A Real Option Approach to Telecommunications. Network Optimization

Transcription

1 A Real Opton Approach to Telecommuncatons Network Optmzaton Dohyun Pak and Juss Keppo Department of Industral and Operatons Engneerng, The Unversty of Mchgan 05 Beal Avenue, Ann Arbor, Mchgan , USA E-mal: and We optmze network flow by mnmzng network blockng and/or delay and by modelng network routng possbltes as real optons. The uncertantes n the network are drven by stochastc pont-to-pont demands and we consder correlatons among them n a general network structure. We derve an analytcal approxmaton for the blockng/delay probabltes and solve the optmal network flows by usng a global optmzaton technque. We llustrate the model wth examples. Key words: Optmzaton, network, basket opton.. Introducton The basc goal of network routng s to respond randomly fluctuatng network demands by reroutng traffcs and reallocatng resources. Nowadays many network systems, for nstance telecommuncatons networks, are able to do ths so well that n many respects large-scale networks appear as coherent and almost ntellgent organsms. However, as network structures become more complcated and new network servces and products are developed, the more effcent methodologes for network routng selecton and the estmaton of network blockng and delay probabltes are needed. The development of such methodologes presents challenges of a mathematcal, engneerng, and economc nature.

2 In ths paper we optmze network flows by mnmzng network blockng/delay n a general network and under stochastc pont-to-pont demands. Our network optmzaton method s based on real opton modelng. We assume that pont-to-pont transmsson tmes are ndependent of network routngs and the network capactes are constant. That s, regardless of the selected pontto-pont connecton, transmsson tme s constant and we do not optmze the network capactes. Ths s the case n telecommuncatons networks and, thus, ths paper s a real opton applcaton to telecommuncatons network routng. If the network under consderaton s a callng network then our method mnmzes the network blockng, and f the network has buffers, such as Internet network, then we mnmze the network delays. The network blockng/delay can be approxmated analytcally by usng a fnancal basket opton formula. The network has an opton to change ts routng and ths flexblty s a real opton. That s, routng alternatves are real optons and, therefore, a routng change corresponds to the exercse of one of these optons. Basc real opton results mply that the network routers have hgh value f there s a hgh uncertanty n the network demands snce n ths case there s a hgh probablty that network routng s changed. Further, f a network has lot of nodes and routers then the value of the network s hgh because t has lot of routng optons. The network optmzaton model of ths paper maxmzes the value of network and, hence, the value of the routng optons. Real opton theory s summarzed, e.g., n Dxt and Pndyck (994) and telecommuncatons applcatons are consdered, for nstance, n Alleman and Noam (999) and Keppo (00, 003). After the real opton modelng we solve optmal network routng by usng routng probabltes and the analytcal representaton of network blockng/delay. The optmzaton of routng probabltes s a nonconvex optmzaton problem and we utlze global optmzaton technques [for global optmzaton methods see, e.g., Horst and Pardalos (995) and Neumaer (004)] Many papers have analyzed network traffcs. Caceres, Danzg, Jamn, and Mtzel (99), Leland, Taqqu, Wllnger, and Wlson (994), and Feldmann (996) have shown that telecommuncatons traffc s qute complex, exhbtng phenomena such as long-tal probablty dstrbutons, long-range dependence, and self-smlarty. Therefore, varous assumptons on traffc processes are made to smplfy the network routng models. For nstance, Kelly (99, 996) assumes an ndependent Posson process for actual network traffc demand and Norros (994)

