A Traffic-pricing Model for the Packet-switching Network. with Prioritized Round-robin Queueing 1

Transcription

1 A Traffic-pricing Mode for te Pacet-switcing Networ wit Prioritized Round-robin Queueing 1 Zangxi Lin *, Dae O. Sta + and Andrew B. Winston ** Extended Abstract Dynaic networ traffic pricing as been considered a proising econoic approac to soving networ traffic congestion probe tat as been diinising te benefit of te Internet fro e-businesses. Tis paper presents a traffic-pricing ode for te pacet-switcing networ tat uses prioritized round-robin bandwidt aocation. We are exporing te scee in wic ocaized pricing can optiize goba networ perforance. Tis researc is aied at industriaappication deands and intended to provide bot te teoretica derivation and te practica scee to te ipeentation of a traffic pricing syste. We foow te etodoogy of stocastic genera euiibriu used in te GSW ode (Gupta, Sta and Winston, 1996, 1997) and focus te discussion on te service wefare optiization probe in eastic traffic contro were te service deay is counted as te ain constraint. One of te iportant advantages of pacet switcing networs is te tota deay a data fow, or aternativey a job, encounters fro one ost to anoter is ess tan te suation of te waiting and servicing tie in a nodes aong its pat. Tis feature aows pacet switcing networs to provide better bandwidt efficiency tan oter types of networ, suc as circuitswitced networs. However, te pipeine effect in a pacet switcing networ copicates 1 Tis researc as been sponsored by Inte Corp., Lucent Tecnoogies Inc., and Mannesan Ag. * Departent of ISQS, Texas Tec University, TX 79409, USA, zin@ba.ttu.edu + Departent of Econoics, Te University of Texas at Austin, TX 7871, USA, sta@eco.utexas.edu ** Departent of MSIS, Te University of Texas at Austin, TX 7871, USA, abw@uts.cc.utexas.edu 1

2 traffic pricing processes because te tota job trougput tie is not a suation of trougput ties of cannes in a route. A pacet-switcing networ is defined as a set of distributed data transission services enabing utipe routes siutaneousy servicing different data fows. Structuray, a pacetswitcing networ consists of a set of nodes and te cannes between te nodes. A route forwarding a data fow fro a sender to a receiver can be odeed as a series of ueue service stations, eac of wic is ade up of a ueue in a networ node and a bandwidt server by an outbound canne fro te node. Assue tat tere is a pacet switcing networ consisting of a set of G {g} nodes. A canne c s C is between two nodes, wit service capacity featured as bandwidt B s, s S. A route R j transits data fow j fro a sender to a receiver (Figure (a)). R j is a set of nodes and cannes. It can be denoted as R j {G j, C j }, were G j {g j 1, g j,, g j n,, g j N} G, C j {c j 1, c j,, c j n,, c j N} C, and N indicates te nuber of networ nodes in te route and n 1,,..., N. R j can be odeed as a series of ueueing systes. An individua ueueing syste in te series is ade up of a ueue in a networ node and te bandwidt service by an outbound canne fro te node (Figure (b)). Wen a canne c j n is busy, a ueue fors at node g j n. Sender g j 1 g j g j 3 g j N c j 1 c j c j 3 c j N-1 c j N (a) Receiver Sender g j 1 c j 1 c j c j 3 c j N g j g j 3 g j N (b) Receiver Figure: A route in a pacet switcing networ is a series of ueueing systes

