A Customers Discrete Choice Model for Competition over Incomes in Telecommunications Market

Transcription

1 Internatonal Arab Conference on Informaton Technology (ACIT'2016) A Customers Dscrete Choce Model for Competton over Incomes n Telecommuncatons Market M hamed Outanoute 1, Mohamed Baslam 1, Belad Boukhalene 2 1 Sultan MoulaySlmane Unversty, Faculty of Scences and Technques, BenMellal, Morocco 2 Sultan MoulaySlmane Unversty, Polydscplnary Faculty, BenMellal, Morocco mhamed.outanoute@gmal.com, balsam.med@gmal.com, bboukhalene@yahoo.fr Abstract :The customers churn between servce provders (SPs) due to better prces, better qualtes of servces or better reputaton. Each provder s assumed to look for a maxmzed revenue, whch depends on the strategy of the compettor. Several works present n the lterature were based on the prce as the only decson parameter, whereas others parameters such as qualty of servce (QoS) have a decsve mpact to regster to an operator rather than the others. We formulate the nteracton between SPs as a non-cooperatve game. Frst, each SP chooses QoS to guarantee and the correspondng prce. Second, each customer chooses hs SP and may churn to another or alternatvely swtch to -no subscrpton state- dependng on the observed prce/qos. In ths work, we operate Markov chan to model user's decsons, whch depend on the strategc actons of SPs. We adopt logt model to represent transton rates. Fnally, we provde extensve numercal results to show the mportance of takng prce and QoS as jont decson parameters. Keywords :Prcng, QoS, Mgratng customers, Servce provders competton, Logt model. 1. Introducton Thelberalzatonoftelecommuncatonsservcesandtheprolfe ratonofservceprovdershasledtorelevanceofcustomermgrat onphenomenonbetweenoperatorsandtoaveryatrocouscompet tonntelecommuncatonsmarket.churnsespecallylargen moblenetworks,whereyearlymgratonratesashghas25%are notuncommon[15][22].provdersseekretanthercustomersan dattractnewones,countngonpreventon[1][11]andonreactves trateges[18][14]. Among factorsthatnfluencethecustomers decsonstochangeoperator,prceplays anmportantrole[15][8][9]. Today,wthchangngtraffccharacterstcs,communcatonsar emorestrngentntherqualtyrequrements(delveryrateandt me)thanrawdata.hence,theneedofsomereconsderatonofthe mplcatonsofarchtecture,servceclasses,anddesgnprncples ontheprcngmodelsofservceprovders[7].fromcustomers po ntofvew,moretheguaranteedqualtyofservcetendsto100%se rvcequalty(fulldelveryrateandreasonabledelay),theservcep rovderbecomesmoreattractve.fromservceprovderspontof vew,generatesuchhghservcequaltyreducesthernetrevenue. Now,tsclearthatprceandservcequaltystmulateandnflu encethedecsonsofcustomerstochurnfromoneservceprovder toanother.servceprovdersareseekngtoattractnewcustomers, butmostlyretanther exstngcustomers. Thelossofonecustomermplesthelossoffuturerevenuesassoc ated tothat customer(a zero-sum game canmodelze suchstuatons).ths phenomenons largelymodeled nseveral worksbasngonnon-cooperatve gametheoryandassumngasnglecharacterstc,suchasprce[15 ][9][16][21],QoSntermsofdelay[19],or evenlossprobabltes[4].however,andtodevelopamodelsml artorealty,other worksncorporatejontlyprceandsomemeasureofqualtyofser vce[7][10]. Formoredetalsonsurveystechnquesofcompettve gamentelecommuncatons,see[3][5]. Ths work s organzed as follows. In Secton 2, we formulate a model of customers behavor usng a Markov chan and a descrpton of probabltes to churn. Secton 3 presents the descrpton of the non-cooperatve game between servce provders and the strateges that defne the best polcy for both of them. In Secton 4, we gve a defnton of the logt model used to expresses the rate of mgraton n terms of prce and qualty of servce. In Secton 5, we provde extensve numercal results to assst our analyss. Fnally, n Secton 6, we conclude our paper. 2. Costumers' Behavor Model We consder a system wth N customers and 2 servce provders (SPs). Each SP decdes a QoS q to guarantee and the correspondng prce p. To choose hs operator, the customer seeks to fnd the SP that proposes a good compromse prce/qos. Then, each costumer decdes to regster to one of the two operators or to stay at no subscrpton state, see fgure.1.

