A model for market share distribution between high-speed and conventional rail services in a transportation corridor

A model for market share distribution

between high-speed and conventional rail services

in a transportation corridor

Chaug-Ing Hsu *, Wen-Ming Chung **

Department of Transportation Engineering and Management, National Chiao Tung University, Hsinchu, 1001 Ta Hsueh Road, Taiwan, 30050, Republic of China (Fax: 03-5720844) Received: June 1996 / Accepted: December 1996

The research fund through grant NSC 85-2211-E-009-021 from the National Science Council, R.O.C. is acknowledged.

Abstract. High-speed rail (HSR) lines are usually planned to serve corri-dors with existing conventional rail (CR) lines, since these corricorri-dors typi-cally have large markets concentrated around major cities. This paper for-mulates a new analytical model to estimate market shares of HSR and CR in a fundamental way, and from an individual behavior point of view. Pas-sengers are divided into those who can take an HSR train directly to their destination stations and those who cannot. Optimal route choices are as-sumed by minimizing the “generalized total travel time”. The relationship among demand-supply attributes such as value of time, train departure time, speed, trip length and fares is explored to identify market boundaries by comparing different routing strategies for each type of passenger. Individual route choices are aggregated by accumulating a transformation probability density function of value of time to estimate the spatial distribu-tion of markets for two types of rail lines. The result estimates detail mar-ket distributions for passengers alighting at stations along the corridor. HSRs are shown to best serve medium- to long-trip markets, while CRs are shown to serve best for commuter travel and as feeders for the HSRs.

1. Introduction

The excellent performance of high-speed rail (HSR) lines is a significant improvement over the disadvantage of conventional rail (CR) lines for speed. However, HSRs are usually planned for corridors presently served by CRs, since these corridors typically have large markets around and

be-* Associate Professor, corresponding author ** Graduate Research Assistant

tween major cities. Therefore, addressing the problem of how rail passen-gers will make use of the two rail systems becomes important. This paper develops a new analytical model for exploring this problem in a fundamen-tal way. The problem addressed here involves both mode-choice and route-choice problems. Passengers may choose between the two rail lines for dif-ferences in speed and fare, or use both rail services to complete a single in-tercity trip. For instance, CR may compete with HSR in short to medium range markets to serve passengers whose destinations can be reached direct-ly via CR or HSR, or complement HSR as a collection/distribution mode in medium to long range markets in serving passengers whose destinations can’t be reached directly via HSR alone.

The traditional approach to transportation planning for path ridership dis-tribution or transit route choice usually assumes that the traveler’s choice is based on the waiting time and the riding time (e.g., Spiess 1983; Jansson and Ridderstolpe 1992). Individual differences in fares and values of time are usually ignored in transit-route-choice problems. However, since the con-struction cost of HSR line is high and must usually be recovered by expensive fares on travel, and substantial riding time saved due to the high speed of HSR service is the most important factor affecting traveler’s choice, a tradeoff be-tween time and fare differences bebe-tween HSR and CR exists. Furthermore, this tradeoff may be perceived differently by a variety of passengers who have different values of time. Thus, traditional transit-route-choice models are probably not appropriate for solving this problem.

On the other hand, HSR ridership is forecasted intensively in the literature by using multinomial logit or nested logit models that focus on estimating the entire market for HSR, as compared with all other existing modes (e.g., Brand et al. 1992; Mandel et al. 1994). Instead of using a logit model to estimate the entire market, this paper develops a new analytical model for estimating de-tailed market shares of HSR and CR along different segments of a rail corri-dor. This model incorporates rail passengers’ mode choices as well as route choices into a framework by exploring the relationship among key variables of concern. Passengers are divided into two types according to whether or not they can take an HSR train directly to their destinations. Passengers’ optimal route choices are assumed by minimizing their generalized travel time, which is composed of access/egress time, waiting/transfer time, riding time, and fare conversion time. Instead of assuming an “average” value of time, which is commonly used in route choice or zonal design literature for transit or rail corridors (e.g., Wirasinghe and Seneviratne 1986; Furth 1986; Ghoneim and Wirasinghe 1987; Janjsson and Ridderstople 1992), this paper treats an individual’s value of time as a random variable and uses it to convert fare into an individual-dependent fare time. Passengers’ route choices are formu-lated as depending on trip distance, value of time, departure time, and fare and service characteristics of HSR and CR services. The relationships among these variables are explored to identify market boundaries by comparing dif-ferent routing strategies for each type of passenger. Consequently, individual route choices are aggregated so as to estimate the distribution of market shares along different segments of the corridor.

