An Economic Cost Model for Network Deployment and Spectrum in Wireless Networks

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Gerald Dixon
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1 24th Europea Regioal Coferece of the Iteratioal Telecommuicatios Society (ITS) Title: A Ecoomic Cost Model for Netork Deploymet ad Spectrum i Wireless Netorks Sag-ook Ha, Ki Wo Sug, Jes Zader School of Iformatio ad Commuicatio Techology KTH Royal Istitute of Techology Electrum 229 S Kista Sede sha@kthse

2 ABSTRACT We describe the basic ecoomic theory of cost model for etork deploymet ad spectrum i ireless etorks I particular, e develop a productio fuctio for ireless etorks With this productio fuctio model, e explore the techical rate of sutitutio ad the elasticity of sutitutio i the productio fuctio for ireless etork ad fid its isight for ireless etork Fially, e compare the egieerig value of spectrum ad ecoomic value of spectrum I INTRODUCTION Wireless tems have bee evolvig to support orealtime data service Hoever, as the data rates for the service icrease, the umber of users that oe basestatio ca support become smaller Therefore, the umber of basestatio per service area is eeded to expad, hich leads to a high degree of base-statio desity The icreased base-statio desity, i tur, results i a high ifrastructure cost, hich ill be directly trasferred to a high service cost Aother bottleeck is the radio spectrum available for ireless data service ad tems It is extremely scarce, hile demad for this service is groig at a rapid pace Base-statio desity ad the spectrum are therefore of primary cocer i the desig of future ireless etork deploymet A iitial ork [1] of Zader deals ith these issues, the cost structure of future ireless access, basestatio desity ad spectrum cost problem It itroduces some simple models to aalyze the cost structure of both ifrastructure costs ad spectrum costs I this paper, e develop the cost aalysis of both ifrastructure costs ad spectrum costs of [1] usig firm theory i microecoomics By usig productio fuctio i firm theory, e try to aalyze the cost structure of ireless etork deploymet The theory of microecoomics is idely used to aalyze commuicatio tems but researches have focused o cosumer theory such as the utility maximizatio of each user I terms of ireless operator s reveue maximizatio, productio fuctio, cost miimizatio, ad the techical rate of sutitutio i firm theory are much more importat ad iterestig for ireless etork operator After e formulate the productio fuctio of ireless etork operator usig area spectral efficiecy (ASE) [2], e ill address the issues of firm theory such as the techical rate of sutitutio ad cost miimizatio i this paper Also, by usig the techical rate of sutitutio (TRS), e derive the ecoomic value of spectrum ad compare ith the egieerig value of spectrum The rest of this paper is structured as follos Sectio II presets a tem model for ireless etorks ad defies the productio fuctio for deploymet ad spectrum i ireless etorks Sectio III ivestigates the characteristic of ecoomic cost fuctio Especially it deals ith explore the techical rate of sutitutio ad the elasticity of sutitutio i the productio fuctio for ireless etork Sectio VI itroduces a cost miimizatio model

3 based o this productio fuctio Sectio V compares the egieerig value of spectrum ad ecoomic value of spectrum Sectio VI dras some coclusios II SYSTEM MODEL I this sectio e itroduce the cocept of area spectral efficiecy (ASE) for fully loaded tems i hich the cell s resource (service chaels) are fully used ad the umber of iterferers is costat [1] The ASE of a cell is defied as the sum of the maximum bit rates/hz/uit area supported by a cell s BS Whe the cell radius is R, the area covered by oe of this cells is π ( R) 2 The ASE, 2 A [ / / / ] e bits s Hz m, is therefore approximated by A e N s Ck k = 1 = (1) 2 πw ( R) here N is the al umber of active serviced chaels per cell, s maximum data rate of the K th user, ad We defie the maximum rate C [bits/s] is the K W [Hz] is the al allocated badidth per cell C to be the Shao capacity of the k th user i the cell, k hich depeds o γ k, the receives SINR of that user, ad W, the badidth allocated to k that user The Shao capacity formula assumes that the iterferece has Gaussia characteristics Whe e assume that there are the coverage service area, W, the al tem capacity, R, is as belo by [1], A ad al tem badidth, svr he N s k k = 1 R A 2 srvw W ( π R ) 2 A is the cell coverage of a BS,( π R ), C Therefore, N is svr A, A Ns Ns Ns C C C A R A W A W = = = W k k k k 1 k 1 k 1 srv srv = srv = W A BS A BS A N s C R N W k k = 1 BS