2 Internatonal Arab Conference on Informaton Technology (ACIT'2016) (1) Q 0, 2 1, 0 0, 0,1, 2. Fgure 1: Customer s behavour as a contnuous tme Markov chan. From SP pont of vew, t shall to defne the best prcng strategy and the best amount of bandwdth, to request from the network owner, for ncreasng hs market share. From customer pont of vew,the queston s to set the best probabltes vector to regster to an operator. The problem s thus conceved as astackelberg game [2] where the operators are the leaders and the customers are the followers. Usng backward nducton, the provders play frst and antcpate the resultng strategy of end users. Each costumer may stay wth hs operator or leave t to regster under servces of another. We assume that the contnuous tme Markov chan that s depcted n fgure 1 represents the behavor of a customer. Here state 1 means that the customer s wth provder 1, state 2 that he s wth provder 2 and state 0 that he does not use any servce. The transton from state to state j depends not only on parameters of current SP that are p and q, but also on the prce and QoS offered by ts compettors, that s, π depends upon the entre prce vector p = (p 1, p 2 ) and the entre QoS vector q = (q 1, q 2 ). To avod negatve profts, we consder here that both prce and QoS ndcator are bounded and postves,.e., 0<p mn p p max and 0 <q mn q q max. Let us denote by λ j the transton probablty from state to state j, the transton probabltes are depcted n fgure 1. The resultng nfntesmal generator of Markov chan correspondng to the fgure 1 s the followng Let (p,q) { (p,q), {0,1,2}} denotes the steady state of the Markovan system, where the probablty that a gven customer s wth servce provder s. From non-cooperatve game theory perspectve, π s theequlbrum mxed strategy of any gven customer. Equvalently, can also be seen as the market share of SP, tfollows that the average number of customers that are wth SP s N. The customers problem s then a soluton ofthe system: For two SPs, the soluton of ths system s gven by: ( ) C, ( ) C, ( ) C, Where: C Servce Provders Strategc Decson After modelng the behavor of customers, wth takng nto consderaton both the unt prce and the guaranteed QoS, we turn now to defne the best polcy (prce and QoS) for each SP. We frst defne the utlty functon and then analyze the equlbrum concept. As t s known, each SP seeks to attract the maxmum possble number of customers among a populaton ofn customers. On one hand, ncreasng the prce (and/or decreasng the qualty to guarantee) wll ncrease the revenue per customer. On the other hand, ths polcy may potentally reduce the number of customers (equvalently the market share). Henceforth, there s a tradeoff to be analyzed. Havng a market share π, the total revenue of SP s then N π. We assume that we have a sngle network owner, ths latter charges each SP a cost ϑ per unt of requested bandwdth. In order to nsure the customers loyalty, the amount of bandwdth µ requred by SP should depend on π, N and on the QoS q t wshes to offer to ts customers. Therefore, the net proft of SP s smply the dfference between the total revenue and the fee pad to the network owner: U (p,q) N p F(q, ), {1,2}. (2) The fee pad by SP can be wrtten as [10]: F (N,,q ) where (N,,q ) stheamountofbandwdthrequredbysptoguaranteethean nouncedqosq,whchhasthefollowngform: (N,,q ) N g (q ) h (q ) (3) where g (q ) and h (q ) are postve functons, whch mean that the profle functon of SP becomes: U (p,q) N (p g (q )) h (q ), {1,2}. (4)

3 Internatonal Arab Conference on Informaton Technology (ACIT'2016) Expected Delay as QoSIndcator: In the rest of ths paper, we assume that the measure defnng the QoS corresponds to some functon of the expected delay. We consder the Klenrock delay functon, whch s a common delay functon used n networkng games, see [10] and [19]. In ths way, the maxmzaton of QoS requres mnmzaton of the delay. For ths reason, and nstead of mnmzng delays, we consder the maxmzaton of the recprocal of ts square root: q 1/ Delay N.