Financial Derivatives for Computer Network Capacity Markets with Quality-of-Service Guarantees

Transcription

1 Financial Deivatives fo Compute Netwok Capacity Makets with Quality-of-Sevice Guaantees Pette Pettesson Febuay 2003 SICS Technical Repot T2003:03 Keywods Netwoking and Intenet Achitectue. Abstact Five netwok sevices that meet the needs of new Intenet applications ae fomulated as deivatives on the spot pice of the capacity in netwok connections. The deivatives ae piced using the Black-Scholes model. The sevices suggested ae: a multcast sevice fo one-to-many connections, a video on-demand sevice, a sevice that gives the use peceived highe quality fo many applications, a sevice that allows the equested capacity to vay with time and a sevice that does not specify the exact time of delivey of capacity. ISSN ISRN: SICS-T 2003/03-SE 1

2 Acknowledgement This wok was caied out at SICS, Swedish Institute of Compute Science within the famewok of poject AMRAM. I thank Eik Auell and Las Rasmusson fo help and diection. 2

3 Contents 1 Intoduction Spotmaketsfooutecapacity Quality of sevice Netwok capacity Stuctueofthisthesis Model Netwokachitectue Risk-neutal valuation and the Black-Scholes model Rainbow options Seveal undelying assets The n-dimensional Gisanov tansfom of E Q [Sg(S) Definitions The cost of esouces The send fee New Intenet applications Auditoy applications Video-based applications Remote contol of instuments Gidcomputing VitualEnvionments Netwok sevices Reliable multicast Video on-demand IntenetCapacityBoost Time-dependent capacity Taffic sometime duing the night Discussion 27 A Netwok deivatives 27 A.1 Picingexample Netwokfowad A.2 Picing example Netwok cash-o-nothing option B Basket options 29 3

4 B.1 Valuation of basket options using Edgewoth-seies expansion.. 29 B.2 Sumoftwolog-nomalvaiables B.3 Distibution expansion Index 34 Refeences 35 4

5 1 Intoduction The usefulness of a compute is geatly inceased when it is connected to othe computes by a netwok. A netwok of netwoks is called an intenet and the lagest intenet in the wold is called the Intenet. This thesis is based on the idea that the ownes of the netwoks that ae pat of the Intenet should be able to sell thei spae capacity on a spot maket to individual uses. This will enable new Intenet applications that has high capacity equiements. The capacity ights can be taded in the fom of sevices designed to povide capacity fo the applications. The sevices ae piced as financial deivatives of the undelying asset netwok capacity. 1.1 Spot makets fo oute capacity The most common way of handling data steams though netwok outes is to use best-effot outing. When best-effot outing is used all steams of netwok taffic expeience equal loss. Thee ae howeve many applications that may not be accommodated by the best-effot Intenet sevice model. Thee ae some data packets, such as audio/video steams, that have deadlines. Ovecowding, o congestion, can cause unacceptable delays due to packet losses and etansmission if best-effot outing is used. As netwok outes become congested some uses will thus want to eseve netwok capacity in the outes. When uses ae able to eseve capacity the netwok is said to be able to povide guaanteed quality of sevice, o QoS. It is desiable that a esevation of capacity does not block othe esevations and that the esevation scheme doesn t equie extensive negotiation. Tading oute capacity in spot makets is a way to meet these equiements. In this maket uses buy o sell oute capacity depending on thei needs. As pices inceases with demand altenative paths in the netwok become competitive and uses tend to move thei bandwidth usage away fom congested outes. The woking hypothesis is that a suitable numbe of contingent claims, o deivatives, on oute capacity will impove the efficiency of the oute capacity maket (see Rasmusson et. al.[20). The objective of this thesis is to design a numbe of sevices that ae piced as financial deivatives. We assume that deivative pices ae functions of cuent maket pices and the statistical model. The model of pice dynamics in a spot maket used in this thesis is the one poposed by Rasmusson and Auell[19. Rasmusson has povided some useful theoems unde the assumption that pices ae log-nomal[16 and a way of evaluating the CDF fo weighted sums of coelated log-nomal andom vaiables[ Quality of sevice Netwok Quality-of-Sevice can be descibed using diffeent metics. Netwokes and application developes have diffeent pespectives and do not measue quality in the same way[12. Application pefomance is expessed in tems that focus on use-peceivable effects. A use may not peceive e.g. data loss in the netwok as loss of audio claity. This means that data loss on the netwok level is not the same as fom data loss on the application level. Using the netwok pespective these ae some of the most impotant metics 5

6 that chaacteize pefomance: Bandwidth. The amount of data in bits-pe-second (bps) that can be sent though a given communications cicuit. Delay. The time it takes fo data units to be caied by the netwok to the destination. Delay vaiation. This is often due to buffeing on outes duing peiods of inceased taffic. Packet loss. A packet is the unit of data sent acoss a netwok. Lost packets ae usually a esult of congestion on the netwok. Loss patten. Peiods and othe pattens can be helpful when designing applications to handle packet loss. Thee ae othe also othe factos, such as e-odeing and secuity, that can be egaded as QoS metics. One wod that is often used to descibe netwok quality is eliability. It is not completely clea what this means, but the intepetation advocated hee is that it summaizes the effect of all the QoS metics except bandwidth and delay. Delay vaiation, packet loss loss patten, e-odeing and secuity ae thus all factos that affect eliability. When descibing the guaanteed QoS sold on spot makets we will howeve simply use the metic netwok capacity: Netwok capacity In many situations it is convenient to use only one vaiable to descibe Qualityof-Sevice netwok capacity. Real netwok capacity is dependent in a complex way on seveal netwok quality factos, such as bandwidth, delay, jitte, packet loss and loss patten. How the quality is affected by these factos depend on the application. A sevice that is valuable to one use may be wothless to anothe use if e.g. the delays ae to long, egadless of the bandwidth. It is not feasible to allow uses to specify evey QoS facto each time a sevice contact is bought since this would make the pice negotiation too complicated. The way of getting aound this poblem will depend on the applications. A possible solution is to intepet capacity as bandwidth with some guaantees egading delay and packet loss. A use will be able to buy moe bandwidth, but will pehaps not be able to decease delays o incease the eliability. Fo moe on the quality levels in the model used hee, see 2.1 on the following page. 1.3 Stuctue of this thesis In chapte 2 the netwok achitectue and mathematical models used in this thesis is pesented. Chapte 3 suveys some of the most impotant new Intenet applications. In chapte 4 netwok sevices ae suggested to meet the capacity needs of some of the applications. Chapte 5 concludes the thesis with a discussion of sevices. 6