3 models traffc wth a fractonal Brownan moton. In ths paper we consder Brownan moton drven network demands and ths way we are able to apply real opton framework and derve the analytcal approxmaton for network s blockngs/delays. A smlar demand model s used, e.g., n Zhao and Kockelman (00), Ryan (00), and Gune and Keppo (00). Gune and Keppo show emprcally that dal-up demand s usually dstrbuted accordng to a log-normal dstrbuton. They also show that there are cycles n the demand and hence the parameters of the dstrbuton depend on tme. In the present paper we utlze ths result and model network pont-to-pont demands wth log normal dstrbutons. Network routng s carred out along varous routes. Routng has been studed extensvely over the last few decades. For nstance, some telecommuncaton companes have extended tradtonal statc call routng methods to dynamc strateges. These strateges route calls dependng on the gven network load and, therefore, they guarantee better qualty [see e.g. Ash and Oberer (989), Ash, Chen, Frey, and Huang (99), Ash (998), and Gune and Keppo (00)]. For nstance, DAR (Dynamc Alternate Routng) by Brtsh Telecom routes calls along the drect routng between the start and end ponts as long as there s free capacty and f the drect connecton s blocked, the call s routed along an alternatve route. Further, f there exst several alternatve routng possbltes, the alternatve routng s selected by usng the hstorcal blockng data. Our model can be seen as an extenson to DAR snce we use a smlar routng strategy. However, our routng selecton s based on the future network blockng that depends on the current network demands, ther stochastc processes, and the correlatons between the demands. For nstance, one pont-to-pont blockng probablty mght be hgh even though there have not been any blockngs n the hstory f the pont-to-pont demand s currently hgh frst tme. Further, n contrast to DAR we also consder explctly the nteractons between the pont-to-pont routng decsons. That s, f one pont-to-pont routng s changed then t affects the blockng probabltes of the other pont-to-pont connectons and n order to fnd the optmal routng these effects have to be consdered. Mtra, Morrson and Ramakrshnan (996, 999) consder network optmzaton n mult-servce broadband networks. They maxmze network revenue by usng Posson processes and the correspondng end-to-end loss probabltes for each servce and route. They assume no buffer,.e., f there s nsuffcent bandwdth on a lnk an arrvng call s blocked and lost. In ths 3

4 paper we also maxmze the revenues of the network but we use Brownan moton drven network demands n order to model the correlatons between the demands and nteractons between the routng decsons. In addton to callng networks, our network model can also be used n the optmzaton of Internet s backbone network. Currently the routng of ths network s statc. However, for nstance, Ra and Samaddar (998) and Beler and Stevensen (998) predct that the amount of transmtted data and the number of Internet hosts and connectons ncrease exponentally. Ths creates network delays n the future and, because routng optmzaton s a cheap alternatve to capacty nvestments, backbone network optmzaton becomes mportant. Ths paper suggests a framework to the backbone network optmzaton by mnmzng the network delays. The rest of the paper s dvded as follows: Secton ntroduces the underlyng models used n the paper and derves a representaton for the pont-to-pont traffc. The stochastc processes for the demands are defned and these processes are then used n the network optmzaton n Secton 3. Secton 4 llustrates the model wth examples. Secton 5 dscusses the mplementaton of the optmzaton method and fnally Secton 6 concludes.. Network flow representaton Telecommuncatons and Internet networks are n general modeled as graphs, whch have nodes (vertcals) and edges. Graphs are a natural choce for telecommuncatons and Internet networks, because the networks are not fully meshed. Fully or almost fully meshed networks can be modeled wth transton matrces as s done, e.g. n Harrson (988) wth processng networks. In the graph, nodes act as endponts for pont-to-pont connectons. A drect pont-to-pont connecton consttutes an edge also known as a lnk. Other connectons are modeled as a sequence of lnks, often referred as routes, and correspond to paths n the graph. In ths research, we consder blockng probabltes, capactes, and traffc on network lnks. We make the followng assumpton on network capactes. ASSUMPTION. Network capactes are constant. 4

5 Accordng to Assumpton. n our model, we are not able to optmze the capactes. An nvestment model where also capactes are optmzed s consdered, e.g., n Keppo (003). The pont-to-pont demands follow stochastc processes and we specfy these processes n Secton 3.. Smple three pont-to-pont network structure In ths secton we consder a smple network of three fxed pont-to-pont capactes (C) and demands (D). For example, the ponts could be New York, Los Angeles, and Atlanta, and all the pont-to-pont capactes are OC-3 (55.5 Mbps). Fgure. llustrates the stuaton. Lnk S, C, D Lnk S, C, D Lnk 3 S 3, C 3, D 3 Fgure. Network costs (S), capactes (C), and demands (D). The S-costs n Fgure. are costs from the blockng/delay of the lnks. For nstance, S s the blockng/delay cost between the up and left ponts by usng the drect routng between them. Note that S does not necessarly equal the realzed cost between the up and left ponts because f we have S + S 3 S then t s optmal to use the longer routng and, therefore, the realzed cost between the up and left ponts equals S + S 3. Accordng to Assumpton. each lnk s capacty s constant. However, the demands are stochastc and, therefore, the network routng mght be changng all the tme. For nstance, part of demand D mght be routed va the alternatve routng due to the lmted capacty on the drect 5