4 Whe e ormalize Here, the basestatio desity R ith the coverage service area, Ns Ns A, srv Ck Ck R k 1 1 k 1 N = = W = W Asrv W A W Asrv 2 K [ BSs/ m ], umber of basestatio N 1 1 K = coverage are = A = A = a cell coverage Therefore, e defie the tem capacity fuctio of basestatio desity ad spectrum, srv R N η W Asrv Asrv here η is the sum of maximum average data rates per uit badidth by a cell s basestatio, N s ( Ck)/( W), [ b/ s]/[ Hz BS] Simply, R is the icreasig fuctio of the umber of k= 1 basestaio, N, ad amout of spectrum, W, as belo [1], R η N W (2) C η η R ηw = NBSW = NW c A A A BS III ECONOMIC MODEL FOR PRODUCTION FUNCTION AND COST The simplest ad most commo ay to describe the techology of a firm is the productio fuctio A firm produces output from various combiatio of iput I ireless operators, these iputs are poer, spectrum, ad umber of basestatios, deploymet to icrease capacity I order to research firm choices, e eed a coveiet ay to summarize the productio possibility of the firm, ie, hich is the combiatio of iputs ad outputs are techologically feasible [3] Based o (2), e ca defie the productio fuctio of ireless tems We ca say iputs as amout of spectrum, W, ad umber of basestatios, N, ad a output as capacity, R as belo, R = R(, N ) = R(, ), here, is the amout of spectrum ad is the umber of the basestatios With the above equatio, e ca defie the margial product (MP) of productio fuctio for ireless tems R dr(, ) = = MP = p, d

5 R dr(, ) = = MP = p, d here MP is the margial product of spectrum, price of spectrum ( margial product of a basestatio, price of a basestatio ( A The techical rate of sutitutio (TRS) p ) p ) ad MP is the A aalogue to the margial rate of sutitutio i cosumer theory is the techical rate of sutitutio (TRS) i productio theory [3]This measures the rate at hich oe iput ca be sutituted for aother ithout chagig the amout of output produced Assume that e have some techology summarized by a smooth productio fuctio ad that e are producig at a particular poit R = R(, ) Suppose that e at to icrease the amout of spectrum ad decrease the umber of basestatios so as to maitai a costat level of output Let ( ) be the (implicit) fuctio that tells us ho much of it takes to produce R if e are usig uits of the other iput The by defiitio, the fuctio ( ) has to be satisfy the idetity, R (, ( )) R We are after a expressio o ( ) Differetiatig the above idetity, e fid: R (, ) R (, ) ( ) + = 0, R (, ) ( ) MP = = = TRS R (, ) MP This gives a explicit expressio for the techical rate of sutitutio Here is aother ay to derive the techical rate of sutitutio Thik of a vector of small chage i the iput levels hich e rite dx = ( d, d) The associated chage i the output is approximated by, f f dy = d + d This expressio is ko as the al differetial of the fuctio f(, ) Cosider a particular chage i hich oly spectrum ad basestatio chage, ad the chage is such that output remais costat That is, d, ad d adjust "alog a isoquat" Sice the output remais costat, e have, f f 0 = d + d,

