Revenue Management for Online Advertising: Impatient Advertisers

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1 Reveue Maagemet fo Ole Advetsg: Impatet Advetses Kst Fdgesdott Maagemet Scece ad Opeatos, Lodo Busess School, Reget s Pak, Lodo, NW 4SA, Uted Kgdom, kst@lodo.edu Sam Naaf Asadolah Maagemet Scece ad Opeatos, Lodo Busess School, Reget s Pak, Lodo, NW 4SA, Uted Kgdom, saaf@lodo.edu The Iteet s cuetly the fastest gowg advetsg medum. Ole advetsg bgs ew oppotutes ad has may dffeet chaactestcs fom advetsg tadtoal meda that suppot effcet ad mechazed decso makg. We cosde a web publshe that geeates eveues fom dsplayg advetsemets o ts webste. The advetses appoach the web publshe, equest the ad to be dsplayed to a ceta umbe of vstos to the webste, ad ae chaged accodg to the so-called pay-pe-mpesso pcg scheme. The advetses ae mpatet ad wat the ads to be posted ght away othewse they appoach aothe web publshe. We model the advetsg opeato of the web publshe as a queueg system wth o watg spaces (a loss system) whee advetsg slots coespod to seves. Ths system s dffeet fom kow mult-seve systems. We deve a closed-fom soluto fo ts steady state pobabltes ad aalyze the system popetes. We deteme the optmal advetsg pce ad povde maageal sghts such that the optmal pce s ceasg the umbe of mpessos made to the vewes, whch goes agast the ecoomy-of-scale tuto. The queueg model s compaed to kow queueg systems such as the bulk system ad we povde addtoal esults fo those. Key wods : maketg: advetsg ad meda, pcg; queues; vetoy polces: maketg/pcg. Itoducto The Iteet s cuetly the fastest gowg advetsg medum (TNS (2007)). It povdes a access to a eomous cosume base ad compaes ae ceasgly allocatg a lage poto of the maketg budget towads ole advetsg (IAB (2007)). Ole advetsg bgs ew oppotutes ad has dffeet chaactestcs fom advetsg tadtoal meda such as televso ad ewspapes. I ole advetsg t s possble to detect mmedately how a vsto to a

2 2 Reveue Maagemet fo Ole Advetsg: Impatet Advetses webste espods to advetsemets (ads) ad hs actos ca be kept tack of. Futhemoe, wth the Iteet s capabltes to detfy webste s vstos t s feasble to taget ads o a level ot possble befoe. These chaactestcs ad othes make ole advetsg sutable fo moe quattatve ad automated decso makg. Ole advetsg ca be dvded to two domas: sposoed seach advetsg ad dsplay advetsg. Sposoed seach advetsg volves advetses payg a fee to appea ext to seach esults fo patcula seach wods. Seach eges such as Google ad Yahoo geeate a lage potos of the eveues fom ths type of advetsg ad the pcg ad allocato mechasm ae usually based o bddg (fo a ovevew ad modelg see Feg et. al (2008)). Dsplay advetsg volves a web publshe, e.g., Yahoo ad MSN, that povdes sevces o cotet ad stead of chagg a subscpto fee t geeates eveues fom postg ads o ts ste. The most popula dsplay ad s the bae (othe ad types ae, e.g., pop-ups, pop-udes, ad -steam vdeo). I ths pape we focus o dsplay advetsg ad cosde a web publshe that sells advetsg space o ts webste. The web publshe has to maage the uceta demad of the advetses fo the advetsg space as well as the uceta supply of the vstos to ts webste. I ode to maxmze ts eveues t detemes the optmal pce to chage to attact a sutable amout of advetses gve the steam of vstos. Ofte pactse, the pcg s detemed va estmatos ad egotatos. Howeve, the atue of the Iteet suppots a moe effcet ad automated pcg appoach. The obectve of ths pape s to povde a decso makg tool as well as sghts based o a stylzed model of the opeato of the web publshe to fll ths gap. Maagg the advetsg opeato ca be a challegg task. The web publshe usually maages multple webpages wth the webste. The web publshe ot oly eeds to decde o the pce to offe fo a ceta umbe of mpessos but s also faced by decsos such as how may advetsg slots to have ad how may advetses to seve at the same tme. The web publshe ca fact seve moe advetses tha thee ae slots. Oe ca otce ths whe efeshg a webpage that a ew set of ads s ofte dsplayed. Also f a vsto stays fo a log tme o a webste sometmes a ew set of ads s dsplayed.

3 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 3 Web publshes use dffeet pcg schemes to chage advetses. The most commo oe s pay-pe-mpesso whee the advetse s chaged fo each tme a vewe vsts the ste whee the ad s dsplayed. The fee s usually quoted as cost pe thousad mpessos o CPM (the M stads fo the Roma M fo mll). The cost vaes wth web publshes but oe ca commoly see a CPM of $4 - $60. Aothe fequetly used pcg scheme s pay-pe-clck. I ths type of a cotact the web publshe chages a ceta pce (cost pe clck o CPC) fo evey tme a vstoclcksoaad.usuallythecpcsbelowadolla.othelessfequetlyusedschemesae pay-pe-lead, pay-pe-acqusto ad pay-pe-sale. I ths pape we develop a stylzed model that captues may of the ssues faced by a web publshe ug a advetsg opeato. The model ca also be used as a buldg block fo othe ssues such as teactos betwee web publshes. We buld o the ovel modellg famewok developed by Aama ad Fdgesdott (2007) fo a web publshe that geeates eveues fom postg ads o ts webste ad chages accodg to the most commo pay-pe-mpesso pcg scheme. As Aama ad Fdgesdott (2007) we cosde avals of advetses teested postg the ads ad avals of vewes vstg the webste. Gve the dyamcs of how the advetses ae seved a sychozed way by the vewes (llustated detal late) the esultg system s ot a tadtoal mult-seve system. I ths pape, cotast wth Aama ad Fdgesdott (2007), the advetses ae assumed to be mpatet ad ot wllg to wat fo the ads to be posted athe they would choose to go to aothe web publshe. The opeato of the web publshe s modeled as a queueg system wth o watg spaces (ofte called a loss system). Usually pactse, thee s a lage umbe of web publshes that offe smla sevces ad theefoe, the advetses do ot have a easo to wat f the ad caot be posted whe they make the equest. Hece, assumg a advetse s lost f hs demad s ot met s a easoable assumpto the compettve ole advetsg wold. (Fo the same assumpto aothe settg see Sav et. al (2005).) Fom a techcal pot of vew t mpoves the tactablty of the poblem eablg closed fom solutos that ca gve futhe sghts.

