Tradable Network Permits: A New Scheme for the Most Efficient Use of Network Capacity

Transcription

1 adable Netwok Pemits: A New Scheme fo the Most Efficient Use of Netwok Capacity akashi Akamatsu Gaduate School of Infomation Sciences, ohoku Univesity, Sendai, Miyagi, , Japan Akamatsu et al.(26) poposed a new tanspotation demand management scheme called tadable bottleneck pemits (BP), and poved its efficiency popeties fo a single bottleneck model. his pape exploes the popeties of a BP system fo geneal netwoks. An equilibium model is fist constucted to descibe the states unde the BP system. It is poved that equilibium esouce allocation is efficient in the sense that the total tanspotation cost in a netwok is minimized. heoetical elationships between BP and congestion picing (CP) ae also discussed. It is demonstated that BP has definite advantages ove CP when demand infomation is not pefect, wheeas both BP and CP ae equivalent fo the pefect infomation case. Finally, it is shown that the self-financing pinciple also holds fo the BP system. Key Wods : bottleneck congestion, dynamic taffic assignment, time-space netwok, tadable pemit, IS Recent advances in infomation and communication technology (IC) have led to apid changes in the vitual wold epesented by the Intenet. he inceasing capabilities and deceasing cost of IC is now becoming the impetus fo changing the eal wold. he effects of IC on tanspotation systems ae no exception. he boadly defined Intelligent anspotation Systems (IS) that exploits IC has a lage potential fo damatically impoving efficiency of oad tanspotation systems the systems ae implemented togethe with appopiate tanspotation demand management (DM) schemes. As an example of such futuistic DM schemes making the most of IC/IS, Akamatsu, Sato and Nguyen (26), and Akamatsu (27a) poposed the tadable bottleneck pemit system. hei poposed scheme is designed fo esolving the poblem of congestion duing the moning ush hou at a single bottleneck, and consists of the following two pats: a) the oad manage issues a ight that allows a pemit holde to pass though the bottleneck at a pe-specied time peiod ( bottleneck pemits ), b) a new tading maket is established fo bottleneck pemits dfeentiated by a pe-specied time Note hee that both pats a) and b) of this scheme ae feasible fo implementation fom a technical point of view, even at the pesent time. he system fo handling pat a) may be constucted as an application of the dedicated shot ange communication (DSRC) system that is used in the cuent electic toll collection (EC) system; the tading makets in pat b) also can be ealized inexpensively by using Intenet auction makets. It is, theefoe, easonable to assume that implementing this scheme will become technically easie when we take into account the futue advances of IC/IS. 1

2 Pat a) of this scheme is almost the same as the concept of advance highway booking (esevations o quotas)that has been peviously poposed by seveal authos (e.g., Akahane and Kuwahaa (1996), Wong (1997), eodoovic and Edaa (25)). Unde this scheme, the aival flow ate at a bottleneck at any time peiod is, fom the definition of the scheme, equal to the numbe of pemits issued fo that time peiod. his implies that we can completely eliminate the occuence of queuing congestion by setting the numbe of pemits issued pe unit time to be less than o equal to the bottleneck capacity. Howeve, thee may be cases in which oad uses cannot choose thei desied time fo using the oad the pemits ae assigned accoding to some unefined ule (e.g., a simple quota scheme). Such an infingement on feedom of choice necessaily causes economic losses and should not be socially acceptable. In ode to cicumvent this poblem that aises in employing only pat a) of the scheme, we need to add an appopiate mechanism in which each use can choose his o he desied pemit. It is pat b) of the scheme that gives the foundation fo this choice mechanism by a maket system fo buying and selling pemits. With the complementay popeties of pats a) and b) in the combined scheme above, we can expect that this is the most efficient scheme of using the limited esouce of oad capacity. Indeed, fo a depatue-time choice equilibium poblem with a single bottleneck a la Vickey (1969), Akamatsu (27a) showed that the poposed system has the following desiable popeties: (1) Compaing equilibium states with and without the poposed system, we can achieve Paeto impovement fo both the oad manage and all oad uses, (2) the equilibium with the poposed system achieves the most efficient (i.e., Paeto optimal) esouce allocation, (3) the self-financing pinciple holds fo the equilibium with the poposed system that is, the total evenue (maket value) of selling the pemits is equal to the investment cost equied fo inceasing the bottleneck capacity to a socially optimal level. hese popeties of the tadable pemit system ae poved only fo a oad with a single bottleneck. Specically, the poof is based on the isomophism between the commutes depatuetime equilibium in a single bottleneck model and an equilibium model of an uban esidential location (see, fo example, Fujita (1989)). Since such isomophism cannot be extended to a case with multiple bottlenecks, the popeties of the tadable pemit system fo geneal netwoks ae lagely unknown. Futhemoe, we should note hee that thee ae no studies dealing with tadable pemits in the tanspotation field, othe than Vehoef, Nkamp and Rietveld (1997), Akamatsu et al. (26) and Akamatsu (27a). Although Vehoef et al. (1997) give an excellent discussion on the possibilities of tadable pemits in oad tanspotation, they do not mention time-dependent tadable pemits fo eliminating bottleneck congestion. hus, whethe o not the above desiable popeties hold fo geneal netwoks is a poblem yet to be studied. he pupose of this pape is to exploe some of the popeties of a system of tadable bottleneck pemits fo geneal netwoks (we call this a system of tadable netwok pemits. ). Specically, afte defining the system of a tadable netwok pemit, we pesent a mathematical model that descibes the equilibium that aises unde the tadable pemit system. We then pove that the equilibium esouce allocation unde the system is efficient in the sense that the total tanspotation cost in a netwok is minimized: fomulating a dynamic system optimal assignment, we show that the equilibium assignment coincides with the optimal assignment; we also show that the feasibility of the optimal assignment can be easily confimed by constucting a time-space netwok. We futhe show the theoetical elationship between the tadable pemit system and congestion picing: we demonstate the definite advantages of the tadable pemit system ove 2

