Algorithms for Wireless Network Design : A Cell Breathing Heuristic Algorithms for Wireless Network Design : A Cell Breathing Heuristic MohammadTaghi HajiAghayi U. of Maryland, College Park Joint work with Bahl, Jain, Mirrokni, Qui, Saberi

Application of Market Equilibrium in Distributed Load Balancing Wireless devices Wireless devices Cell-phones, laptops with WiFi cards Cell-phones, laptops with WiFi cards Referred as clients or users interchangeably Referred as clients or users interchangeably Demand connections to access points Demand connections to access points Uniform for cell-phones (voice connection) Uniform for cell-phones (voice connection) Non-uniform for laptops (application dependent) Non-uniform for laptops (application dependent)

Application of Market Equilibrium in Distributed Load Balancing Access points Access points Cell-towers, Base stations, Wireless routers Cell-towers, Base stations, Wireless routers Capacities Capacities Total traffic they can serve Total traffic they can serve Integer for cell-towers Integer for cell-towers Variable transmission power Variable transmission power Capable of operating at various power levels Capable of operating at various power levels Assume levels are continuous real numbers Assume levels are continuous real numbers

Clients to APs assignment Assign clients to APs in an efficient way Assign clients to APs in an efficient way No over-loading of APs No over-loading of APs Assigning the maximum number of clients and thus satisfying maximum demand Assigning the maximum number of clients and thus satisfying maximum demand

One Heuristic Solution A client connects to the AP with reasonable signal and then the lightest load A client connects to the AP with reasonable signal and then the lightest load Requires support both from AP and Clients Requires support both from AP and Clients APs have to communicate their current load APs have to communicate their current load Clients have WiFi cards from various vendors running legacy software Clients have WiFi cards from various vendors running legacy software Overall it has limited benefit in practice Overall it has limited benefit in practice

Ideal Case We would like a client connects to the AP with the best received signal strength We would like a client connects to the AP with the best received signal strength If an AP j transmitting at power level P j then a client i at distance d ij receives signal with strength If 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 -c P ij = a.P j.d ij -c where a and c are constants capturing various models of power attenuation

Cell Breathing Heuristic An overloaded AP decreases its communication radius by decreasing power An overloaded AP decreases its communication radius by decreasing power A lightly loaded AP increases its communication radius by increasing power A lightly loaded AP increases its communication radius by increasing power Hopefully an equilibrium would be reached Hopefully an equilibrium would be reached Will show that an equilibrium exist Will show that an equilibrium exist Can be computed in polynomial time Can be computed in polynomial time Can be reached by a tatonnement process Can be reached by a tatonnement process Let’s start with economics and game theory Let’s start with economics and game theory

Market Equilibrium – A distributed load balancing mechanism. Fisher setting with linear Utilities: Fisher setting with linear Utilities: m buyers (each with budget B i ) and n goods for sale m buyers (each with budget B i ) and n goods for sale (each with quantity q j ) Each buyer has linear utility u i, i.e. utility of i is Each buyer has linear utility u i, i.e. utility of i is sum j u ij x ij where u ij >= 0 is the utility of buyer i for good j and x ij is the amount of good j bought by i. A market equilibrium or market clearance is a price vector p that A market equilibrium or market clearance is a price vector p that maximizes utility sum j u ij x ij of buyer i subject to his budget sum j p j x ij <= B i maximizes utility sum j u ij x ij of buyer i subject to his budget sum j p j x ij <= B i The demand and supply for each good j are equal The demand and supply for each good j are equal sum j x ij = q j (and thus the budgets are totally spent).

