Optimal Pricing for Integrated-Services Networks. with Guaranteed Quality of Service &

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1 Optmal Prcng for Integrated-Servces Networks wth Guaranteed Qualty of Servce & y Qong Wang * Jon M. Peha^ Marvn A. Sru # Carnege Mellon Unversty Chapter n Internet Economcs, edted y Joseph Baley and Lee McKnght, MIT Press, 1996 Also avalale at & The research reported n ths paper was support n part y the Natonal Scence Foundaton under grants NCR and NCR. Vews and concluson contaned n ths document are those of the authors and should not e nterpreted as representng the offcal polces, ether expressed or mpled, of the Natonal Scence Foundaton or the U.S. Government. * Doctoral Canddate. Department of Engneerng and Pulc Polcy, Carnege Mellon Unversty, Pttsurgh, PA Tel: Emal: qw22@andrew.cmu.edu. ^ Assstant Professor of Electrcal and Computer Engneerng, Engneerng and Pulc Polcy. Department of Electrcal and Computer Engneerng, Carnege Mellon Unversty, Pttsurgh, PA Tel: E-mal: peha@ece.cmu.edu. # Professor of Engneerng and Pulc Polcy, Industral Admnstraton and Electrcal and Computer Engneerng. Department of Engneerng and Pulc Polcy, Carnege Mellon Unversty, Pttsurgh, PA Tel: E-mal: sru@cmu.edu. 1

2 Astract Integratng multple servces nto a sngle network s ecomng ncreasngly common n today s telecommuncatons ndustry. Drven y the emergence of new applcatons, many of these servces wll e offered wth guaranteed qualty of servce. Whle there are extensve studes of the engneerng prolems of desgnng ntegratedservces networks wth guaranteed qualty of servce, related economc prolems, such as how to prce servces offered y ths type of network, are not well understood. In ths chapter, we analyze the prolem of prcng and capacty nvestment for an ntegrated-servces network wth guaranteed qualty of servce. Based on an optmal control model formulaton, we develop a 3-stage procedure to determne the optmal amount of capacty and the optmal prce schedule. We show that prcng a network servce s smlar to prcng a tangle product, except that the margnal cost of producng the product s replaced y the opportunty cost of provdng the servce, whch ncludes oth the opportunty cost of reservng and the opportunty cost of usng network capacty. Our fndngs lays out a framework for makng nvestment and prcng decsons, as well as for the analyss of related economc tradeoffs. The analyss n ths chapter assumes an ntegrated-servce network wth fxedlength data unts such as Asynchronous Transfer Mode (ATM) network. The same approach can e used to analyze varale packet length IP networks offerng guaranteed qualty of servce through the use of protocols such as Resources reservaton Protocol (RSVP). Keywords. ntegrated-servces, opportunty cost, optmal prce, ATM 2

3 1. Introducton The economcs of provdng multple types of servces through a sngle network s a queston of growng sgnfcance to network operators and users. As a result of the rapd development of packet-swtchng technology, t s ecomng ncreasngly effcent to provde dfferent telecommuncaton servces through one ntegrated-servces network nstead of multple sngle-servce networks, such as telephone networks for voce communcatons, cale networks for roadcastng vdeo, and the Internet for data transfer. In a packet-swtched ntegrated servces network, any pece of nformaton, regardless of whether t s voce, mage, or text, s organzed as a stream of packets and transmtted over the network. By controllng the packet transmsson rate and packet delay dstruton of each packet stream, the network can use a sngle packet transmsson technology to provde a varety of transmsson servces, such as telephony, vdeo, and fle transfer. Whle ntegratng multple servces nto a sngle network generates economes of scope, heterogeneous servces complcate prcng decsons. For example, users watchng Hgh-Defnton Televson (HDTV) through the network requre up to tens of megats per second (Mps) transmsson capacty whle users who make phone calls only send/receve tens of klots per second (kps); telnet users requre mean cell transmsson delay to e kept elow a few tens of mllseconds ut e-mal senders wll tolerate longer delay; we rowsng generates a very ursty cell stream whle constant-t-rate fle transfer results n a very smooth cell stream; to carry a voce conversaton wth acceptale qualty, under certan encodng schemes, packet loss rate,.e. the percentage of packet that are allowed to mss a maxmum delay ound (usually ms for voce conversaton), should not exceed 5%, whle to carry vdeo servce, packet loss rate should e kept much lower (Peha,1991). Asynchronous Transfer Mode (ATM) technology emerges as an approprate ass for ntegrated-servces networks. ATM networks have the capalty to meet the strct performance requrements of applcatons lke voce and vdeo, and the flexlty to make 3