6 routng. Ths means that usng Fgure. we can represent the routng possbltes wth the followng table. Table. Routng alternatves. The numbers represent the routng numbers. Lnk Lnk Lnk 3 Demand Demand Demand 3 In Table., the frst row mples that Lnk s used as D s frst routng possblty (routng number ) and lnks and 3 construct the second routng possblty (routng number ). In the same way, the second row mples that Lnk s D s frst routng possblty and lnks and 3 are the second routng possblty. On the other hand, the frst column mples that the traffc on Lnk conssts of D s frst routng, D s second routng, and D 3 s second routng. We denote β as the proporton of demand D that s routed through ts frst routng D possblty and, therefore, ( β = β as the proporton that s routed va the second routng. ) Thus, demand β s for the frst routng and β D for the second. For demands D and D 3 we have smlar representaton. Therefore, β and β 3 D are for ther frst routngs and β D and β 3 D3 D 3 for the second routngs. Then we can represent the flow on the lnks as follows F() t = β D () t + β D () t + β D () t 3 3 (.) F () t = β D () t + β D () t + β D () t, 3 3 F () t = β D () t + β D () t + β D () t where F (t) s the flow on Lnk at tme t and β = for all {,,3}. = Accordng to (.) F conssts of the frst routng part of D and the second routng parts of D and D 3. Thus, we ust read the frst column of Table.. In the same way, F and F 3 are gven by the second and thrd columns of Table.. Note that D-processes are demands between the 6

7 start and end nodes over all possble routngs and F-processes are total flows on the physcal lnks between the nodes.. General network structure Wth a more complex network structure, the flow representaton on each lnk s smlar as n the prevous smple model. We frst construct the routng table and then from that table we get the flow representaton for each lnk as a functon of betas and demands. We consder a general telecommuncatons network wth n pont-to-pont connectons. Therefore, there exst n pont-to-pont capactes (C) and demands (D) and the problem s to construct the correspondng flow representaton. As expected, ths flow representaton depends on the structure of the network,.e., even though there s the same number of pont-to-pont connectons, the representaton can be dfferent wth dfferent network structures. Frst, we consder a seven pont-to-pont connectons network example as an extenson to Secton.. The network structure for ths example s llustrated n Fgure.. S 3, C 3, D 3 S 7, C 7, D 7 S, C, D S 6, C 6, D 6 S 5, C 5, D 5 S, C, D S 4, C 4, D 4 Fgure. Network costs (S), capactes (C), and demands (D). 7

8 From Fgure., we can construct the routng table, Table.. We use the followng rule n numberng the routngs. The less lnks n the route the smaller s the routng number and the routngs wth the same number of lnks have an arbtrary sequence among them. For nstance, for D, Lnk gets the routng number, Lnk + Lnk 3 and Lnk 4 + Lnk 5 get the routng number or 3, and Lnk 4 + Lnk 6 + Lnk 7 gets the routng number 4. Table. Routng alternatves. The numbers represent the routng numbers. Lnk Lnk Lnk 3 Lnk 4 Lnk 5 Lnk 6 Lnk 7 D 3, D, 3 3, D 3, 3, 4 3, D 4, 4 3, 5 3, 5, 3 4, 5 4, 5 D 5 4 4, D , 4, 3, 4 D , 4, 3, 4 The frst row of Table. ndcates that Lnk s used as the frst routng of D. Lnks and 3 are the second routng of D, lnks 4 and 5 are the thrd, and lnks 4, 6, and 7 are the fourth routng. The other demands can be analyzed n the same way. On the other hand, the frst column of Table. mples that the flow on Lnk conssts of the D s routng, D s routng, D 3 s routng, D 4 s routngs and 4, D 5 s routng, D 6 s routng 3, and D 7 s routng 3. Thus, the amount of flow on Lnk can be represented as a lnear combnaton of the demands. The parameters of the lnear mappng are our decson parameters, betas. The constructon of the routng table s agan the frst step n the flow representaton. Ths table s created row by row by usng the pont-to-pont demand representatons and then the correspondng flows are the columns of that table. 8