6 hich ca be solve for d f / = d f / Either the implicit fuctio method or the al differetial method may be used to calculate the techical rate of sutitutio The implicit fuctio method is a bit rigorous, but the al differece method is perhaps more ituitive B The elasticity of sutitutio The techical rate of sutitutio measures the slope of a isoquat The elasticity of sutitutio measures the curvature of a isoquat More specifically, the elasticity of sutitutio measures the percetage chage i the factor ratio divided by the percetage chages i TRS, ith output beig held fixed If e let Δ ( / ) be the chage i the factor ratio ad Δ TRS be the chage i the techical rate of sutitutio, e ca be expressed this as, Δ( / ) ( / ) θ = Δ TRS TRS This is a relatively atural measure of curvature: it asks ho the ratio of factor iputs chages as the slope of the isoquat chages This quatity measures the extet to hich operators ca sutitute basestatios for spectrum as the relative productivity or relative cost of to factors chages Whe θ is large, it meas that the operator ca easily sutitute betee spectrum ad basestatios If a small chage i slope gives us a large chage i the factor iput ratio, the isoquat is relatively flat hich meas the elasticity of sutitutio is large I practice, e thik of the percet chage as beig very small ad take the limit of this expressio as Δ goes to zero Hece the expressio for θ becomes, θ = TRS d( / ) ( / ) dtrs It is ofte coveiet to calculate θ usig the logarithmic derivative I geeral, if y= g( x), the elasticity of y ith respect to x refers to the percetage i y iduced by a small percetage chage i x That is,

7 dy y dy x ε = = dx y dx x Provided that x ad y are positive, this derivative ca be ritte as, ε = dl y dl x Applyig this to the elasticity of sutitutio, e ca rite, θ = dl( / ) dltrs I geeral, the closer θ is to zero, the model L-shaped the isoquats are ad the more difficult sutitutio betee iputs; the larger θ is, the flatter the isoquats ad easier sutitutio betee them IV COST MINIMIZATION I this Sectio, e ill ivestigate the behavior of a cost-miimizig firm This is of iterest for to reasos: Firs it gives us aother ay to look at the supply behavior of a firm facig competitive output markets, ad the secod, the cost fuctio allos us to model the productio behavior of firms that do ot face competitive output markets I additio, the aalysis of cost miimizatio gives us a taste of the aalytic methods used i costraied optimizatio problems A Calculus aalysis of cost miimizatio Here e itroduce cost miimizatio model as belo, miimize pr subject to R () r = R here p is price vector ad r is resource vector, ad R is the required tem capacity We aalyze this costraied miimizatio problem usig the method of Lagrage multipliers Begi by ritig the Lagragia, L R R ( λ, r) = pr λ( ( r ) )

8 ad differetiate it ith respect to each of the choice variables, r i ad the Lagrage multiplier, λ The first-order coditios characterizig ad iterior solutio r are, R( ) pi λ r = 0 for i= 1, r R( r ) = R i These coditios ca also be ritte i vector otatio Lettig ΔR(r) be the gradiet vector, the vector of partial derivative of R() r, e ca rite the derivative coditio as p = λδr(r ) We ca iterpret the first order coditios by dividig the i th coditio by the j th coditio to get, pi p j R( r ) ri = i, j = 1, R( r ) r i (3) The right-had side of this expressio is the techical rate of sutitutio, the rate at hich factor j ca be sutitute for factor i hile maitaiig a costat level of output The lefthad side of this expressio is the ecoomic rate of sutitutio at hat rate factor j ca be sutituted for factor i hile maitaiig a costat cost The coditios give above require that the techical rate of sutitutio ca be equal to the ecoomic rate of sutitutio If this ere ot so, there ould be some kid of adjustmet that ould result i a loer cost ay of producig the same output For example, suppose pi p j R( r ) 2 1 ri = = 1 1 R( r ) r i The if e use oe uit less of factor i ad oe uit more of factor j, output remais essetially uchaged but cost have goe do For e have saved to dollars by hirig oe