4 effcent use of network capacty for applcatons lke e-mal and we rowsng. The use of fxed-length packets (cells) also facltate the mplementaton of hgh-speed swtches. As a result, telephone and cale TV networks wll adopt cell swtchng technology, as they expand the range of servces that they offer. The Internet has already egun to offer new servces lke telephony, ut wthout the guarantee of adequate performance that telephone customers have come to expect. Eventually, the Internet wll also employ protocols that dfferentate packets ased on the type of traffc that they carry, and guarantee adequate qualty of servce approprate for each servce. Ths could e done y adoptng ATM technology, or y addng the capalty to guarantee performance through use of protocols lke the Resource reservaton Protocol [RSVP] (Zhang et al, 1993). Ths paper wll focus on ATM-ased ntegrated-servces networks, as the technology s avalale today, ut trval extensons would enale the same approach to e appled to any ntegrated-servces networks whch offers qualty of servce guarantees. Snce there are great dfferences among the servces offered y ATM networks, one mght ask whether the prces of these servce should also dffer, and f so, how? There s some lterature on how to prce a network that offers heterogeneous servces. Cocch et al (Cocch et al, 1993) study the prcng of a sngle network whch provdes multple servces at dfferent performance levels. They gve a very mpressve example whch shows that n comparson wth flat-rate prcng for all servces, a prce schedule ased on performance ojectves can enale every customer to derve a hgher surplus from the servce, and at the same tme, generate greater profts for the servce provder. Dewan, Whang and Mendelson et al (Dewan and Mendelson, 1990; Whang and Mendelson,1990) developed a sngle queung model n whch the network s formulated as a server (or servers) wth lmted capacty, and consumers demand the same servce from the server ut vary n oth wllngness to pay for the servce and tolerance for delay. Based on that model, they dscussed the optmal prcng polcy and capacty nvestment strategy. MacKe-Mason and Varan (Macke-Mason and Varan, 1994) suggest a spot-prce model for Internet prcng. 4

5 In ther model, every Internet packet s marked wth the consumer's wllngness to pay for sendng t. The network always transmts packets wth hgher wllngness to pay and drops packets wth lower wllngness to pay. The network charges a spot prce that equals the lowest wllngness to pay among all packets sent durng each short perod. The major eneft of ths approach s t provdes consumers wth an ncentve to reveal ther true wllngness to pay, and ased on that nformaton, the network can resolve capacty contenton n transmttng packets n a way that maxmzes socal welfare. In the work y Gupta et al (Gupta, Stahl, and Whnston, 1996), prorty-ased prcng and congestonased prcng are ntegrated. In ther prcng model, servces are dvded nto dfferent prorty classes. Packets from a hgh-prorty class always have precedence over packets from a low-prorty class. The prce for each packet depends not only on the packet s prorty level, ut also on the current network load. In the optmal prcng models mentoned aove, the fact that dfferent applcatons may have dfferent performance ojectves was usually not consdered. For example, Dewan, Whang and Mendelson s work (Dewan and Mendelson; 1990l; Whang and Mendelson, 1990) assumes that the consumer s wllngness to pay depends only on expected mean delay, and Macke-Mason (Macke-Mason,1996) assumes that consumers do not care aout delay only whether or not ther packets are eventually transmtted. There s no way, for example, to accommodate a servce that would mpose a maxmum delay lmt. These formulatons also do not consder the case of heterogeneous data rate and urstness. Consequently, prcng polces developed n these studes can not e appled n ATM ntegrated-servces networks n whch servces dffer from each other n terms of performance ojectves and traffc pattern (data rate and urstness). Some of these factors are dscussed n the paper y Cocch et al (Cocch et al, 1993), however, they do not dscuss procedures for desgnng an optmal prcng scheme. Gupta et al (Gupta, Stahl, and Whnston, 1996) consder dfferent servces whch are dvded nto dfferent 5

6 prorty classes, however, none of these servces can e guaranteed a gven performance ojectve under ther prcng scheme. In ths paper, we examne the optmal prcng prolem for ATM ntegrated-servces networks. In our approach, the optmal prce for each servce s determned from the demand elastcty for the servce, as well as the opportunty cost of provdng the servce. The opportunty cost s determned y the requred performance ojectves and traffc pattern of each servce. Snce demand for network servces usually changes wth tme of day, we wll develop a tme-varyng prce schedule (.e. prce as a functon of tme of day) nstead of gvng a sngle prce for each servce. The rest of our paper s organzed as follows: n secton 2, we present servce models for dfferent servces offered y an ATM ntegrated-servces network. In secton 3, we formulate an optmal prcng model and dscuss how to solve t usng a 3-stage procedure. We dscuss the procedure n detal n secton 4. Conclusons and future work are dscussed n Secton The Network Servce Model Network capacty s often su-addtve, leadng to condtons of natural monopoly for an ntegrated-servces network operator. In the model whch follows, we assume a sngle proft-maxmzng monopolst s operatng the network. In ths chapter, we consder only a pont-to-pont sngle lnk network. Ths frees us from network routng detals and allows us to focus our attenton on the economc prncples for desgnng prcng polcy. The capacty of that lnk s denoted as C T, whose unt s the maxmum numer of cells that can e transmtted over the lnk per unt of tme. The network s used for provdng multple servces. Qualty of servce s measured y the dstruton of cell delay tme, where lost cells are consdered as eng delayed nfntely. A servce wll e laeled as a guaranteed servce f durng each sesson, the 6