3 congestion picing when the demand infomation is not pefect, wheeas they ae equivalent fo the pefect infomation case. Finally, we pove that the self-financing pinciple holds not only fo the single bottleneck case but also fo the geneal netwok case. he oganization of this pape is as follows. In Section 1, we outline the famewok of the tadable pemits system analyzed in this pape. In Section 2, we pesent a model of the equilibium unde the tadable netwok pemit system. In Section 3, we analyze the efficiency of the equilibium allocation. In Section 4, we discuss the theoetical elationship between the tadable pemit system and congestion picing. Section 5 shows that the self-financing pinciple holds fo tadable netwok pemits. Finally, Section 6 concludes the pape. 1. A System of adable Bottleneck Pemits in anspotation Netwoks 1.1. Netwoks In this pape, we conside dynamic taffic flows on a geneal netwok G (i.e., a tanspotation netwok with geneal topology). he netwok consists of a set N of nodes, and a set L of diected links. he node set N includes an oigin node fom which uses stat thei tips, and a destination node at which uses teminate thei tips. o avoid notational complexity and to outline essential aspects of the theoy, we deal with netwoks with a single oigin-destination (OD) pai. Each element of N (i.e., each node) is identied by a sequential natual numbe i, and each element of L (i.e., each link) is denoted by a pai (i, j) of the upsteam node i and the downsteam node j. he time inteval I = [, ] fo which we assign the dynamic taffic flow is fixed. We assume that the tavel demand Q that makes tips fo the time inteval I is a given constant. We also assume, without any loss of geneality, that each link in a netwok consists of a fee flow segment and a single bottleneck segment. he tavel time to pass though the fee flow segment of link (i, j) is a constant t (i.e., t is independent of time and flow). he bottleneck of each link is epesented by a point queue model with constant capacity μ. Note that this modeling appoach can deal with any numbe of bottlenecks in a oad segment. When we wish to conside a segment without a bottleneck, we set the capacity as infinity; when we conside a segment with multiple bottlenecks, we just set up multiple links coesponding to each bottleneck Agents he model pesented in this pape has two types of agents: a oad netwok manage and oad netwok uses. he oad manage aims to estain taffic congestion on the netwok and minimize the social tanspotation cost. o achieve this, the manage egulates the taffic flow ates enteing into each bottleneck in the netwok by using time-dependent bottleneck pemits. he pecise definition and setup of the bottleneck pemit system ae descibed in Subsection 1.3 below. Each use makes a single tip (fo the time inteval I) fom an oigin (e.g., esidential zone) to a destination (e.g., cental business distict (CBD)) in the netwok. he use chooses a destination aival time and a path between the oigin and the destination so as to minimize his o he disutility (o genealized tanspotation cost ). he detailed definition of disutility is mentioned in Section 2. Unde the bottleneck pemits system, each use must puchase a set of pemits coesponding to a set of links included in the use s chosen path. his implies that choosing a destination aival time and a path diectly links to puchasing time-dependent bottleneck pemits in the pemits makets. 3

4 Moe detailed explanations on the puchase scheme and the pemit makets ae given in Subsections 1.3 and Bottleneck Pemits ime-dependent bottleneck pemits is a ight that allows the pemit holde to pass though a pe-specied bottleneck at a pe-specied time. In this pape, we assume that the oad manage can issue time-dependent bottleneck pemits fo all bottlenecks (i.e., links) in the netwok. his implies that the taffic flow enteing into link (i, j) at time t consists of only uses who have a time t pemit fo link (i, j), and uses without this pemit cannot pass though this link at this time. houghout this pape, we assume that the numbe of pemits issued fo each link fo each unit time is equal to o less than the taffic capacity of each link in the netwok. his means that queuing congestion neve occus in the netwok unde this pemits-issue scheme. his may be easily seen fom this explanation of pemits: the inflow ate of each link is equal to (o less than) the numbe of pemits issued, and hence the inflow ate cannot exceed the taffic capacity of each link, which implies that queuing congestion at each link can neve occu ading Makets of Bottleneck Pemits Fo assigning the bottleneck pemits to uses, we can conside two epesentative schemes: maket selling scheme and fee distibution scheme (Akamatsu (27a)). In the maket selling scheme, the oad manage sells all the bottleneck pemits to uses in bottleneck pemits makets. All sales fom selling the pemits esult in evenue fo the oad manage in this scheme. In the fee distibution scheme, the oad manage distibutes all the pemits to uses fo fee accoding to methods that conside the equity among uses. In this scheme, the time designated by the pemits assigned fo each use does not necessaily match one s own desied aival time. Fo that case, uses can mutually tade a wide vaiety of pemits in the bottleneck pemits makets: each use can buy the necessay pemits while selling the unnecessay pemits. As a esult of such tading activities, pemits ae e-distibuted among uses though a system of pices emeging in the maket. hus, all income tansfes take place only among the uses in this scheme. his pape deals with only the fome scheme ( maket selling scheme ) to make the exposition as simple as possible. he pemits issued fo each link (bottleneck) ae put on sale by the oad manage. hee ae as many makets fo tading pemits as thee ae links, and each maket is dedicated fo tading the pemits fo each link. he pemits fo each link ae futhe distinguished by a specied time allowable to use the link. Unde the bottleneck pemits system, each use who would like to use a path must have a set of pemits coesponding to a set of links included in the path befoe making a tip. o fulfill this equiement, each use is assumed to puchase the needed set of pemits in the tading makets. he pice of each pemit is detemined by an auction system, which implies that the pice is adjusted so as to clea the excess demand fo each type of pemit. We also assume that the makets ae pefectly competitive; that is, neithe a monopoly no oligopoly occus. 2. Equilibium unde a System of adable Netwok Pemits his section povides a model of equilibium that takes place afte intoducing the tadable netwok pemits system. We fist descibe the conditions that should be satisfied by seveal 4

