Algorithms for Algorithms for Wireless Network DesignWireless Network Design:: A A Cell Breathing HeuristicCell Breathing Heuristic

MohammadTaghi HajiAghayiMohammadTaghi HajiAghayi U. of Maryland, College ParkU. of Maryland, College Park

Joint work with Bahl, Jain, Mirrokni, Qui, SaberiJoint work with Bahl, Jain, Mirrokni, Qui, Saberi

Application of Market Equilibrium in Distributed Load Balancing

Wireless devicesWireless devices Cell-phones, laptops with WiFi cardsCell-phones, laptops with WiFi cards Referred as clients or users Referred as clients or users

interchangeablyinterchangeably Demand connections to access pointsDemand connections to access points

Uniform for cell-phones (voice connection)Uniform for cell-phones (voice connection) Non-uniform for laptops (application Non-uniform for laptops (application

dependent)dependent)

Application of Market Equilibrium in Distributed Load Balancing

Access pointsAccess points Cell-towers, Base stations, Wireless routersCell-towers, Base stations, Wireless routers

CapacitiesCapacities Total traffic they can serveTotal traffic they can serve Integer for cell-towersInteger for cell-towers

Variable transmission powerVariable transmission power Capable of operating at various power levelsCapable of operating at various power levels Assume levels are continuous real numbersAssume levels are continuous real numbers

Clients to APs assignmentClients to APs assignment

Assign clients to APs in an efficient Assign clients to APs in an efficient wayway No over-loading of APsNo over-loading of APs Assigning the maximum number of Assigning the maximum number of

clients and thus satisfying maximum clients and thus satisfying maximum demanddemand

One Heuristic SolutionOne Heuristic Solution

A client connects to the AP with A client connects to the AP with reasonable signal and then the lightest reasonable signal and then the lightest loadload Requires support both from AP and ClientsRequires support both from AP and Clients APs have to communicate their current loadAPs have to communicate their current load Clients have WiFi cards from various vendors Clients have WiFi cards from various vendors

running legacy softwarerunning legacy software Overall it has limited benefit in practiceOverall it has limited benefit in practice

Ideal CaseIdeal Case

We would like a client connects to the AP We would like a client connects to the AP with the best received signal strengthwith the best received signal strength

If an AP If an AP jj transmitting at power level transmitting at power level PPjj then a client then a client ii at distance d at distance dijij receives receives signal with strengthsignal with strength

PPijij = = a.Pa.Pjj.d.dijij-c-c

where where aa and and cc are constants capturing are constants capturing various models of power attenuation various models of power attenuation

Cell Breathing HeuristicCell Breathing Heuristic

An overloaded AP decreases its An overloaded AP decreases its communication radius by decreasing powercommunication radius by decreasing power

A lightly loaded AP increases its A lightly loaded AP increases its communication radius by increasing powercommunication 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 timeCan be computed in polynomial time Can be reached by a tatonnement processCan be reached by a tatonnement process

Let’s start withLet’s start with economics and game theory economics and game theory

Market Equilibrium – A Market Equilibrium – A distributed load balancing distributed load balancing

mechanism.mechanism. Fisher setting with linear Utilities:Fisher setting with linear Utilities: m buyers (each with budget Bm buyers (each with budget Bii) and n goods for sale ) and n goods for sale

(each with quantity q(each with quantity qjj)) Each buyer has linear utility Each buyer has linear utility uuii, i.e. utility of i is , i.e. utility of i is

sumsumjj u uij ij xxijij where u where uijij>= 0 is the utility of buyer i for >= 0 is the utility of buyer i for good j and xgood j and xijij is the amount of good j bought by i. is the amount of good j bought by i.

A A market equilibriummarket equilibrium or or market clearancemarket clearance is a is a price vector price vector p p that that maximizes utility maximizes utility sumsumjj u uij ij xxijij of buyer i subject to his of buyer i subject to his

budgetbudget sum sumjj p pj j xxijij <= B <= Bii The demand and supply for each good j are equalThe demand and supply for each good j are equal

sumsumjj x xij ij = q= qj j (and thus the budgets are totally spent).(and thus the budgets are totally spent).