4 4 Reveue Maagemet fo Ole Advetsg: Impatet Advetses The fst cotbuto of the pape s the developmet ad aalyss of a ew queueg model fo a opeato of a web publshe that deals wth mpatet advetses. A closed fom soluto s deved fo the pobablty dstbuto of the umbe of advetses the system fo a geeal umbe of advetsg slots ad mpessos. Ths eables us to solve the eveue maxmzg poblem of the web publshe ad we deve the optmal pce to chage pe mpesso. The model also seves as a buldg block fo moe othe settgs such as competto ad advetsg etwoks sevg multple advetses ad webste (see a dscusso o extesos Secto 7). The secod cotbuto s the maageal sghts deved fo a web publshe o the optmal eveue ad pce. Fo example, we show that the optmal pce deceases wth the umbe of advetsg slots o the page, whch s qute tutve. Howeve, the optmal pce ceases wth the umbe of mpessos equested by the advetses, whch s coute tutve compaed to the usual ecoomes-of-scale pecepto. Oveall, we povde a pcg tool fo a web publshe to deteme the pce to chage based o the demad fo the advetsg space ad the web taffc, whch ca eplace the cuet ofte ad-hoc appoaches. Thdly, we compae ou queueg system to kow systems. Fo the Elag loss system we show that the aveage umbe of obs s geate tha fo ou system. Fo the bulk system we geeate some addtoal esults ad sghts. Fally, we povde addtoal sghts though umecal examples. The pape s ogazed as follows. I the followg secto the elevat lteatue s evewed. Secto 3 pesets the model developed of the opeato of a web publshe. The optmal pce to chage fo advetsg s deved Secto 4 ad umecal examples wth futhe sghts ae povded Secto 5. I Secto 6 we compae the queueg model developed to kow queueg models. Fally, we coclude Secto 7 ad gve some sghts fo umeous futue dectos. 2. Lteatue Revew The lteatue o ole advetsg wth the maketg aea s qute extesve. Novak ad Hoffma (2000) povde a ovevew of advetsg pcg models fo the teet. Howeve, thee s lmted lteatue o aalytcal models fo pcg ad othe decso makg fo a web publshe

5 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 5 wth a advetsg opeato. (Fo ssues faced by advetses such as pedctg audece fo advetsg campags see, e.g., Daahe (2007) ad papes efeeced wth.) Sav et. al (2005) cosde eveue maagemet fo etal busesses wth two custome classes. As ou case tadtoal eveue maagemet models ae ot adequate the settg as those assume capacty to be sold by a fte hozo. Futhemoe, ou case the capacty s adom (the comg vewes), whch makes tadtoal eveue maagemet appoaches eve less applcable. Eve though the dyamcs of the poblem s dffeet fom ous esultg dffeet models thee ae some smlates such as the avals of the customes. The pape by Aama ad Popescu (2007) s coceptually elated to ou pape the cotext of meda. They study eveue maagemet fo tadtoal meda, moe specfcally boadcastg. The ssues cosdeed thee ae of smla atue as ths pape. Howeve, the settg s qute dffeet. They cosde the challege faced by a meda boadcastg compay of allocatg lmted advetsg space betwee up-fot cotacts ad the scatte maket ode to maxmze eveues ad meet cotactual commtmets. Eve though the demad sde of the poblem s somewhat smla to ous the supply sde of fxed commecal tmes wth uceta umbe of vewes s dffeet fom ou aval steam of vewes. May web publshes ot oly geeate eveues fom advetsg but also fom subscptos. Seveal papes cosde the tade-off betwee those two eveue steams. Baye ad Moga (2000) develop a smple ecoomc model of ole advetsg ad subscpto fees. Pasad et. al (2003) model two offegs to vewes of a webste: A lowe fee wth moe ads ad a hghe fee wth fewe ads. Kuma ad Seth (2006) study the poblem of dyamcally detemg the subscpto fee ad the sze of advetsg space o a webste. They use optmal cotol theoy to solve the poblem ad obta the optmal subscpto fee ad the optmal advetsemet level ove tme. The optmal pce to chage pe mpesso of a ad s ot detemed. I ths pape we cosde the pay-pe-mpesso pcg scheme. Aothe pcg scheme s paype-clck. Baye ad Moga (2003) develop a aalytcal appoach to model cosume espose to ad exposues at a webste. Maga (2003) compaes the expected eveues fom the pcg

6 6 Reveue Maagemet fo Ole Advetsg: Impatet Advetses stategy of pay-pe-clck wth pay-pe-mpesso usg a smple detemstc model. Chckeg ad Heckema (2003) develop a delvey system that maxmzes clck-though ate gve vetoymaagemet costats the fom of advetsemet quotas. Noe of these papes cosde pcg decsos. I Fdgesdott ad Naaf (2008) we deteme optmal pces fo pay-pe-clck pcg schemes, whch eques dffeet models as the clck-though pobablty depeds o the umbe of ads dsplayed. Schedulg of ads o a webste s oe of the most popula topcs fo usg opeatos eseach tools ole advetsg. Kuma et. al (2006) develop a model that detemes how ads o a webste should be scheduled a plag hozo to maxmze eveue. They cosde geomety ad dsplay fequecy as the two mpotat factos specfyg the ads. The poblem belogs to the class of NP-had poblems ad they develop a heustc to solve t. They also povde a good ovevew of othe papes o schedulg. The model developed ths pape uses the modelg famewok developed by Aama ad Fdgesdott (2007). Howeve, we cosde the advetses to be mpatet,.e., they ae ot wllg to wat fo a advetsg slot to be avalable. (Sav et. al (2005) make the same assumpto fo customes of a etal busess.) I addto to beg a ealstc assumpto fo the compettve advetsg wold ad fo most types of advetses that eque geec web publshes, t s aalytcally appealg. I ths settg we ca develop closed fom solutos fo the systems that does ot seem possble whe watg s allowed. 3. Ole Advetsg Model We cosde a web publshe that geeates eveues fom postg ads o ts webste. We assume the web publshe s maagg a sgle webpage but multple webpages wth the same webste ca be hadled as poposed Aama ad Fdgesdott (2007). The web publshe faces uceta demad fom advetses watg to post the ad ad uceta taffc fomvewesvstgthe webste. Thee ae slots fo ads ad the pce fo each s the same. (Dffeet pces ae befly dscussed Secto 7). The advetses ae assumed to appoach the web publshe accodg to a

7 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 7 Possopocesswthateλ a ad each advetse equests the ad to be show to x vewes. The vewes ae assumed to vst the webste accodg to a Posso pocess wth ate λ v. Both Posso assumptos ae equed fo tactablty. The fst s commo the lteatue (see e.g. Sav et. al (2005)). The secod oe ca be ctczed based o the fact that some eseach suppots that web taffc shows self smlaty, log age depedece ad heavy taled dstbuto (see Gog et. al (2005)), whch ae ot popetes of the Posso pocess. Howeve, othe studes ecogze that a Posso dstbuto s a easoable assumpto (see Cao et. al (2002)) ad we wll make that assumpto hee. Whe a advetse appoaches the web publshe two thgs typcally happe. Ethe a slot s avalable fo hs ad o all slots ae occuped by othe advetses. (Note a web publshe does ot usually leave a slot empty athe t places ts ow ad thee. Howeve, t would mmedately fee ths slot up fo a eveue geeatg advetse.) If a slot s avalable the ad s posted ad t stays o the webste utl x vewes have vsted. If o slot s avalable the advetse appoaches aothe web publshe ad s lost. Gve the fact that the advetses ae ot wllg to wat we efe to them as mpatet. Hee we ae assumg that the umbe of advetses that ca be seved caot exceed the umbe of slots. Howeve, ealty thee ae ofte moe advetses beg seved tha thee ae slots. Oe ca otce ths whe efeshg a webpage a ew set of ads s ofte dsplayed. Aama ad Fdgesdott (2007) dscuss how ths ca be copoated to the modelg famewok by toducg vesos of the webste. The same could be doe fo ou model. Howeve, we wll be focusg hee o the fudametal dyamcs of a sgle veso of a webste. The settg descbed above s qute smple ad at fst sght t seems equvalet to tadtoal queueg settgs wth slots coespodg to seves. Howeve, what makes the system dffeet s the sevce mechasm. Whe a vewe vsts the webste all the advetses that have the ads posted ae seved wth the aval of the vewe. Ths meas that the mpessos left fo all the ads go dow by oe at the same tme. Hece, we ca cosde the system to have sychozed seves as Aama ad Fdgesdott (2007).

