A Theoretical Analysis on Dynamic Marginal Cost Pricing. Masao Kuwahara Institute of Industrial Science, University of Tokyo.

Transcription

1 A Theoreical Analyi on Dynamic Marginal Co Pricing Maao Kuahara Iniue of Indurial Science, Univeriy of Tokyo Abrac Thi udy exend he aic marginal co analyi o he dynamic one o ha he dynamic boleneck phenomena are properly included in i upply curve. Convenionally in he economic field, he equilibrium analyi of demand-upply ha been udied and he marginal co pricing raegy a propoed o a o maximize conumer urplu. Hoever, he aic analyi doe no ell conider he ime-dependen queue evoluion in he upply funcion. On he oher hand, in raffic engineering field, alhough he ime-dependen queueing analyi ha been exenively udied, he demand funcion ha no been deal ih in mo analyi. Thi udy aemp o exend he demand-upply analyi o incorporae dynamic queueing phenomena. For he problem ihou deparure ime choice, i i hon ha dynamic marginal co i equivalen o he duraion of congeion. For he problem ih deparure ime choice, he demand-upply analyi i combined ih he exiing heory of deparure ime choice ih ime conrain a deinaion.

2 . INTRODUCTION Thi udy exend he curren aic marginal co analyi o he dynamic one o ha he dynamic boleneck phenomena are properly included in i upply curve. Convenionally in he economic field, he equilibrium analyi of demand-upply ha been udied uing demand a ell a upply funcion and he marginal co pricing raegy a propoed o a o maximize conumer urplu. Hoever, he aic analyi doe no ell conider he imedependen queue evoluion in he upply funcion. On he oher hand, in raffic engineering field, alhough he ime-dependen queueing analyi ha been exenively udied given cerain amoun of demand, he demand funcion ha no been explicily deal ih in mo analyi. Thu, hi udy aemp o exend he equilibrium demand-upply analyi o a o incorporae dynamic queueing phenomena. 2. REVIEW OF STATIC MARGINAL COST ANALYSIS Le u quickly revie he derivaion of he marginal co pricing in he aic frameork. The demand and upply funcion are fir defined a follo: ρ(p) = demand generaed a rip co p () S(x) = rip co hen demand i x (2) Thee funcion are decribed in Fig. and he inerecion of o curve i he equilibrium poin a demand x*. The conumer urplu, he haded are in Fig., i hen rien a F = x ρ ( y) dy x S( x). (3) Co p MC( x ) = d{ x S( x ) dx Taking derivaive ih repec o demand x yield ρ(p) d ρ ( x ) = { x S( x ) = MC( x ). (4) dx S(x) Thi implie ha he righ hand ide of he marginal co MC(x), he derivaive of oal co regarding x, mu be equal o he invere of demand funcion ρ - (x). x x* Demand x 3. DYNAMIC BOTTLENECK PHENOMENA Fig. Saic Demand-Supply Equilibrium Queue evoluion a a boleneck i baically ime-dependen phenomena, hich canno be decribed by he above aic analyi. Le u conider a ingle boleneck. If raffic demand rae exceeding he boleneck capaciy µ ener, a queue ould gro a hon in Fig.2. The arrival and deparure rae λ() and µ(), and aiing ime a he boleneck () are defined: λ() = arrival rae a he boleneck a ime, µ() = deparure rae from he boleneck a ime, () = aiing ime of a uer arriving a he boleneck a ime. And he cumulaive arrival and deparure are alo defined a follo: 2