2. Basic assumptions

As in many theoretical modeling studies, this model will initially be formu-lated as a rather idealized one. However, while intentionally ignoring some factors to obtain mathematical tractability, we have tried to retain enough of the salient features for a reasonable approximation of the real problem. In Sect. 6, we discuss and describe the computational consequences of re-laxing some assumptions.

We assume HSR and CR trains travel the full length of the corridor, and make, respectively, all HSR and CR stops. In reality, HSR and CR trains may not stop at all their stations. In Sect. 6.3, we will describe how to re-lax this assumption. Wherever there are HSR stations they are assumed to be joined to CR stations, so passengers may transfer easily between the two rail services. The passengers are assumed to be well informed about departure times, itineraries and farces associated with all routes, either in advance or after their arrival at departure stations. The riding time for a train traveling between two consecutive stations with a spacing of L, T can be found in studies that use similar assumptions (e.g., Campbell 1993, Cook and Russell 1980, etc.) and is expressed asT ˆ L=V ‡ V=a, where a ˆ V2_{=2 d. In this expression, the train is assumed to accelerate from the}

station over a distance d at an average acceleration of a, cruise over a dis-tance of L ÿ 2 d after reaching cruise speed V, decelerate over a distance d at an average rate of a, and then stop at the next station.

Following the above expression for T, the riding time for traveling in a HSR train between two consecutive HSR stations,t_h, can be expressed as L=Vh‡ Vh=a. Similarly, the riding time for traveling in a CR train

be-tween two consecutive CR stations,t_r, isl=V_r‡ V_r=b. In these two expres-sions, L and l are average station spacings,V_h and V_r are cruise speeds, a and b are average rates of acceleration, respectively, for HSR and CR trains. To simplify mathematical expressions, we assume spacings between any two consecutive stations are equivalent. This assumption will be re-laxed in Sect. 6 so as to capture the real problem reasonably.

Figure 1 illustrates the station labels used in this paper. Along the HSR corridor, we assume there are n ‡ 1 stations, labeled as i=0,1,...,n. All of these stations are joint HSR-CR stations. Using these stations as markers, the CR line is divided into n non-overlapping zones. Each zone is labeled with a number which is the same as that of the HSR-CR joint station lo-cated at the right end of the zone. Based on the equivalent spacing assump-tion, there are nL_r…ˆ L=l† ÿ 1 CR stations, labeled as j ˆ 1 ,...; nL_r ÿ 1, in each zone.

Passengers are assumed to select routes so as to minimize their general-ized travel times. The generalgeneral-ized travel time includes a variety of time components, i.e. access time, waiting time, riding time, transfer time, and egress time, and fare (converted to time according to a formula of value of time). We classify passengers as Type I passengers who may ride either HSR or CR trains directly to their destination stations from which they then walk or use modes of transportation other than rail services to reach

their final destinations, and Type II passengers who cannot ride HSR trains directly to their destination stations, but who may either take an HSR train to a joint station, then transfer to a CR train that progresses or backtracks to their destination stations, or take a CR train directly to their destination stations. That is, the destinations of Type I passengers have more-or-less di-rect HSR train services without any transfers between the two rail services, while those of Type II passengers do not.