Fisher Setting with Linear Fisher Setting with Linear UtilitiesUtilitiesBuyers Goods

1 1 1 1, j jj

M u u x

, i i ij ijj

M u u x

, n n nj njj

M u u x

1q

jq

mq

ijx

ij iju x

Market Equilibrium – A Market Equilibrium – A distributed load balancing distributed load balancing

mechanism.mechanism. Static supplyStatic supply

corresponding to capacities of APscorresponding to capacities of APs PricesPrices

corresponding to powers at APscorresponding to powers at APs Utilities Utilities

Analogous to received signal strength functionAnalogous to received signal strength function Either all clients are served or all APs are Either all clients are served or all APs are

saturatedsaturated Analogous to the market clearance(equiblirum) Analogous to the market clearance(equiblirum)

conditioncondition Thus our situation is analogous to Fisher setting Thus our situation is analogous to Fisher setting

with linear utilitieswith linear utilities

Clients assignment to APsClients assignment to APsClients APs

1 1 1 1, max j jjD u P x

, maxi i ij ijjD u P x

, maxn n nj njjD u P x

1C

jC

mC

ijx

ij ijP x

Analogousness Is Only Analogousness Is Only InspirationalInspirational

Get inspiration from various Get inspiration from various algorithms for the Fisher setting and algorithms for the Fisher setting and develop algorithms for our settingdevelop algorithms for our setting

Though we do not know any reduction Though we do not know any reduction – in fact there are some key – in fact there are some key differencesdifferences

Differences from the Market Differences from the Market Equilibrium settingEquilibrium setting

DemandDemand Price dependent in Market equilibrium settingPrice dependent in Market equilibrium setting Power independent in our settingPower independent in our setting

Is demand splittable?Is demand splittable? Yes for the Market equilibrium settingYes for the Market equilibrium setting No for our settingNo for our setting

Market equilibrium clears both sides but our Market equilibrium clears both sides but our solution requires clearance on either client side solution requires clearance on either client side or AP sideor AP side This also means two separate linear programs This also means two separate linear programs

for these two separate casesfor these two separate cases

Three Approaches for Three Approaches for Market EquilibriumMarket Equilibrium

Convex Programming BasedConvex Programming Based Eisenberg, Gale 1957Eisenberg, Gale 1957

Primal-Dual BasedPrimal-Dual Based Devanur, Papadimitriou, Saberi, Vazirani Devanur, Papadimitriou, Saberi, Vazirani

20042004 Auction BasedAuction Based

Garg, Kapoor 2003Garg, Kapoor 2003

Three Approaches for Three Approaches for Load BalancingLoad Balancing

Linear ProgrammingLinear Programming Minimum weight complete matchingMinimum weight complete matching

Primal-DualPrimal-Dual Uses properties of bipartite graph Uses properties of bipartite graph

matchingmatching AuctionAuction

Useful in dynamically changing situationUseful in dynamically changing situation

Linear Programming Based Linear Programming Based Solution Solution

Create a complete bipartite graphCreate a complete bipartite graph One side is the set of all clientsOne side is the set of all clients The other side is the set of all APs, The other side is the set of all APs,

conceptually each AP is repeated as conceptually each AP is repeated as many times as its capacity (unit many times as its capacity (unit demand)demand)

The weight between client The weight between client i i and AP and AP jj is is wwijij = = c.c.ln(ln(ddijij) – ln() – ln(aa)= -ln()= -ln(PPijij/P/Pjj )) Find the minimum weight complete Find the minimum weight complete

matchingmatching

TheoremTheorem

Minimum weight matching is supported Minimum weight matching is supported by a power assignment to APsby a power assignment to APs

Power assignment are the dual Power assignment are the dual variablesvariables

Two cases for the primal program Two cases for the primal program which is known at the beginningwhich is known at the beginning Solution can satisfy all clientsSolution can satisfy all clients Solution can saturate all APsSolution can saturate all APs

Case 1 – Complete Case 1 – Complete matching covers all clientsmatching covers all clients

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

ij ji C

ij

w x

x

j A x C

i C j A x

Case 1 – Pick Dual VariablesCase 1 – Pick Dual Variables

,

ijj A

minimize

subject toi C 1

, 0

ij iji C j A

i

ij j ji C

ij

w x

x

j A x C

i C j A x

Write Dual ProgramWrite Dual Program

maximize

subject to,

0

i j ji C j A

i j ij

j

C

i C j A w

j A

xxijij

Optimize the dual programOptimize the dual program

Choose Choose PPjj = e = e ππjj

Using the complementary slackness Using the complementary slackness condition we will show that the condition we will show that the minimum weight complete matching minimum weight complete matching is supported by these power levelsis supported by these power levels