7 network makes a commtment to meet some pre-specfed delay ojectves. These guarantees are typcally expressed n stochastc, not asolute terms, e.g. no more than 5% of the cells wll e delayed for more than 30 mllseconds; or the average delay wll e less than 200 mllseconds. If no such guarantee s made, the servce s consdered as esteffort servce. Telephone calls, Hgh Defnton Televson (HDTV), and nteractve games typcally requre some type of guaranteed servce, whle e-mal s usually specfed as a est-effort servce. In our prcng model, the network servce provder attempts to maxmze proft whch s the sum of profts from guaranteed servces and est-effort servce. In estalshng a tarff for network servce, one mght charge for access ndependent of any usage; capacty reservaton for guaranteed servces, and actual usage. In ths chapter, we assume dedcated access s prced at average cost, and the cost of all shared network facltes s recovered through a comnaton of reservaton and usage prces. Ths assumpton allows us to consder reservaton and usage prces ndependent of access prces. 2.1 Servce Model for Guaranteed Servces Guaranteed servces dffer from each other sgnfcantly n terms of performance ojectves, traffc pattern (data rate and urstness), and call duraton dstruton. For example, HDTV servce has a much strcter performance ojectve and 500-tmes hgher mean data rate than telephone servce. An HDTV sesson can take hours to complete, whle telephone calls usually last only mnutes. The transmsson rate of the former (f the data stream s compressed) s also much urster than that of the latter, whch may not e compressed. 7

8 In ths chapter, we assume the network offers N guaranteed servces. Wthn the same servce category (=1,N), calls requre the same performance ojectve, exht the same nter-cell arrval statstcs, and have call duraton drawn from the same dstruton. We assume the prce for a call usng guaranteed servce s determned y servce type, call startng tme, and servce duraton. For a call of servce whch egns at tme t, p (t) s the prce whch wll e charged for each unt of tme that the call lasts. A consumer wll e charged a prce equal to p (t) tmes the call duraton f the call starts at tme t. We shall also assume that for calls of a gven servce, call duraton s ndependent of prce. We defne λ [p (t),t] as the arrval rate of calls for servce gven that the prce of a call whch starts at t wll e p (t) throughout the call 1. We also assume that at any gven prce, p (t), and any gven tme t, call arrvals are Posson,.e. the numer of calls arrvng wthn any perod s ndependent of the numer of calls whch arrved wthn prevous perods. Note that we have also assumed no cross-elastcty of demand etween dfferent servces, whch may not e realstc. We leave that enhancement for future paper. To meet guaranteed performance ojectves, the network can only carry lmted numers of calls smultaneously. These numers are determned y performance ojectves and traffc patterns of each servce. To avod acceptng more calls than t can handle, ATM ntegrated-servces networks enforce an admsson polcy y whch the 1 The consumer thus expected to pay p t ( ) 1 f call length has a mean value of. It s more typcal for a r r provder to defne a prce schedule R (t) where a call s charged R (t) at each nstant t s n progress. Our formulaton of p (t)s related to R (t) y p t + ( ) r ( t ) = R ( ) e d r τ τ τ t when call length s exponentally dstruted. 8

9 network montors the current network load and decdes whether an ncomng call should e admtted or rejected (Peha, 1993). Ths process s shown n Fgure 1. For the purpose of ths chapter, we assume calls are not queued f they can not e admtted mmedately. prce admsson polcy call admtted demand for servce call arrval call locked Fgure 1 Call Admsson Process for Guaranteed Servces For each servce (=1,N), we assume the call duraton s exponentally dstruted wth departure rate r. Defne q (t) as the numer of calls underway of servce at tme t, and q ( t ) as the expected value of q (t). Under the assumptons we made aout call arrval and departure processes, the rate of change of q ( t) should follow: dq dt = ( 1 β ) λ ( p, t) r q ( t) q =1, N (2-1) where β s the lockng proalty. Snce oth call arrval and departure are stochastc, unless the network has an nfnte amount of capacty, there wll always e a posslty of lockng calls. A hgh lockng proalty gves consumers an unpleasant experence wth the network and reduces the demand eventually, ut a lower lockng proalty also means more capacty wll lay dle most of the tme. From a network operator's perspectve, the lockng proalty should e kept at a desred level at whch any margnal revenue ncrease from ncreasng demand y reducng lockng proalty can no longer offset the margnal loss 9

10 from lettng more capacty lay dle. Values of desred lockng proalty are usually determned durng the process of makng long-term capacty nvestment decson. In the followng, we wll show how keepng lockng proalty at the desred level wll affect short-term prcng decsons: Suppose the network only offers one servce; then the lockng proalty at each tme can e determned y: β = H ρ H! H ρ = 0! (2-2) where β s the lockng proalty, H s the maxmum numer of calls that can e carred y the network, and ρ s the product of call arrval rate and expected call duraton. Let ~ β e the lockng proalty that the network operator desres to mantan. From (2-2), H can e unquely determned y the desred lockng proalty ~ β and the network load ρ,.e. H=d(ρ, ~ β ). In other words, to keep lockng proalty at a desred level under gven load ρ, the network should e desgned to carry H calls. Ths requrement can e translated nto a demand for network capacty: defne θ(h) as the amount of capacty needed to carry H calls, and α(ρ, ~ β )=θ[d(ρ, ~ β )]. α(ρ, ~ β ) can e nterpreted as the amount of capacty needed to keep lockng proalty at ~ β when the network load s ρ. Snce at each tme, the network load s related to the expected numer of calls n q progress y ρ = ~, we can also express the amount of capacty needed as a functon of 1 β q expected numer of calls n progress as A( q, ~ β ) = α( ~, ~ ) 1 β β. A( q, ~ β ) ncreases wth q. A( q, ~ β ) s defned as the amount of capacty requred to carry an average of q calls wth lockng proalty ~ β. If capacty requred exceeds total capacty C T, the network 10