For a Type I passenger, whose origin and destination station are, respec-tively, station 0 and station i, the generalized travel time for taking an HSR train alone,T_H, and the generalized travel time for taking a CR train alone, TR, are given by:

TH ˆ ta‡ thw‡ i th‡ …i ÿ 1†ts‡ te‡ thp …1†

TRˆ ta‡ trw‡ i nLrtr‡ …i nLr ÿ 1†ts‡ te‡ trp …2†

where t_a and t_e are, respectively, access and egress times for origin and destination, which are joint HSR-CR stations, th_w and tr_w are waiting times for HSR and CR trains,t_s is the average stop time for any station,t_h is the riding time in a HSR train between two consecutive joint stations,t_r is the riding time in a CR train between two consecutive CR stations, th_p and tr_p are fare conversion times for taking HSR and CR trains on this trip.

For a Type II passenger, whose origin and destination stations are, re-spectively, station 0 and the jth CR station in zone i, the generalized travel time for taking an HSR train to the i–1th station, then transferring to a CR train that progresses to the jth station of zone i,T_H1, can be expressed as

TH1ˆ ta‡ thw‡ …i ÿ 1†th‡ …i ÿ 2†ts‡ thp1‡ ‰trw…iÿ1†‡ j tr

‡ …j ÿ 1†ts‡ trp1Š ‡ te …3†

Similarly, for the same passenger, the generalized travel time for taking an HSR train to the ith station then transfer to a CR train that backtracks to the jth station of zone i,T_H2, is

TH2ˆ ta‡ thw‡ i th‡ …i ÿ 1†ts‡ thp2

‡ ‰t0r

wi‡ …nLr ÿ j†tr‡ …nrLÿ j ÿ 1†ts‡ trp2Š ‡ te …4†

In (3) and (4), tr_w…iÿ1† and t0r_wi are the transfer times at station i and station

i–1, andtr_p1 and tr_p2 are the fare conversion times, respectively, for traveling in a CR train from joint station i–1 to station j of zone i, and from joint station i to station j of zone i. If this Type II passenger directly takes a CR train to the jth station of zone i, then the generalized travel time,T_R0, is

T0

Rˆ ta‡ trw‡ ‰…i ÿ 1†nLr ‡ jŠtr‡ ‰…i ÿ 1†nLr ‡ j ÿ 1Šts‡ trp‡ te …5†

The generalized travel time formulation in (1)–(5) do not assume different weights for various time components in order to simplify mathematical ex-pressions. This assumption is discussed in Sect. 6.

3. Individual route choices and market shares for Type I passengers We assume that a Type I passenger selects his route so as to minimize the generalized travel time. That is, the choice of traveling in a high-speed train depends on whether or not T_H  T_R. Let T_Hÿ T_Rˆ 0, then, from (1) and (2), it is obtained that

THÿ TR ˆ …thwÿ trw† ‡ i…th‡ ts† ÿ i nLr…tr‡ ts† ‡ …thpÿ trp† ˆ 0 …6†

Lett_dˆ th_wÿ tr_w, which stands for the waiting time differences between the latest HSR train and the latest CR train which depart after the passenger’s arrival at joint station 0; then, t_d> 0 represents the latest departure train is a CR train, t_d< 0 represents the latest depature train is a HSR train. Let th

L ˆ th‡ ts and trl ˆ tr‡ ts; suppose dh and dr represent, respectively, the

unit distance fares for HSR and CR trains, and c represents the value of time of an individual passenger, then th_pÿ tr_pˆ …i L d_hÿ i nL_rl d_r†= c ˆ i L…dhÿ dr†=c: Let K ˆ L…dhÿ dr†=c, then (6) can be rewritten as:

td ˆ i…nLrtrl ÿ thLÿ K† …7†

LetX ˆ nL_rtr_lÿ th_Lÿ K, then, if t_d iX, i.e. T_Hÿ T_R 0, so, passengers will choose to travel via HSR, otherwise they will choose to travel via CR. (7) shows that for a passenger with a specific value of time, i.e. a specific

K value, the travel distance, which is represented by destination label i, is

the most important factor affecting the choice between two kinds of trains for a givent_d value and given riding times.

As travel distance i increases, the difference in fare conversion time, iK, and the difference in total riding time between these two trains, i …nL

rtrlÿ thL†, will simultaneously expand, while the waiting time

differ-ence, t_d, holds, thereby yielding a relatively smaller effect on the passen-ger’s choice. In other words, the extent of t_d’s effect on the choice be-tween HSR and CR trains depends on the travel distance. The value of time of an individual passenger is also an important factor affecting the choice providing other variables in (7) are held constant. When passen-gers’ values of time are higher, K is smaller and X is larger, they will then tend to choose HSR trains; otherwise, they will tend to choose CR trains.

