An Economic Analysis of Interconnection Arrangements between Internet Backbone Providers

Transcription

1 ONLINE SUPPLEMENT TO An Eonom Analyss of Interonneton Arrangements between Internet Bakbone Provders Yong Tan Unversty of Washngton Busness Shool Box 353 Seattle Washngton I Robert Chang Aenture One Fnanal Plaza Hartford Connetut 63 Vay S Mookeree Unversty of Texas at Dallas Shool of Management SOM 349 Rhardson Texas vaym@utdallasedu Aendx: Proofs Oeratons Researh Proof of Prooston r r FIGURE A State transton dagram for entralzed IBPs Here we assume that when a aket arrves and both networks are dle the entral admnster wll assgn t to IBP- wth robablty r We an reresent the transton dagram wth a set of balane euatons n the steady state In the matrx form r r The above euatons an be readly solved to gve the loss rato for the entralzed IBPs s r r r r The senstvty of the loss rato wth reset to the routng deson r s / r Ths

2 ndates that r mnmzes f The akets thus should always be routed to the faster ath when both are dle The mnmal loss rato an be aheved by settng r f Proof of Corollary Frst gnorng the seond term n the denomnator of ths gves < Then we examne the dfferene Corollary follows Proof of Prooston FIGURE A State transton dagram for nteronneted IBPs The state transton dagram s deted n Fgure A When IBP- s busy and IBP- dles e n the state IBP- wll beome busy f ts own aket arrves wth robablty or f t dedes to aet a aket routed from IBP- wth robablty Ths leads to a state where both IBPs are busy or The rest of the transton ars are self-exlanatory We an agan wrte all the balane euatons n a matrx form State robabltes for the ase of nteronneton where an be obtaned by normalzaton Exltly

3 3 Proof of Corollary The frst neualty follows sne mn The seond neualty an be shown followng the same ste n the roof of Corollary Proof of Corollary 3 We start wth the frst dervatves of state robabltes wth reset to where the non-negatve term s gven by Due to the symmetry the frst dervatves wth reset to an be found by swang the ndes for arameters and also The sgns of these frst dervatves are summarzed n Table It s obvous that the frst neualty holds as the frst dervatve of both terms s ostve To rove the seond neualty we exltly arry out the dfferentaton Ths gves L where the ndes and take values of and but The above exresson s strtly negatve sne Proof of Prooston 3 The frst dervatve of IBP- s roft funton wth reset to ts nteronneton deson s A where Note that the rght hand sde of A s ndeendent of suggestng extreme-ont solutons that deend on the routng fees as well as the nteronneton strategy of IBP- Prooston 3 follows an examnaton to the sgn of A s the value of when A

4 4 beomes zero Proof of Lemma To dentfy the domnant strategy n the Regon- the rofts for a gven re ar are and By adotng the mxed state nteronneton strategy eah IBP s roft dereases wth the transfer re t harges Nevertheless n the Regon- the eual sgn holds only when or for IBP- We frst show that Ths s evdent as namely strtly nreases wth along the lne Next we have sne Therefore Proof of Lemma 3 We exltly alulate Therefore when s hanged from to nreases Proof of Lemma 4 From we fnd that / α R Along Lne-R we have α R The roof for IBP- s smlar Proof of Prooston 5 We frst note that wthn eah of the four regons defned n Prooston 4 and drawn n Fgure 3 e Φ Φ Φ and Φ resetvely

5 / A Here s the roft funton for IBP- gven a re ar and the orresondng otmal nteronneton strateges Ths neualty s vald from the roft funtons whh nrease lnearly wth holdng other fators onstant Euaton A mles that the best resonse funton wll be loated on the boundares In the followng we derve the best resonse funton for IBP- For any gven we nrease from and try to fnd a value of where s the maxmum Suose that that s we start n Φ an be nreased due to A to reah the boundary wth Φ On that boundary s ndfferent to the value of Ths s ndated by Prooston 3 sne on the boundary between Φ and Φ We an ontnue nrease aross the boundary and eventually end u n Smlarly for low value of the otmal resonse re s When κ wll be nreased to enter Φ and reah ts bottom border e the Lne-R drawn n Fgure 3 an be further nreased to ross Lne-R enter Φ and reah For ths to haen the neualty R < has to be vald However aordng to Lemmas 3 and 4 for lose to the above neualty does not hold Therefore for max η κ the best re resonse funton s gven by R Here η solves R η η η Proof of Prooston 6 Prooston 6 follows Prooston 5 dretly Euatons 4 and 5 guarantee that two best resonse funtons ross at κ κ Euaton 3 s obtaned by smultaneously solvng κ R and κ R κ Proof of Lemma 5 κ For a gven we solve η η Euaton 6 η η R for η Ths gves orresondng to α ϕ α η α From the roof of Lemma 3 we know the denomnator n the above two exressons s ostve It an be verfed by substtutng n the exlt exressons of state robabltes that the numerator of η s ostve Proof of Prooston 7 The eulbrum res are straghtforward followng the best re resonse funtons deted n Fgure 6 Lemma 5 states that η nreases as s redued The threshold value L an be found by settng η to 5