5 dynamic tanspotation cost vaiables, and a model of uses behavios fo choosing pemits is shown. We then fomulate the equilibium unde the tadable pemits system Dynamic avel Costs in Geneal Netwoks he tanspotation cost fo a single tip of a netwok use consists of the following thee types of costs: a) schedule cost, b) tavel cost, c) pemit puchase cost. a) he schedule cost fo a use is the cost due to the dfeence between the use s desied aival time and the actual aival time t. he desied aival time is assumed to be the same fo all uses and is equal to s. he schedule cost is epesented by the function w ( of destination aival time t, which is common to all uses, and is assumed to be a stictly convex function with a minimum at s = t, following pevious studies on the depatue time equilibium (e.g., Smith (1984), Daganzo (1985), Kuwahaa (199)). b) he tavel cost is the monetay equivalent of the tavel time needed fo a tip fom the oigin to the destination. he tavel times ae dfeent among paths. he tavel time of a path between the oigin-destination pai is defined as the sum of tavel times of the links included in the path. Note that the tavel time of each link (i, j) is a constant t at equilibium unde the pemits system, in which no queuing occus. Hence, at equilibium, the tavel time of path between the oigin-destination pai aival time is also constant: = t, ( o, d ) L δ, (1) whee δ, (o,d) is a typical element of the path-link incidence matix fo node pai (o, d): it is 1 link (i, j) is on path connecting OD pai (o, d); othewise, it is zeo. c) he pemit puchase cost is the total payment fo puchasing a set of link pemits equied fo going though a path fom the oigin to the destination. o put it anothe way, the pemit puchase cost of a use is defined as the sum of pemit pices of the links included in the path used. his cost vaies depending on what path is taken and at what time because the pemits fo each link ae futhe dfeentiated by the specied time and each pemit is piced depending on the time and the link. o see this cost moe pecisely, conside a use who uses path and aives at the destination at time t. Suppose hee that he o she uses a path that contains link (i, j) and, is the tavel time equied fo aiving at the destination fom node i:, = t δ (2) kl kl, ( i, d ) kl L whee δ kl, (i, d) is a typical element of the path-link incidence matix fo node-pai (i,d): it is 1 link (k, l) is on path fom node i to the destination d; othewise, it is zeo. hen, the use should ente into the link at time t,, which implies that this use has to obtain the time t, pemit fo link (i, j), whose pice is p (t, ). It follows fom this that the pemit puchase cost (i.e., the total payment fo puchasing the set of link pemits equied fo going though path and aiving at the destination at time is given by P = p ( t, ), ( o, d ) L ( δ (3) 5

6 We call the sum of the tavel cost and pemit puchase cost as the path tanspotation cost; that is, the tanspotation cost C ( of path fo a use aiving at the destination at time t is given by C = P + α, (4) whee α is a coefficient that convets tavel time to the monetay equivalent Uses Behavios Each use chooses an aival time t at the destination and a path between the oigin and the destination so as to minimize the genealized tanspotation cost, defined as the sum of the schedule cost and tanspotation cost. hat is, the use solves the following poblem: min. w( + C (5) t, his optimization poblem can be solved by backwad induction. hat is, the optimal choice pai (t, ) can be obtained by solving a two-stage (hieachical) choice poblem, in which one chooses sepaately the destination aival time t (the uppe-level choice) and the optimal path (the lowelevel choice). Moe specically, we fist solve the lowe-level poblem of the path choice fo a given aival time t; we then obtain the optimal oute choice ( and the optimal value π( conditional on aival time t: π = min. C (6a) = ag.min. C (6b) By using this optimal choice function, we can educe the uppe-level poblem of the aival time choice to t min. w( + π (7a) t = ag.min. w( + π (7b) t 2.3. Equilibium Conditions Fo the settings above, we constuct a model that descibes the equilibium that aises in the pesence of the poposed system. At equilibium, the following six conditions as well as the definitional equations (1)-(4) fo time-dependent cost vaiables should hold. 1) Flow consevation fo path flows and OD flows: OD flow ate q ( fo a use aiving at the destination at time t should be the sum of the path flow ates: R f q(, t I (8) = whee f ( is the flow ate of path aiving at the destination at time t; R is the set of paths connecting the oigin-destination pai. 2) Equilibium conditions fo the path choice: As mentioned in Subsection 2.2, the use s optimal choice poblem can be solved by backwad induction. By fist obtaining the solution of the path choice poblem as a function of aival time t, we then solve the aival time choice poblem. o obtain the equilibium condition fo the fist poblem, conside a use aiving at the destination at time t. At equilibium, no use can impove his o he own cost by changing the path choice unilateally (i.e., no use has the incentive 6