9 As n Secton., we can create the flow representaton from Table. s columns. For nstance, the total flow on Lnk s demand D multpled by the drect routng probablty ( β ) plus the flow from other demands,.e., (.) F() t = β D () t + β D () t + β D () t + ( β + β ) D () t + β D () t + β D () t + β D () t, where 0 for all {,, 7} and {,, 4}, and β Smlarly wth the other lnks we have from Table. 4 β = for all {,, 7}. = (.3) F () t = β D () t + β D () t + ( β + β + β ) D () t + ( β + β ) D () t + β D () t + β D () t + β D () t F () t = β D () t + β D () t + β D () t + β D () t + ( β + β ) D () t + β D () t + ( β + β + β ) D () t where 0 for all {,,7} and {,, m }, and β m β = for all {,,7}. Varable m s the number of possble routngs for demand, and n ths case, all the demands except D 4 have four possble routngs and D 4 has fve, therefore m 4 = 5 and m = 4 for {,,7}-{4} n ths example. as follows For the general network problem, let I, be a network structure ndcator and t s defned, f Lnk k s used n D ' s ' th routng, (.4) Ik =. 0, otherwse Note that ths ndcator corresponds to the routng tables, tables. and.. follows k Usng ths notaton the total flow on the k th lnk n a general network s represented as m m m,, n n m n,, k k k n k n k = = = = = (.5) F () t = β I D () t + β I D () t + + β I D () t = β I D() t, where n s the number of lnks n the network, 0 for all {,,n} and {,, m }, β = and m β = for all {,, n}. = 9

10 Accordng to (.5) the total flow on each lnk can be represented as a lnear combnaton of the pont-to-pont demands. In Secton 5, we consder m = max[ m ] as a parameter to ncrease the effcency of the network optmzaton. 3. Network Routng Optmzaton In ths secton we optmze the network routng by usng the flow representaton of Secton. The obectve n the network routng s to maxmze the network revenues mnus the costs. That s, T n (3.) sup E rt e [ Pk( t) Sk( t) ] dt, 0 k= where n s the number of lnks n the network, [0,T] s the plannng horzon, r s a constant dscount rate, P k and S k are the k th pont-to-pont connecton s prce and blockng/delay cost. The prce P k corresponds to the demand D k and the cost S k corresponds the network flow F k. That s, P k does not depend on the network structure because D k s the total demand between the start and end ponts. Therefore, we make the followng assumpton. ASSUMPTION 3. The k th pont-to-pont prce P k s ndependent of the network routng for all k {,,n}. Assumpton 3. mples that P k depends on network capacty C = ( C,, Cn ) that s constant and on demand D = ( D,, Dn ). Hence, we can wrte Pk(t) = P k (D(t),C) and, therefore, the uncertantes n P k are only from the demand process D. Further, because only the capactes and the demands affect the pont-to-pont prce, the capacty prces are ndependent of the network routng and they do not affect the routng optmzaton. Prcng of the network has been studed n many papers. For nstance, Kelly (997) consders the optmal prcng model of a network by maxmzng users aggregated utlty and shows a compettve equlbrum. Johar and Tstskls (004) extend Kelly s research to a game model, where selfsh users antcpate the prce effects of ther actons. If the network under consderaton s a network wth buffers, for nstance Internet, then the blocked demands are added to the demand of the next perod. That s, 0

11 Dt () = Dt () + Bt ( ), where Dt () s the new demand at tme t and B(t-) s the buffer pror to tme t. Mkosch et al. (99) showed that ths knd of cumulatve broadband network traffc s well modeled by fractonal Brownan moton. If we consder callng network then B(t-) = 0 snce there are no buffers. In both cases, we make the followng assumpton on the stochastc process for demands n order to use the real opton framework. ASSUMPTION 3. The process of the expected th demand D(, t T) s gven by the followng Itô stochastc dfferental equaton (3.) dd (, t T) = D (, t T) σ (, t T) db () t for all {,, n}, t [ 0, T ], T [ 0, τ] where D(, t T) = E D( T) F t s the expected total demand of the th pont-to-pont connecton at tme T calculated wth respect to the nformaton at tme t, σ (, T) s determnstc and bounded, B( ) s the Brownan moton correspondng to the th pont-to-pont connecton on the probablty space (Ω, F, P) along wth the standard fltraton {F t : t [0,τ]}, and we denote by ρ the correlaton between the th and z th Brownan motons. Accordng to equaton (3.), the stochastc processes for the expected pont-to-pont demand over all possble routngs follow an exponental process. The boundedness of the volatlty parameter guarantees the exstence and unqueness of the soluton to (3.). Assumpton 3. s vald, e.g., f we can model the number of network users wth a lognormal dstrbuton and assume that each user receves the same amount of capacty. In ths case D ( T ) s dstrbuted accordng to T a lognormal dstrbuton wth mean D(, t T) and varance D (, t T)exp σ( y, T) dy. Snce we t model expected value D( t, T ), the demand process D( t ) can be e.g. geometrc Brownan moton or mean-revertng [see for nstance Schwartz (997)]. Smlar demand models are used, e.g., n Zhao and Kockelman (00), Ryan (00), and Gune and Keppo (00). Gune and Keppo show emprcally that Assumpton 3. usually holds wth dal-up data that can be seen as a callng pont-to-pont demand. Therefore, Assumpton 3. s most convenent for callng networks. However, for smplcty n ths paper we use ths log-normal assumpton for all telecommuncatons z,