(5) Therefore, (N,,q ) N g (q ) h (q ) We wll focus on the rest of ths paper to the smple case 2 where g (q ) 1and h (q ) q. Thus,equaton(4)becomes: 2 U (p,q) N (p ) q, {1,2}. (6) Each servce provder strves to fnd ts best strategy,.e., ts prce and QoS to guarantee maxmzng ts revenue, whch can be modfed by the strategy of the compettor. The soluton concept to adopt here s naturally that ofa Nash equlbrum. Defnton1(NashEqulbrum):Sncethecustomersstead ystatedependsonlyonthefxedprcesandtheofferedqosbyt * * * * * * hetwoadversaralsp,then (p,q ) (p 1,p 2,q 1,q 2) saservceprovdersnashequlbrumft satsfes: * * * * a) (p,q ) argmax U (p,p,q,q ), 1,2 and b) p,q * * (p,q ) s a feasble strategy profle. Algorthm1:Numercally fndng the Nash equlbrum of the jont Prce-QoS game. a) Input:TranstonratesoftheMarkovchanasfunctonofpand q, b) For each SP Forallfeasblecoupleofvalues (p,q ) fndtheset BR (p,q ), * * * * c) Each 4-tuple (p 1,p 2,q 1,q 2) suchthat * * * * * * * * (p 1,q 1) BR 1(p 2,q 2) and (p 2,q 2) BR 2(p 1,q 1), sanash equlbrum. In order to solve ths non-cooperatve game we use a backward nducton technque. We start wth thecustomers regstraton and derve ther steady behavour as a functon of the prce set and the QoS offered by the servce provders. Wthout extra assumptons, exstence of a Nash equlbrum cannot be ensured, nor ts unqueness when exstence s shown. In the case where utltes and rates functons are smple enough n terms of prces and QoS, we may fnd the form of the Nash equlbrum analytcally. Otherwse, the computatons can be performed numercally usng the followng algorthm 1. We defne the best response of each provder as a functon of the strategy of tsopponent by: 4. Mgraton Rate Model BR (p,q ) argmax U (p,q), (7) p 1,q1 BR (p,q ) argmax U (p,q). (8) p 2,q2 The transton rate descrbng the customer mgraton probabltes from one operator to another, depend on both prces and QoSs defned by provders. The choce of smplfed cases has shown that the Nash equlbrum can be obtaned analytcally, but not easly [15][7]. In ths secton, we are nterested n usng a model as close as possble to realty, whch expresses the rate of mgraton n terms of prces and QoSs ChurnRatesntheLterature Several studes have been conducted to dentfy churn determnants, whch are the most relevant factors n determnng churn. These works are based on both parametrc and non-parametrc approaches, for quanttatve expresson of churn rates n terms of prces and QoSs. We cte, among the nonparametrc approaches, the use of neural networks and decson trees [17], and a learnng evolutonary algorthm [6]. In ths work, we focus on parametrc approaches to obtan a closed form relatonshp. The logt model s the most wdespread adopted n the lterature to represent ths relatonshp. Ths model employs a logstc probablty dstrbuton functon [12][13][20]. Ths functon s a lnear functon of a number of determnants churn. The probablty that a user churns n the next perod s gven by the expresson: 1 churn, (9) I 1 e where I s the logt factor, gven by n I X, (10) 1 where X, 1,...,n arethechurndetermnantsand, 1,...,n are the coeffcents representng therelatvemportanceofthosedetermnants Transton Rates

4 Internatonal Arab Conference on Informaton Technology (ACIT'2016) InthspaperwehavefocussedontheprceandQoS,sothatwecang roupthempactoftheotherchurndetermnantsntheoverallterm,arrvngatthesmplerexpresson: 1 churn, (11) ( 1p /pj 2q j /q ) 1 e for the probablty that n the specfed perod the user swtches from state to state j. We may employ that expresson for a tme perod of any duraton, so we can adopt t n the Markov chan model descrbed n Secton 2. We note that, accordng to expresson (11), there s a non 1 zero probablty, namely churn, that the user 1 swtches provder due to the ensemble of other dssatsfacton factors, even when the servce offered by the losng provder s free and/or ts QoS s very hgh. The lterature consders dscretzed tme, whereas we focus here on a contnuous tme model. That latter dffculty s addressed here by assumng that tmeperods consdered n Subsecton 4.1, are short wth respect to the mean sojourn tme n a gven state. Ths mples that the dscrete tme transton probabltes are approxmately the contnuous-tme transton rates multpled by the perod duraton. Consequently, we would lke to consder transton rates from state {0,1,2} to state j {0,1, 2} of the form: k churn, (12) ( 1p /pj 2q j /q ) 1 e where k 0representsthenverseoftheperodduraton[15]. Snce 1and 2 representrespectvelytheusersenstvtytoprcesandtheusers enstvtytoqos,weconsdertsthesameforthedfferentstat esofthemodel.wentroduceasymmetryamongprovdersthrou ghtheparameter asexplanedbefore,thsparameterencompassesthereasonsothe rthanprceandqos(e.g.,reputaton...),whyausershouldleavestate. Themodelparametersthatweconsderarethen: theusersenstvtytoprceandqos 1 and 2, thelkelness tostayncurrentstate, {0,1, 2}, thenverse of perod duratonk(ths parameter should not playa role n our model,snce by a tme