7 2 Model In this chapte the netwok and deivatives model used in this thesis ae descibed. Fist, the netwok achitectue is descibed. Then financial mathematical models ae intoduced and applied to the netwok capacity maket. The main eason fo intoducing a send fee as a pat of the netwok achitectue is explained. 2.1 Netwok achitectue The netwok achitectue is one of the most impotant concepts when discussing compute communications. Stallings descibes the achitectue as the stuctued set of subtasks that implements communication[22. The TCP/PI efeence model is one example of an netwok achitectue. TCP/IP (Tansmission Contol Potocol ove Intenet Potocol.) is divided into fou layes: Link, Netwok, Tanspot and Application. They can be thought of as a stack with the Application laye at the top. The Link laye (a k a: Host to netwok laye, Laye 2) enables the uppe layes to communicate with the hadwae. The Netwok laye (a k a Intenet laye, Laye 3) figues out how to get the data to its destination. The eliability of the tansmission is guaanteed by the Tanspot laye and the Application laye povides a use inteface. Anothe example is the OSI (Open Systems Inteconnection) seven-layes model that among othe things defines a Physical laye, which is the physical medium used to tansmit data. Hee follows a shot desciption of Rasmusson s netwok achitectue. A complete desciption can be found in Rasmusson s dissetation[18. Rasmusson s achitectue This Intenet consists of seveal subnetwoks (figue 1 on the next page). In this achitectue, the netwok taffic is switched, monitoed and measued by neutal manages at exchange locations, oex- change points. The exchange locations ae nodes in the gaph that descibe the netwok layout (figue 2 on the following page). They connect subnetwoks owned and un by netwok ownes such as opeatos, companies and univesities. The ownes detemine how much taffic that can be sent between the exchange locations at the edges of the subnetwok. This capacity is then announced as available and sold in shaes at a maket-place. The ownes may change the flows though the subnetwok (figue 3 on the next page) as they want, as long as the taffic is deliveed as pomised. The netwok capacity shaes ae sold in bundles called netwok sevices. Netwok uses ae pesons, companies o automatic agents. An automatic agent is a softwae component that acts on ou behalf, with ou authoity, and that is intended to do so in ou best inteest.[10 All uses ae assumed to be selfinteested in the sense that they pefe bette pefomance fo themselves to bette pefomance fo someone else. The netwok ownes must be able to handle two taffic classes, poviding diffeent QoS levels: 1. The Fist class. Fist class taffic is guaanteed to be handled by the netwok. To send taffic in this class a use must pay fo the eseved 7

8 End use Access node Exchange node Subnetwok The Intenet Figue 1: The Intenet consists of a lage numbe of subnetwoks. In this example thee ae five subnetwoks connected by exchange locations. The uses access to netwok capacity is contolled by access points. Figue by the autho basedonafiguein[18. Figue 2: The exchange locations ae nodes in the gaph that descibe the netwok layout. The uses at the edge of the netwok have seveal diffeent paths (solid black lines) to choose fom. Configued path Intenal outes and links Bode oute Figue 3: A close look at one of the subnetwoks. The netwok owne of the subnetwok configues paths fo esevable capacity between bode outes. The configuation of intenal outes and links need only be known by the netwok owne. Figue by the autho based on a figue in [18. 8

9 capacity. Guaanteed Quality-of-Sevice, o guaanteed netwok capacity, is povided. Not only the amount of bandwidth, but also the level and chaacte of othe quality measues, such as delay and packet loss, is specified. 2. The Best-effot class. Uses can send taffic in this class fo fee, but the class offes no guaantee at all. Packets may be thown away when the netwok becomes congested. By combining these two taffic classes in diffeent ways one can implement any level of Quality-of-Sevice. The pice of a sevice is affected only by the amount of taffic sent in the fist class, since taffic in the best-effot class is fee. The shaes of fist class taffic that ae sold at the maket specify the capacity and the enty and exit gateway addesses, and the exchange locations between which the shae povides capacity. A shae owne is guaanteed to send packets at the specified ate indefinitely. Capacity tokens ae used to veify that the use has eseved capacity. The tokens ae bit-stings with cyptogaphically signed contacts. The use pays not only fo the shaes, but also has to pay a pesecond send fee (see on page 12) to the netwok owne. The send fee is necessay to pice netwok capacity deivatives. Some of the exchange points ae also access points that uses connect to. The access points shape the taffic fom the use so that it does not exceed the allowed amount anywhee along the path. Taffic that exceeds the allowed ate is maked as best-effot taffic. The access point admits taffic only if the use can send capacity tokens to pove that he owns sufficient shaes. Thee will be one maket fo each kind of capacity, that is, each exchange node. End-uses may buy capacity on the spot makets, but it is assumed that most uses will buy the capacity in the fom of sevices. The sevices ae financial deivatives of netwok capacity assets. The deivatives ae piced and sold to the end-uses by middle-men, o bokes. The next section descibes the picing model used to pice deivatives of netwok capacity. 2.2 Risk-neutal valuation and the Black-Scholes model Desciptions of the Black-Scholes[3 method of picing options and othe deivatives ae given in e.g. Hull[8, Bingham-Kiesel[1 and Bjök[2. The Black-Scholes wold unde the pobability measue P is ds(t) =S(t)(µdt + σdw(t)), S(0) = S 0 (0, ) (1) db(t) = B(t)dt, B(0) = 1 (2) whee, µ and σ ae constant coefficients. The constant denotes the isk-fee inteest ate, σ is the volatility. W (t) is Bownian motion, a stochastic pocess with nomal distibution such that E[W (t) = 0 and E[W (t) 2 =t. The measue P can be thought of as the odinay wold, a wold whee investos do cae about isk. The constant µ is a measue of isk avesion by investos, a highe µ means a highe isk avesion. In ode to pice deivatives we move into a wold whee investos ae isk-neutal. This means changing measue to 9