8 8 Reveue Maagemet fo Ole Advetsg: Impatet Advetses It takes x vstos wth expoetal teaval tme to sevce oe advetse, whch meas the sevce tme of a advetse has a Elag(λ v,x) dstbuto. The fact that thee ae sepaate advetsg slots gves ths system the flavou of a M/E x // queueg system (wth depedet seves ad o watg space). Howeve, the fact that the slots opeate a sychozed mae ad the sevce s tated by oe vewe at a tme povdes a dstctve featue. We compae ou system to kow queueg systems Secto 6. The goal of the web publshe s to maxmze the eveues fom the ads. Each advetse pays apcep pe mpesso made to a vsto. Wth x mpessos equested the total paymet pe advetse s px. To captue the pce sestvty of the advetse we assume a pce demad elatoshp p(λ a ) (moe detals to follow Secto 4). The the eveue ate geeated by the web publshe s: R(λ a )=λ a ( P (λ a ))p(λ a )x () whee λ a ( P (λ a )) s the actual aval ate cosstg of λ a, the aval ate of advetses checkg whethe a slot s avalable, ad P (λ a ) the pobablty of the equest beg accepted. P (λ a ) s the pobablty that advetses ae beg seved,.e., the system s full. I ode to deteme the optmal pce fo the web publshe to chage we eed to deve P (λ a ). The ext sectos ae dedcated to devg P. The case of a sgle advetsg slot s tval as t s equvalet to the M/E x // queueg system. Wth two slots ad moe, the sychozed sevce stats playg a ole. λ a λ v x P π, 2,..., π (, 2,..., ) L L p Summay of Notatos Aval ate of advetses Aval ate of vewes = λa λ v Numbe of advetsg slots Numbe of mpessos Pobablty of havg advetses the system Pobablty of havg k mpessos left to satsfy slot k Pobablty of havg, 2,..., mpessos left ay of the slots Aveage umbe of advetses the system Aveage umbe of advetses slot o occupato ate of slot The pce chaged pe mpesso

9 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 9 Bae Slot Advetses Vewes ( λ ) ( ) a Bae Slot 2 λ v Fgue System wth two slots 3.. The Case of a Sgle Advetsg Slot As metoed eale the case of a sgle slot s tval. Wth advetses avg accodg to a Posso pocess wth ate λ a watg x mpessos, ad vewes vstg accodg to a Posso pocess wth ate λ v, the esultg queueg system s a tadtoal M/E x // queue wth oe seve ad o watg space. Note that wth oly oe slot fo a ad thee s o oto of sychozed o depedet seves. Fo ths case the pobablty of havg the system full s P = x +x ad Has (998)) whee = λa λ v. (see Goss 3.2. The Case of Advetsg Slots Whe thee ae two o moe slots o the webste the cocept of sychozato stats playg a ole ad esults fo kow queueg systems o loge apply. The system has a oto of a queueg system wth sgals (see Chao et. al (999)) whee the aval of a vewe tgges a sgal fo the slots to wok togethe. Howeve, ths lteatue cludes maly sgals acoss odes etwoks ad to ou kowledge t does ot povde us wth soluto appoaches fo ou system. Havg Makova aval ad sevce pocesses we model the system usg Makov Chas. Eve though we ae ultmately teested keepg tack of the umbe of advetses the system, ode to fom a Makov cha we eed to keep tack of the system at a moe detaled level;.e., of theumbeofmpessosleftfoeachslot.theslotsaecosdeedtobeequvalet(temsof pce) ad whe a advetse aves to the system he s adomly assged to ay of the avalable slots wth equal pobablty. (I Secto 3.2. we cosde aothe type of a ad-to-slot allocato.)

10 0 Reveue Maagemet fo Ole Advetsg: Impatet Advetses We defe the state of the system as the umbe of mpessos left to satsfy each slot. The advetsg slots ae labeled fom to ad π k,k 2,...,k s the pobablty that thee ae k,k 2,..., k mpesso left slots to. The adom ad-to-slot allocato meas that the slots ae symmetcal ad we ca keep tack of the dyamcs of the system wthout dstgushg betwee them. Wth ths md we toduce the cocept of pemutato pobabltes. Defto. Fo a gve vecto (k,k 2,..., k ) whee each elemet epesets the umbe of mpessos left to satsfy a slot we defe π (k,k 2,...,k ), the pemutato pobablty, as the sum ove the pobabltes of all possble pemutatos of k,k 2,..., ad k. To llustate let us look at a example of the pemutato pobabltes defed above fo the case of thee slots. If k,k 2 ad k 3 ae dffeet umbes the the umbe of pemutatos ae 3!=6.Theefoeπ (k,k 2,k 3 ) = π k,k 2,k 3 + π k,k 3,k 2 + π k3,k 2,k + π k2,k,k 3 + π k3,k,k 2 +π k2,k 3,k.Howeve, f k = k 2 = a the the umbe of pemutatos amog k,k 2 ad k 3 ae 3! =3 ad theefoe 2! π (k,k 2,k 3 ) = π (a,a,k3 ) = π a,a,k3 + π a,k3,a + π k3,a,a. Remak. The pemutato pobablty π (k,k 2,...,k ) s the pobablty of fdg k,k 2,..., k mpessos left the slots. I the followg poposto we state the closed fom soluto of the pobablstc popetes of the system. Gve the complexty of the system (see the flow balace equato the poof of the poposto below) t s qute attactve ad pehaps ot expected to get closed fom esults. Theoem. The pobablty of havg k,k 2,..., ad k mpessos left to satsfy slots ad empty slots s: π (k,k 2,...,k,0,...,0) = π (k,k 2,...,k ) = ( + ) P =0 P =0 fo =0,, 2,..., (2) (3) Futhemoe, the steady-state pobablty of havg advetses the system s: P = x+ ( + ) P =0 fo =0,, 2,..., (4)

11 Reveue Maagemet fo Ole Advetsg: Impatet Advetses P = P =0 (5) All poofs ae povded the techcal appedx. Note that eve though π (k,k 2,...,k,0,...,0) does ot deped o the actual umbe of mpessos left each slot oly the umbe of flled slots, we eed to keep tack of the umbe of mpessos left each slot whe devg the fomula. Theefoe, we keep ths otato. To gve a smplfed ovevew of the poof t volves lstg the flow-balace equatos ad the vefyg that Equatos (2) ad (3) satsfy them. The the elevat tems fom Equatos (2) ad (3) ae added to pove Equatos (4) ad (5). (The poof of Poposto gves futhe sghts to the pocess of detfyg equatos such as Equatos (2) ad (3).) Wth the poposto above we have fully chaactezed the pobablstc popetes of the system wth the closed fom soluto of the steady state pobabltes. Usg Lttle s law we ca fd the aveage umbe of advetses the system: L = λ a ( P )x/λ v (6) Note that sce the slots ae equvalet,.e., we do ot dstgush betwee them, the aveage umbe of advetses each slot (the occupato ate) s smply L/. The popostos below state some stuctual popetes of the aveage umbe of advetses the system ad the busy pobablty that wll be useful whe cosdeg the pcg poblem of the web publshe. Coollay. The pobablty of all slots beg occuped, P,sdeceasg. Coollay 2. The pobablty of all slots beg occuped, P,sceasgλ a. Coollay 3. The pobablty of all slots beg occuped, P,sceasgx. Coollay 4. The aveage umbe of advetses the system, L, s cocave ceasg λ a. Coollay 5. The aveage umbe of advetses the system, L s covex ceasg x. Coollay 6. The aveage umbe of advetses the system, L s ceasg. Numecal examples llustate that P s ot ecessaly cocave x.