7 to deviate fom his o he stategy). hat is, tanspotation cost C ( of path (fom the oigin to the destination) that is chosen at equilibium is equal to the minimum path tavel cost π (, and the costs of othe (unused) paths ae geate than π (: π = C π C f > f t I, R (9) 3) Flow consevation fo OD flow ates and OD tavel demands: All OD tavel demands Q have to be assigned to each time point (in tems of the aival time at the destination) in the inteval [, ]; that is, the time-dependent OD flow ates {q(} should satisfy q( u) du = Q. (1) 4) Equilibium conditions fo destination aival time choice: At equilibium, no one can impove his o he own genealized tanspotation cost by changing the destination aival time unilateally. It follows fom the path choice equilibium condition in 2) that the genealized tanspotation cost fo a use aiving at time t is w ( + π (, whee w ( is the schedule cost fo a use aiving at the destination at time t. heefoe, the equilibium condition fo the use s aival time choice can be expessed as ρ = π + w( q( > ρ π + w( q( whee ρ epesents the minimum (equilibium) genealized tanspotation cost. t I (11) 5) Demand-supply equilibium (maket cleaing) conditions in each link pemit maket: As mentioned in Section 1, the pemits fo each link ae distinguished by a specied time at which each pemit makes it allowable to use the link. Let p ( denote the pice of the pemit fo link (i, j) with a specied allowance time t. Since the tading makets ae assumed to be pefectly competitive (i.e., neithe a monopoly no oligopoly occus), the pice p ( of each pemit type is adjusted to clea the excess demand fo each type of pemit. Moe pecisely, at equilibium, the pice of a cetain type of pemit is positive, the quantities supplied and the quantities demanded fo the pemit ae equal; fo the pemit whose supply quantity exceeds the quantity demanded, the pice is zeo. Note hee that, fo each link (i, j) and each allowance time t, the demand of the time t pemit of the link is equal to the inflow ate y (. On the othe hand, the maximum supply (uppe bound) of the time t pemit of link (i, j) is given by the link capacity μ. heefoe, the demandsupply equilibium condition fo the pemits maket is epesented as y y = μ μ p > p t I, L (12) 6) Flow consevation fo link flows and path flows: he inflow ate on each link at each time point should be consistent with the timedependent path flows. A use going though path and enteing into link (i, j) at time t aives at the destination at time t +,. heefoe, the inflow ate y ( enteing into link (i, j) at time t is the sum of the flows on all paths going though that link and aiving at the destination at t +, : y = f ( t +, ), ( o, d ) R ( δ t I, L (13) 7

8 2.4. Ac-Node Fomulation he path-based fomulation pesented above is convenient fo pesenting the use s oute choice behavio in a staightfowad manne; howeve, this fomulation is not necessaily convenient fo analyzing the demand-supply equilibium condition (12) at each link due to the complexity of the elationship (13) between time-dependent path flows and link flows. In ode to alleviate this complexity, we tansfom the model above into a fomulation expessed in tems of link-node vaiables, which enables us to analyze the efficiency of the equilibium, as is pesented in late sections. he ac-node fomulation of the model can be summaized by the following five conditions. 1a) Flow consevation at each node: Consevation of the dynamic taffic flow in a netwok is epesented as the equality of inflow and outflow at each node at each time point. o fomalize this, let y ( be the flow ate aiving at link (i, j) at time t, and z ( be the flow ate depating fom link (i, j) at time t. hen the flow consevation is epesented as y ik k NO( i) z = q( δ ki k NI ( i) id, t I, i N (14) whee δ id is Konecke s delta (i.e., 1 i = d, zeo othewise ); NO (i) is a set of downsteam nodes of the links incident fom node i; NI (i) is a set of upsteam nodes of the links incident to node i. 1b) Fist-In-Fist-Out conditions on each link: We assume that the dynamic taffic flow in ou model should satisfy the Fist-In-Fist-Out (FIFO) popety on each link (i.e., we assume that passing can be neglected). As shown in the liteatue (see, fo example, Kuwahaa and Akamatsu (1993), Akamatsu and Kuwahaa (1994)), the FIFO condition fo each link can be witten as A = D ( t t ), (15) + whee A ( and D ( ae the cumulative numbes of vehicles enteing into and leaving fom link (i, j) at time t,espectively. Using the flow ate vaiables, we can equivalently ewite this as y = z ( t + t ) (1 dt / d, (16) + whee t ( is the tavel time of link (i, j) fo a use enteing into the link at time t. Note hee that t ( is a constant egadless of the aival time when thee is no queue. Hence, at equilibium unde the pemits system (i.e., when thee is no queue in the netwok), the FIFO condition (16) educes to the following simple epesentation: y = z ( t + t ) t I, L (17) ( 2) Equilibium conditions fo path choice: Conside a use aiving at node i at time t. If the use chooses link (i, j), the aival time at node j is t + t. Hence, at equilibium, link (i, j) should be on the minimum path fo a use aiving at node j at time t + t thee exists a use enteing into link (i, j) at time t. Denoting π i ( as the minimum path cost fom the oigin to node i fo a use aiving at the node at time t, we can epesent this condition as π j ( t + t ) = c + π i y > t I, L (18) π j ( t + t ) c + π i y whee c ( is the tanspotation cost fo a use who entes into link ( i, j) at time t : 8

9 c p + α t (19) he condition (18) is equivalent to the minimum cost path choice condition (9) epesented by the path vaiables since (18) can be deived by applying the dynamic pogamming pinciple to (9). 3) Flow consevation fo OD flow ates and OD tavel demands: his is the same as the conditions in 3) in Subsection 2.3, that is, q( u) du = Q (2) 4) Equilibium conditions fo destination aival time choice: Minimum path cost π ( defined in Subsection 2.3 is the same as π d ( defined in (18) above. Hence, in a simila manne to the conditions in 4) in Subsection 2.3, the equilibium conditions fo the use s choice of destination aival times is given by ρ = π d + w( ρ π d + w( q( > q( 5) Demand-supply equilibium (maket cleaing) conditions in each link pemit maket: his is the same as the conditions in 3) in Subsection 2.3, that is, y y = μ μ p > p t I (21) t I, L (22) 3. Efficiency of the Equilibium with adable Netwok Pemits 3.1. Dynamic System Optimal Assignment without Queuing In ode to examine the efficiency of the equilibium allocation pattens defined in (14)- (22), conside the following optimization poblem [P-1]: min. ( q, y) FP ( q, y) q( w( dt + α y t dt (23) L subject to q( u) du = Q (24) y t ik k NO( i) ( ) μ t I, L (25) y y ki k NI ( i) ( t t ki ) = q( δ id t I, i N (26) his is the poblem of finding a dynamic taffic flow patten that minimizes the total genealized tanspotation cost in the netwok, subject to the physical constaints of flows epesenting the netwok pefomance. Specically, the fist tem of the objective function F P is the total schedule cost expensed by all uses, and the second tem is the monetay equivalent of the total tavel time paid by all uses. he fist constaint (24) is consevation of the OD demand, the second constaint (25) is the taffic capacity constaints on each link. he final constaint (26) is the consevation of 9