3 A() = D() = ( ) d λ (5) ( ) d µ (6) Noe ha in Fig.2, he cumulaive curve are dran baed on he poin queue concep. Under he FIFO queue dicipline, he horizonal diance beeen A() and D() ho he aiing ime () = D - (A()) - a hon in he figure. From hi figure, he aiing ime a ime, (), i clearly depend upon he hiory of λ() unil ime. Waiing ime (), hich correpond o rip co, herefore canno be imply a funcion of he demand rae a ha ime, bu () may change if λ( ), < varie. cumulaive rip lope=λ() A( ) D( ) () ( -)-() µ lope= µ() ime Fig.2 Cumulaive Arrival and Deparure a a Boleneck 4. EXTENSION TO DYNAMIC MARGINAL COST ANALYSIS 4.. Dynamic Marginal Co Le conider he dynamic marginal co by adding he ime axi. A hon in Fig.3, co p() i defined a he co of a uer arriving a he boleneck a ime and rien a a funcion of (): p()=f {(). On he oher hand, he demand during ~+d i rien a x()=λ()d. Similar o he aic analyi, he demand funcion i defined: ρ(p(),) = x(), (7) ρ - (x(),) = p(). (8) The demand funcion ρ(p(),) i aumed given for all, and he upper figure of Fig.3 ho ha ρ(p(),) are changing over ime. Thi imedependen demand funcion i inroduced o reflec he fac ha he generaed demand rae may change over ime, for inance peak and offpeak, even if he ame amoun of co i impoed o uer. A ime, aiing ime () i evaluaed from he cumulaive curve a in he loer figure. Then, he co a ime, p()=f {(), deermine he generaed demand of ρ(f {(),)=λ()d during ~+d. The demand rae λ() mu be equal o he lope of A() a ime. Thi equilibrium i here called a demand-upply equilibrium. Employing he dicree ime incremen of d, he conumer urplu F i rien a Cum. Trip lope=λ() -d Co p() f {() A( ) () D( ) λ()d ime ime ρ(p(),) ρ(,) Demand x() Fig.3 Time-Dependen Demand Plane and Queue Evoluion 3

4 F = F F λ( ) d = ρ ( x, ) dx f 2 { ( ) λ( ) d. (9) The opimaliy condiion o maximize F i obained by aking derivaive ih repec o x()=λ()d. The derivaive of he fir erm F i F λ( ) d = ρ = ρ ( λ( ) d, ) + ( λ( ) d, ) u= + λ( u) ρ λ( ) ( λ( u) d, u) Since λ(u), u> can be conrolled independen of λ() a ime, he econd erm on he righ hand ide, λ( u ) / λ( ), become zero. On he oher hand, he derivaive of F 2 mean MC() becaue i i he derivaive of he oal co: f { ( u ) λ( u )d F2 MC( ) = = λ( )d λ( )d = u= = f λ( u )d f λ( )d { ( u ) + u= { ( u ) + u= df { ( u ) d λ( u ) d( u ) µ df { ( u ) ( u ) λ( u ) d( u ) λ( ) Hoever, he above reul i applicable only hen he boleneck i buy, ()> or λ() µ. When he boleneck i idle ju like before, marginal co MC() i obviouly equal o f {, ince () canno be affeced by he demand change and conequenly he econd erm of he righ hand ide become zero. A a hole, he opimaliy condiion i ummarized a follo: df { ( u ) d f { ( u ) + > = = λ( u ), ( ) or λ( ) µ ρ ( λ( )d, ) MC( ) u= d( u ) µ (2) f {, oherie For a pecial cae, if he aiing co funcion i linear: f {=b, b >, (3) he marginal co i imply rien a b( MC( ) = ), if ( ) >, or λ( ) µ oherie, () (), (4) here - mean he duraion of he congeion afer ime becaue ime i he queue vanihing ime. Since uer co i f {(), he dynamic oll hile he boleneck i buy i equal o he fir erm of Eq.(): df { ( u ) d λ( u ) = b{ d( u ) µ ( ), for linear aiing co funcion. (5) 4