9 uit of loss of factor i ad icurred a additioal cost of oly oe dollar by hirig more of factor j This first order coditio ca also be represeted graphically I Figure 1, the curved lies represet isoquats ad the straight lies represet costat cost curves Whe R is fixed, the problem of the firm is to fid a cost-miimizig poit o a give isoquat The equatio of a costat cost curve, C = p+ p, ca be ritte as = C/ p ( p / p) For fixed p ad p, the firm ats to fid a poit o a give isoquat here the associated costat cost curve has miimal vertical itercept It is clear that such a poit ill be characterized by the tagecy coditio that the slope of the costat cost curve must be equal to the slope of the isoquat Sutitutig the algebraic expressios for these to slopes gives us equatio (3) C = p + p R = R(, ) Figure 1 Cost miimizatio At a poit that miimize costs, the isoquat must be taget to the costat lie EXAMPLE 1 Based o the above productio fuctio cocept, e fially formulate ad solve the belo cost miimizatio problem usig firm theory i microecoomic ad deal ith the reveue maximizatio problem mi, 0 subject to R (, ) R p+ p Where p ad tem capacity p are the price of spectrum ad a basestatio, ad R is the required

10 V ENGINEERING AND ECONOMIC VALUES OF SPECTRUM By [3], e ca calculate the egieerig value of spectrum as belo, Providig a certai average data rate R bits Hz Km a certai area of size A i a highly 2 ( / / ) loaded (iterferece limited) cellular data tem requires the folloig umber of basestatios N ad spectrum BS W as e metioed [3] R η N W A BS The costat η ca be iterpreted as spectral efficiecy of the tem (i bit/s/hz) ad is a property of the desig of the radio trasmissio techology used No assumig a operator providig the data rate R ad iterested i icreasig his data rate by Δ R, η η η R +ΔR NBSW + Δ NW + NBSΔW A A A The secod term correspods to icreasig the umber of base statio by Δ N, here as the last term represets icreasig the spectrum allocatio by Δ W Usig this, the additioal cost Δ C as belo, here c is the cost of a (additioal) basestatio ad BS Δ c is the cost of refittig the BS existig basestatio ith equipmet for the e spectrum ad c sp is the cost(per Hz) for the e spectrum So e ca compare the ext equatio to miimize cost, We ca get the ext equatios ΔR ΔR mi Δ C = mi cbs A, ( Δ cbs NBS + csp ) A ηwsys ηnbs 1 1 cbs = ( Δ cbs NBS + csp ) W N SYS ( ) c N = Δ c N + c W BS BS BS BS sp SYS ( ), C +ΔC C + c Δ N+ Δ c N + c ΔW BS BS BS sp c N c = Δc N BS BS sp BS BS W BS

11 The above equatio is called the egieerig value of spectrum Usig the techical rate of sutitutio (TRS), also e ca calculate the ecoomic value of spectrum as belo, R (, ) R (, ) Δ W + Δ N = 0 p = p ΔN ΔW I the egieerig value of spectrum, if e igore the refittig cost egieerig value of spectrum equals the ecoomic value of spectrum VI CONCLUSION R (, ) R (, ) ΔN = ΔW Δ, or set Δ = 0, the We deal ith the basic ecoomic theory of cost model for etork deploymet ad spectrum i ireless etorks I particular, e develop a productio fuctio for ireless etorks With this productio fuctio model, e explore the techical rate of sutitutio ad the elasticity of sutitutio i the productio fuctio for ireless etork ad fid its isight for ireless etork Fially, e compare the egieerig value of spectrum ad ecoomic value of spectrum cbs c BS REFERENCES [1] J Zader, O the cost structure of future idebad ireless access, i Proc IEEE VTC Sprig 1997, pp , May 1997 [2] M-S Alouii, ad A J Goldsmith, Area Spectral Efficiecy of Cellular Mobile Radio Systems, IEEE Tras Vehic Tech, vol 48, o 4, pp , July 1999 [3] A Mas-Colell, M D Whisto, ad J R Gree, Microecoomic Theory Oxford, UK: Oxford Uiv Press, 1995 [4] K Johasso Cost Effective Deploymet Strategies for Heterogeeous Wireless Neorks, PhD Thesis, KTH, Stockholm, Dec 2007

John Riley 30 August 2016

John Riley 30 August 2016 Joh Riley 3 August 6 Basic mathematics of ecoomic models Fuctios ad derivatives Limit of a fuctio Cotiuity 3 Level ad superlevel sets 3 4 Cost fuctio ad margial cost 4 5 Derivative of a fuctio 5 6 Higher