10 Q, often called the equivalent matingale measue. The pice we will obtain is, howeve, valid in all wolds. Unde the pobability measue Q the wold is ds(t) =S(t) ( dt + σdw Q (t) ), S(0) = S 0 (0, ) (3) db(t) = B(t)dt, B(0) = 1 whee W Q (t) is Bownian motion. Unde this measue S(t)/B(t) is a matingale. Equation 3, a linea SDE, is known as geometic Bownian motion and has the (stong) solution S(t) =S 0 exp ( ( σ 2 /2)t + σw Q (t) ). (4) In e.g. Hull[8 the Black-Scholes PDE is deived fom equations 1 and 2 and a set of assumptions. The equation is f(t, S) = 1 2 σ2 S 2 2 f f + S S2 S + f t f(t,s)=φ(s) [0,T) R, (5) whee S = S(t). Pat of the Black-Scholes PDE (equation 5) is ecognized as being the geneato of geometic Bownian motion. Let f C0(R), 2 i. e. f is a twice continuously diffeentiable function with compact suppot. The (infinitesimal) geneato A of the geometic Bownian motion unde Q is (see e.g. Øksendal[14) Af(x) = 1 2 σ2 x 2 f (x)+xf (x). The tem f(t, S) in equation 5 epesents the possibility of investos to invest in the isk-less asset B(t). If the Black-Scholes equation is used with the Feynman-Kac fomula we get a epesentation of f(t,x) which is the pice of an attainable contingent claim Φ(S(T )) at time t f(t, S(t)) = e T E Q [Φ(S(T )) F t (6) This is the Black-Scholes vesion of the isk-neutal valuation fomula. Thee ae moe geneal vesions of this fomula, whee the numéaie (in the Black- Scholes case e T ) and the asset ae substituted fo moe geneal ones. See e.g. Bingham-Kiesel[ Rainbow options Options involving moe than one isky asset ae often efeed to as ainbow options. One example of a ainbow option is the basket option, an option whose 10

11 payoff depends on the value of a potfolio of assets. Anothe example is options on the minimum o maximum of two o moe assets. The netwok deivatives in this thesis ae all ainbow options, since the payoff geneally involve moe than one isky asset. The cost of esouces, defined by equation 10 and path send fee 12 ae both examples of baskets of assets. Rasmusson has suggested a method of picing netwok capacity deivatives by using Monte Calo simulation to evaluate the PDF of the asset basket[20. Basket options can also be piced e.g. by using Edgewoth seies expansion[9. This is descibed biefly in the appendix of this thesis (appendix B.1 on page 29) Seveal undelying assets In Bjök[2 the Black-Scholes model is genealized to the case whee we have seveal isky assets apat fom the isk-fee asset. If thee ae n isky assets, the asset pice vecto is S(t) =[S 1 (t)...s n (t) T The contingent claims ae of the fom Φ(S(T )) whee T is the fixed execise time. The pice vecto is assumed to be diven by n independent Wiene pocesses. But, since it is easonable to assume that the pices S i ae coelated, each pice pocess is assumed to be dependent on all n Wiene pocesses. Unde the objective pobability measue P, the S-dynamics is given by ds i (t) =µ i S i (t)dt + S i (t) n σ ij dw j (t), S i (0) = S 0,i (7) j=1 whee µ i and σ ij ae assumed to be known constants. The volatility matix Σ={σ ij } n i,j=1 is assumed to be nonsingula. We also have the isk fee asset defined by equation 2 on page 9. The volatilities ae the same unde Q as unde P and the isk-neutal valuation fomula (equation 6 on the page befoe) still holds The n-dimensional Gisanov tansfom of E Q [Sg(S) The Gisanov tansfom can be used to simplify expessions of the fom E Q [Sg(S). If S(T )isann-dimensional log-nomal pice pocess with coelation {D} ij = Co[dW i,dw j unde pobability measue Q. Then E Q [S m (T )g(s(t )) F 0 =S m,0 e T E Q [g((ξ T m1s 1 (T ),...,ξ T mns N (T )) F 0, (8) whee ξ mi =exp(σ i σ m {D} im ) = exp( 1 dt Cov[log ds i(t ), log ds m (T )). Details and a poof can be found in Rasmusson s dissetation[ Definitions The netwok is modelled as an undiected weighted gaph. The connections ae edges in the gaph and the exchange and access points ae vetices. The 11

12 Figue 4: The example fom figues 1 to 2 on page 8 continued. The access and exchange points ae nodes in the gaph and the subnetwok connections between access points ae edges in the gaph. weights ae netwok capacity. Routes fom one point in the netwok to anothe ae paths in the gaph. A set of paths in the netwok can be descibed as a netwok capacity matix, V(t) with elements {v im }. Each edge m is given a capacity weight v im fo each path i. In figue 4 an example of a netwok with seven connections is given. If capacity c A is sent along path A = {1, 2, 7} and capacity c B is sent along path B = {1, 4, 5, 7}, the capacity matix is [ ca c V = A c A. (9) c B 0 0 c B c B 0 c B The cost of esouces The capacity pices in a netwok with N connections ae {S m (t)} N. Thiscan be witten as an asset pice vecto S(t). Let S m (t) be the pice of capacity on oute m at time t, S m (0) = S 0,m. The cost of esouces along path i in the netwok is C i (t) = = C im (t) v im S m (t) (10) whee N is the numbe of connections in the netwok and v im is the amount of capacity needed. The set of M cost of esouces can be witten as a vecto C(t) =V(t)S(t) with elements {C i (t)} M i=1 : C 1 (t). C M (t) = v 11 v 1N S 1 (t) v M1 v MN S N (t) The send fee Even though uses have to buy and sell esouce shaes in ode to send taffic some othe payment is necessay to pice netwok deivatives. Rasmusson[16 12