11 ether has to admt more calls than t can handle, thus falng to meet some qualty of servce guarantee, or exceed the desred lockng proalty. At each tme t, q( t ), the expected numer of calls n progress s a functon of prevous and current prces. Therefore, n the short term, prces should e set such that the reserved capacty can never go aove total capacty,.e.: A[ q ( t), ~ β ] at all t (2-3) C T (2-3) defnes the admssle regon constrant (see Hyman et al, 1993; Tewar and Peha, 1995), whch specfes the maxmum numer of calls that can e carred under a gven amount of network capacty and a gven lockng proalty. The defnton of the admssle regon constrant can e extended to a multple servces scenaro, n whch the reserved capacty s a functon of the expected numers of calls n progress for all servces, whch s shown elow: A[ q ( t),..., q ; ~ β ( t ),..., ~ β ( t)] C (2-4) 1 N 1 N T 2.2 Servce Model for Best-effort Servce: Wthout a performance guarantee, cells of est-effort servce wll e put n a uffer and transmtted only when there s remanng capacty after the needs of guaranteed servces have een met. If there s not enough uffer space for all ncomng cells, some of them wll e dropped. In our model, users of est-effort servce are charged on a per-cell ass. We assume all cells of est-effort servce share a uffer of sze B s. The wllngness to pay for sendng each cell s revealed to the network. At each tme t, the network sets a cut-off prce, p (t), whch s a functon of oth current uffer occupancy and predcted wllngness to pay values of future ncomng cells. A cell wll e accepted f and only f the wllngness 11

12 to pay for that cell s hgher than p (t), and p (t) wll also e the prce charged for sendng that cell. Accepted cells wll e admtted nto the uffer as long as the uffer s not full. Once admtted nto the uffer, cells wll e eventually transmtted accordng to a sequence dctated y some schedulng algorthm, such as frst-come-frst-serve, or cost-asedschedulng (Peha, 1996). arrvng cells spot prce wllngness to pay greater than the spot prce? no cell dropped cell accepted yes uffer full? yes cell dropped cell admtted no uffer cell transmtted Fgure 2 Servce Model for Best-effort Servce If we assume that at tme t, the arrval process of cells of est-effort servce s Posson wth expected value λ (0,t); the acceptance of cells s also Posson wth expected value λ [p (t),t]. Defne s (t) as the nstantaneous transmsson rate of est-effort servce at that tme, then: s ( t) C s[ q 1( t ), q2 ( t ),..., q ( t)] (2-5) T N where s[q 1 (t), q 2 (t),..., q N (t)] s the nstantaneous transmsson rate of all guaranteed servces, whch s a functon of numers of calls n progress. Equaton (2-5) mples the nstantaneous transmsson rate of est-effort servce can not exceed the total andwdth left after transmttng all guaranteed servces. 12

13 If one accepts the assumptons that: 1) accepted cells consttute a Posson random process; 2) the nstantaneous transmsson rate depends on the andwdth left y guaranteed servces, whch s also random; and 3) the uffer sze s lmted, there s a posslty that even accepted cells (.e. cells wth wllngness to pay hgher than the cut-off prce) can e dropped ecause the uffer can ecome temporarly full. Defne υ(t, t) as the numer of cells actually admtted nto the uffer durng the nterval [t,t+ t); then the nstantaneous admsson rate can e defned as: υ t t ω ( t) lm (, = ) t > 0 t (2-6) ω (t) s a random varale and we assume ts expected value s ϖ ( t), then ϖ ( t) λ [ p ( t), t] (2-7) Defne q (t) as the numer of cells n the uffer at tme t, then: dq ( t ) dt = ϖ ( t) s ( t) (2-8) and s ( t) q ( t ) B (2-9) s 3. The Optmal Prcng Polcy In ths secton, we wll dscuss the proft-maxmzng prcng polcy for network operators. We formulate an optmal control model to derve the prcng polcy, and dscuss how to solve ths model through a 3-stage procedure. 3.1 The Optmal Prcng Model Assume a network operator wants to maxmze total proft over a perod composed of multple dentcal usness cycles (such as days). The cycle length s T. Her ratonal ehavor would e to choose a prce schedule for each type of guaranteed servce p (t), and 13