In reality, each passenger has a value of time,c, so c can be viewed as a random variable by considering the market of interest as a whole. c usually varies from individual to individual in light of differences in socioe-conomic characteristics, such as income and trip purpose. c reflects how passengers weigh fares with travel time. Normally, c cannot be negative although its value can be very small. Letc be a random variable with con-tinuous type, having the probability density function (p.d.f.), f_c…c†, in the space fc ; 0 < c < 1g and 0 elsewhere. The probability distribution of c may be different for various markets of interest. For instance, passengers in North America and Japan may have different weights of fares and travel time. The estimation for probability distribution ofc based on actual data is necessary for future application in specific corridor. In the following discus-sion, we assumec for all passengers in the same corridor of interest belong to the same probability distribution without loss of generosity.

X is also a random variable transformed from value of time, c, because

all variables other than c are exogenous in the definition of X. Those Type I passengers who depart from joint station 0 and are confronted with the same t_d will have different probability distributions of iX that vary as the travel distance i increases, as shown in (7) and illustrated in Fig. 2. Let XI

i ˆ iX; ÿ1 < XiI < i…nLrtrl ÿ trL† then the p.d.f. of XiI, fi…XI†, can be

written as: fi…XI† ˆ fc  _ÿiC 2 XI _{ÿ iC} 1 
iC2 …XI _{ÿ iC1†}2 …8†

whereC₁ˆ nL_rtr_l ÿ th_L, and C₂ˆ L …d_hÿ d_r†. The derivations of (8) are de-tailed in Appendix A.

Passengers departing from the same station 0 but alighting at different destinations i, i = 1, 2, 3, belong to various curves with different probability density functions as shown in Fig. 2. For a givent_d, passengers whose des-tinations are station 1 will choose to travel in a HSR train if their X_iIs are larger thant_d, i.e. they belong to the X curve and are beyond t_d as shown in Fig. 2 a. On the other hand, if theirX_iIs are smaller than t_d, i.e. they be-long to the X curve but do not exceed t_d as shown in Fig. 2 a, then they will choose to travel in a CR train. Passengers who alight at station i,

It is shown in Fig. 2 b that when t_d is negative, i.e. the latest departure train is a HSR train, it is very hard for CR to attract passengers. Conse-quently, for a given t_d, the proportion of Type I passengers, who alight at station, i, i = 1, ..., n, and choose a CR train,CIi_{, can be obtained from:}

CIi ˆ

Ztd

ÿ1

fi…XI† dXI for i ˆ 1 ; ::: ; n …9†

and the proportion of Type I passengers who alight at station i and choose an HSR train is1 ÿ CIi_.

We can use departure schedules for HSR and CR trains departing from station 0 in a day as cut points and divide a day into many time zones la-beled s, s = 1, ..., 10, ..., as shown in Fig. 3. All passengers who arrive at sta-tion 0 during time zone s are confronted with the same waiting time

ence t_ds. For example, passengers arriving during time zone 1 are con-fronted witht_d1ˆ th₂ÿ tr₁, and t_d1> 0; and passengers arriving during time zone 2 are confronted with t_d2ˆ th₂ÿ tr₂, and t_d2< 0, and so on. Conse-quently, we can calculate CIi _and 1 ÿ CIi _{for passengers confronted with}

differentt_ds using similar analysis of Fig. 2 and (9).

4. Route choices and market shares for Type II passengers

The destination stations of Type II passengers are served by CR alone, so they may either take CR trains directly or use an HSR train and a CR train that progresses or backtracks to their destination stations. We now examine all combinations of these routes by comparing two routes at a time, then in-tegrate these results so as to arrive at overall market shares for each of the three routes.