ProofProof Dual feasibility gives:Dual feasibility gives: --λλi i ≥ ≥ ππjj – w – wijij= = ln(ln(PPjj) – ) – c.c.ln(ln(ddijij) + ln() + ln(aa) = ln() = ln(a.Pa.Pjj.d.dijij

-c-c)) Complementary slackness gives:Complementary slackness gives: xxijij=1=1 implies implies --λλi i = ln(= ln(a.Pa.Pjj.d.dijij

-c-c))

(Remember if an AP (Remember if an AP jj transmitting at power level transmitting at power level PPjj then then a client a client ii at distance d at distance dijij receives signal with strength receives signal with strength PPijij = = a.Pa.Pjj.d.dijij

-c)-c)

Together they imply that Together they imply that ii is connected to the AP is connected to the AP with the strongest received signal strengthwith the strongest received signal strength

Case 2 – Complete Case 2 – Complete matching saturates all APsmatching saturates all APs

,

ijj A

minimize

subject toi C 1

, 0

ij iji C j A

ij ji C

ij

w x

x

j A x C

i C j A x

Case 2 – The rest of the proof Case 2 – The rest of the proof is similaris similar

Unsplittable DemandUnsplittable Demand

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

i ij ji C

ij

w x

x

j A D x C

i C j A x

Unsplittable DemandUnsplittable Demand

The integer program is APX-hard in The integer program is APX-hard in general (because of knapsack)general (because of knapsack)

Assuming that the number of clients is Assuming that the number of clients is much larger than the number of APs, a much larger than the number of APs, a realistic assumption, we can obtain a realistic assumption, we can obtain a nice approximation heuristic.nice approximation heuristic.

First we compute a basic feasible First we compute a basic feasible solutionsolution

Analysis of Basic Feasible Analysis of Basic Feasible SolutionSolution

,

ijj A

minimize

subject to

i C 1

, 0

ij iji C j A

i ij ji C

ij

w x

x

j A D x C

i C j A x

Approximate SolutionApproximate Solution

All All xxijij’s but a small number of ’s but a small number of xxijij’s are integral’s are integral Theorem: Number of Theorem: Number of xxijij which are integral is which are integral is

at least the number of clients – the number at least the number of clients – the number APsAPs

Most clients are served unsplittablyMost clients are served unsplittably Clients which are served splittably – do not Clients which are served splittably – do not

serve themserve them The algorithm is almost optimalThe algorithm is almost optimal

Discrete Power LevelsDiscrete Power Levels Over the shelf APs have only fixed number of Over the shelf APs have only fixed number of

discrete power levelsdiscrete power levels Equilibrium may not existEquilibrium may not exist

In fact it is NP-hard to test whether it exists or notIn fact it is NP-hard to test whether it exists or not If every client has a deterministic tie If every client has a deterministic tie

breaking rule then we can compute the breaking rule then we can compute the equilibrium in which every client is served – if equilibrium in which every client is served – if exists under the tie breaking ruleexists under the tie breaking rule

Discrete Power LevelsDiscrete Power Levels Start with the maximum power levels for Start with the maximum power levels for

each APeach AP Take any overloaded AP and decrease its Take any overloaded AP and decrease its

power level by one notchpower level by one notch If an equilibrium exist then it will be If an equilibrium exist then it will be

computed in time computed in time mkmk, where , where mm is the is the number of APs and number of APs and kk is the number of is the number of power levelspower levels

This is a distributed tatonnement process!This is a distributed tatonnement process!

ProofProof

Suppose Suppose PPjj is an equilibrium power level is an equilibrium power level for the for the jjthth AP. AP.

Inductively prove that when Inductively prove that when jj reaches the reaches the power level power level PPjj then it will not be then it will not be overloaded again.overloaded again. Here we use the deterministic tie breaking Here we use the deterministic tie breaking

rule.rule.

ConclusionConclusion

Theory of market equilibrium is a Theory of market equilibrium is a good way of synchronizing good way of synchronizing independent entity’s to do independent entity’s to do distributed load balancing.distributed load balancing.

We simulated these algorithm and We simulated these algorithm and observed meaningful results.observed meaningful results.

Thanks for your attention…Thanks for your attention…

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