14 est-effort servce, p (t), and the amount of andwdth C T to maxmze the followng ojectve: T N ~ [ ( ), ] { ( 1 β ) λ p t t p ( t) + ϖ ( t ) p} dt K( C T ) (3-1) = 1 r 0 under constrants: dq dt ~ = ( 1 β ) λ ( p, t ) rq, q 0 =1,N (3-2) A[ q ( t),..., q ; ~ β ( t ),..., ~ β ( t)] C (3-3) 1 N 1 N T dq ( t ) = ω dt (t) - s (t) (3-4) when q ( t ) = Bs ω ( t) s ( t) (3-5) 0 q ( t) B (3-6) s s ( t) C s[ q 1( t ), q2 ( t ),..., q ( t)] (3-7) T N when q ( t ) = 0 ω ( t) s ( t) (3-8) q (0)=q 0, =1,N (3-9) Interpretatons of these constrants are the same as dscussed n secton 2, and defntons of varales can found n oth secton 2 and n the followng lst: Varales of guaranteed servces: N numer of dfferent servces; p (t) unt prce for servce, as a functon of call startng tme t; λ (p,t) call arrval rate of servce at tme t, when prce s p ; r call departure rate of servce ; q (t) numer of calls of servce n progress at tme t; q t expected value of q (t); s[q 1 (t), q 2 (t),..., q N (t)] total data rate of all guaranteed servces at tme t; s[ q1 ( t), q 2 ( t ),..., q N ( t)] average total data rate of all guaranteed servces at tme t; ~ β ( t ) desred lockng proalty for servce at tme t; Varales descrng est-effort servce: p (t) prce for admttng one cell nto the uffer at tme t; q (t) queue length of est-effort servce at tme t; 14

15 s (t) cell transmsson rate at tme t; λ [p (t),t] cell acceptng rate,.e. arrval rate of cells wth wllngness to pay hgher than p (t); ω (t) admsson rate of cells at tme t; ( t) expected value of ω [p (t), t]; ϖ Other varales: T C T K(C T ) B S duraton of usness cycle; total andwdth; amortzaton of capacty nvestment cost over one cycle; uffer sze. In (3-1), ( 1 β ) λ ( p, t) dt s the expected numer of calls of servce that wll e admtted durng the perod [t,t+dt). Multplyng ths numer y the unt prce, p (t), and 1 expected call duraton,, yelds the expected revenue from all calls of servce r admtted n that nterval. At tme t, the network also charges a prce for each cell of esteffort servce that enters the uffer, and ϖ ( t) dt s the expected numer of cells that wll enter the uffer at that tme. Thus ϖ ( t) p ( t ) dt s the expected revenue from est-effort servce at t. The total expected proft s calculated y summng up expected revenue from all servces, accumulated over all tme n [0,T], mnus the amortzed capacty cost. At ths pont, we assume zero dscount rate for smplcty. 3.2 The Soluton: A 3-stage Procedure Though t would e deal to solve the model defned n (3-1) - (3-8) drectly to get the analytcal form of the optmal prcng trajectory (p (t),p (t)) and the optmal amount of andwdth (C T ), t s mathematcally ntractale. Therefore, we construct a three-stage procedure to fnd a near-optmal soluton. At each stage, we wll make some smplfyng assumptons, or treat some varales as constants, and solve part of the prolem. The soluton otaned at one stage wll e used ether as an nput to the next stage or as a 15

16 feedack for modfyng assumptons made n the prevous stage. Ths process s terated untl prces stalze at a near-optmal level. Stage 1 Optmal Investment Decson optmal amount of capacty desred capacty Stage 2 Optmal Prcng for Guaranteed Servces load from guaranteed servces Stage 3 Spot Prcng for Best-effort Servce shadow prce of usng andwdth Fgure 3 The 3-stage Procedure The 3-stage procedure s defned as follows: at stage 1, we solve a long-term optmal nvestment prolem to fnd the optmal amount of total andwdth (C T ), as well as the desred lockng proalty, ~ β ( t ), whch we expect wll vary wth tme of day. Usng these values as nputs, we develop the optmal prcng polcy at the second stage. The result shows that the optmal prce for a servce should e a functon of the opportunty cost of provdng that servce. The opportunty cost s determned y oth the servce characterstcs and the shadow prces of reservng/usng network andwdth. We gve tral values to shadow prces and set up a prce schedule for each guaranteed servce accordngly. Based on these prce schedules, the traffc load from guaranteed servces can e determned. Under a gven traffc load from guaranteed servces, at the thrd stage, we 16

17 formulate a more precse model to descre the cell flow of est-effort servce at each moment. The spot prce for est-effort servce s then derved to maxmze the revenue from est-effort servce. From these spot prces, we can then decde the nstantaneous value of usng network andwdth. Ths nformaton s used as feedack to the second stage for adjustng the tral value of shadow prces we prevously calculated, so the prce schedule for guaranteed servces can e refned. The process s terated untl oth the prce schedule for guaranteed servces and the spot prce for est-effort servce stalze. In the next secton, we wll dscuss the mplementaton detals at each stage, and nterpret the economc mplcatons of our results. 4 Implementaton of the 3-stage Procedure 4.1 Stage 1: Optmal Investment At ths stage, we formulate and solve an optmzaton prolem to determne the optmal amount of total andwdth, C T, and the desred lockng proalty of each guaranteed servce at each tme, ~ β ( t ), =1,N. The formulaton of the prolem s as follows: Dvde [0,T] nto M tme ntervals, each lastng w m, (m=1,m). Take the average λ( τ ) dτ [ m m arrval rate λm = 1, ) as the arrval rate for all tme durng that nterval. λ w m s m determned y prce p m. We also assume that calls admtted durng the nterval [m-2,m-1] wll have no nfluence on traffc load wthn the nterval [m-1,m]. β m s the lockng proalty durng the nterval [m-1,m], whch s a functon of network loads wthn that nterval. At ths stage we gnore lockng due to fnte uffer space for est-effort traffc. Then the expected cell acceptance rate equals the expected cell admsson rate,.e. λ m (p m )=ϖ m. In other words, all cells wth wllngness to pay hgher than the cut-off 17