13 shows that the pice of a futue to buy esouces on the cheapest path between two netwok nodes at T 1 that ae esold at T 2 is zeo. The buying and selling of esouces that ae a pat of the deivative payoffs will always amount to a bundle futue o a sum of bundle futues, since esouce holdes can be assumed to want to sell the esouces when they ae done sending. In ode to give esouce holdes an incentive to elease esouces Rasmusson uses a send fee. The owne of a esouce is allowed to send an amount v of taffic ove connection m fo a shot amount of time if he pays εvs m (t) t, whee ε R. If the use wants to send fo a longe duation he should pay the send fee at evey instant while sending. This may be had to do in pactise, so instead he may pay the discounted expected value of the total send fee at T 1. Using this elation fom on page 28 in the appendix we get [ T2 E Q S m (t)dt F T1 T 1 [ T2 e (T2 T1) E Q S m (t)dt F T1 T 1 = S m (T 1 ) (e(t2 T1) 1). (25) = S m (T 1 ) (1 e (T2 T1) ), (11) which appoaches (T 2 T 1 )S m (T 1 )as 0. The pice of sending pepetually stating at T 1 is S m (T 1 )/. Thepathsendfee duation t is The send fee fo sending taffic along path i fo a shot Π i (t) =C i (t) t = v im S m (t) t (12) i=1 We define the path send fee vecto as Π(t) = V(t)S(t) t. Π has elements { Π i } M i=1. We also define the total path send fee at time t as [ t+τ Π i (t, t + τ) =e τ E Q dπ i F t = (1 e τ ) t v im S m (t) (13) fo which (1 e τ )/ τ as 0. The total path send fee is what a use will have to pay at t to send taffic along path i fo a duation τ. 13

14 Send fee example To illustate why a send fee of some sot is necessay, hee is an example based on one of one of Rasmusson s theoems[16. Conside the pice of sending taffic ove connection m between times T 1 and T 2.AtT 1 the pice m vs m (T 1 ) is paid to get the needed capacity. The send fee of sending taffic with this capacity T2 T 1 v m εs m (t)dt is paid continuously between times T 1 and T 2. At T 2 the esouce is sold fo S m (T 2 ). A esouce holde who wishes to avoid isk would be inteested in buying a contact today, a time t = 0 that gives him the cash flows in the pevious paagaph. The isk neutal picing fomula gives us a pice of this contact: Π 0 = e T1 E Q [v m S(T 1 ) F 0 +e T2 E Q [ T2 T 1 v m εs m (t)dt F 0 e T2 E Q [v m S(T 2 ) F 0 [ T2 = v m s 0 + e T2 E Q v m εs m (t)dt F 0 v m s 0 T 1 [ [ T2 = e T2 E Q E Q v m εs m (t)dt F T1 F 0 T 1 [ (e = e T2 v m ε E Q (T 2 T 1) 1) S m (T 1 ) F 0 = e T2 v m ε (e(t2 T1) 1) E Q [S m (T 1 ) F 0 e = v m ε (e T1 T2 ) e T1 S m,0 1 e (T2 T1) = v m εs m,0. Thus, if ε = 0, that is, if thee wee no send fee, the pice of the contact would be zeo. Fom hee on ε is assumed to be 1. 3 New Intenet applications The Intenet2 QoS Woking Goup is cuently doing a suvey on QoS needs fo new Intenet applications[12. I will only focus on a few applications since a complete suvey of the sevices equied fo all the new applications is out of scope of this thesis. Hee, howeve is a bief suvey of applications based on the classes of applications discussed in the Intenet2 suvey, and thei QoS equiements. 3.1 Auditoy applications Applications elated to sound can be divided into inteactive and non-inteactive auditoy applications. 14

15 Inteactive The most common fom of communication on eath is voice communication. Convesational audio is an inteactive auditoy application that enables voice communication ove long distances. The inteactivity of these applications equies some level of QoS, but not as high as some of the moe advanced applications. Non-inteactive Pofessional quality audio steaming applications will be used to distibute music. It has to be high-sampling, multichannel audio with CD-equivalent o bette quality. The steams may need to be tansmitted uncompessed o losslessly compessed to maintain quality. Fo this application delays in the ode of seconds ae acceptable. When demands on timing ae highe, such as when sending a live concet, high quality audio ochestation applications ae used. They equie highe levels of QoS since end-to-end delay, jitte and packet-loss ae cucial factos. 3.2 Video-based applications Just like auditoy applications, video-based applications come in inteactive and non-inteactive vaieties. Inteactive applications ae geneally moe sensitive to delays than non-inteactive applications. On the othe hand uses will pobably toleate cetain video distotion in long-distance collaboation, so equiements on packet loss and bandwidth ae less stingent. Inteactive The quality equiements of video-based applications diffe fo inteactive and non-inteactive applications. High quality audiovisual confeencing, o simply video-confeencing, is an inteactive video-based application. The quality equiements ae dependent on the exact natue of the confeencing application. To achieve a collaboative confeence expeience this may involve ochestating both audio and video, which equies a high level of QoS. Non-inteactive Examples of non-inteactive video-based applications ae video steaming and high definition TV. Video steaming can eithe be Real-time dissemination of live events, such as news o spots. This involves tansmitting video to a lage goup of uses. These applications ae best seviced in a scalable fashion by multicast netwoks. (Fo moe on the quality equiements of multicasting, see section 4.1 on page 17.) Steaming of on-demand pe-ecoded, stoed video mateial fom a emote seve. The latency demands may be slightly elaxed which means that low levels of jitte can be alleviated with appopiate buffeing algoithms and lost packets can etansmitted. This in tun elaxes the netwok quality equiements. The objective of High definition TV, o HDTV, is to povide high-esolution, high-quality moving images at qualities compaable o bette than any contempoay digital equivalent such as DVD, by fa supassing today s TV expeience. HDTV comes in diffeent vaieties fo diffeent taget uses, so the equiements also vay. Bandwidth equiements ange fom 19.2Mbps fo consume 15