5 If you apply he marginal co pricing o uer in Fig.2, no oll hould be impoed before ime, becaue he boleneck i idle and MC()=f (). A ime, arrival rae λ() become equal o capaciy µ. If you impoe he oll deermined from Eq.(5), clearly he arrival rae ould be belo he capaciy; ha i, he dynamic oll i oo much. The be pricing i o impoe oll hich conrol he arrival rae ju equal o i capaciy. Tha i, from Eq.(2), MC() i diconinuou hen ()= a ell a λ()=µ uch a ime. Namely, hen he demand increae by one uni, MC() i equal o he upper value of Eq.(2); on he oher hand, hen i decreae, MC() jump o he loer value. Apparenly, a he idle boleneck due o he pricing doe no improve he conumer urplu, he be raegy for hi iuaion i o impoe he oll o a o conrol he arrival rae ju equal o i capaciy. Therefore, during ome ime afer ime, he arrival rae mu be conrolled a capaciy µ. Meanhile, he poenial demand ould become ufficien o ha demand larger han µ could be generaed even if oll of Eq.(5) i fully impoed An Example Le u conider a boleneck ih i capaciy of 2[veh/uni ime]. The demand funcion ρ - (x,), hich give demand x[veh] during a hor inerval of d, i aumed in he folloing linear form: ρ a ( x, ) = { x ρ(, ) [uni co], ρ(, ) here ρ(,) a in Fig.3 mean he maximum demand hen he uer co i zero and i cumulaive value i hon in he upper figure of Fig.4. Alo, he aiing co funcion i aumed linear: f {=b, b=. [uni co/uni ime]. Uing hee linear funcion, he cumulaive arrival curve ih he marginal co pricing can be eaily dran in he folloing manner. Suppoe ha e have evaluaed A() unil ime and ry o exend A() hereafer. Sep : Aume queue vanihing ime. Sep 2 : From A() unil ime, D() can be deermined unil ime +(). From A() Cumulaive Trip [veh] and D(), aiing ime of a uer arriving a ime, (), i evaluaed and hence he dynamic oll i alo deermined a b{( -)-(). Sep 3 : From he dynamic oll and (), demand generaed during +d i deermined from he given demand funcion: λ()d=ρ(b( -),). Then, he arrival curve i exended up o ime +d: A(+d) = A() + λ()d. Sep 4 : Updae ime a +d, and reurn o Sep 2 if ime +d i ill ihin he udy period. Sep 5 : From A() dran, find queue vanihing ime. If i equal o he aumed ime Co [uni co] oll Cum.ρ(,) () 2 2 Fig.4 Marginal Co Pricing a a Single Boleneck A( ) : no pricing A( ) : pricing D( ) --() ime [uni ime] oll +aiing co = MC()=b( -) oll={b( --()) aiing co =b() arrival ime [uni ime] 5

6 , end; oherie reurn o Sep. In Sep, mu be modified uing i characeriic ha i monoonically decreaing ih repec o aumed in Sep. Fig.4 ho he cumulaive curve and he ime-dependen oll. We noice ha queue aring ime doe no change before and afer pricing. Thi i due o no pricing before ime becaue of no ufficien demand. A explained above, from ime o 2, he dynamic oll i adjued o ha he generaed demand rae i ju equal o he capaciy. Afer ime 2, ince he poenial demand become larger and he amoun of b{( -)-() obained from Eq.(5) i fully impoed unil he queue vanihe a ime. A hon in he dahed line, he marginal co from Eq.(4) i linearly decreaing from 2 o : MC() = b( -). In hi example, he conumer urplu increae from 395 [veh*uni co] o 3663 [veh*uni co] by he pricing. 5. ANALYSIS WITH DEPARTURE TIME CHOICE Dicuion o far ha been limied ihin cae here a uer ha only one choice of heher he/he make a rip. Hoever, normally uer ould have oher choice a ell uch a mode, deparure ime, ec. In hi ecion, among hee choice, e ould like o add deparure ime choice ino he dynamic marginal co analyi. 5.. Deparure Time Choice Problem ih Time Conrain Mainly from 98, everal udie on deparure ime choice have been repored moly for rip ih deired arrival ime a deinaion (Vickrey(969), Hendrickon (98), and Kuahara (987)). The ypical applicaion a morning commue rip, in hich commuer have ime conrain of heir ork chedule. Trip of oher rip purpoe may more or le have ome ime conrain a deinaion; for inance in recreaional rip, raveler migh have approximae deired arrival ime a heir home. Thu, e ould like o conider he previou reearch o include deparure ime choice o he marginal co analyi. Le revie he deparure ime choice problem a a ingle boleneck ih i capaciy µ. Everyone i aumed o pa hrough he boleneck o reach he deinaion. Since each uer ha he deired arrival ime a he deinaion, he rip co of a uer coni of aiing co a he boleneck and chedule co aociaed ih ime difference beeen he acual and deired arrival ime. The rip co of a uer ih deired arrival ime i herefore rien a p( d, ) = f {( d ) + f {( d, ) (6) d = deparure ime from he boleneck = deired arrival ime a he deinaion (given) p( d, ) = rip co of a uer ih deire arrival ime deparing from he boleneck a 6 Cumulaive Trip lope = λ() lope = µ Fig.5 ( d ) D( ) A( ) d W( ) 2 ( d, ) = - d Queue Evoluion a a Single Boleneck, and Waiing Time and Schedule Delay ime