Fisher Setting with Linear Utilities Buyers Goods

Market Equilibrium – A distributed load balancing mechanism. Static supply Static supply corresponding to capacities of APs corresponding to capacities of APs Prices Prices corresponding to powers at APs corresponding to powers at APs Utilities Utilities Analogous to received signal strength function Analogous to received signal strength function Either all clients are served or all APs are saturated Either all clients are served or all APs are saturated Analogous to the market clearance(equiblirum) condition Analogous to the market clearance(equiblirum) condition Thus our situation is analogous to Fisher setting with linear utilities Thus our situation is analogous to Fisher setting with linear utilities

Clients assignment to APs Clients APs

Analogousness Is Only Inspirational Get inspiration from various algorithms for the Fisher setting and develop algorithms for our setting Get inspiration from various algorithms for the Fisher setting and develop algorithms for our setting Though we do not know any reduction – in fact there are some key differences Though we do not know any reduction – in fact there are some key differences

Differences from the Market Equilibrium setting Demand Demand Price dependent in Market equilibrium setting Price dependent in Market equilibrium setting Power independent in our setting Power independent in our setting Is demand splittable? Is demand splittable? Yes for the Market equilibrium setting Yes for the Market equilibrium setting No for our setting No for our setting Market equilibrium clears both sides but our solution requires clearance on either client side or AP side Market equilibrium clears both sides but our solution requires clearance on either client side or AP side This also means two separate linear programs for these two separate cases This also means two separate linear programs for these two separate cases

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

Three Approaches for Load Balancing Linear Programming Linear Programming Minimum weight complete matching Minimum weight complete matching Primal-Dual Primal-Dual Uses properties of bipartite graph matching Uses properties of bipartite graph matching Auction Auction Useful in dynamically changing situation 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 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 This is essentially a distributed load balancing i.e., distributed congestion control problem

Linear Programming Based Solution Create a complete bipartite graph Create a complete bipartite graph One side is the set of all clients 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 (unit demand) The other side is the set of all APs, conceptually each AP is repeated as many times as its capacity (unit demand) The weight between client i and AP j is The weight between client i and AP j is w ij = c.ln(d ij ) – ln(a)= -ln(P ij /P j ) w ij = c.ln(d ij ) – ln(a)= -ln(P ij /P j ) Find the minimum weight complete matching Find the minimum weight complete matching

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

Case 1 – Complete matching covers all clients

Case 1 – Pick Dual Variables

Write Dual Program x ij

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

Proof Dual feasibility gives: Dual feasibility gives: -λ i ≥ π j – w ij = ln(P j ) – c.ln(d ij ) + ln(a) = ln(a.P j.d ij -c ) -λ i ≥ π j – w ij = ln(P j ) – c.ln(d ij ) + ln(a) = ln(a.P j.d ij -c ) Complementary slackness gives: Complementary slackness gives: x ij =1 implies -λ i = ln(a.P j.d ij -c ) x ij =1 implies -λ i = ln(a.P j.d ij -c ) (Remember if 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 -c) Together they imply that i is connected to the AP with the strongest received signal strength 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 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 Previous algorithm needs the input upfront In practice, we need a tatonnement process In practice, we need a tatonnement process The received signal strength formula does not work in case there are obstructions 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 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

The integer program is APX-hard in general (because of knapsack) The integer program is APX-hard in general (because of knapsack) Assuming that the number of clients is much larger than the number of APs, a realistic assumption, we can obtain a nice approximation heuristic. Assuming that the number of clients is much larger than the number of APs, a realistic assumption, we can obtain a nice approximation heuristic. First we compute a basic feasible solution First we compute a basic feasible solution

Analysis of Basic Feasible Solution

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

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

Discrete Power Levels Start with the maximum power levels for each AP Start with the maximum power levels for each AP Take any overloaded AP and decrease its power level by one notch 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 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 tatonnement process! This is a distributed tatonnement process!

Proof Suppose P j is an equilibrium power level for the j th AP. 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. 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. 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. Theory of market equilibrium is a good way of synchronizing independent entity’s to do distributed load balancing. We simulated these algorithm and observed meaningful results. We simulated these algorithm and observed meaningful results.

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