Title Page An Application of Market Equilibrium in Distributed Load Balancing in Wireless Networking 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  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 P j then a client i at distance d ij receives signal with strength P ij = a.P j.d ij -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 Buyers Goods

Clients assignment to APs Clients APs

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 w ij = α.ln(d ij ) – 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

Case 1 – Complete matching covers all clients

Case 1 – Pick Dual Variables

Write Dual Program

Optimize the dual program Choose P j = e π j Using the complementary slackness condition we will show that the minimum weight complete matching is supported by these power levels

Proof Dual feasibility gives: -λ i ≥ π j – w ij = ln(P j ) – α.ln(d ij ) + ln(a) = ln(a.P j.d ij -α ) Complementary slackness gives: x ij =1 implies -λ i = ln(a.P j.d ij -α ) Together they imply that i is connected to the AP with the strongest received signal strength

Case 2 – Complete matching saturates all APs

Case 2 – The rest of the proof is similar

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-phones Cell-towers

Start with arbitrary non-zero powers

Powers and Received Signal Strength RSS

Equality Edges Max RSS

Equality Graph Desirable APs for each Client

Maximum Matching Maximum Matching, Deficiency =

Neighborhood Set S T

Deficiency of a Set Deficiency of S = Capacities on T - |S| S T

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

Maximum Matching Maximum Matching, Deficiency =

Most Deficient Sets Two Most Deficient Sets

Smallest Most Deficient Set Pick the smallest. Use Super-modularity! S

Neighborhood Set S T

Complement of the Neighborhood Set S TcTc

Initialize r. Initialize r = r 30r S TcTc

About to start raising r. Start Raising r r 30r S TcTc

Equality edges about to be lost. Equality edge which will be lost r 30r S TcTc

Useless equality edges. Did not belong to any maximum matching r 30r S TcTc

Equality edges deleted. Let it go r 30r S TcTc

All other equality edges remain. All other equality edges are preserved! r 30r S TcTc

A new equality edge added At some point a new equality appears. r = S TcTc

Subcase A – Deficiency Decreases New equality edge gives an augmenting path S TcTc

Subcase B – Deficiency does not decrease New edge does not create any augmenting path S TcTc

Smallest most deficient set increases New S is a strict super set of old S! S

Eventually Subcase A will happen Eventually the size of the matching increases S

Case 1 – Deficiency Reaches Zero All Clients are served! S

All APs are saturated Or the algorithm will prove that none exist! S

Unsplittable Demand Solve the splittable case by solving the minimum weight matching linear program

Unsplittable Demand

Solve the splittable case by solving the minimum weight matching linear program In fact compute a basic feasible solution Assume that the number of clients is much larger than the number of APs – a realistic assumption

Approximate Solution All x ij ’s but a small number of x ij ’s are integral

Analysis of Basic Feasible Solution

Approximate Solution All x ij ’s but a small number of x ij ’s are integral Number of x ij which are not integral is at most the number of APs Most clients are served unsplittably Clients which are served splittably – do not serve them The algorithm is still almost optimal

Discrete Power Levels Over the shelf APs have only fixed number of discrete power levels Equilibrium may not exist  In fact it is NP-hard to test whether it exist or not If every client has a deterministic tie breaking rule then we can compute the equilibrium – if exist under the tie breaking rule

Discrete Power Levels Start with the maximum power levels for each AP Take any overloaded AP and decrease its power level by one notch If an equilibrium exist then it will be computed in time mk, where m is the number of APs and k is the number of power levels This is a distributed tatonement process!

Proof. Suppose P j is an equilibrium power level for the j th AP. Inductively prove that when j reaches the power level P j then it will not be overloaded again.  Here we use the deterministic tie breaking rule.

Conclusion. Theory of market equilibrium is a good way of synchronizing independent entity’s to do distributed load balancing. We simulated these algorithm. Observed meaningful results. Thanks Kamal Jain for the main part of slides.