3 We prove tat te tota trougput tie a data fow is transitted troug a route can be expressed as te tie spent at te bottenec canne pus a reativey trivia aount of deay on oter nodes of te pat. Specificay, te residua transission tie is te tie for a singe data segent to go troug a non-bottenec nodes. Tis is because tere is aost a tota overay in transission tie of te bottenec canne and te oter cannes in te route. Since te size of a data segent is independent of te size of te data fow, te probabiity distribution of te extra trougput tie on te reaining section of te pat for a data fow is independent of te size of te data fow. Te above "pipeine" effect of pacet-switcing networs adds copexity to traffic-pricing odeing. To te pacet-switcing networ wit prioritized transission services, we adopt a preeptive prioritization, i.e., a data fow joining te ueue of te oentariy igest priority obtains te bandwidt iediatey regardess of te service status of te data fows at ower priority casses. Data fows at te sae priority sare te bandwidt euay wen te priority becoes te igest one aong oters. A priced pacet-switcing networ is an econoy wit data transission services as te product and networ users as te consuer. Te prioritization resuts in differentiated product uaities wit different eves of prices. Te foowing econoic conditions are assued for estabising a traffic pricing ecanis: Data fows in te networ, referred to as jobs, are inerent wit vaues perceived by users as utiities; A job's net vaue to te user depends on two factors: te service price te user pays and te deay cost proportiona to te trougput tie; 3

4 If jobs arrive at a canne oentariy in a rate tat exceeds te canne's bandwidt tey are ueued; Prices are differentiay set for te bandwidt of cannes in accordance wit service priorities and te traffic status of eac priority; Te inforation of bot te bandwidt price and te expected trougput tie are periodicay disseinated to users; Users are rationa, i.e., tey wi coose priority eves accordingy for teir jobs to optiize te net benefit of te subission, and tey subit teir jobs ony if expected net gains fro job subissions are nonnegative. Tere exist natura deands, i.e., axiu arriva rates of different types of jobs if tere is no cost, to te networ transission service. However, te deay cost and te price paid for transission services resut in restricted job subissions. Te objective of te traffic pricing is to dynaicay coose a set of prices for te prioritized services at eac networ canne to suppress te traffic of ower vaued jobs to a certain eve. Tus, te subitted jobs resut in an aggregate rate tat turns out statisticay te best service wefare rate. Tis issue can be foraized as a service-wefare rate axiization probe wit benefit contributions fro job subissions troug various networ routes ade up of nodes and cannes. Soving te probe, we obtain two benefit-axiization unit prices for jobs of size tat are subitted to priority- service of canne s: r s * K s s y if canne s is te bottenec of te route, and r s ' 1 K s s y if canne s is not te bottenec of te route, 4

5 were J is te set of job types, and K is te set of priority casses, is te subission rate weigted deay cost coefficient for type- jobs, y is te arriva rate of type- jobs at priority- service, Ω s is te expected trougput tie of type- jobs subitted to priority- service at canne s, and s is te arriva rate of jobs of size subitted to priority- service at canne s. Hence te necessary condition tat te user subits a job can be expressed as: V ij ij T j j (r jh * + ' rjs j s R, s H ), (*) were V ij is te gross vaue of type-j job subitted by user i, ij is te deay cost coefficient for te job, T j is te expect tota trougput tie of te route for type-j jobs subitted to te priority- service, j is job size, r jh * is te bottenec price for te priority- service of canne H, and r js * is te non-bottenec price for te priority- service of canne s. Canne H and oter canne s are in te route R j transporting te type-j job. Since te bottenec price is uc iger tan prices for non-bottenec cannes, te criterion (*) can be approxiatey expressed as V ij ij T j j r jh * (**) In tis way, pricing for a route usage becoes for te bottenec canne usage. Tis is obvious because te bottenec canne is te doinant factor affecting te tota trougput tie of te route. If tere is ore tan one service cass tat eet (**), te user wi subit te job to a best service cass * sup { V ij ij T j - j r * jh } (***) Tat is, te optiu service cass axiizes te net vaue of te job. 5