18 prce are assumed to e ale to enter the uffer. To keep the queue length n the uffer reasonaly short, we assume the expected cell admsson rate equals the expected cell transmsson rate,.e. ϖ m = s m. Consequently, λ m p m s m ( ) =. The network operator controls p m, p m, and C T to maxmze total proft,.e. M max w [ pm, pm, CT m= 1 m N = 1 ( 1 β ) λ ( p ) p r m m m m + pmλ m ( pm )] K( CT ) (4-1) s.t. q m = ( 1 βm ) λm ( pm) r (4-2) r β m = A q m = 1 N C T (,,, ), (4-3) where v β = ( β,..., β ) m 1m Nm s[( 1 β ) q, = 1, N ] + λ C, m=1, M (4-4) m m m T Ths s an optmzaton prolem wth (N+1)M+1 controllng varales. It can e solved ether y non-lner optmzaton technques or generc algorthms such as smulated annealng. The resultng C T and β m wll e consdered as optmal values for the total amount of capacty and for lockng proalty n each perod. The soluton we have otaned so far s not truly optmal ecause we have made several smplfcatons. One smplfcaton s that we assume the traffc load n any perod has no nfluence on the traffc load n succeedng perods. We have also gnored the fact that the arrval rate may change contnuously over tme wthn each perod y usng a sngle value λ as the arrval rate for all tme n a perod [m-1,m). Both smplfcatons wll cause naccuracy n our results. Interestngly, the effects of these two smplfcatons depend on how we dvde [0,T) nto dfferent ntervals. If we dvde [0,T) nto longer ntervals,.e. w m s larger, the effect of not consderng the relatonshp etween traffc load n dfferent perods wll e smaller and the effect of gnorng the change of arrval rate wthn a perod s more serous. If we choose a smaller w m, the effects wll go n the opposte drecton. 18

19 Therefore, w m should e chosen to mnmze the total negatve effect of these two smplfcatons Stage 2: Optmal Prcng We now allow the arrval rate to change contnuously over tme, consder the dependency of traffc load at dfferent tmes, and derve the optmal prcng polcy at ths stage. We wll stll keep the assumpton that for est-effort servce, cell admsson rate equals cell transmsson rate at all tmes, and gnore lockng of est-effort traffc. As a result, λ (p,t), the arrval rate of cells for whch the wllngness to pay s aove the cut-off prce p (t) at tme t, s used oth as the average rate of cell admsson nto the uffer and the average rate of cell transmsson out of the uffer at tme t for est-effort servce n the prolem formulaton. Gven the amount of andwdth (C T ) and optmal lockng proalty ( ~ β ( t ), =1,N) calculated at stage 1, we can smplfy the optmal prcng model defned n (3-1) - (3-9) as follows: T N ~ p maxmze p t p t t p t p t p dt ( ), ( ) { [ 1 β ( )] λ (, ) + λ (, ) } (4-5) 0 = 1 r dq ~ suject to: = [ 1 β ( t )] λ ( p, t ) rq ( t ) =1,N, (4-6) dt A[ q ( t),..., q ; ~ β ( t),..., ~ β ( t)] C (4-7) 1 N 1 N T λ ( p, t) + s[ q ( t),..., q N ( t)] C T 1 (4-8) q (0)=q 0, =1,N (4-9) 2 It s preferale to choose a larger w m f call arrval rate s stale over tme, and call duraton s long, and a smaller w m f arrval rate s sporadc and call duraton s short. 19

20 We assume that the optmal soluton exsts for ths prcng model. The optmal soluton to equaton (4-5) through (4-9) must oey the followng proposton, whch yelds the optmal prcng polcy: Proposton: The optmal prcng polcy: Suppose p * (t), p * (t) are the optmal solutons to the prcng model defned n (4-5)-(4-9), then: (1) p * (t) = * ε ( p, t ) h t ε ( p, t ) * ( ) * 1+ f l 2 (t) > 0 and h (t) > 0, =1,N or (2) p * (t) = * ε ( p, t ) h t ε ( p, t ) * ( ) * 1+ and p * (t) = * ε ( p, t) l t ε ( p, t) * ( ) * 2 1+ (4-10) and p * (t) = p 0 (t) (4-11) f l 2 (t) = 0 and h (t) > 0, =1,N or (3) p * (t) = p 0 (t) and p * (t) = p 0 (t) (4-12) f h (t) = 0, =1,N where: p 0 (t)maxmzes p (t)λ (p,t), p 0 (t)maxmzes p (t)λ (p,t), * λ p λ * p * ε ( p, t ) = *, ε ( p, t ) = * (4-13) p λ p λ T A h t q l s q re r ( t ) d ( ) = [ τ 1 ( τ ) + τ =1,N (4-14) t l 1 (t) s the Lagrangan multpler of constrant (4-7), l 2 (t) s the Lagrangan multpler of constrant (4-8). In elow, we dscuss the economc mplcatons of ths polcy. How to decde the optmal prcng schedule for guaranteed servces ased on the polcy s dscussed n Economc mplcatons The prcng polcy shown n (4-10) s desgned for stuatons n whch the network capacty s tghtly constraned. If the network operator prces servces wthout consderng capacty constrants, for guaranteed servces, ether the network can not meet performance requrements, or some servces wll experence a lockng rate eyond the desgned value. 20