16 gade quality (boadcast quality MPEG-2) to 1.5Gbps fo aw, o uncompessed, HDTV used by studios in some stages of poduction. If nothing else, HDTV equies a lot of bandwidth and multicasting seems to be the only solution. 3.3 Remote contol of instuments To be able to contol instuments ove a geat distance is useful in many aeas of science, especially medicine whee it can be used in e.g. obotic sugey. Quality equiements vay with applications. Robotic sugey The main eason fo using obots in medicine is to educe the people needed to opeate. Today nealy a dozen people ae needed to pefom an opeation. In the futue, sugey may equie only one sugeon, an anesthesiologist and one o two nuses. The sugeon pefoms the opeation at a console. This console could be in the opeating oom o, if the netwok quality is sufficient, at a location a long distance fom the patient[4. The quality equiements ae high. High esolution images and/o hi-fi video feeds will be tansmitted to the use. This will equie quality simila to that of inteactive video. The emote contol data sent sent to the emote instuments also has high quality equiements, but of a diffeent kind. The bandwidth is elatively low, but the equiements on quality measuements elated to eliability, e.g. delay vaiation and packet loss, ae high. Othe examples ae emote contol of lage telescopes (e.g. the SOAR telescope) and emote access of poweful micoscopes (e.g. the MAGIC cente at Cal State Haywad)[ Gid computing Gid computing is a way fo computational scientists to shae both computational esouces and data ove a netwok. Many complex systems demand computational, stoage and netwoking esouces on a vey lage scale. Cuent gid computing pojects ae e.g. the EU sponsoed DataGid Poject, GiPhyN Gid Physics Netwok, funded by NSF and NEESgid fo the eathquake engineeing community, also funded by NSF. Gid systems involves many kinds of applications with diffeent quality equiements. The Euopean DataGid Poject is intended to enable thee vitual oganizations: the High Enegy Physics community led by CERN in Switzeland, the Biology and Medical Image pocessing community led by CNRS in Fance and the Eath Obsevations community led by the ESA/ESRIN in Italy. A Vitual Oganization is a distibuted community of institutions and individuals willing to shae thei esouces in ode to achieve common goals. The idea is that community membes should have access to poweful esouces without having to know wee the esouces come fom. Uses simply submit equests to the Gid specifying application-level equiements and povides input data. The Gid then finds and allocates suitable esouces to satisfy these equests. The pocessing is monitoed by the Gid, and the use is notified when the esults ae eady to be pesented[15. 16

17 3.5 Vitual Envionments A vitual envionment is a compute geneated simulation, often with an inteface that uses 3D compute gaphics. Thee ae many appoaches to developing applications that utilize compute geneated 3D wolds. Hee ae a few examples of concepts elated to eseach on vitual envionments. The netwok capacity equiements ae simila to those of inteactive visual applications. DVE s A Distibuted Vitual Envionment, o DVE, is a vitual envionment that is multi-use and distibuted i. e. a vitual envionment suppoting multiple inteacting uses and unning on seveal computes connected by a netwok[7. DIVE, developed by SICS in Kista, is an example of a platfom fo DVE s. The DIVE platfom uses multicast and patitioning of the vitual univese to allow a lage numbe of simultaneous paticipants[5. Mixed Reality Mixed Reality is an inteface that ovelays digital images onto those of the eal wold. 3-D objects ae meged into the eal envionment placing gaphical infomation diectly in the viewpoint of the use. The National Univesity of Singapoe has done eseach in this aea and claim that the technology will be available in one to two yeas[11. Tele-immesion Tele-immesion ceates the illusion of two people, possibly on opposite sides of the globe, being in the same physical space as each othe. It enables uses at geogaphically distibuted sites to collaboate in eal time in a simulated envionment as if they wee in the same oom. The ultimate goal of tele-immesive eseach is to constuct a holodeck simila to the one seen on the Sta Tek TV-seies. E.g. the National Tele-immesion Initiative - NTII did eseach on this until ecently[13. 4 Netwok sevices In this chapte five sevices will be descibed and piced as financial deivatives. As mentioned ealie (2.4.2) the cost of esouces (equation 10) pat of the deivative contacts will always amount to a bundle futue, which costs nothing. This is why the send fee is the only pat of the cost mentioned when picing sevices in this chapte. 4.1 Reliable multicast The use gets the capacity needed to multicast data fo some dua- Sevice tion. What multicast is Multicasting is a bandwidth conseving technology that educes taffic by simultaneously deliveing a single steam of infomation to thousands of ecipients. This is possible today, but multicasting equies high sevice quality in ode to be useful fo all applications. Since netwok capacity 17