untchangewecanassumek=1), theuserpercevedcostp 0 fornotbenefttngfromtheservc eandq 0 thesupposedqosnnon-subscrpton state. Consderng the prevous expressons of the transton rates, we suggest to study the game proposed n ths work. The dependence of those rates on provder prces and QoS are too complcated to solve the problem analytcally, therefore we perform here a numercal study. The parameter values consdered n ths secton are the followng: N=10000,p 0 =1,q 0 =1,β 1 =0.5,β 2 =0.3,k=1, γ 0 =3,γ 1 =1,γ 2 = Prce Game wth Fxed QoS Inthssubsecton,wewllanalyzethebehavorofthesystemft hequaltyofservcesfxedandtheprcesconsderedtheonly actonofthegame.expresson(12)wouldmplythatalltranst onratesbenanntervalk / (1 ),k regardlessoftheprceandqosvalues.inthsgamewthfxedq os,thssnotrealstc,sncetwouldmplythataprovdercould ensureanarbtrarlylargerevenuebysettngaverylargeprce. Wethereforeneedthatthetranstonratestoaprovdertendto 0and/orthattheratesfromprovdertendto,where p tendto and/or q tendto0. Tothatend,weslghtlymodfytheprevousexpresson,andtak etranstonratesoftheform: k k 1p /pj 2q j /q j (p,p j,q,q j ) e e ( 1p /pj 2q j /q ) e (13) Fgure 2 plots the utlty U 1 of provder 1 when ts prce p 1 vares, for dfferent values of the adversary prce p 2. We remark that the net revenue of provder 1 s frst ncreasng, and then decreasng n p 1. Also that the utlty of provder 1 decreases when ts prce p 1 tends to nfnty, whch mples exstence of a fnte prce p 0 maxmzng U 1. Ths prce value consttutes the best response of provder 1 aganst the prce set by provder 2. We remark also that the net revenue of SP 1 ncrease when adversary prce p 2 ncrease, whch s lkely ntutve. Indeed, f the opponentchooses a hgher prcng strategy, then t may lose a fracton of ts customers. Moreover, servce provder SP 1 may attract those latter and make ncte them to churn. Consequently, ts net revenue s ncreased. Surprsngly, we observed an nterestng feature characterzng our proposed prce game: Through hundreds of numercal example runs, we 1) checked exstence of a unque best response, and 2) checked that as the prce defned by SP 2 ncreases, the best response of SP 1 tends to ncrease also! Ths motvated us to work about provng super-modularty of ths game. 5. Numercal Results and Dscussons

5 Internatonal Arab Conference on Informaton Technology (ACIT'2016) Fgure 4 : Net revenue of provder 1 versus p 1 for dfferent values of q 2, wth q 1 = 10 and p 2 = 4. Fgure 2 : Netrevenueofprovder1versusp1fordfferentvaluesofp2,wthq1=15and q2=15. Fgure 3 and 4 present respectvely the nfluence of QoS q1 and q2 of both provders on the net revenue U1 of SP 1. In terms of servce provder s own qualty of servce, the net revenue behaves n two ways. When SP 1 sets a low prce, t tends to be more benefcal to request relatvely low amount of bandwdth provdng low qualty of servce. Ths s a qute surprsng behavour, but can have a realstc good extent. Indeed, when an SP offers some gven servce at a low cost, ts market share goes to 1. Absorbng the total number of customers, the tagged SP tends to no nvestng anymore and the offered servce qualty becomes poor. Another mportant result s that as the offered QOS ncreases as the best response ncrease respectng by the way: If you want more, you have to pay more. Next, we depct the best response of the two adversaral servce provders. We recall that ths fgure s obtaned usng algorthm 1 and the ntersecton between the two graphs represents the Nash equlbrum pont of the game. Through several examples, we always obtaned a unque Nash equlbrum. Another vsble feature s that the fxed prce should evolve n the same drecton of the nvestment. In other words, the end prce ncreases as the offered QoS ncreases. Fgure 5 : Best response prce for both provders for dfferent values of q 1 = q 2. Fgure 6 : Best responses prce for both provders for dfferent values of q 1 q 2. Fgure 3 : Net revenue of provder 1 versus p1 for dfferent values of q1, wth q2 = 15 and p2 = QoS Game wth Fxed Prce In ths subsecton, we wll analyze the behavor of the system f the prce s fxed and the qualty of servce s consdered the only acton of the game. We frst notce that the net revenue s decreasng wth the QoS. Ths s qute ntutve snce t means for servce provder to nvest more and more whle chargng customers a constant prce, see fgure 7. It s clear from the same fgure that, as the opponent SP 2 ncreases ts qualty of servce, t s benefcal for SP 1 to ncrease ts QoS. Ths latter result s well known n economcs.