12 networks. Note that f the above assumpton does not hold we can extend our real opton modelng to other demands by changng the process n (3.), e.g., to a Posson process. Gven Assumpton 3. we can consder a QoS prcng example that s based on Keppo, Rnaz, and Shah (00). In equaton (3.) we assumed that P k (t) s ndependent of the routng at tme t. Ths s the case, e.g., f the pont-to-pont prces are leasng contract prces (or forward prces),.e., f the prces are fxed at tme 0. Let us frst assume that T s small and the network structure s gven by Fgure.. In ths case β f D(T) C and f D (T) > C then β DT ( ) C. Further, the blockng ndcator of demand D s gven by { D( T) C} { Dk( T) Ck ( D( T) C } ) k {,3}, f X Y where, { X Y} =. Thus, f the above equaton s equal to one then part of D s 0, otherwse blocked. Based on ths blockng equaton we get that T-maturty prce for the frst pont-to-pont connecton at tme 0 s gven by [see detals and extensons from Keppo, Rnaz, and Shah (00)] ( ) 0 P(0, T ) = P N( d) + G d, d, d3, ρ,, ρ,3, ρ 0 where P s a constant, N() s a cumulatve standard normal dstrbuton, G( d, d, d3, ρ,, ρ,3, ρ,3) s the area under a standard trvarate normal dstrbuton functon coverng the regon from - to -d, - to d, and - to d 3, the three random varables have correlatons ρ,, ρ, and ρ,3, the varables of the cumulatve dstrbutons d C ( D t) ),3 ln + (0, σ ( TT ) =, σ ( T) T ( d d k ln = Ck r T ( D T ),3 (0, ) + σ ( TT ) (0, ) k k σ ( T) T k for all k {,3}, and r(0, T ) = D (0, T) N + σ ( T) T) C N( d ). Note that the other pont-to-pont prces and other maturtes are gven n the same way. Thus, we can calculate pont-to-pont prces for all maturtes and, because these prces are fxed at tme 0, the future routng decsons do not affect the prces,.e., Assumpton 3. holds. The next assumpton gves the blockng/delay cost functon. ASSUMPTION 3.3 Blockng/Delay cost S k s gven by

13 S () t = max[ F () t C,0] for all t [0, T], k {,, n}. k k k We assume that the nvestment cost for the fxed capacty s a sunk cost that s already pad. Therefore, our model consders only the costs from the blockng/delay of the network lnks that are due to the overflow of the lnks. Assumpton 3.3 mples that ths blockng/delay depends on the network demands and capactes as well as on the routng probabltes through F k. Usng (.5) the cost S k can be represented as follows n m, Sk(, t β) = max [ Fk() t Ck,0] = max βik D() t C,0 k (3.3) = = n = max wk, ( β) D( t) Ck, 0, = m, m where wk, ( β) = βi k, β = { β,, βn}, and β = { β,, β }. = Equaton (3.3) s smlar as the payoff functon of a fnancal basket call opton where the demand processes are consdered as the underlyng assets, w s the weght of the th underlyng asset, and Ck s the strke prce. By usng the basket opton prcng model of Gentle (993) we get (3.4) ES [ ( T, β)] = F( β) c( β) N( l( T t, β) ) ( K( β) + c( β) ) N( l( T t, )) k k k k k where n n c exp k = ρ, wk, wk, σσ wk, σ ( ) T t,, = = l lnc ln( K + c ) ± 0.5v = t k k k,() t vk n vk = ρ, wk, ( β) wk, ( β) σσ,, = k t, k, β, w k, wk, ( β) S ( β) = n wk, ( β) S =, rt ( t) e Ck K k = n, wk, ( β) S = and N( ) s a cumulatve standard normal dstrbuton. 3