6 Te trougput tie of type-j jobs subitted to priority- service of canne s is Ω js f(l s, j, B s ) depending on te ueueing agorit, were L s is te average nuber of jobs in priority-, and B s is te average bandwidt of canne s to priority ueue. We can obtain L s 1 s s, were s is te bandwidt utiization ratio of priority- service of canne s, and B s B s (1 - < ), were B s is te bandwidt of canne s. Te appendix provides te derivation of te anaytica for of te bandwidt-renta price for a singe node situation, wic can been easiy generaized to uti-node route situations. We appied te above traffic-pricing ode to virtua private networs (VPNs) (Kosiur, 1998) to expore te ipeentation issues for a networ traffic pricing syste witout prioritization (Lin, Ow, Sta, and Winston, 1999 and 000). VPNs are referred to as private networs buit upon a pubic networ, specificay te Internet, to provide secured transission services. VPN traffic pricing is bot feasibe and necessary because it is controabe to te organization operating te VPN, wic is a socia wefare axiizer. A VPN route consists of two cannes between two VPN gateways and te Internet tunne, and a virtua circuit in te Internet, wic is ade up of unnown nuber of cannes between Internet routers. Te Internet tunne is not priced because it is assued not te bottenec in VPN data transissions and is not controabe to te organization. Tus, te organization can buid up a traffic pricing syste for te VPN to optiize te bandwidt utiization of its Internet connections. Te VPN job subission criterion is custoized fro (*) as: V ij ij T j j (r jh * + r jl ' ), 6

7 were r * jh is te unit price for te priority- service of te VPN-Internet bottenec canne and r ' jl is te unit price for te priority- service of anoter VPN-Internet canne tat is not a bottenec. We furter propose a transaction-eve pricing arcitecture for te VPN traffic-pricing syste to cope wit issues suc as digita contracting, VPN traffic pricing efficiency, ogistic syste construction, integration of te traffic-pricing syste wit existing traffic contro tecniues, and user acceptabiity. Wit transaction eve ipeentation, te VPN aocates bandwidt reuests on te per-transaction basis in regard to te appication needs and priorities. We cai tat tis process provides te possibiity to incorporate te pricing ecanis witin enterprise resource panning appications. We deveoped a prototype syste naed VPN Traffic-Pricing Experient Syste (VTPES), wic is buit on a test networ patfor in te Center for Researc in Eectronic Coerce (CREC). VTPES is a scaabe, distributed syste operated wit rea-tie data fows. Its dua-ueue structure aows us to copare different experient scees using identica data traffic patterns. Te experients ave sowed te effectiveness of te pricing foruas derived fro te traffic-pricing ode. VTPES as been used for a series of diversified experients, suc as approxiate job subission decision-aing and networ traffic pricing wit uan subjects. Currenty, VTPES is being enanced wit te prioritized round-robin sceduing scee. Te outcoes are to be furter reported. 7

8 Appendix: Te Optiu Pricing Forua for A Singe Bandwidt Service wit Prioritized Round-robin Queueing (PRR) Te prioritization of PRR is assued preeptive. Wen tere is no job in iger priority ueues te round-robin agorit is appied to eac priority cass. In tis case, te avaiabe bandwidt to a priority cass is independent to te status of te ower priority casses, but reevant to iger priority casses. Denote te service bandwidt as B and te set of priority casses as K wit a saer nuber for a iger priority. Ten te avaiabe bandwidt to cass-1 jobs is B and te avaiabe bandwidt to reaining casses is B (1-1 ), were B is a notation for te bandwidt utiization ratio of cass- jobs. Since cass- jobs aways get service prior to oter jobs at ower casses, teir avaiabe bandwidt is B (1-1 ). In genera, te avaiabe bandwidt of cass- jobs is B B (1 - < ). According to Keinroc (1975), te average nuber of jobs in a non-prioritized round- robin ueue is:. Tis is appicabe to te nuber of cass-1 jobs in te case of prioritized 1 round-robin ueueing. Denote ' B B B. In te siiar way, te nuber of jobs in a prioritized round-robin ueue is: L ' ' 1 /(1 1 /(1 < < ) ) 1, for 1,,, K 8

9 Te expected trougput tie of type-j job subitted to cass- priority service wit prioritized round-robin sceduing becoes: τ j Ω j (L (, j), j, B) L j / B j, were { } and j B( 1 )(1 ) < { }, wit 1,,, J and 1,,, K. Te first order condition turns out: j j L [(1 B (1 ) ) ], if < j jl B, if j 0, if >. Te price for type-j job subitted to cass- is r j I K j x j J > j B L [(1 (1 ) ) ] j L + B j L B (1 ) + (1 (1 ) ) J > Unit price 9