6 Internatonal Arab Conference on Informaton Technology (ACIT'2016) 10) and where they set adopt dfferent prcng strateges (fgure 11). Fgure 7 : Net revenue of provder 1 versus q 1 for several values of q 2, wth p 1 = 4 and p 2 = 4. Fgure 8 (respectvely 9) shows that the utlty functon of SP 1 s concave w.r.t q 1 for dfferent prces p 1 (respectvely p 2 ). Ths means that the best response n terms of QoS s unque and therefore the exstence of the equlbrum n term of QoS s assured. On the other hand, the same fgures show the need to defne a hgher prce as the own QoS mproves and the need nvestment (mprove promsed QoS) as the compettor reduces ts prce. Fgure 10 : Best response QoS for both provders for dfferent values of p 1 = p 2. Fgure 11 : Best responses QoS for both provders for dfferent values of p 1 p 2. Fgure 8 : Net revenue of provder 1 versus q 1 for several values of p 1, wth q 2 = 15 and p 2 = 4. Fgure 9 : Net revenue of provder 1 versus q 1 for several values of p 2, wth q 2 = 15 and p 1 = 4. Later, we plot the best response n terms of QoS whle the two servce provders set the same prce (fgure 5.3. Convergence to Nash Equlbrum Ifservceprovdersallowfreecostservcewthpossblynullqual ty,.e.,prceandqualtymaybe0,wenotcethat ( p1, p2) (0,0) salsoasatsfyngstuatonforbothsps,thustrepresentsalsoane qulbrumpont.sncetbrngsanegatverevenuetotheprovder s,andmoreovertsnotastablenashequlbrum:fanyofthetwo provdersslghtlydevatesfromthatstuatonbysettngastrctlyp ostveprce,thenanteratvebestresponsebasedalgorthmleadstotheother(stable)nashequlbrum.b ymposngnon-nullprceandnonnullqualty,thenashequlbrumwouldbeaprorunqueandth esngleparametergamewouldbesupermodular.wewllconsequently focusonthatequlbrumnthefollowng,whentexsts.algor thm2nsuresconvergenceofbothspstothenashequlbrum pontshghlghtednprevousfgures,seefgure12andfgure 13.Itsclearthatthespeedofconvergence srelatvelylow(around6teratonstoconvergetoprcebasedequlbrumandqos-basedequlbrum). Algorthm 2:ConvergencetoNashequlbrum(whenexsts)

7 Internatonal Arab Conference on Informaton Technology (ACIT'2016) 1: z BR (z ), {1,2}. t1 t where BR (.) sthebestresponsefunctonofsp,and z sagenercvarablerepresentng p or q (whlemantanngtheotherparameterconstant). Fgure 12: Prcng game: Convergence to NE under ( q, q ) (30,30). 1 2 Fgure 13: QoS game: Convergence to NE under 6. Concluson (p,p ) (5,8). 1 2 In ths work, we have presented a contrbuton to the study of the competton between telecommuncatons operators. These take nto account the customer mgraton behavor to decde the prce and qualty of servce to offer n the am of maxmzng ther revenue. The problem s presented as a non-cooperatve game based on the prce and qualty of servce as jont decson parameters. The customer mgraton behavor s modeled by a Markov chan where the transton rates are expressed usng the logt model. We have shown through numercal analyss the exstence of many Nash equlbrum or none for ths game, whch can be unque f the game s symmetrcal and the mnmum prce p mn and mnmum qualty of servce q are not zero. References [1] Ahn,J.-H.,Han,S.-P.,andLee, Y.-S.