18 is shaed by many uses when multicasting data, the capacity costs ae lowe than it of unicasting the same infomation. The conventional way of netwoking, sometimes called unicast, is by sending data fom one compute to anothe. This is inefficient when thee ae many ecipients of the same infomation since sepaate identical data steams has to be geneated fo each ecipient. A copy of the data is sent to each client, egadless of whethe the paths to clients ae simila. When multicasting data the sending is done in such a way that the data is sent in just one copy along any pat of the netwok. A copy is made only when the paths to two ecipients diffe. The most cost-effective way of doing this is by constucting a tee that spans evey membe and minimizes the costs of sending. Such a tee is called an optimal spanning tee. Packets ae eplicated by outes at the tee nodes and sent to a host-goup consisting of zeo o moe hosts. The paticipants of the host-goup has to be listed as membes in ode to eceive data. This sevice is intended fo wide-aea multicast tansmission on tansit netwoks. Tansit netwoks ae netwoks that tansfes taffic to othe netwoks in addition to caying taffic fo thei own hosts. Wide-aea multicast is implemented on the Netwok laye (see 2.1 on page 7). Local-aea multicast does not equie the use of multicast outes, but instead uses the multicast tansmission capabilities of the Physical laye, in othe wods, woks on a lowe level than wide-aea multicast. In addition, the sevice only guaantees capacity on the Intenet-wide pat of the wide-aea netwok. At the leaves of the spanning tee multicast outes use a potocol such as IGMP to communicate with eceives on leaf netwoks[21. Poblems with multicast In ode to be used fo tansmission of e.g. video, multicasting has to be moe eliable than it is today. Multicasting is a push technology the seve sends data to the client without the client equesting it. Clients cannot equest etansmissions of lost data, since this would mean that othe membes of the host-goup had to wait fo a etansmission that they didn t need. Receiving applications cannot adapt to packet loss by equesting lost data. This is a poblem fo applications that ae sensitive to lost data. Spamming is anothe poblem. Today s system, also a push technology, suffes fom spamming. Data sent by must howeve be copied, wheeas multicast data need only be sent in one copy. Thus, multicasting could be used to spam millions of eceives with lage amounts of data they have not equested. The send fee and cost of esouces does not povide the sende with an incentive to avoid unnecessay multicasting, since the sende can send data in the best effot class fo fee. The poblem could be solved e.g. by intoducing a new multicast fee, o by equiing most of the multicast data to be sent in the fist-class class. Applications Multicasting can be used wheneve data is sent fom one use to many uses one-to-many. Non-inteactive video-based tansmissions with many eceives (in othe wods, TV) is the obvious application and pehaps also the most impotant one. Multicasting can also be used in a numbe of applications such as video con- 18

19 feencing, vitual envionments and tele-immesion, when seveal uses want to both send data to and eceive data fom many uses a many-to-many situation. We will focus on sevices fo situations when data is sent fom one sende to many ecipients. v Use v v Route v Use Data is sent fom hee Route v Use Figue 5: This is a sevice fo a one-to-many multicast connection. Payment and delivey The use pays fo the contact at time 0 and the capacity is deliveed at T. AtT the use gets enough capacity to multicast taffic with capacity v fo a duation τ along the path in the tee that connects the use with the sende. Figue 5 shows thee uses connected to access points at the leaves of a multicast spanning tee. Although the capacity is v fo each connection in the tee, the capacity needed fo one of the ecipients of a one-tomany tansmission vaies along the path. The fist pat of the path is shaed among all the membes of the multicast goup. At each node of the tee, each use s shae of the capacity inceases, until a leaf of the tee is eached. At each leaf the capacity is possibly still shaed by a lage numbe of uses. Picing The pice is, as usual, the discounted expected value of the capacity needed. The path of one use is not necessaily the minimum cost path. It will pobably be beneficial to all uses in the host-goup if the paths ae chosen so that the total cost of all the paths in the spanning tee is minimized. This means that we need to know something about the expected spanning tee in ode to pice the sevice. The stuctue of the spanning tee is dependent on the pices of capacity and the othe uses of the multicast sevice. S 1 S 2 S n 1 S n Figue 6: A multicast path. Hee s an example of an appoach to multicast sevice picing. Let the equied capacity be constant and the same fo all uses and paths: v. The capacity 19

20 matix is defined as on page 12 and descibes the paths of one use. Figue 6 on the peceding page shows one of the paths, path i of one of the uses. The path consists of connections 1 to n i = n. The send fee fo of multicasting data fo a vey shot duation, t stating now, at t = 0, along this path, assuming that this path is pat of the multicast spanning tee, is n i k=1 vs k (t) N ik (t) t, whee N ik is the numbe of uses shaing connection k on path i if the path is pat of the multicast spanning tee. The send fee of sending fo a longe duation is n i τ k=1 0 vs k (t) N ik (t) dt, The vaiables S k (t) andn ik (t) ae stochastic fo t>0. In addition, we do not know which of the paths will be pat of the multicast spanning tee. This uncetainty can be epesented by a switch function: 1 {Mi}, which is 1 if M i, the event that path i is pat of the multicast spanning tee, occus and 0 if not. The pice of a multicast fowad contact with delivey date T can then be witten as v e (T +τ) E Q [ ni k=1 T +τ T S k (t) N ik (t) dt 1 {M i} F 0, (14) a fomula that may be had to evaluate. The above is one of many appoaches to picing multicast sevices. Fo example, the switch function, 1 {Mi} and the vaiable N ik (t) in the equation above ae both dependent on the stuctue of the tee, including the numbe of uses at each leaf. They could be substituted fo one vaiable that has a diffeent value of the fom 1/N fo each possible multicast spanning tee. The poblem with this is that the numbe of possible spanning tees fo a netwok with millions of uses could be huge, much lage than the numbe of possible paths to one use. It may be necessay to simplify the poblem in ode to find a feasible picing system. An example of that is the Backbone picing that follows, wee we confine ouselves to a small but impotant pat of the tee. Backbone picing The exact stuctue of the futue multicast spanning tee is unknown. Howeve, the stuctue of the fist main banches of the tee, which is pat of a backbone netwok will be easy to pedict o even specified in advance. Picing multicast contacts on the backbone tee is easie than the multicast picing poblem stated above, if the stuctue of the tee is known. Let s look at the pice of multicasting along one of these connections, connection k, fo one use. With simila notation as above, the pice is 20