21 For est-effort servce, f the numer of cells admtted exceeds the numer of cells transmtted, the queue would grow wthout ound. Our proposton shows that under these scenaros, the network operator's optmal strategy s to attach an opportunty cost to each servce (h (t) for guaranteed servce, and l 2 (t) for est effort servce), and prce a network servce n the same way as prcng a tangle product, except that the margnal producton cost should e replaced y opportunty costs. We now explan the ratonale for usng h (t) as the opportunty cost for provdng guaranteed servce, and l 2 (t) as the opportunty cost for provdng est-effort servce, startng y explanng the Lagrangan multplers of the two capacty constrants. The economc mplcaton of the Lagrangan multpler of a resource constrant s the maxmum value that can e derved from havng one more unt of the constraned resource,.e. the shadow prce of consumng one unt of that resource. In our case, l 1 (t), l 2 (t) are shadow prces of reservng and usng one unt of andwdth, respectvely. Snce we measure the andwdth n terms of the numer of cells that can e sent per unt of tme, at tme t, when one cell of est-effort servce s sent, one unt of andwdth s consumed. Therefore, the unt opportunty cost for est-effort servce at tme t s just the shadow prce of usng one unt of andwdth at that tme,.e. l 2 (t). To meet performance requrements for guaranteed servces, the network needs to reserve some capacty each tme a call s admtted. At each moment, part or all of reserved andwdth wll actually e used y guaranteed servces. Consequently, the opportunty cost should nclude two components: the opportunty cost of reservng the andwdth, and the opportunty cost of usng t. In our formulaton, at tme t, the former equals the shadow prce for reservng one unt of andwdth, l 1 (t), tmes the margnal ncrease of the amount of reserved andwdth for admttng one more call, A, and the latter equals the shadow prce for usng one unt of andwdth, l 2 (t), tmes the margnal ncrease of andwdth usage q 21

22 whch results from admttng one more call, s. The total opportunty cost for a call s q thus the sum of these two components, accumulated over all tme. Snce the servce duraton s an exponentally-dstruted random varale, the total cost, h (t), s estmated y takng mathematcal expectaton, usng the dstruton functon of the call duraton ( r e r t ). Equaton (4-10) s approprate when the numer of guaranteed calls that can e admtted whle meetng performance requrements s stll lmted, ut there s more than enough capacty to carry the cells from all guaranteed calls that are admtted, as well as all of the est-effort traffc that the network wants to carry. Ths stuaton mght occur, for example, f the guaranteed calls are extremely ursty, or ther performance requrements are extremely strct..e. λ ( p, t) + s[ q1( t ),..., q N ( t)] < C T As a result, at tme t, the shadow prce of usng the andwdth, l 2 (t), equals 0, and the optmal prcng polcy should follow (4-10),.e. the network operator should set prce to maxmze total revenue from est-effort servce wthout consderng the constrant on data rate. Equaton (4-11) specfes the prcng polcy for the stuaton when there s an excessve amount of andwdth. In ths case, even f the network operator maxmzes revenue wthout consderng capacty constrants, she can stll meet performance ojectves for all servces, keep lockng proalty elow the desred level, and have more transmsson capacty for est-effort servce than what s needed. As a result, oth the opportunty costs for guaranteed servces and the opportunty cost for est-effort servce equal zero (.e. h (t) =0, l 2 (t) = 0). Ths only happens when capacty s not constraned for oth reservaton and use for all tme, or n other words, the capacty s over provsoned. Snce we have assumed that the capacty, C T, s set at the optmal level n stage 1, ths cannot occur. 22