(2006). Customerchurnanalyss: Churndetermnantsandmedatoneffectsofpartaldefect mn onnthekoreanmobletelecommuncatonsservcendust ry. Telecommuncatons polcy,30(10): [2] Altman, E., Boulogne, T., El-Azouz, R., Jme nez, T., and Wynter, L. (2004). A survey on networkng games. Computers and Operatons Research. [3] Altman, E., Boulogne, T., El-Azouz, R., Jmee nez, T., and Wynter, L. (2006). A survey on networkng games n telecommuncatons. Computers & Operatons Research, 33(2): [4] Altman, E., El Azouz, R., and Abramov, V. (2002). Non-cooperatve routng n loss networks. Performance Evaluaton, 49(1): [5] Altman, E. and Wynter, L. (2004). Equlbrum, games, and prcng n transportaton and telecommuncaton networks. Networks and Spatal Economcs, 4(1):7 21. [6] Au, W.-H., Chan, K. C., and Yao, X. (2003). A novel evolutonary data-mnng algorthm wth applcatons to churn predcton. Evolutonary Computaton, IEEE Transactons on, 7(6): [7] Baslam, M., Echabb, L., El-Azouz, R., and Sabr, E. (2012). Jont prce and qos market share game wth adversaral servce provders and mgratng customers. In Game Theory for Networks, pages Sprnger. [8] Buehler, S., Dewenter, R., and Haucap, J. (2006). Moble number portablty n Europe. Telecommuncatons Polcy, 30(7): [9] Danaher, P. J. (2002). Optmal prcng of new subscrpton servces: Analyss of a market experment. Marketng Scence, 21(2): [10] El Azouz, R., Altman, E., and Wynter, L. (2003). Telecommuncatons network equlbrum wth prce and qualty-of-servce characterstcs. Teletraffc Scence and Engneerng, 5: [11] Eshgh, A., Haughton, D., and Top, H. (2007). Determnants of customer loyalty n the wreless telecommuncatons ndustry. Telecommuncatons polcy, 31(2): [12] Km, H.-S. and Yoon, C.-H. (2004). Determnants of subscrber churn and customer loyalty n the Korean moble telephony market. Telecommuncatons polcy, 28(9): [13] Km, M.-K., Park, J., Pak, J., Seol, S., and Park, H. (2004). An emprcal study on the nfluencng factors of customer churnng ntenton after ntroducng moble number portablty n korea. In 15th ITS Bennal Conference, Berln, pages 4 7. [14] Mallé, P., Nald, M., and Tuffn, B. (2008). Competton for mgratng customers: a gametheoretc analyss n a regulated regme. In Global Telecommuncatons Conference, IEEE GLOBECOM IEEE, pages 1 5. IEEE. [15] Mallé, P., Nald, M., and Tuffn, B. (2009). Prce war wth mgratng customers. In Modelng, Analyss &

8 Internatonal Arab Conference on Informaton Technology (ACIT'2016) Smulaton of Computer and Telecommuncaton Systems, MASCOTS 09. IEEE Internatonal Symposum on, pages 1 8. IEEE. [16] Mallé, P. and Tuffn, B. (2010). Prce war n heterogeneous wreless networks. Computer Networks, 54(13): [17] Mozer, M. C., Wolnewcz, R., Grmes, D. B., Johnson, E., and Kaushansky, H. (2000). Predctng subscrber dssatsfacton and mprovng retenton n the wreless telecommuncatons ndustry. Neural Networks, IEEE Trans- actons on, 11(3): [18] Nald, M. (2008). A smple model for the effectveness of delayng strateges for telecommuncatons churn reducton. In Computer Modelng and Smulaton, UKSIM Tenth Internatonal Conference on, pages IEEE. [19] Orda, A., Rom, R., and Shmkn, N. (1993). Compettve routng n mult-user envronments. IEEE/ACM Transactons on Networkng, pages [20] Q, J., Zhang, Y., Zhang, Y., and Sh, S. (2006). Treelogt model for customer churn predcton. In Servces Computng, APSCC 06. IEEE Asa- Pacfc Conference on, pages IEEE.