14 Next we make the followng assumpton on the routng probabltes. ASSUMPTION 3.4 The routng probabltes are constants over the routng optmzaton tme nterval [0,T]. Assumpton 3.4 mples that our routng optmzaton problem s statc on [0,T]. Ths mples that we calculate and change the routng probabltes n dscrete tme e.g., every hour or every day. Ths s true n practce snce network routng probabltes are changed n dscrete tme due to, e.g., data collecton. Usng assumptons and equaton (3.) we get that the optmzaton problem can be represented as follows n T rt (3.5) nf E e Sk ( t, β) β dt k = 0 such that 0 and β m β = for all {,,n} and {,, m }. = Equatons (3.) and (3.5) mply that because the pont-to-pont prces are ndependent of the network routng, the optmal network routng s solved by mnmzng the pont-to-pont blockng/delay costs. Note that f we optmzed each routng as follows n T rt (3.6) nf E e Sk ( t, β), β dt k = 0 where β β β m = { }, then we would get a game equlbrum where each pont-to-pont demand s optmzed accordng to (3.6) and the optmal network routng would correspond to a mxed strategy equlbrum. Note that we would have a game because the routng decson of a pont-topont demand affects the routng of the other network demands and each pont-to-pont demand mnmzes ts own costs. There would exst a mxed strategy equlbrum for ths problem [see Nash (950)] snce there s a fnte number of demands and possble routngs. However, the soluton would not necessarly be unque and numercal technques would be requred to solve the game. For the network routng by usng a game model see Lambert, Epelman, and Smth (00). 4

15 Snce we use (3.5) we do not have a game and we solve the global optmum. Ths s a nonconvex optmzaton problem and numercal global optmzaton technques have to be used. We use the generalzed reduced gradent optmzaton method developed by Lasdon and Waren (978). Fylstra, Lasdon, Watson, and Waren (998) have analyzed ths method n more detal. 4. Examples In ths secton we llustrate our optmzaton model wth two examples, whch are based on the network structures of fgures. and.. 4. Three pont-to-pont network We consder frst the smple three pont-to-pont network structure of Fgure.. The network flows are as follows (4.) F( t, β) = β D ( t) + β D ( t) + β D ( t) 3 F (, t β) = β D () t + β D () t + β D () t 3 F (, t β) = β D () t + β D () t + β D (). t Usng (3.4), we get that the network costs are gven by 3 3 (4.) Sk( t, β) = Fk( β) ck( β) N( l( T t, β) ) ( Kk( β) + ck( β) ) N( l( T t, β) ) k= k= where β = for all {,,3}. = Let us assume that we can approxmate equaton (3.5) as follows n rt (4.3) nf E Sk ( T, β) e T. β k= That s, the cumulatve dscounted blockng/delay cost on [0,T] s approxmated by the tme T dscounted cost. In order to solve ths optmzaton problem we assume the followng parameter values: tme horzon T =, dscount rate r = 0, expected demand D (0,T) = 0, and capacty C = for all 5

16 {,,3}. Note that wth these values we have S k ( 0, β ) = 0 for all k and, therefore, equaton (4.3) s a good approxmaton f T s small. We analyze the effect of correlatons and the volatltes of D, D and D 3 on the optmal routng probabltes from (4.) (4.3). Drect routng probablty r = r = 0 r = - - Demand Volatltes Fgure 4. Relatonshps between demand volatltes (σ =σ =σ 3 ) and the optmal drect routng probabltes of D wth dfferent correlatons (network structure of Fgure., r = r,, r,3 ). Parameter values: T =, r = 0, D (0,T) = 0, and C = for all {,,3}. Fgure 4. shows the optmal drect routng probabltes for dfferent demand volatltes (σ, σ, σ 3 ) and correlatons wth demand and 3 (r,, r,3 ). We assume that demand and 3 are perfectly correlated (r,3 = ). Note that the lnes n Fgure 4. descrbe the fractons of D that s routed drectly n dfferent cases. We can see that there s a small probablty to use the alternatve routng when the correlatons are postve because n ths case the alternatve routng and the drect routng are full at the same tme. Thus, the lower the correlatons and the hgher 6