12 2 Reveue Maagemet fo Ole Advetsg: Impatet Advetses Odeed Ad-to-Slot Allocato Oe ca thk of othe ealstc mechasm to allocate ads to slots addto to the adom allocato cosdeed utl ow. Fo example, the advetses mght have pefeeces fo some of the slots. We do ot model the pefeeces explctly ths pape as we ae cosdeg homogeeous advetses. Howeve, t s teestg to detfy how the pobablstc popetes of the system chages. We cosde a odeed allocato to slots such that the slots ae ot cosdeed to be equvalet. Rathe f slots ad, >,aeavalable the a ad s placed slot. Ths lack of symmety makes the poblem less tactable. We ca o loge avod keepg tack of each slot specfcally wth a aggegated appoach as we dd fo the adom ad-to-slot allocato usg the pemutato pobabltes. We focus o the case of two slots ad deve π,m, the steady-state pobablty of fdg mpessos left slot ad m mpessos left slot 2. A smla appoach ca be used fo thee slots but fo a lage umbe of slots the poblem seems otactable. Note howeve, that o matte what the allocato of ads to slots s the pobablty of havg ads o the webste, P,sthesame. I the followg poposto we state the closed fom soluto fo the pobablty π,m. Poposto. I a system wth two advetsg slots ad a odeed ad-to-slot allocato, the steady-state pobabltes fo the umbe of mpessos left to satsfy each slot ae: π 0,0 = π,0 = π 0, = π, = π, = π, = + +(x +) + x+ 2 2 ( + ) ( + x) +(x +) + x+ 2 2 (x ) 2 ( + x) +(x +) + x+ 2 2 ( + (x + )) 2 ( + x) +(x +) + 0 << x x+ 2 2 ( ) 3 ( + x) +(x +) + 0 << x x (x +) + x+ 2 2 whee, =, 2,..., x. The poof of Poposto, povded the techcal appedx, gves though smple examples

13 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 3 thoough sghts to how the equatos above ae deved addto to vefyg them usg the flow-balace equatos. Havg the pobablty dstbuto o had we ca deve futhe chaactestcs of the system. I the followg coollay we obta the occupato ate of each slot, whch ca also be cosdeed as the aveage umbe of advetses a slot. Coollay 7. Let L ad L 2 be the occupato ate of slots ad 2 espectvely. The L = x +x ad L 2 = x 2 2 +x( x+ 2 ) 3 (+x)(+(x+)+( x+ 2 ) 2 ). We ote that whe a advetse aves ad thee ae two empty slots the default s to allocate the ad to slot. That makes the slots asymmetc ad we have L 6= L 2.Howeve,thesumofthe occupato ates, L + L 2, s the same as the aveage umbe of advetses the system fo the adom allocato mechasm, L. Note that the occupato ate of the fst slot s equal to the occupato ate a oe slot system. Hece, the fst slot opeates as the oly slot the sgle slot system ad the secod slot deals wth some of the advetses that would have bee eected the sgle slot case. Ths obsevato ca be geealzed to the slot system wth the odeed ad-to-slot allocato. I othe wods a slot system the occupato ate of slot s the same as the occupato ate fo slot ay m slot system such that m. As metoed befoe the pobablty of advetses the system, P, does ot deped o how the ads ae allocated to slots. Theefoe, those pobabltes fo the odeed ad-to-slot allocato s the same as fo the adom allocato eve though the pobabltes of fdg ceta umbe of mpessos each slot ad the occupato ates of the slots ae ot the same Sevce Polcy We have made the atual assumpto that each advetse does ot occupy moe tha oe slot o a webste. I addto to ths sevce aagemet beg moe effectve fom the advetsg pot of vew we ca show that t s also moe attactve fom a sevce pot of vew. Note that based o the obectve fucto Equato () ad Equato (6) hghe aveage umbe of advetses, L, gves moe eveues.

14 4 Reveue Maagemet fo Ole Advetsg: Impatet Advetses Let us focus o a webste wth two slots. The web publshe could gve both slots to oe advetse ad stead of x mpessos pe slot, x/2 mpessos would be made fo each of the two ads. Ths system s equvalet to a sgle slot system wth x/2 mpessos. We deote the aveage umbe of advetses (the occupato ate) of ths settg as L (x/2). We compae ths sevce polcy to the oe we have cosdeed as the atual oe of havg oly oe advetse each slot wth x mpessos each. We deote the aveage umbe of advetses of ths settg as L 2 (x). Usgthe fact that L s ceasg x ad (seecoollaes5ad6)weotethatl (x/2) L (x) L 2 (x). Futhemoe, usg Equatos () ad (6) we ca show that the optmal eveue s hghe whe the same advetse occupes oly a sgle slot. Hece, thee ae ot oly maketg easos fo the sevce polcy we have used so fa but also opeatoal easos. 4. The Optmal Pce Havg fully chaactezed the pobablstc popetes of the web publshe s opeato we ow focus o the decso makg poblem of the web publshe. The web publshe s obectve s to deteme the pce to chage pe mpesso ode to maxmze the eveue ate. (As most cost compoets ae fxed we do ot cosde those. Howeve, loss-of-goodwll cost fo eected advetses ca be copoated to the model. ) As stated Secto 3 the eveue ate geeated s R(λ a )=λ a ( P (λ a ))px. Accodg to Equato (6) t s equvalet to R(λ a )=L(λ a )λ v p. Advetses ae usually sestve to the pce offeed by the web publshe. Theefoe, a web publshe offeg lowe pces ca expect moe teest fom advetses tha a web publshe offeg hgh pces. We captue ths behavo by defg a cotuous pce-demad fucto p(λ a ) that s assumed to be deceasg. Fo the momet we assume the pce does ot deped o the umbe of slots o the webste o the umbe of mpessos. I Secto 5 we llustate examples wth these depedeces. As metoed befoe the obectve of the web publshe s to deteme the pce that maxmzes the eveue ate. Sce, we have a oe-to-oe elatoshp betwee the pce ad the aval ate of the advetses we ca as well optmze wth espect to λ a ad the