7 ( d ) ( d, ) f { f { ime d. = aiing ime a he boleneck for a uer deparing a ime d. (In he previou ecion, aiing ime i defined by referring arrival ime a he boleneck. Hoever, under FIFO, aiing ime can be defined alo ih repec o deparure ime from he boleneck.) = chedule delay for a uer ih deired arrival ime deparing from he boleneck a ime d = - d = aiing co funcion hich conver aiing ime o he co = chedule co funcion hich conver chedule delay o he co A uer i aumed o ravel from he origin o he boleneck a a aic ravel peed, ai a he boleneck, and again ravel o he deinaion a a aic ravel peed. Therefore, only aiing ime and chedule delay are ime-dependen. Thu, for our convenience, he deired arrival ime a he deinaion can be iched o he deired deparure ime from he boleneck, ince e can eaily obain he deired deparure ime by ubracing aic ravel ime from he boleneck o he deinaion from he deired arrival ime a he deinaion. The chedule delay can be herefore rien a ( d, ) = - d. Each uer i aumed o chooe he be deparure ime from he boleneck o a o minimize hi/her on rip co. An imporan propery of hi problem i he Fir In Fir Work (FIFW) dicipline in hich he order of arrival ime a he boleneck i he ame a he order of deired arrival ime, provided ha chedule co funcion f { i convex in (Smih (98) and Daganzo (985)). Since, under FIFW ogeher ih FIFO, A()=D( d )=W( ) i valid, here W( ) i he cumulaive uer ih deired arrival ime before. Arrival ime a he boleneck and deparure ime from he boleneck d can be rien a a funcion of deired arrival ime : d ( ) = D - (W( )), ( )=A - (W( )). (7) Alo, ( d ), λ(), and p( d, ) can be decribed ih repec o : ( ) = ( d ( )), λ( ) = λ(( )), and p( ) =p( d ( ), )= f {( ) + f { - d ( ). (8) I ha been alo knon ha he folloing emporal equilibrium condiion mu be aified a he equilibrium iuaion (Kuahara e.al.(987)): dp( d( d here ), f '{ = ) p( d( = d = f '{ ( ( d f{. d ), ) p( d( ), + ), ) = f '{ ( d ) ) dd( d, ) p(d ( ), ) ( Q = ), (9) By olving he above differenial equaion, he uer co a deired arrival ime, p( ), can be rien imply a a funcion of chedule delay: p( ) = f ' { ( ) d. (2) d 7

8 5.2. Demand Funcion and Dynamic Marginal Co Wih deparure ime choice, e have o imulaneouly eablih emporal equilibrium in addiion o demand-upply equilibrium dicued in ecion Dynamic Marginal Co For he problem ih deparure ime choice, he diribuion of deired arrival ime, W( ), can be conidered a he generaed demand, ince he cumulaive arrival curve A() i obained a he reul of deparure ime choice given W( ). Therefore, defining η( ) = dw( )/d, he demand funcion i rien a belo ih repec o : ρ(p( ), ) = η( )d (2) ρ - (η( )d, ) = p( ) (22) The conumer urplu F i hen rien F = F F 2 η( ) d = ρ ( x, ) dx p( ) η( ) d The opimaliy condiion i obained by differeniaing ih repec o demand η( )d : p(u ) p( ) + > = = η( u ), ( ) or λ( ) µ ρ ( λ( )d, ) MC( ) u η( ) (24) p( ), oherie The econd erm on he righ hand ide canno be generally rien in he explici form, ince if demand η( )d increae by one uni a ime, W( ) a ell a A() and D() ould hif for an enire ime period a hon in dahed line in Fig.6. Conequenly uer co p( ) ould alo change for all ime. (23) Conan Deired Arrival Time for Everyone Alhough he evaluaion of he righ hand ide of Eq.(24) i difficul in general, e could relaively eaily analyze a pecial cae here everyone ha he ame deired arrival ime. Since everyone ha he ame deired ime, e canno diinguih uer and herefore everyone ha he ame co under he emporal equilibrium condiion. From Eq.(2), e ee ha he co for everyone i given by f { -. A hon in Appendix, ih hi aumpion, marginal co MC( ) i explicily rien a Cumulaive Trip lope = λ( ) ( ) A() D() W() ( ) Fig.6 Impac on A() and D() by Demand hif lope = η( ) ime MC( ) = f { here N + [ ], if ( ) > or λ( ) µ µ f ' { f ' { (25) N = W ( ) W ( ). 8