10 dynamic link flows at each node (14) combined with the FIFO condition on each link (17). his poblem does not necessaily have a feasible solution (satisfying constaints (24), (25) and (26)) due to the capacity constaint (25) on each link. he most exteme case fo the nonexistence of a solution is when the time inteval I fo the assignment is limited to an extemely shot inteval Δt. In this case, the OD tavel demand Q cannot be distibuted into a time space and is foced to be distibuted into only a netwok space (i.e., paths) while satisfying all the link capacity constaints. his implies that no feasible solution exists Q /Δt, which can take an extemely lage value, exceeds the maximum netwok capacity. As shown late in Subsection 3.2, howeve, we can easily examine whethe o not the poblem [P-1] has a feasible solution fo any netwok. Also, [P-1] always has feasible solutions (and an optimal solution) the assignment time inteval I is lage enough that we can make OD flow ates smalle than the maximum capacity of the undelying netwok. hus, we concentate ou discussions on the elation between the optimal assignment and the equilibium assignment. Fo the discussions in late sections, we note that any feasible solutions of [P-1] imply taffic flow pattens with no queuing congestion on the netwok. Unde this assumption, the most impotant popety that chaacteizes the equilibium assignment is that [P-1] is an equivalent optimization poblem to the equilibium conditions (14)- (22). hat is, Poposition 1: Fo any netwoks with a single OD pai in which [P-1] has feasible solutions, the equilibium assignment unde the system of time-dependent tadable link pemits minimizes the social tanspotation cost defined by (23). (Poof: see Appendix) On the suface, this poposition states that the equilibium taffic pattens always coincide with the optimal taffic pattens fo [P-1]. We should note, howeve, that poblem [P-1] futhe povides infomation on the equilibium pices/costs as well as the equilibium flow pattens. Specically, it is clea fom the poof of Poposition 1 that a) the optimal Lagange multiplies {p (} fo constaint (25) coincide with the time-dependent link pemit pices at equilibium, b) the optimal Lagange multiplie ρ fo constaint (24) gives the genealized tanspotation cost at equilibium, and c) the optimal Lagange multiplies {π (} fo (26) yield the tanspotation cost fom the oigin to each node at equilibium. In ode to see moe diectly the fact that these Lagange multiplies (ρ, π (, p () epesent the equilibium pices unde the system of tadable pemits, conside the dual poblem, [D-1], of the optimal assignment poblem [P-1]: max. ( ρ, π, p) F D subject to ( ρ, π, p) ρ Q L p μ dt (27) ρ w + π t I (28) d π ( t + t ) π + ( α t p ) t I, L (29) j i + 1

11 Fom the duality theoem, the optimal value of the objective function of [D-1], F D, coincides with the optimal value of the objective function of [P-1], F P, that is, q w( dt + α y t dt = Q L L ρ p μ dt. (3) he ight-hand side of this equation, F P, is the social tanspotation cost at the equilibium unde the system of tadable pemits: [total schedule cost] + [total tavel time cost] (31a) On the othe hand, fom the identity elation of tanspotation costs defined in Subsection 1.1, this should coincide with the value of [total genealized tanspotation cost] - [total pemits payments] (31b) at equilibium. heefoe, the ight-hand side of (3), F D, should be the value of (31b) at equilibium. Indeed, we can easily vey that F D epesents the social tanspotation cost at equilibium with the fom of (31b) we intepet the optimal solution (ρ, π (, p () of [D-1] as the equilibium pices unde the tadable pemits system. Moe specically, the fist tem of F D on the ight-hand side of (3) denotes the equilibium genealized tanspotation cost multiplied by the numbe of uses, which is the total genealized tanspotation cost paid by uses in the netwok. Note that this cost includes uses payments to the oad manage to puchase pemits, but these payments should not be counted as the social cost because they ae just income tansfes between the uses and oad manage. he total amount of these income tansfes is given by the total maket values of all the pemits (i.e., the sum of the numbe of each pemit μ multiplied by its pice p ( fo all links and times), which is given by the second tem of F D. hus, we see that F D epesents exactly the social tanspotation cost at the equilibium with the fom of (31b) Feasibility of the Assignment without Queuing We can examine whethe o not the poblem [P-1] has a feasible solution fo any netwoks by simply constucting time-space netwoks. he concept of time-space netwok may be best explained by the example of a tain diagam. Denoting x i as the position of the i-th tain in a 1- dimensional guideway as a function of time t, we can depict the tajectoies of tains, {x i (}, in a 2- dimensional (time, space) plane. Fo a set of appopiate time points {t k }, the tajectoies povide a set of points {(t k, x i (t k ))} in the plane; egading these points as a set N of nodes, we can define a set L of links by teating each segment of tajectoies as a link connecting two nodes in N. he two sets N and L thus obtained constitute a time-space netwok G(N, L) fo the tain diagam. By applying a pocedue simila to that fo the tain diagam above, we can constuct the time-space netwok fo analyzing dynamic flows in a geneal netwok. his is a poweful appoach fo solving a ange of netwok poblems with constant link tavel times (fo moe detailed examples of this appoach, see Ieda and Akamatsu (1988), Cascetta (21)), although it is dficult to apply this to the poblems with state-dependent link tavel times. We can take advantage of this appoach to check the feasibility of [P-1] since we do not need to conside queuing fo the feasible solution of [P-1], which implies that the link tavel times in [P-1] ae constant. Befoe descibing the pecise pocedue fo constucting the time-space netwok fo [P-1], we fist discetize time into small intevals of length Δt : each time point is epesented by t = m Δt, whee m,1,2,, M. We then assume that tavel time t of each link is epesented by a natual multiple of Δt (i.e., an intege n satisfying t = n Δt is given fo each link). Unde this setting, the 11