15 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 5 deteme the pce fom the pce-demad fucto, p(λ a ). The optmzato poblem of the web publshe ca be wtte as follows: max λ a R(λ a )=λ a ( P )p(λ a )x = L(λ a )λ v p(λ a ) st λ a 0 The followg poposto esues the uqueess of the optmal soluto ad gves the optmal pce. Poposto 2. If p(λ a ) s cocave the the eveue ate, R(λ a ),scocaveλ a ad the optmal pce s p = p(λ a) whee λ a satsfes the equato L 0 (λ a)p(λ a)+l(λ a)p 0 (λ a)=0. Note that ode to esue cocavty we eed p(λ a ) to be cocave. Eve though ths mght seem a estctve assumpto t cludes a lea pce ad umecal aalyss dcates that may covex pcg fucto gve a umodal eveue fucto. (Othe weake codtos such assumg cocave paymet ate λ a p(λ a ) o mootocty of the pce elastcty dλa dp p λ a do ot seem suffcet.) The poposto below gves the tutve esults that the web publshe s bette off wth havg moe umbe of slots, offeg hghe umbe of mpessos ad havg moe taffc to ts webste. Note that hee the advetses ae ot cosdeed to be sestve to the umbe of slots ad umbe of mpessos, whch we cosde Secto 5. Poposto 3. The optmal eveue ate, R(λ a), s ceasg the umbe of slots,, the umbe of mpessos, x, ad the aval ate of the vewes, λ v. The followg poposto states the coute-tutve esult that the optmal pce does ot follow the ecoomes-of-scale popety wth espect to x. Poposto 4. The optmal pce, p, s ceasg the umbe of mpessos equested, x ad the optmal eveues ae ceasg x. The poposto above s qute teestg as oe could expect the opposte,.e., the pce to be lowe whe moe mpessos ae offeed. I ode to udestad what dves these esults we

16 6 Reveue Maagemet fo Ole Advetsg: Impatet Advetses c =.2 c = c = Optmal eveue Slots Fgue 2 Optmal eveue vs. slots otce that thee ae two competg foces. Fst, the hghe the umbe of mpessos the loge t takes to sevce each advetse, whch meas that the web publshe does ot eed as may advetses as befoe ad ca theefoe chage a hghe pce. Moe mpessos mea moe demad o the capacty,.e., the vewes, ad theefoe, less advetses ae eeded. Secod, geeal a hgh demad helps to fll quckly ay slot that becomes avalable. Howeve, the fst effect seems domatg, whch esults hghe pce wth lowe demad whe the umbe of mpessos s hgh. Pactcally speakg, the web publshe should ot offe quatty dscouts fom a opeatoal pot of vew. Howeve, thee could be maketg easos fo offeg quatty dscout. We wll exploe those the followg secto. 5. Numecal Examples 5.. Advetsg Slots We fst cosde sestvty wth espect to the umbe of advetsg slots. The vewes aval ate s set λ v = 000 (whch ca be cosdeed as umbe of vewes pe hou). Each advetse equests x = mpessos. The pce-demad fucto (pe mpesso) fo the advetses s chose to be p(λ a )= λ c a,wheec =0.8, o.2,.e., the pce fucto s covex, lea o cocave. We exploe the optmal pce ad the optmal eveues. As llustated Poposto 3 ad o Fgue 2 thee s a deceasg elatoshp betwee the

17 Reveue Maagemet fo Ole Advetsg: Impatet Advetses c =.2 c = c = 0.8 Optmal pce pe 000 mpessos Slots Fgue 3 Optmalpcevs.slots optmal eveues ad the umbe of slots. The tuto behd ths s that by ceasg the umbe of slots we ae fact ceasg the capacty of the system. I that case the web publshe ca hadle moe advetses at a tme ad to attact moe of them he chages a lowe pce as dcated o Fgue 3. Howeve, ths effect levels off as dcated fo c =0.8 Fgues 2 ad 3. Note that ou pce demad fucto we have ot take to accout that advetses mght ot be wllg to pay as much fo the ad to be posted o a webste wth fve ads compaed to a webste wth oly oe ad. I geeal, advetses ae lkely to be wllg to pay less whe webstes have a hgh umbe of ads as the effectve mpact of each ad o the comg vewes s expected to be less. Theefoe, t s ealstc to model the pce pe mpesso to be deceasg wth the umbe of slots. The same apples to umbe of mpessos (see Secto 5.2). To captue ths effect we set the pce to deped ot oly o the aval ate of advetses λ a but also the umbe of slots. We cosde the followg pce fucto: p(λ a )= λ c a 0.00 We cotue to exploe the sestvty wth espect to the umbe of advetsg slots. We set the umbe of mpessos as x = Fgues 4 ad 5 show the optmal eveue ad the optmal pce vs. umbe of advetsg slots.

18 8 Reveue Maagemet fo Ole Advetsg: Impatet Advetses c =.2 c = c = Optmal eveue Slots Fgue 4 Optmal eveue vs. slots wth pce depedg o umbe of slots Optmal pce pe 000 mpessos c =.2 c = c = Slots Fgue 5 Optmal pce vs. slots wth pce depedg o umbe of slots Compag Fgues 2 ad 4 we ca see that the optmal eveue does ot cotue to cease wth the umbe of slots as befoe. Istead afte a ceta umbe of slots the mpact of the pce sestvty wth espect to umbe of slots stats playg a ole ad the eveue stats deceasg. Hece, hee the optmal umbe of slots to choose would be thee o fou slots depedg o the pce-demad elatoshp. The optmal pce deceases ow faste wth the umbe of slots;.e., the web publshe has to lowe the pce to attact the customes lost due to the mpact of the umbe of slots.

19 Reveue Maagemet fo Ole Advetsg: Impatet Advetses c =.2 c = c = Optmal eveue Impessos (000s) Fgue 6 Optmal eveue vs. mpessos 6 5 c =.2 c = c = 0.8 Optmal pce pe 000 mpessos Impessos (000s) Fgue 7 Optmalpcevs.mpessos 5.2. Impessos Next we cosde the sestvty wth espect to the umbe of mpessos equested by the advetses. We suppose that thee ae two slots o the webste ad the pce-demad fucto fo the advetsesschoseasbefoetobep(λ a )= λ c a,wheec =0.8, o.2. Fgues 6 ad 7 show how the umbe of mpessos affects the optmal eveue ad the optmal pce. As show Poposto 4 ad llustated Fgues 6 ad 7 the optmal eveue ad the optmal pce cease wth the umbe of mpessos. Hece, the effect of ecoomes-of-scale does ot apply hee. The tuto behd ths sght s, as we metoed befoe, that moe mpessos

20 20 Reveue Maagemet fo Ole Advetsg: Impatet Advetses c =.2 c = c = Optmal eveue Impessos (000s) Fgue 8 Optmal eveue vs. mpessos wth pce depedg o umbe of mpessos 0 c =.2 c = c = 0.8 Optmal pce pe 000 mpessos Impessos (000s) Fgue 9 Optmal pce vs. mpessos wth pce depedg o umbe of mpessos meas the web publshe eeds less advetses ad thus ca chage hghe pce. Note that hee we have ot take ay quatty dscouts to accout ou pcg fucto that advetses mght expect ad ae ofte offeed ealty,.e., the web publshe could offe lowe pce pe mpesso fo hghe total umbe of mpessos. Ths pheomeo s copoated the pce-demad fucto, p(λ a )= λ c a 0 7 x. Usg ths fucto we exploe how the optmal pce ad eveues chage wth the umbe of mpessos wth umbe of slots as =4. By copoatg quatty dscouts the pcg, the optmal eveue does ot cotue to cease as befoe, stead t stats deceasg, dcatg a optmal value fo the umbe of