9 The econd erm correpond o he dynamic oll, hich i he ame for everyone. For he more implificaion, le u conider he folloing linear chedule co funcion: c,, { = c > and c. (26) c2, <, f 2 > Then, MC( ) become MC( ) = p( N ) + ( µ c + c 2 ), if ( ) > or λ( ) µ. (27) On he oher hand, from Eq.(2), N p( ) = ( + ). (28) c c µ 2 Thu, MC( )=2p( ) ih he linear funcion, hich mean he oll i p( ) ame for everyone. Under hi implificaion, ince everyone ha he ame rip co a ell a he ame amoun of oll, he demand-upply balance can be ummarized in one graph a hon in Fig.7. From Eq.(27) and (28), he uer co and he marginal co can be rien a funcion of he demand N: p(n) and MC(N). The MC(N) ha a lope ice a much a one for p(n). The opimal demandupply equilibrium i he inerecion of demand curve ρ(p(n)) ih MC(N) a he marginal co pricing of p(n*) Dicuion Baed on he above analyi under everal aumpion for implificaion, le u dicu he implicaion. () Addiion of he aic co We have eliminaed he aic ravel co uch a aic porion of ravel ime and aic oll from our analyi. If you ould like o include hem, you imply add he aic co o he dynamic co e analyzed. For inance in Fig.7, e may add ome conan co o uer co p(n) a ell a MC(N). (2) Differen deired arrival ime for each uer In realiy, W( ) ould be diribued over ome ime inerval a in Figure 5 and 6. For he diribued W( ), uer co depend upon 9 Co Co p(n) Saic Co N ρ(p(n)) N* p(n*) p(n*) b-plane MC(N)=2p(N) MC( ) a-plane p( ) Demand N Fig.8 Three-Dimenional Illuraion of Uer and Marginal Co p(n) Demand N Fig.7 Demand-Supply Equilibrium hen Everyone ha he Same Deired Arrival Time (Linear Schedule Co Funcion)

10 hi/her deired arrival ime, and e canno ummarize he demand-upply equilibrium ihin one graph a Fig.7. Inead, p( ) and MC( ) ould form plane a hon in Fig.8, hich include he aic co menioned above. Fig.9 illurae ho uer co change in relaion o he deired arrival ime ih fixed demand N (a-plane in Fig.8). From Eq.(2), queueing delay ar from ime, reache o i maximum value a for zero chedule delay, and finally vanihe a ime. If he demand N ge maller, ime and ould approach each oher and evenually coincide. Namely, in Fig. 8, no queue form hen he demand i le han N. On he oher hand, Fig. illurae ho p( ) and MC( ) vary depending on demand N by ih fixed deired arrival ime (b-plane in Fig.8). Noe ha Fig.7 i a pecial cae of Fig. hen i he ame for everyone. Alhough Fig. look imilar o he demandupply curve under he aic frameork a in Fig., he implicaion i quie differen. Fir, he hape of p( ) and MC( ) depend upon. Second, he locaion N here p( ) ar increaing depend on he diribuion of, W( ). For he ame deired arrival ime, W( ) ha he infinie lope a ime. Hence, a queue mu form even for a very mall demand N, hich i he reaon hy p(n) ar increaing for N>. Hoever, hen are diribued over ome ime period, i ould be quie poible ha no queue form for mall N < N. Third, he endency of increaing MC( ) for N > N i relaed o he chedule co funcion a ell a ime and a explained in Eq.(25) and (27). For a linear chedule co funcion ih he conan deired arrival ime, e have een ha p( ) and MC( ) become linear a in Eq.(27) and (28). Bu, for a general cae, explici form of hem canno be decribed. Co MC( ) p( ) Co MC( ) p( ) Saic Co Saic Co N Demand N Fig.9 Uer and Marginal Co in Profile (Fixed oal demand N) Fig. Uer and Marginal Co in Relaion o Toal Demand N (Fixed ) (3) Convering aiing ime o oll Even ih he dynamic marginal co pricing, a queue mu form o eablih he emporal equilibrium for ufficienly large demand. The aiing ime i required o compenae he chedule delay changing over ime for he emporal equilibrium. Hoever, from heoreical poin of vie, e could impoe addiional dynamic oll equivalen o he aiing ime value o eliminae he queueing delay. The emporal equilibrium ould be mainained afer he addiional oll, ince uer pay he ame amoun of co a he aiing co. (Noe ha for he problem ihou deparure ime choice dicued in ecion 4, he converion of aiing co o oll i impoible. Since he demand rae i he lope of arrival curve A(), he A() and D() do no necearily coincide each oher afer he converion under he demand-upply equilibrium.) (4) Demand-Supply equilibrium Figure 8, 9, and are dran if chedule co funcion f { a ell a W( ) are aumed,