17 the demand uncertantes the more the alternatve routng s used and, therefore, the more valuable the network routng optons. 4. Seven pont-to-pont network Next we consder the seven pont-to-pont network structure of Fgure.. From Table., the network flows are gven as follows F() t = β D () t + β D () t + β D () t + ( β + β ) D () t + β D () t + β D () t + β D () t F () t = β D () t + β D () t + ( β + β + β ) D () t + ( β + β ) D () t + β D () t + β D () t + β D () t F () t = β D () t + β D () t + β D () t + β D () t + ( β + β ) D () t + β D () t + ( β + β + β ) D () t where 0 and β m β = for all {,,7} and {,,m }. = We have 7 demands and 9 betas (m 4 = 5 and m = 4 for {,,7}-{4}) n ths example. We consder drect routng probabltes for D on Fgure 4. and assume that D and D 3 are perfectly correlated wth each other and D 4, D 6 and D 7 are perfectly correlated wth each other. That s, n addton to D we assume three sets of demands {D, D 3 }, {D 5 }, and {D 4, D 6, D 7 }, where the demands n a same set are perfectly correlated. Then we consder the effects of the volatltes of D, D,, D 7 and the correlatons between the demand sets to the optmal drect routng probabltes of D. 7

18 Drect routng probablty r = r = 0 r = Demand Volatltes Fgure 4. Relatonshp between demand volatltes (s = s = = s 7 ) and the optmal drect routng probabltes wth dfferent correlatons (network structure of Fgure., r = r,, r,3,, r,7 ). Parameter values: T =, r = 0, D (0,T) = 0, and C = for all {,,7}. As n Secton 4., we calculate the optmal routng probabltes for D. We assume the same parameter values as n Secton 4.. Fgure 4. shows the optmal drect routng probablty of D wth dfferent demand volatltes (s, s,, s 7 ) and correlatons. Smlarly as wth the three pont-to-pont network, the hgher (the lower) the volatltes (the correlatons) the lower s the drect routng probablty. Comparng fgures 4. and 4. we see that the alternatve routngs are used more n the complex network. Ths s smply because there are more routng possbltes n the complex network. 8

19 Average blockng/delay cost r = 0 r = 0 - r = - Demand Volatlty Fgure 4.3 Relatonshp between demand volatltes (s =s =s 3 ) and the average blockng/delay costs of all the lnks wth dfferent correlaton structures (network structure of Fgure., r = r,, r,3 ). Parameter values: T =, r = 0, D (0,T) = 0, and C = for all {,,3}. Let us consder the average one lnk blockng/delay costs of the two network examples. Fgure 4.3 shows the average blockng/delay cost n the three pont-to-pont network. It mples that the hgher the demand uncertantes and the correlatons the hgher the average blockng cost. Smlar results are obtaned for the seven pont-to-pont network n Fgure 4.4. However, comparng fgures 4.3 and 4.4 the average blockng/delay costs n the seven pont-to-pont network are lower than those n the three pont-to-pont case. The average dfferences correspondng to correlatons r =, r = 0 and r = - are 0.%, 0.7% and 6.%. Thus, the value of routng optons ncreases as a functon of the network complexty. Ths s because the network complexty ncreases the reroutng possbltes and reduces the blockng/delay costs. 9

20 Average blockng/delay cost r = r = 0 Demand Volatlty r = - Fgure 4.4 Relatonshp between demand volatltes (s = s = = s 7 ) and the average blockng/delay costs of all the lnks wth dfferent correlatons (network structure of Fgure., r = r,, r,3,, r,7 ). Parameter values: T =, r = 0, D (0,T) = 0, and C = for all {,,7}. Note that accordng to fgures 4.3 and 4.4, when r = - the cost can decrease when the demand volatlty s hgh enough. The reason s that, n ths case f a drect routng s full then the probablty that there s free capacty on the alternatve routngs ncreases, and wth volatltes hgher than 0.9 ths probablty reduces the average blockng/delay costs. Ths cost decreasng threshold of demand volatlty depends on the ntal values of demand and capacty,.e., f ntal demand s hgher than the capacty n our example then there s blockng/delay cost at tme 0 and the threshold s observed wth the demand volatlty lower than Dscussons In the general routng problem, the number of parameters ncreases as a functon of the number of nodes and paths. In Fgure 5. we have four nodes and fve lnks (demands). There are three betas for each demand and 5 betas total (m = 3 for {,, 5}). However, f the structure of the 0