12 time-space netwok is constucted by the following pocedue: Step 1-a. Replicates M copies of each node in the oiginal netwok G(N, L). Step 1-b. Fo the expanded node set, add a single dummy node fo the oigin in G, and similaly, add a single new dummy node fo the destination in G. he set of nodes ceated by Step 1-a and Step 1-b is the node set N of the time-space netwok. o identy each node in N, we denote by i (m) the node in N that epesents node i at time m Δt. We denote by o the dummy node in N fo the oigin o, and denote by o in N the dummy node fo the destination d Step 2-a. Fo evey m, ceate a link going fom node i (m) towad node j (m + n ) thee exists a link with tavel time t = n Δt in the oiginal netwok G(N, L). he attibutes (i.e., tavel cost and capacity) of each link ceated above is the same as those of the coesponding oiginal link in L. Repeat this pocedue fo all links in L. Step 2-b. Fo evey m, ceate a link going fom dummy node o towad oigin copy node o(m) in N. Fo these new links, the tavel cost is set to zeo, and the capacity is infinity. Fo evey time point m, ceate a link incident fom destination copy node d(m) towad dummy node d. Fo these new links, the tavel cost is set to w d (m), and the capacity is infinity. A set of links ceated by Step 2-a and Step 2-b is the link set L of the time-space netwok. he dynamic system optimal assignment poblem [P-1] (o its equivalent of the equilibium poblem (14)-(22)) educes to a static minimum cost flow poblem defined on the time-space netwok G (N, L ) constucted by the pocedue above. Moe specically, conside a static minimum cost flow poblem defined on G, in which the oigin node is given by the dummy node o, the destination node is the dummy node d, and the OD tavel demand is given as Q. We then egad the flow on link (i (m), j (m + n )) in G as the dynamic link flow ate y (mδ in the oiginal netwok G, and the flow on link ( d(m), d ) in G as the dynamic OD flow ate q(mδ in G. Fo this setting, the optimal solution of [P-1] is given by the link flow patten in G obtained as the solution of the minimum cost flow poblem above. o undestand this fact, it is sufficient to check that all conditions of [P-1] ae epesented as the conditions fo the minimum cost flow poblem defined on G. We fist see that consevation of the dynamic flow in [P-1] is automatically epesented by consevation of the static flow at each node on G. Constaint (26) is satisfied by the flow consevation at each node i (m) on G ; constaint (24) is satisfied by the flow consevation at each dummy node d in Step 2-a of the pocedue to constuct G. We also see that the capacity constaint of each link (25) fo each time peiod coesponds to the capacity constaint of each link on G. Futhemoe, the objective function of [P- 1] is consistently epesented by the total tavel cost in the minimum cost flow poblem defined fo G : the fist tem of the objective function is epesented by the sum of tavel costs fo the set of links (d (m), d ) constucted in Step 2-b, and the second tem is the sum of tavel costs fo the set of links constucted in Step 2-a. hus, the dynamic optimal assignment poblem [P-1] is educed to solving a static minimum cost flow poblem on G. he static minimum cost flow poblem has been well studied in compute science (see, fo example, Kennington and Helgason (198), Ahuja, Magnanti and Olin 12

13 (1993)), and vey efficient (polynomial ode) algoithms ae available fo solving this poblem. heefoe, we see that the feasibility of [P-1] can be easily veied by using these algoithms. 4. Congestion Picing vs. adable Pemits 4.1. Pefect Infomation Case he equilibium pemit pices fomulated in Section 2 can be intepeted as the optimal toll levels fo a congestion picing scheme. o see this, conside a dynamic congestion picing scheme in which the oad manage imposes a time-dependent toll (congestion tax) fo each link in the netwok. Denoting by p ( the toll of link (i, j) fo a use aiving at the link at time t, we easily see that all tanspotation costs equied fo a use to make a tip have pecisely the same fom as the tanspotation costs unde the pemit system in Section 2. Consequently, it is obvious that the equilibium states fo the dynamic congestion picing with a toll patten {p(} satisfy the conditions (14)-(21). In othe wods, we can intepet the conditions (14)-(21) as those fo defining the equilibium flow patten {y(} that aises when the oad manage sets a dynamic toll patten {p(}. With this in mind, suppose that the oad manage wishes to pevent the occuence of queuing congestion in the netwok by setting an appopiate toll patten {p (} such that the equilibium inflow ate y ( of each link neve exceeds the capacity μ.( i.e., queuing congestion neve occus). hen, it follows that the conditions fo achieving this goal ae given by the same fom as (22). hat is, condition (22), which epesents the maket cleaing (demand-supply equilibium) condition fo the tadable pemit scheme, is now the condition fo the oad manage to achieve the optimal flow patten unde the congestion picing scheme. hus, the system of equations (14)-(22) can be viewed as the conditions fo obtaining the optimal toll patten when the equilibium flow patten does not cause queuing congestion. As we have seen in Poposition 1, the system of equations (14)-(22) ae also equivalent to the optimal assignment poblem [P-1], and theefoe, the pai {p (, y (} of the optimal toll patten and the esulting equilibium flow patten exist [P-1] has feasible solutions. heefoe, we obtain the following poposition on the equivalence of the tadable pemit scheme and the congestion picing scheme: Poposition 2: Conside any netwoks with a single OD pai in which [P-1] has feasible solutions, and futhe suppose that the oad manage has pefect infomation about the tanspotation demands. hen, the equilibium assignment unde the system of time-dependent tadable link pemits coincides with the equilibium assignment unde the system of optimal dynamic congestion picing Impefect Infomation Cases: Economic Losses due to Congestion Picing he poposition above states that a congestion picing scheme unde a cetain condition attains the same ideal state as does the tadable pemit scheme. Howeve, we should pay attention to the fact that the equied conditions fo attaining the ideal state in the tadable pemit scheme ae lagely dfeent fom those in the congestion picing scheme. he dfeences lie in the amount and accuacy of the infomation needed fo the authoity (i.e., the oad manage) to implement the egulation. In the tadable pemit scheme, what the oad manage should know is just the taffic capacity of each link (and queuing congestion can be eliminated just by issuing the numbe of pemits fo each link that is equal to o less than the link capacity). In the congestion picing scheme, 13