11 ince uer co p( ) under he emporal equilibrium can be deermined. Hoever, for demand-upply equilibrium, generaed demand from demand funcion ρ(p( ), ) mu be equal o η( )d hich i evaluaed from he aumed W( ). For he problem ihou deparure ime choice, he uer co a ime depend only on arrival rae λ() before under FIFO. A explained in ecion 4.2, e can deermine generaed demand rae λ(), hich aifie he demand-upply equilibrium, equenially from he earlier ime. Hoever, co p( ) here depend upon he demand rae of he enire ime period. An efficien algorihm o find he demand-upply equilibrium i no propoed ye. 6. SUMMARY AND FUTURE RESEARCH NEEDS Thi reearch analyze he dynamic marginal co by combining he demand-upply analyi in economic ih he ime-dependen queueing analyi in raffic engineering. Main concluion are lied belo: () For he problem ihou deparure ime choice, he dynamic marginal co a ime, MC(), i equal o he duraion of congeion afer ime. (2) Through a imple example, e ho ha he conumer urplu can be improved by inroducing he dynamic marginal co pricing. (3) For he problem ih deparure ime choice, he demand-upply analyi i combined ih he exiing heory of deparure ime choice. When deired arrival ime i diribued over ime, MC( ) canno be explicily rien. Hoever, under he implificaion ha everyone ha he ame deired arrival ime, uer co p( ) and marginal co MC( ) are explicily decribed a a funcion of he oal generaed demand during he congeed ime period. (4) Waiing ime a a boleneck can be convered o he dynamic oll for he problem ih deparure ime choice. Several fuure reearch need are ummarized: () Thi reearch mainly focue on he upply funcion o ha i reflec dynamic boleneck phenomena. Hoever, for he demand-upply analyi, he demand funcion mu be evaluaed. The demand funcion e have inroduced here form a hree-dimenional plane a in Fig.3, and herefore i ould be quie difficul o evaluae he hape. (2) In hi analyi, e aume ha he generaed demand a a cerain ime depend only on uer co a ha ime. Several differen definiion on he demand funcion, hoever, could be poible. For inance, one can propoe ha he demand may depend on he average uer co for ome ime period or for he enire udy period. A menioned in () above, he definiion and evaluaion of he demand funcion are he major reearch need. (3) For he problem ih deparure ime choice, e hould propoe an algorihm o find he oluion hich aifie boh he emporal a ell a demand-upply equilibrium. APPENDIX The p( u ) η( ) in Eq.(24) i rien a p( u ) df { d = = f ' { ( ) dn dn η (A) On he oher hand, in general, queue aring ime i deermined o ha p( )= hen he queue vanihe a ime. From Eq.(2),

12 p( ) = f ' { ( )d = f { f { = (A2) Plugging - =N/µ ino he above, N f { = f {( ) (A3) µ The differeniaion of Eq.(A3) yield d = [ dn µ Thu, from Eq.(A), p η (u ) ( ) f ' { f ' { = [ µ f ' { ] f ' { ] (A4) (A5) MC( ) = p( if ( ) + u p( u ) η( u ) = f η( ) ) > or λ( ) µ { N + [ µ f ' { f ' { ], REFERENCES Daganzo C.F. (985) The Uniquene of a Time-Dependen Equilibrium Diribuion of Arrival a a Single Boleneck, Tranporaion Science, Vol.9, pp Hendrickon C., and Kocur G. (98) Schedule Delay and Deparure Time Deciion in a Deerminiic Model, Tranporaion Science, Vol.5, No.. Kuahara M., and Neell G.F. (987) Queue Evoluion on Freeay Leading o a Single Core Ciy during he Morning Peak, Proceeding of he h Inernaional Sympoium on Tranporaion and Traffic Theory, pp.2-4, Boon. Kuahara M. (99) Equilibrium Queueing Paern a a To-Tandem Boleneck during he Morning Peak, Tranporaion Science Vol.24, No.3, pp Smih M.J. (98) The Exience of a Time-Dependen Equilibrium Diribuion of Arrival a a Single Boleneck, Inernaional Sympoium on Fronier in Tranporaion Equilibrium and Supply Model, Monreal. Vickrey W.S (969) Congeion Theory and Tranporaion Invemen, American Economic Revie 59. 2