21 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 2 mpessos to offe. 6. Compaso Wth Kow Queug Models I ths secto we compae ou model of the web publshe s opeato wth some elated models fom the exstg lteatue. 6.. Elag Loss System We fst compae the web publshe s model wth the M/E x // queue, whch s the so-called Elag s loss model. I the Elag loss model the system does ot povde ay space fo obs to wat ad the oly obs the system ae the oes beg seved by oe of the seves. Fom ths espect ths system s vey smla to ou system. Howeve, thee s a substatal dffeece. I the Elag system the sevce chaels opeate depedetly. Howeve, ou system the slots ae sychozed,.e., the advetses eceve sevce smultaeously. The Elag s loss fomula that epesets the pobablty dstbuto of the umbe of obs the system s the followg: P E = (x)! P =0 (x)!, 0 whch we ca compae to the dstbuto fo the web publshe s system: P = P = x+ ( + ) P, 0 =0 P =0 If = the two fomulas yeld the same esults as we have exploed befoe. Ths s because wth oly oe slot the system thee s o oto of tedepedece amog slots ad ou model s educed to the M/E x // model. Howeve, whe thee s moe tha oe slot, >, thetwo systems ae dffeet. It s teestg to otce how the teacto betwee the empty slots ad the occuped slots comes though the ole system. I the fomula fo P, seems to play the ole of the occuped slots whle (+) plays the ole of the empty slots. The multplcato of those two tems

22 22 Reveue Maagemet fo Ole Advetsg: Impatet Advetses captues some sese the effect of the teacto of occuped slots wth empty slots. Sce P all the slots ae occuped thee s o teacto betwee the empty ad the occuped slots. Theefoe, P does ot have a tem of the fom ( + ). Thssdffeet fom the M/E x // model whee the fomula fo P, 0, has the same fomat eve though thee ae empty seves. I the followg poposto we specfcally compae the pobablty of the system beg full fo the Elag loss system ad the web publshe s system Poposto 5. The pobablty of a fully occuped system s hghe fo the web publshe tha fo the Elag loss system,.e., P P E. I addto, the aveage umbe of people the web publshe s system s less tha the aveage umbe of obs the Elag loss system, L L E. The poposto above dcates that the ole system s some sese less effcet tha the Elag loss system wth depedet seves. Ths meas that sychozed seves ae less effcet tha depedet seves, whch s tutve as the sychozato mposes a estcto Bulk Sevce A bulk sevce system, ofte deoted M/M [] /, has some smlates to the web publshe s system. The set up of ths system s the followg. The avals ae Posso ad the sevce tme s expoetal. Thee ae slots fo sevce ad fte watg space. Whe o less obs ae the system they ae all seved at the same tme ad f a ob aves dug the sevce ad a slot s empty that ob s also seved ad fshes at the same tme as the othes (memoyless sevce popety). If thee ae moe tha obs the system oly ae seved (all at the same tme) ad the est wats. Ths system wth the addtoal assumpto of o watg space s the same as the ole system wth oe mpesso. We ca deote t by M/M [] //. Sce havg oe mpesso s ot ealstc fo the ole settg thee does ot seem to be much to ga fo us fom the bulk sevce lteatue. Howeve, let us fd out whethe we ca use the esults fom the ole system to lea moe about the bulk sevce system. Fst, the soluto of the system M/M [] / (Goss ad Has (998)) s the followg: P 0 =( x 0 )

23 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 23 P =( x 0 )x 0 =, 2, 3,... whee x 0 s the uque soluto (betwee zeo ad oe) of the chaactestc equato: μx + (λ + μ)x + λ =0 wth λ as the customes aval ate ad μ as the sevce ate. The chaactestc equato s of ode ( +) ad has at most ( +) oots. Howeve, most applcatos t has a uque soluto. The dawback of ths fomula s whe s lage o appoaches fty,.e., the system has fte capacty, the chaactestc equato wll be had to solve. Howeve, whe the assumpto of o watg space does ot play a ole ad the esult fom the ole model ca be used as see the followg poposto. Poposto 6. If the sevce capacty the bulk sevce system s fte the the pobablty dstbuto of the umbe of obs s: P = ( + ) + fo 0 I addto, the aveage umbe of obs the system s. We ca potetally take advatage of the poposto above to appoxmate bulk sevce systems wth lage. Let us exploe ths a umecal example whee we cosde two systems M/M [] / ad M/M [] // (whch s the same as the ole system wth slots ad oe mpesso) wth λ =5 ad μ =0 ad dffeet values fo. Whe calculatg accuately the aveage umbe of obs M/M [] / (usg the chaactestc equatos above fom Goss ad Has (998)) ad M/M [] // (usg Equatos (5) ad (6) wth x =)weobtaadffeece L of less tha.2% wth 0. Fgue 0 llustates the dffeece. The speed of the coveges depeds o the systems paametes. Note that L fo the M/M [] / system s hghe as could be expected sce thee ca be obs watg a queue eady to go to sevce whle the M/M [] // system eeds to wat fo the ext aval. The full state pobablty s llustated Fgue. The covegece of P seems to be a bt slowe ad thee s less tha.4% dffeece fo 0.

24 24 Reveue Maagemet fo Ole Advetsg: Impatet Advetses Aveage umbe of obs the system, L Sevce Capacty, M/M[]// M/M[]/ Fgue 0 Aveage umbe of obs vs. the sevce capacty The full state pobablty, P Sevce Capacty, M/M[]// M/M[]/ Fgue The full state pobablty vs. the sevce capacty Note that the bulk sevce system wth Elag sevce tme s ot the same as ou ole advetsg system. I the ole system the obs ca leave ad ete the bulk ;.e., the obs beg seved smultaeously ae ot ecessaly the same phase of the Elag dstbuto. Howeve, the bulk sevce system all obs belogg to the same bulk have the same sevce tme.

25 Reveue Maagemet fo Ole Advetsg: Impatet Advetses Cocluso I ths pape we cosde a web publshe that geeates eveue by dsplayg ads o ts webste ad chages accodg to the pay-pe-mpesso pcg scheme. The web publshe s opeato s modeled as a queueg system whee the aval pocess coespods to the advetses teested postg the ads ad the sevce pocess coespods to the vewes vstg the webste. The advetses ae assumed to be mpatet ad ot wllg to wat fo the ads to be posted athe they would choose to go to aothe web publshe. I queueg tems ths coespods to a system wth o watg spaces (a loss system). Gve the dyamcsofhowtheadvetsesaesevedthe sevce s sychozed. Desptethecomplextyofthemodelweaeabletodeve a closed fom soluto of the pobablty dstbuto of the umbe of advetses the system fo ay umbe of advetsg slots ad ay umbe of mpessos made of each ad. We deteme the optmal pce to chage pe mpesso ad show, e.g., that t s ceasg the umbe of mpessos, whch goes agast the ecoomyof-scale tuto. The model developed s dffeet fom models exstg the lteatue ad we compae ths model to some kow queueg models. It has smlates to bulk sevce models ad we use ou esults to get futhe sghts fo specal cases of the bulk sevce settgs. Ths pape povdes a modelg famewok fo a web publshe s opeatos that ca be expaded beyod the esults of the pape. We have povded closed fom solutos ad maageal sghts fo the case of homogeeous advetses. Addg some heteogeety to the chaactestcs of the advetses, such as the umbe of mpessos equested, would be teestg ad would eable pce dffeetato. Ths mght ot be feasble fo a geeal umbe of advetsg slots but mght be possble fo two slots. Eve though pay-pe-mpesso s the most commo pcg scheme othes exst such as pay-pe-clck o eve a mxtue of these two. Pay-pe-clck cotacts eque dffeet models ad compaso of them to pay-pe-mpesso cotacts ca be foud Fdgesdott ad Naaf (2008). As thee ca be may web publshes fo a advetse to choose fom,