14 on the othe hand, it is not just taffic capacity that the oad manage is equied to know; it is obvious fom the discussion above that the oad manage cannot calculate an appopiate toll patten that pevents congestion without having accuate infomation on the uses behavios (i.e., pecise demands). Consideing the dfeences between the two schemes, the desiable tanspotation demand management scheme must be found. In genealized tems, this becomes the poblem of compaing between quantity-based egulation and pice-based egulation. In the field of economics, thee have been many studies concening the compaison of the two egulation schemes. Accoding to the standad theoy (see, fo example, Weizman (1974), Laffont (1977)), quantity-based egulation poduces moe efficient outcomes than pice-based egulation a egulation authoity has only impefect infomation on the demand side conditions (i.e., demand functions) while having pefect infomation on the supply side conditions (i.e., supply functions). Fo the poblem in this pape, we obtain a simila conclusion, although the undelying assumptions of ou poblem ae dfeent fom those in conventional economic theoy. We show this fact below in moe concete tems. In ou poblem, what coesponds to supply side conditions is the netwok pefomance (taffic capacity of each link), fo which infomation can be obtained accuately with elatively small effot. On the othe hand, the infomation on the demand side conditions is the OD demand Q, the schedule cost function w (, the tavel time patten t and the value of time α. It is not easy fo the oad manage to obtain this infomation accuately. It is theefoe natual to think of the events that happen in the dynamic congestion picing scheme the demand side infomation is not pefect. As a simple example, suppose that the oad manage estimates the uses value of time as β, which is dfeent fom the tue value of time α. Unde this assumption, the oad manage solves (14)-(22), and then obtains a toll patten {p (} that is dfeent fom the tue optimal toll patten {p (}. It is obvious that the taffic patten {y (} aising unde the incoect toll patten {p (} does not minimize the objective function of [P-1] (i.e., the sum of the schedule cost and tavel cos, which causes economic loss in the social tanspotation cost. Wose still, thee is no guaantee that the capacity constaint in each link is satisfied in the taffic flow pattens {y (}; that is, queuing congestion may occu in the netwok. he eason is as follows: the manage believes that a taffic flow patten {y (} should aise assuming that the uses value of time is β and the toll patten is {p (}, which is detemined such that {y (} satisfies capacity constaint (22); but the flow patten aising in eality is {y (} (i.e., the flow patten that aises when the uses value of time is α and the toll patten is {p (}), and it follows that thee is no necessity that {y (} satisfies (22). Note hee that the social tanspotation cost in the optimal assignment poblem [P-1] is defined unde the assumption that queuing congestion neve occus. Consequently, in the congestion picing scheme, any incoect toll patten {p (} not only minimizes the sum of the schedule cost and tavel cost but also causes additional economic loss due to the occuence of queuing congestion. On the othe hand, the tadable pemit system neve poduces the above economic loss in theoy, the oad manage just has coect infomation on the link capacities. Of couse, thee may be the possibility in eality that appopiate (theoetically coec pemit pices do not occu in the tadable pemits maket. his coesponds to the possibility of setting the incoect toll patten {p (} in the congestion picing scheme. It may be said fom conventional wisdom that the degee 14