10 L r B (1 ) + (1 (1 ) ) J > L B (1 (1 ) ) were K, is te fow rate weigted deay cost coefficient. References: Bae, S., D. Bac, M. Carson, E. Davies, Z. Wang, and W. Weiss, An Arcitecture for Differentiated Services, Internet RFC475, Deceber 1998, ftp://ftp.isi.edu/innotes/rfc475.txt. Car, D. and J. Wrocawsi, "An Approac to Service Aocation in te Internet," INTERNET- DRAFT, Juy, 1997, ttp://diffserv.cs.it.edu/drafts/draft-car-diff-svc-aoc-00.txt Fisburn, Peter C., and Andrew M. Odyzo, Dynaic Beavior of Differentia Pricing and Quaity of Service Options for te Internet, DIMACS Tecnica Report 99-1, February 1999, ttp:// Fouia, Noria et a, User Beavior for a Pricing Scee in a Muti-provider Scenario, MIT/Tufts Internet Service Quaity Econoics Worsop, Deceber -3, Gibbens, R. and F. Keey, "Resource pricing and te evoution of congestion contro," Autoatica, 1999, ttp:// Gupta, A., Sta, D. and Winston, A., A Stocastic Euiibriu Mode of Internet Pricing, Journa of Econoic Dynaics and Contro, 1, 1997(a), pp Gupta, A., Sta, D. and Winston, A., Econoic Issues in Eectronic Coerce, Capter 8 in Readings in Eectronic Coerce, Kaaota, R., and A.B. Winston eds., Addison- Wesey, Massacusetts, 1997(b). Gupta, A., Sta, D. and Winston, A., A Priority Pricing Approac to Manage Muti-Service Cass Networs in Rea-Tie, L. McKnigt and J. P. Baiey eds., Internet Econoics, MIT Press, Cabridge, Gupta, Ao, Dae O. Sta and Andrew B. Winston, "Te Econoics of Networ Manageent," Counications of te ACM, (4:9), Septeber 1999, pp

11 Jacobson, Van and Micae J. Kares, "Congestion Avoidance and Contro," SIGCOMM 88, ACM, Kayanaroan, S., T. Ravicandran and Randy Norsworty, Dynaic Capacity Contracting: A Fraewor for Pricing te Differentiated Services Internet, May 1998, ttp:// Keinroc, Leonard, Queueing Systes, Vo. I & II, Jon Wiey & Sons, Kosiur, Dave, "Buiding and Managing Virtua Private Netowrs," Jon Wiey & Sons, Lin, Zangxi, Peng Si Ow, Dae O. Sta, and Andrew B. Winston, "Exporing Traffic Pricing for te Virtua Private Networ," WITS 99, Deceber 11-1, 1999, Carotte, Nort Caroina. Lin, Zangxi, Peng Si Ow, Dae O. Sta, and Andrew B. Winston, "Exporing Traffic Pricing for te Virtua Private Networ," under review by Inforation Tecnoogy and Manageent, May, 000. MacKie-Mason, J., and H. Varian, Pricing te Internet. In Pubic Access to te Internet, ed. B. Kain and J. Keer. Engewood Ciffs, NJ:Prentice-Ha, Odyzo, Andrew M., A Modest Proposa for Preventing Internet Congestion, DIMACS Tecnica Report 97-68, October 1997, ttp:// Sener, Scott, David Car, Debora Estrin and Sai Herzog, Pricing in Coputer Networs: Resaping te Researc Agenda, Teecounication Poicy, (0:3), pp183-01, Stevens, W., TCP Sow Start, Congestion Avoidance, Fast Retransit, and Fast Recovery Agorits, Internet RFC001, January 1997, ttp://info.internet.isi.edu/innotes/rfc/fies/rfc001.txt. Varian, Ha R., "Estiating te Deand for Bandwidt," MIT/Tufts Internet Service Quaity Econoics Worsop, Deceber -3,