26 26 Reveue Maagemet fo Ole Advetsg: Impatet Advetses a competto settg would be ealstc. I addto, gve how ofte t ca be easy to keep tack of vewes behavo ad pofle, tageted advetsg s vey attactve to advetses as well as to web publshes that ca chage a hghe pce fo moe tageted audece. Futhemoe, thee exst advetsg etwoks (such as AdEgage.com ad Advetsg.com) that gve advetses access to may dffeet webstes. The models developed ths pape ca seve as buldg blocks fo the opeato of a advetg etwok. We have aalyzed the opeato of the web publshe fom a steady state pot of vew. Dyamc pcg would be teestg ad possble to mplemet as advetses ofte buy the advetsg space ole, whch makes t feasble fo the web publshe to chage the pces dyamcally. I summay, the modelg famewok developed ths pape wth ts closed-fom solutos ca povde a bass fo multple eseach dectos that would exploe aalytcally may elevat ssues ole advetsg. 8. Techcal Appedx Poof of Theoem To pove Equatos (2) ad (3) we lst the flow balace equatos ad show that the pobabltes ae of the fom: π (k,k 2,...,k,0,...,0) = A ( + ) =0,, 2,..., (7) π (k,k 2,...,k ) = A =,..., (8) whee k,k 2,..., k > 0. The we show by summg ove the elevat π 0 s that the pobabltes of fdg a ceta umbe of advetses the system ae of the fom: µ x + P = A ( + ) =0,, 2,..., µ x + P = A =,..., Fallyweusethefactthat P =0 P = ad solve fo A. We fst cosde a Makov cha whee the state of the system s the vecto (k,k 2,...,k ) ad k epesets the umbe of mpessos left to satsfy a slot. (We do ot dstgush betwee the

27 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 27 slots.) Afte detfyg the possble tastos of the system we lst the flow balace equatos. The equatos ae of fve types: ) π (0,0,0,...,0) = π (,0,0,...,0) + π (,,0,...,0) + π (,,,0,...,0) π (,,,...,) ) ( + )π (k,k 2,...,k,0,0,...,0) = π (k +,k 2 +,...,k +,0,0,...,0) + π (k +,k 2 +,...,k +,,0,...,0) π (k +,k 2 +,...,k +,,,...,) k x, =,..., ) ( + )π (x, x,..., x {z }, k,k 2,...,k,0, 0,..., 0 {z } = π ) (x, x,..., x {z }, k,k 2,...,k,0, 0,...,0 {z } l l l l + l, k x, =,...,, l v) π (x, x,..., x {z }, k,k 2,...,k ) = π (x, x,..., x {z }, l l k,k 2,...,k,0) v) π (k,k 2,...,k ) = π (k +,k 2 +,...,k +) k x, =,..., ) l, k x, =,...,, l + = Next we vefy that the fuctoal fom stated Equatos (7) ad (8) satsfes the Flow Balace Equatos ) - v): ) By setg Equatos (7) ad (8) to the flow balace equato we obta a left had sde of A( + ) ad a ght had sde of A[( + ) ( + ) ( + ) 0 + ]. We use ducto to show that both sdes ae equal,.e., P = ( + ) + = ( + ). We stat wth = ad ote that both sdes ae equal to. We ow assume that the equalty holds fo = k,.e., P k = ( + ) k + k = ( + ) k. I ode to show that the equalty the holds fo = k + we eed to show that P k = ( + ) k + k+ = ( + ) k.wehavethat P k = ( + ) k + k+ =(+) P k = ( + ) k + k + k+. Usg the ducto assumpto we obta ( + )[( + ) k k ]+ k + k+ = ( + ) k k+ + k+ = ( + ) k, whch completes the ducto poof. ) Usg a smla appoach as fo Case ) we eed to show that A( P =0 + ( + ) + )=(+)A ( + ),.e., P =0 ( + ) + =(+). To smplfy the otato we set m =. We the eed to show that P m =0 ( + ) m + m =(+) m.we pove ths equalty by ducto. If m = both sdes of the equalty ae +. Let us ow assume that the equalty holds fo m = k,.e., P k =0 ( + ) k + k =(+) k. We ow eed to show that the equalty holds fo m = k +,.e.,that P k =0 ( + ) k + k+ =(+) k+.wehave

28 28 Reveue Maagemet fo Ole Advetsg: Impatet Advetses P k =0 ( + ) k + k+ =(+) P k =0 ( + ) k + k + k+. Usg the ducto assumpto ths equals to ( + )[( + ) k k ]+ k + k+ =(+) k+, whch completes the ducto poof. ) We eed to vefy that ( + )A k+ ( + ) k = A k + ( + ) k,whchalways holds. v) We eed to vefy that A = A, whch always holds. v) Ths equato always holds. Whe devg A we fst eed to fomulate P, the pobablty that thee ae advetses the system, ad the use the fact that P =0 P = to solve fo A. Fst,wekowthatP 0 = π (0,...,0) = A( + ). Let us the cosde, 0 <<. The pobablty of havg advetses the system whee each has k mpessos left whee 0 <k x, =,..., s π (k,k 2,...,k,0,0,...,0). P s the sum ove all possble values of k, =,...,.Notewehavetobecaefulwththecoutg as π (k,k 2,...,k,0,0,...,0) cludes all the pemutatos of havg k mpessos left. We have that P = P x k = P x k 2 =k... P x k =k π (k,k 2,...,k,0,0,...,0) = B π (k,k 2,...,k,0,0,...,0) whee B epesets the umbe of tems the multple sum. Detemg the umbe of tems fo =s staght fowad wth P 2 = P x k = π (k,0,...,0) = xπ (k,0,...,0) ad thus B = x. Fo =2 we have P 2 = P x k = P x k 2 =k π (k,k 2,0,...,0). To deteme the umbe of tems, B 2,wedvdethemtotwopats:Thetemswthk = k 2 ad the tems wth k 6= k 2.Wehavex tems of the fst type ad x 2 of the secod. Theefoe B 2 = x + x 2 = x+ 2. To llustate futhe the coutg of the tems we gve aothe example ad cosde P (5) = xx k = xx k 2 =k... xx k 5 =k 4 π (k,k 2,...,k 5,0,0,...,0). Thecaseofk = k 2 =... = k 5 gves x tems.thecaseofk = k 2 = k 3 = k 4 6= k 5 gves x Thecaseofk = k 2 = k 3 6= k 4 = k 5 gves x takg dffeet values gves x values gves x gves x 2 x 2 x 3 x 2 x x tems. tems.thecaseofk = k 2 = k 3 ad k 4 ad k 5 tems.thecaseofk = k 2 ad k 3, k 4,adk 5 takg dffeet tems. The case of k = k 2,k 3 = k 4 ad the two values beg dffeet fom k 5 tems. Fally, the case of k 6= k 2 6= k 3 6= k 4 6= k 5 gves x 5 tems. Theefoe, B5 wll