21 v e (T +τ) E Q [ T +τ T S k (t) N k (t) dt F 0 whee N k is the total numbe of uses on the connection. We still need to know the distibution of the numbe of uses on each connection in ode to evaluate the pice of a multicast sevice. 4.2 Video on-demand Sevice The use gets the capacity needed to send data fom one of a numbe of seves fo a specified duation. The use chooses the time of delivey, but the seve and path fom seve to use i chosen by the boke. Applications This is a sevice fo situations when the use wants a cetain data steam, but doesn t cae about whee it comes fom o which way it takes to get to him. Video on-demand is one such application. As mentioned ealie, delays may be acceptable when steaming video, but video equie a lot of bandwidth, so it may be necessay to eseve capacity by buying a sevice. Seve Path Path Use Path Seve Figue 7: When the use execises his contact, the boke selects seves and paths and delives the capacity. Payment and delivey The use pays fo the contact at time 0 and may execise it at any time between 0 and T 1. When the contact is execised the use gets capacity to eceive taffic sent along the cheapest path fo a duation τ. Picing Thee is at least one possible path fo each seve, but thee can also be moe than one path to choose fom fom each seve. The paths and the needed capacity ae both known when the contact is witten. At the time of execise the use eceives the send fee fo sending with a specified capacity fo some duation τ along the least cost path. Let the inteval be [0,T 1, the stike pice K and the numbe of paths be L. Even though most uses of the sevice will not choose the time of delivey so as to maximize thei pofit, this is what 21

22 we have to assume when picing the contact. The contact is an Ameican call option with the pice e T1 E Q [ max T [0,T 1 ( ( max L min i=1 [ N )) T +τ v im S m (t)dt K, 0 T F 0 whee T is the actual time of execise and T 1 is the latest time of execise. By using equation 25 this can be ewitten as e T1 (eτ 1) E Q [ max T [0,T 1 ( ( max L min i=1 [ N )) v im S m (T ) K, 0 F 0 fo which lim 0 e T1 (e τ 1)/ = τ. This is simila to an Ameican basket option. The only diffeence between this deivative and Rasmusson s netwok option[19 is that this sevice allows the use to choose the time of delivey. Picing and hedging multi-asset basket options with both high dimensionality and ealy execise is a had poblem. A numeical algoithm fo picing Ameican basket options has been suggested by Wan[23, but this option is moe complicated. The pice of this sevice may possibly be evaluated by using Rasmusson s method of evaluating the PDF[17 o by using Edgewoth expansion[9 (section B.1 in the appendix) to appoximate the distibution of the asset basket. 4.3 Intenet Capacity Boost Sevice The use gets an Intenet Capacity Boost that aises the peceived level of quality when using diffeent Intenet applications. The intention is to give a lot of feedom to the use, but the choices that the use is allowed to make make the sevice moe expensive. Applications This is an attempt to make a sevice that can be used fo many diffeent puposes. The contact can be execised any time the use peceives low quality while using a numbe of diffeent applications. The use should think of the sevice as an Intenet Capacity Boost that inceases the thoughput and educes delays. In figue 8 the use has thee sets to choose fom. If he is sending data fom eithe London o New Yok he chooses one of those sets. If the use is sending video he chooses the seve set, which execises a video on-demand contact like the one descibed above. Payment and delivey The use chooses the time of execise and a set of paths. The capacity is specified in advance. The boke then chooses one of the paths in the set. Picing The picing is simila to the picing of the Video on-demand sevice above. The diffeence is the choice of paths the use is allowed to make. We have to assume that the use selects the most expensive set of paths. This means that we loose some of the effect of allowing the least cost path be chosen at the 22

23 London Subnetwok New Yok Subnetwok London set of paths Use New Yok set of paths Seves Video seve set of paths Figue 8: The use chooses the time of delivey and a set of paths and the boke chooses which of the paths within the set to use. time of delivey. The two effects will offset each othe a choice made by the use makes the sevice moe expensive, a choice by the sevice povide makes the sevice cheape. A picing fomula, with M path sets each with a subset of L k paths, k =1...M: e T1 (eτ 1) E Q [ max T [0,T 1, k=1...m ( ( max L k min i=1 [ N )) v im S m (T ) K, 0 F 0 fo which lim 0 e T1 (e τ 1)/ = τ. This sevice may possibly be evaluated using one of the methods suggested in section 4.2, but the many min and max functions in the fomula may make picing too time-consuming. 4.4 Time-dependent capacity Sevice A use may need diffeent quality levels at diffeent times. This sevice gives the use the ight to send taffic along a specified path fo a specified duation of time, just like the total path fee in section. The diffeence is that the capacity needed is allowed to be diffeent at diffeent times, even duing sending. Applications Many uses ae pobably going to need diffeent quality levels at diffeent times of day o on diffeent days of the week. This sevice can be used as a building block to make new sevices with time-dependent capacity. Payment and delivey The use pays fo the sevice at the same time that the capacity ights ae deliveed. The use gets the ight to send capacity fo a duation τ along path i. The capacity matix V is a pedetemined function of time, V(t) with elements {v ij (t)}. The function could fo example be peiodical with a peiod of a day, week, month o yea. 23

24 Picing The cost of esouces (equation 10 on page 12) becomes C i (t) = v im (t)s m (t) (15) The path send fee (equation 12 ) becomes Π i (t) = v im (t)s m (t) t (16) and the total path send fee at t, when taffic is sent fom t fo at duation τ [ t+τ Π i (t, t + τ) =e τ E Q dπ i F t = e τ E Q [ t+τ t t v im (s)s m (s)ds F t N [ t+τ = e τ E Q v im (s)s m (s)ds F t N = e (t+τ) S m (t) = S m (t) t t+τ t t+τ t v im (s)e s ds (17) v im (s)e (s t τ) ds (18) which appoaches N S m(t) t+τ v t im (s)ds as 0. Equation 18 educes to (1 e τ ) N v ims m (t) fom equation 13 on page 13 when the capacity is constant, v im (t) =v im. Time-dependent capacity does not seem to complicate the picing of sevices based on a send-fee, since the total path send fee can be witten on the fom N S m(t)f im (t, τ). 4.5 Taffic sometime duing the night Sevice The use gets the ight to send taffic at equested capacity fo a equested duation at some time duing a peiod such as a night. Applications The sevice is intended fo situations when the exact time of tansmission is not impotant, but the capacity is. The idea is to lowe the pice of the sevice by letting the povide choose the time of delivey. As it tuns out, this does not make the sevice cheape, due to the Matingale popeties of the total path send fee. Payment and delivey The use pays fo the sevice at time 0. The path, duation and capacity ae specified. The boke chooses the time of delivey. 24