23 4.2.2 The optmal prcng schedule for guaranteed servces As shown n (4-10), (4-11), the optmal prce for guaranteed servces depends on A the ε, whch s the demand elastcty, whch reflects traffc characterstc and performance requrements, as well as l 1 (t), l 2 (t), the shadow prces for reservng and usng andwdth, respectvely,.e. : ε p ( t) = h ( t) 1 + ε T A where h t q l s q l r e r ( τ t ) d ( ) = [ 1 ( τ ) + 2 ( τ )] τ t q To fnd p (t), values of l 1 (t), l 2 (t) need to e determned. At ths pont, we assume the values of l 2 (t) have een estmated and gven as l $ 2 ( t). (Ths pror estmaton wll e modfed y the feedack from stage 3). We then set l 1 (t) to the tral value l $ 1 ( t ),and construct the followng procedure to fnd the optmal value for p (t), as well as to modfy the estmate of l 1 (t) λ^ ( p^ 1) Calculate the optmal prcng schedule for guaranteed servces y : T $ A ( ) [ $ ( ) $ ( )] ( ) h t q l s q l r e r t d ε τ = 1 τ + 2 τ τ and p $ ( t ) $ = h ( t) 1 + ε t 2) The call arrval rate of guaranteed servces at tme t s then $ [ p$( t ), t ]. Gven (t),t) and the total amount of andwdth, C T, the expected numer of calls n progress, q$( t), and the lockng proalty, β $ ( t ), can e determned. 3) If l 1 (t) s underestmated, p$ ( t ) wll e lower than ts optmal value, so call arrvals wll e hgher than the optmal level, whch leads to the stuaton that lockng proalty s hgher than the desred level,.e. $ ~ β ( t) > β ( t ) at some t. If l 1 (t) s overestmated, p^ (t) wll e lower than ts optmal value and $ ~ ( ) β t < β ( t ). 4) Increase or decrease l $ 1 ( t ) y l 1, dependng on whether t s over or under estmated. Go to 1) to calculate p (t). The process s terated untl $ ~ β ( t) = β ( t ) or s wthn a tolerale error and. λ 23

24 The prce schedule for guaranteed servces s ased on the gven estmates of l 2 (t),.e., the shadow prce for usng the andwdth. Ths estmate was gven artrarly at the egnnng, and needs to e modfed y usng feedack from the thrd stage. 4.3 Spot Prcng Gven the prces for guaranteed servces otaned at the second stage, the dstruton of avalale capacty for est-effort servce as a functon of tme can e determned as CT s[ q1 ( t),..., q N ( t)]. At each nstant, the network operator wll set p (t), the spot prce for admttng cells of est-effort servce nto the uffer to maxmze: T t p ( t) * ω ( t) dt (4-15) under constrants: dq dt = ω ( t) s ( t ) (4-16) s ( t) C s[ q 1 ( t ),..., q ( t )] (4-17) T N 0 q ( t) B (4-18) s when q ( t ) = Bs ω ( t) s ( t) (4-19) when q t ( t) s ( t) (4-20) Gven ω (t), s (t) are random varales wth complcated dstrutons, the prolem n (4-15)-(4-20) can not e solved drectly. However, through smulaton, we can desgn heurstc rules that ndcate how the spot prce, p (t), should e set ased on current uffer occupancy and the expected dstruton of wllngness to pay of cells arrvng n the future. As soon as the spot prce, p (t), s determned, a new estmate of l 2 (t) can e constructed. Ths can e done y usng the proposton aove that defnes the optmal * ε prcng polcy. Equaton (4-10),.e. p * ( p, t) (t) = l t ε ( p, t) * ( ) * 2 apples when the 1+ andwdth s fully used, and Equaton (4-11),.e. l 2 (t)=0 apples otherwse. The new estmate can then e used as feedack to revse the optmal prcng schedule for guaranteed servces. 24

25 The optmal prcng polcy s reached y teratng the second and the thrd stages untl oth the prce schedule for guaranteed servces and the expected spot prce for esteffort servce stalze. 5. Conclusons and Future Work In ths chapter, we dscuss the optmal prcng polcy for Integrated-servce networks wth guaranteed qualty of servce ased on ATM technology. By formulatng the prcng decson as a constraned control prolem and developng a three stage procedure to solve that model, we fnd there s great smlarty etween the optmal prcng polcy for network servces and the optmal prcng polcy for conventonal products. We demonstrate that under capacty constrants, the servce provder should consder the opportunty cost ncurred y servng a customer. Ths opportunty cost should e used to determne the prce of a network servce n the same way as the margnal producton cost s used to determne the prce of a conventonal product. We derve the mathematcal expressons that calculate opportunty costs for dfferent servces offered y a sngle ntegrated-servces network, and explan the mplcatons of these expressons. Though our procedure s desgned for maxmzng the servce provder s proft, a smlar approach can as well e used to maxmze other ojectves, such as socal welfare. Note the prcng polcy developed n ths paper optmzes the proft for provdng ntegrated servces under the assumpton that the demand for each servce s ndependent of prces of any other servces. In future work, we wll relax that assumpton and consder the cross-elastcty effect among servces. Even n the asence of cross-elastcty effect, the prce of one servce can also affect the demand for another servce f the network adopts a three-part tarff prcng scheme, under whch users are not only charged for each servce ased on reservaton and usage, ut also pay a flat suscrpton fee (e.g. an access charge). In ths case, the network operator may maxmze proft y settng reservaton or usage 25

26 prces for each servce dfferent from the optmal values derved n ths chapter. As another example, n the presence of postve network externaltes, t can e optmal to prce access elow average cost, recoverng the alance from the ncreased demand for usage whch results from a larger network populaton. Our paper consders nether three-part tarff nor postve demand externaltes. The desgn of an optmal prcng schedule wth the consderaton of these factors s an nterestng ssue that remans to e explored. 26

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