29 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 29 be B 5 = x + x x + x x + x x 2 + x x 3 show that B 5 = 20 x(x +)(x +2)(x +3)(x +4)= x+4 5 Accodg to Lemma B = x+ A.Sce P ad P =, whchgvesa = + x x x 5. Afte some algeba we ca. Theefoe, we have P = x+ A ( + ) fo < P =0 =we have that P x+ =0 A ( + ) + x+ A = =0 ( x+ ) (+) +( x+ ) =0, whch completes the poof. ( x+ ) Lemma. If B s the total umbe of the tems the summato P = xx k = xx k 2 =k... the B = x+,=, 2,...,. x+ xx. Fally usg Lemma 3 we have that A = k =k π (k,k 2,...,k,0,0,...,0) = B π (k,k 2,...,k,0,0,...,0) Poof We pove the lemma wth ducto. Fo the case =, as dscussed eale, B = x =. Now let us assume that the fomula fo B holds fo = s,.e.,b s = x+s s fo ay x. We the eed to show that t holds fo = s +,.e.,b s+ = x+s s+. Let us codto ou coutg of tems o the value of k s+. We fst assume k s+ takes the value. The umbe of the tems ths case wll be exactly the same as fo the poblem wth s flled slots whch s equal to x+s s accodg to the ducto assumpto. If k s+ =2the othe dces ca vay fom 2 to x. Theyca ot take aymoe because all the states wth ae aleady couted fo the case wth k s+ =. The umbe of tems ths case wll be smla as the fstcaseexceptweolyhavex values to choose fom,.e., x+s 2 s. Wth a smla easog fo ks+ =3 we obta x+s 3 s. Repeatg the same easog we ca see that B s+ = x+s s + x+s 2 s + x+s 3 s s s.byusglemma 2 we obta that ths summato s equal to x+s s+, whch completes the poof. Lemma 2. Fo a fxed k, P x+k =k k = x+k k+ fo all x. Poof We pove the lemma by ducto. Fo x =we have both sdes equal to. Let us assume that fo x = s we have P s+k =k P s+k =k k = s+k+ k+ assumpto we have P s+k =k P s+k =k P s+k.wecaseethat =k + s+k k = s+k k+ k = s+k+ k+, whch completes the poof. k = s+k k+. We the eed to show that fo x = s + we have P s+k k = =k k + s+k k ad by usg the ducto k. Usg the Pascal s ule, a b + a b = a b,weobta

30 30 Reveue Maagemet fo Ole Advetsg: Impatet Advetses Lemma 3. Let x, ad be two tege umbes ad be a eal umbe the the followg equalty holds Poof X µ x + X µ x + ( + ) = =0 We pove the lemma by ducto. If = the both sdes ae equal to. Let us =0 assume the equalty holds fo = k,.e., C = P k =0 x+ ( + ) k P k =0 The we eed to show t also holds fo = k +,.e., that C = P k =0 P k =0 x+k =0 We have C =(+) P k =0 x+ the ducto assumpto we get C =(+) P k =0 P k =0 x+k x+k P k =0 x+k ( + ) k + x+k k x+k + + x+k k + x+k k k x+k k settg the dex the fst sum to =, we have C = P k 2 =0 P k =0 x+k + = P k =0 x+k + P k =0 x+ k P k =0 x+k =0. ( + ) k x+k.usg k = k P k x+k =0 x+k k k. Usg Pascal s ule twce ad x+k + + x+k k x+k + =0, whch completes the poof. k Poof of Coollay We pove P + P usg cotadcto. Suppose P + >P the ( x+ +) + + =0 ( > (x+ ) x+ ) =0 ( x+.thsgves P x+ ) + + some smplfcatos we have x+ + P =0 ght had sde by settg = + gves P =0 =0 (x+ )! > P +!(x+ )! =0 > (x+)! > + P (+)!(x+ )! =0 P + =0 x+. Afte (x+)!. Redexg the sum o the!(x+ )! (x+)! (+)!(x+ )!.Bycompag the sums tem by tem we see that each tem o the left had sde s smalle tha the coespodg oe o the ght had sde, whch leads to a cotadcto. Hece, we must have P + P. Poof of Coollay 2 dp = ( x+ ) =0 ( x+ ) ( x+ d =0 ( x+ )( x+ ) + ( ) =0 ( x+ Poof of Coollay 3 By dffeetatg P wth espect to we obta: ) =0 ( x+ ) =0. Afte some calculatos we get ( dp x+ ) ) 0. Hece, P s ceasg. have that P (x +) P (x)= =0 ( x+ Poof of Coollay 4 d = Afte some calculatos (see moe detals the poof of Coollay 5) we )( x+ ) x =0 ( x+ ) =0 ( x+ ). Theefoe, P (x +) P (x) 0. As = λ a /λ v t s equvalet to show that L s cocave ceasg. We kow that L = x( P )=x xp. Hece, dl = x x d(p) d d that dl 0. Wehavethat: d ad d2 L = x d2 (P ). We fst show d 2 d 2

31 Reveue Maagemet fo Ole Advetsg: Impatet Advetses 3 d(p ) d = (x+ )(+) [ =0 ( x+ ) ] [ =0 ( x+ )]( x+ ) + [ =0 ( x+ ) ] 2 = (x+ ) [ =0 ( x+ ) (+ )] [. =0 ( x+ ) ] 2 Hece, ode to esue that dl 0 we eed: d [ P =0 = P =0 P =0 ] 2 [ P =0 + P =0 + ( + )] + ( + ) 0, whch s tue accodg to Lemma 4. Hece, we have poved that L s ceasg. Now let us show that d2 L 0, whch s equvalet to showg d2 (P ) 0. Fom above we have d 2 d 2 that d(p ) d d 2 (P ) d 2 = (x+ ) + [ =0 ( x+ ) (+ )] [ =0 ( x+ =( 2[ P =0. Theefoe, ) ] 2 P [ x+ =0 + ( + )( + )][ P =0 ] algeba we have that d 2 (P ) = x+ P d 2 ([ x+ =0 2[ P =0 = (x+ ) [ =0 ( x+ [ P =0 ] + ( + )])/[ P =0 + ( + )( + )][ P =0 + ( + )][ P =0 ) ] 3 [P =0 P =0 Fom Lemma 6 we kow that P =0 P =0 ])/[ P =0 ] ] 3 + ( + )( + 2)]. esues d2 (P ) d 2 0. Hece, L s cocave ceasg. Lemma 4. Fo x,, we have X =0 =0 X µ µ x + x + + ] 3 ad afte some + ( + )( + 2) 0, whch X µ µ x + x + + ( + ). =0 Poof We pove ths lemma by selectg a few coveet tems fom the double sum o the left had sde ad the show that the sum s geate tha the sum o the ght had sde. We focus o the double sum o the left had sde of the equalty ad otce sce all ts tems ae postve ths double sum s geate tha a sum ove a few of ts tems. We fstlstthetems whee + =2, the the tem wth + =2, etc: P =0 +[ P = [ ] [ + 3 ] 2

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