10 30 Maximum Matching Maximum Matching, Deficiency = 1**10**40 10 30 Most Deficient Sets Two Most Deficient Sets**10**40 10 30 Smallest Most Deficient Set S Pick the smallest. Use Super-modularity!**10**40 10 30 Neighborhood Set T S Neighborhood Set**10**40 10 30 Complement of the Neighborhood Set S Tc Complement of the Neighborhood Set**10**40 10r 30r Initialize r. S Tc Initialize r = 1**10**40 10r 30r About to start raising r. S Tc Start Raising r**10**40 10r 30r Equality edges about to be lost. S Tc Equality edge which will be lost**10**40 10r 30r Useless equality edges. S Tc Did not belong to any maximum matching**10**40 10r 30r Equality edges deleted. S Tc Let it go**10**40 10r 30r All other equality edges remain. S Tc All other equality edges are preserved!**10**40 20 60 A new equality edge added S Tc At some point a new equality appears. r =2