25 The use sends data fom hee... v i1 v in...to hee. Figue 9: The path, duation and capacity ae specified. Picing The exact time of tansmission is not chosen by the buye, but by the selle of the contact, the boke. The boke chooses the time so as to minimize the cost. The picing should be based on the assumption that the selle chooses this time in an optimal way. To pice the sevice we need to know when the boke can be expected to execise the contact. One appoach is dynamic pogamming. Dynamic pogamming solves a poblem step by step, stating at the temination time and woking back to the beginning[6. We will use the dynamic pogamming appoach to see if the sevice povide can educe his expected cost by choosing the time of delivey in an optimal way. Step 2 Step 1 Wait Wait Delive Delive Delive Delive Time T 1 T 2 τ 2 t T 2 τ t T 2 τ Figue 10: Dynamic pogamming solves the poblem by stating at the end and stepping backwad in time. Let the capacity be v fo all connections, the duation be τ and the inteval be [T 1,T 2. We also assume that the path is specified. At evey instant between times T 1 and T 2 τ the selle has two options to eithe delive the capacity o to wait. Step 1. At T 2 τ he has no choice but to delive the capacity in ode to fulfill his obligation. The total path send fee (equation 13 on page 13) that would be paid at T 2 τ fo sending along some path {1, 2...N} fo the duation τ is 25

26 Π T2 τ,t 2 = E Q [ T2 T 2 τ dπ i F T2 τ = (1 e τ ) v im S m (T 2 τ) Step 2. Ealie, at T 2 τ t, the selle still has a choice of eithe deliveing the capacity, which means paying the total path send fee Π T2 τ t,t 2 t = e τ E Q [ T2 t = (1 e τ ) T 2 τ t dπ i F T2 τ t v im S m (T 2 τ t) (19) o waiting, which should be seen as costing the discounted expected value of the total path send fee of equation 19: e t E Q [ (1 e τ ) e t (1 e τ ) e t E Q [Π T2 τ,t 2 F T2 τ t = v im S m (T 2 τ) F T2 τ t = v im E Q [S m (T 2 τ) F T2 τ t = e t (1 e τ ) v im e t S m (T 1 τ t) = (1 e τ ) v im S m (T 2 τ t). (20) The expected cost is the same whethe he decides to delive o not at T 2 τ t. With the ight choice of t this could be any time between T 1 and T 2 τ. We conclude that the povide of the sevice cannot use the choice to lowe his expected cost. This is because the discounted total path send fee has the matingale popety, that is, E Q [ e (T2 T1) Π T2,T 2+τ F T1 =Π T1,T 1+τ (21) fo any T 2 T 1. We may assume that the boke chooses the time abitaily. The pice of the contact at t = 0, assuming that the contact is deliveed at some abitay point in time t, is e t E Q [ (1 e τ ) v im S m (t) F 0 = (1 e τ ) v im S m,0 (22) Note that the pice is independent of the time of delivey. The picing equation appoaches τ N v ims m,0 as 0. The pice is easy to evaluate since it is a linea function of the cuent capacity spot pices along the path. 26

27 5 Discussion We have used Rasmusson s model to suggest and discuss the picing of five netwok capacity sevices. The Black-Scholes model was used to model the sevices as financial deivatives of netwok capacity esouces. A send fee as suggested by Rasmusson et. al.[19 has been used when picing the deivatives. The send fee gives uses an incentive to sell esouces when they ae done sending. The sevices suggested and piced in this thesis tend to involve the sum of assets called the cost of capacity, a weighted sum of log-nomal vaiables. This indicates that Rasmusson s method[17 fo evaluating the PDF fo such sums should be vey useful fo picing netwok sevices. Rasmusson s method does not ely on the assumption that the weighted sum has a log-nomal distibution. An altenative method that does ely on such an assumption is appoximation of the PDF by Edgewoth expansion[9. In section 4.1 a sevice fo multicasting was suggested. The futue stuctue of the tee and the futue numbe of uses on each connection in the tee ae both uncetain. We need to know something about the distibution of vaiables that descibe the two uncetainties. A uses decision whethe to buy a multicast sevice, is not only based on the netwok pices, but also possibly based on exogenous factos such as pices on some othe maket. To pice multicast sevices we need a model that include assumptions about the decisions of the multicast uses. In some of the sevices suggested hee the use and/o the boke has been allowed to make choices. It is a well know featue of the Black-Scholes model that buyes geneally have to pay moe fo choices. In the sevices of section 4.2 and 4.3 the use was allowed to choose the time of delivey and the boke was allowed to chose the path fom a set of paths. The choice of time by the use aises the pice (with an amount 0) and the choice of paths lowes the pice by the use (with an amount 0). These and othe sevices illustate how giving choices to eithe the use o the boke can eithe aise o lowe the pice of sevices. The Taffic sometime duing the night sevice of section 4.5 tuned out to be an example of a situation when giving a choice to the boke does not make the sevice cheape, at least not when using this model. The expected cost of deliveing was the same wheneve he decided to delive the capacity, so he could not use the choice to lowe his cost. This is because the discounted total path send fee has the matingale popety (equation 21 on the page befoe). Allowing the equested capacity to change deteministically ove time does not seem to make the picing of sevices moe difficult. The objective of the sevice in section 4.4 was to see if a total path send fee as defined by equation 13 could be piced when the capacity was time dependent. The esulting equation indicates that time-dependent capacity is not a poblem when picing sevices using a send fee. A Netwok deivatives Cheapest path pice The pice at T 1, of sending taffic along the cheapest path between times T 1 and T 2 is 27