21 network s dfferent then even wth the same number of nodes the number of betas may vary a lot. In Fgure 5. there are four nodes and sx demands, all of whch have fve betas. Ths gves 30 betas total (m = 5 for {,, 6}). S 3, C 3, D 3 S 5, C 5, D 5 S, C, D S, C, D S 4, C 4, D 4 Fgure 5. Network blockng costs (S), capactes (C), and demands (D). S, C, D S 6, C 6, D 6 S 3, C 3, D 3 S 4, C 4, D 4 S 5, C 5, D 5 S, C, D Fgure 5. Network blockng costs (S), capactes (C), and demands (D). By usng the approach of Secton 3 and the routng tables for the network structures, we can calculate the optmal routng probabltes. However, the requred computatonal and data collecton tme for the whole optmzaton method ncreases as a functon of the network complexty. Therefore, n these cases t s mportant to ncrease the effcency of our method. Ths can be done by consderng only the most sgnfcant routng canddates. In order to show ths, let

22 us consder a general network. If there are a drect routng and some alternatve routngs wth one or two ntermedate nodes then the probablty of usng routngs that have more than, for example, three ntermedate nodes s qute small. Ths s because f the cost of at least one lnk goes up then the whole routng cost ncreases. For example, the probablty of usng the ffth routng of D 4 n Fgure. (Lnk +Lnk 3+Lnk 7+Lnk 6) s usually small at the network optmum (t s, n fact, zero n all the cases of Secton 4.). Therefore, n order to decrease the computatonal tme n a general model, we can reduce the number of parameters n the network problem and smplfy the calculatons by reducng the maxmum number of possble routngs. Table 5. Loss of accuracy on delay cost by lowerng the maxmum number of routng possbltes (m). σ Total cost (m=5) Total cost (m=3) Loss of accuracy 0.00% 0.00% 0.00% 0.00% σ Total cost (m=5) Total cost (m=3) Loss of accuracy 0.6% 0.56% 0.969%.479%.059% Table 5. shows the loss of accuracy n the seven pont-to-pont example (Fgure.) by lowerng m = max[ m ] from 5 to 3 and by assumng negatve correlaton between D and the other demands (r = -). All the other parameter values are the same as n Secton 4.. The optmal routng probabltes are the same when m = 4 or 5 snce the probablty of usng the ffth routng of D 4 s zero. The blockng/delay cost n Table 5. s the sum of all lnks costs. From Table 5. we can see that the loss of accuracy s neglgble even though there are negatve correlatons between D and the other demands. These results are, of course, case-specfc. We do not defne an

23 effcent value for m snce the loss of accuracy depends on the network and the parameter values. However, Table 5. ndcates that analyzng m s mportant snce t can reduce the computatonal tme requred to calculate the optmum. 6. Concluson We have suggested a new method for network routng optmzaton by usng real opton modelng. Our approach can be dvded nto three steps. Frst, we represented the total flow on each lnk by usng pont-to-pont demands and routng probabltes. Second, we calculated the blockng/delay costs by usng the flow representaton and a fnancal basket opton model. Fnally, we calculated the optmal network routng by usng global optmzaton technques. The fundamental dea s to use real opton concepts n the modelng of network routngs. Our approach consders demand correlatons and the nteractons between the routng decsons. Because we use routng probabltes n the network optmzaton, ths routng strategy s smlar to DAR (Dynamc Alternate Routng) and can be vewed as an extenson to that. Numercal examples n Secton 4 showed that the hgher the demand uncertantes and correlatons the hgher the network blockng/delay probablty. Further, we showed how the complexty of the network lowers the network blockng/delay and, therefore, ncreases the value of the network. Acknowledgement The authors are grateful to conference partcpants at the INFORMS 003 Annual Meetng. The authors also thank Mngyan Lu, Tudor Stoenescu, Xu Meng and Sophe Shve for useful dscussons. References 3

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