15 of mispicing in the makets is likely to be lowe than that of the authoity s mispicing. Futhemoe, even the degees of mispicing ae at a compaable level, thee is a signicant dfeence in the esulting effects of distoted pices between the two schemes. he effect of the distoted pices in the tadable pemit system is confined to the loss in only the pemit tading makets. hey ae ielevant to the occuence of queuing in the netwok, unlike the congestion picing scheme. his is because the definition of the pemit system ensues that the taffic flow neve exceeds the numbe of pemits issued, no matte how incoect the pemit pices ae; thus, the tadable pemit system neve poduces additional economic losses due to the occuence of queuing, as obseved in the congestion picing system. Still, one might ague that thee may be the possibility in eality that queuing congestion occus in the tadable pemit system the uses aival times at each link ae not accuate. Howeve, the possibility that such a phenomenon occus as well as the esulting losses ae all the same in dynamic congestion picing, too. Hence, this can be ignoed fom the standpoint of the elative compaison of these two schemes. hus, the discussion above can be summaized by the following Poposition: Poposition 3: Conside any netwoks with a single OD pai in which [P-1] has feasible solutions, and futhe suppose that the oad manage does not have pefect infomation about the tanspotation demands. hen, the equilibium assignment unde a system of dynamic congestion picing incus a lage total tanspotation cost than that in the equilibium unde a system of time-dependent tadable link pemits. 5. Self-financing Pinciple In ode fo the maket selling scheme of the tadabale netwok pemits to be socially acceptable, it may be desiable to edistibute the evenue fom selling the pemits to the oad uses. As a benchmak scheme of the edistibution, it is signicant to conside the case in which the evenue is used fo financing the capacity expansion of the netwok. Fo this type of edistibution scheme in congenstion picing, the self-financing pinciple has been well known since the pioneeing wok of Mohing and Hawitz (1962): fo a congestible facility with a constant long-un aveage cost, the evenue fom the optimal congestion picing exactly coves the cost of the optimal capacity. A consideable numbe of studies have been made on whethe o not this pinciple holds fo a vaiety of assumptions (see, fo example, Keele and Small (1977), Anott and Kaus (1995), Anott and Kaus (1998), Yang and Meng (22)). Accodingly, it is wothwhile to investigate whethe this pinciple applies to the equilibium evenue of tadable pemits. As the fist step of this investigation, Akamatsu (27a) showed that the pinciple also applies fo the tadable pemit scheme fo a single bottleneck. As we shall see below, this pinciple can be extended to the scheme of geneal netwoks. Conside the following minimization poblem [P-K]: min. ( q, y. μ) K ( μ) + F P ( q, y) (32) subject to (24), (25) and (26). his poblem is almost the same as [P-1] except that a) the unknown (contol) vaiables in [P-K] ae 15

16 not only the flow patten (q, y) but also the link capacity patten μ that is assumed to be constant in [P-1], b) the investment cost K as a function of μ is added into the objective function F p of [P-1], which means that the social cost we would like to minimize is defined as the sum of the total tanspotation cost and the investment costs equied fo the inceases in netwok capacity. he investment cost function K is assumed to be homogeneous of degee 1 with espect to μ; that is, K( μ) K( μ ) = μ (33) μ L he optimality conditions of [P-K] ae simila to those of [P-1]. he only dfeence is in the addition of the following condition fo the optimality of link capacity μ: L L μ,, μ, L (34) μ μ whee L is the Lagangean function of [P-K], and the deivative with espect to μ is given as L μ K( μ) = μ p dt. (35) Substituting (35) into (34), and summing the fist equation of (34) fo all links, we obtain L K( μ ) = μ p dt, (36) whee we use the homogeneity popety (33) of function K. he left-hand side of (36) is the investment cost equied fo achieving the optimal link capacity patten. he ight-hand side of (36) is the total maket value of the link pemits at equilibium unde the tadable netwok pemit system; this can be seen fom the fact that, fo any capacity patten, the optimal Lagange multiplie {p (} of [P-1] coincides with the equilibium pemit pices, as shown in Section 3. heefoe, (36) means that the self-financing pinciple holds fo the tadable netwok pemit scheme. his can be summaized as follows: Poposition 4: Conside any netwoks with a single OD pai in which [P-1] has feasible solutions, and futhe suppose that the investment cost function K(μ) is homogeneous of degee 1 in link capacities μ. Let the optimal investment cost be the cost equied fo inceasing the link capacities so as to minimize the social cost defined in (32). hen the optimal investment cost is equal to the total maket value of bottleneck pemits (i.e., total evenue of the oad manage); that is, the self-financing pinciple holds fo the tadable netwok pemits system. 6. Concluding Remaks his pape consideed a system of bottleneck tadable pemits fo geneal netwoks ( tadable netwok pemits ) with a single OD pai. We fist povided a model that descibes timedependent flow pattens at equilibium unde the poposed system. We then evealed that the equilibium coincides with the optimal assignment patten that minimizes the social tanspotation cost (i.e., the equilibium unde the poposed system is efficien. We futhe showed the theoetical elationship between the tadable pemit system and congestion picing: we 16

17 demonstated the definite advantages of the tadable pemit system ove the congestion picing system when the demand infomation is not pefect, wheeas they ae equivalent fo the pefect infomation case. Finally, we poved that the self-financing pinciple holds in the poposed scheme (i.e., the total maket value of the pemits is equal to the investment cost equied fo inceasing the netwok capacity up to a socially optimal level). We showed the efficiency of the tadable netwok pemits system fom the viewpoint of aggegated social costs (i.e., minimization of total tanspotation cos. his esult, howeve, does not necessaily mean that the intoduction of the poposed system always leads to Paeto impovement. In a single bottleneck case, the Paeto impovement can be poved since thee is a one-to-one coespondence between the queuing delay (at equilibium without the system) and the pemit pice (at equilibium with the system). Fo netwoks with many bottlenecks, thee ae complex inteactions among queuing delays at equilibium without the system (see, fo example, Kuwahaa and Akamatsu (1993), Akamatsu (21), Akamatsu and Heydecke (23)); the coespondences between the equilibium queuing delays and the pemit pices ae not staightfowad. Futhe exploation on the Paeto impovement popety of tadable netwok pemits is one of the impotant issues that should be addessed in futue eseach. he esults of this pape can be extended in vaious diections. Fo instance, it is staightfowad to genealize the model to allow fo many-to-many OD pattens. In that case, we again find that the equilibium is efficient in the sense that the total tanspotation cost in the netwok is minimized. Anothe staightfowad extension is to conside elastic OD demands. It can be shown that the efficiency of the tadable netwok pemit system applies to this case, too. In addition to the extensions on the demand side, we can also genealize the theoy to include supply side conditions: the equilibium pemit pices can be exploited fo obtaining dynamic capacity allocation policies (e.g., signal contol policies) that minimize the total tanspotation cost. Moe discussion of these issues can be found in Akamatsu (27b). Acknowledgement: he eseach was patially suppoted by the Japan Society fo the Pomotion of Science, Gant-in-Aid fo Exploatoy Reseach