4.1. HSR and progressive CR vs. CR alone

LetT_H1ÿ T_R0 ˆ 0, then, from (3) and (5) in Sect. 2, we obtain TH1ÿ TR0 ˆtd‡ …thp1‡ trp1ÿ trp† ‡ trw…iÿ1†‡ …i ÿ 1†thL

ÿ ‰…i ÿ 1†nL

r ‡ j Štrl ÿ ts ˆ 0 …10†

Since th_p1‡tr_p1ÿtr_pˆ‰…iÿ1†L d_h‡j l d_rÿ……iÿ1†nL_rÿj†l d_rŠ=cˆ…iÿ1†K, (10) can be simplified to:

td‡ trw…iÿ1†ÿ ts ˆ …i ÿ 1†…nrLtrlÿ thLÿ K† ˆ …i ÿ 1† X …11†

(11) shows that Type II passengers who alight at station j in zone i are con-fronted with tr_w…iÿ1† transfer time at joint station i–1, and with a c trans-formed variable distributed with (i–1) X as shown in Fig. 4, so their route choices will depend on travel distance and transfer time. For any given t_d, the optimal routes for these Type II passengers can be found by comparing their (i–1) X values tot_d‡ tr_w…iÿ1†ÿ t_s. Specifically, those passengers whose …i ÿ 1† X > td‡ tr_w…iÿ1†ÿ ts will choose to take an HSR train to station i–1 and then transfer to a CR train progressing towards their destinations, while those passengers whose…i ÿ 1† X < t_d‡ tr_w…iÿ1†ÿ t_s will take CR trains di-rectly to their destinations.

Normally, the location oft_d‡ tr_w…iÿ1†ÿ t_s will move towards the left with an increase of travel distance i as shown in Fig. 4, supposetr_w…iÿ1† does not vary much with i. That is, for longer trips, the time savings of first choosing an HSR train for long-haul travel and transferring to a CR train to local des-tinations are comparatively higher than those of choosing conventional trains alone. In these cases, HSR trains are more attractive than CR trains. Assume XII

i ˆ …i ÿ 1†X; and ÿ 1 < XiII < …i ÿ 1†…nLrtrl ÿ trL†, then, from

Appen-dix A, the p.d.f. ofX_iII,f_i…XII†, can be written as:

fi…XII† ˆ fc  ÿ…i ÿ 1†C2 XII_{ÿ …i ÿ 1†C1} 
…i ÿ 1†C2 …XII_{ÿ …i ÿ 1†C1†}2 …12†

where C₁ˆ nL_rtr_l ÿ th_L, and C₂ˆ L…d_hÿ d_r†. Consequently, in situations where only two routings, i.e. traveling via CR alone and traveling via HSR and progressive CR, are chosen, the proportion of Type II passengers who alight at station j of zone i, i = 2, ..., n, and choose to travel via CR alone, CIIi_{, can be written as:}

CIIi _ˆ

Z

td‡tr_w…iÿ1†ÿts

ÿ1

fi…XII†dXII for i ˆ 2; ::: ; n …13†

and the proportion of Type II passengers who choose to travel via HSR and progressive CR is 1 ÿ CIIi_{. In zone 1, it is noted here that passengers}

are unable to travel via HSR and progressive CR.

4.2. CR alone vs. HSR and bracktracking CR

Similarly, letT_H2ÿ T_R0 ˆ 0, then, from (4) and (5) in Sect. 2, we obtain TH2ÿ TR0 ˆ td‡ …thp2‡ trp2ÿ trp† ‡ trwi‡ i thLÿ ‰……i ÿ 1†nLr ‡ j†

ÿ …nL

r ÿ j†Štrl ÿ ts ˆ 0 …14†

Since th_p2‡ tr_p2ÿ tr_pˆ ‰i L d_h‡ …nL_r ÿ j†l d_rÿ ……i ÿ 1†nL_r ‡ j†l d_rŠ=c ˆ i K ‡ 2‰…i ÿ 1†nL

r ‡ jŠdr=c, then (14) can be simplified to:

…td‡ t0rwiÿ ts† ‡ 2…nLr ÿ j†  l d_cr‡ tr l  ˆ i …nL rtrl ÿ thLÿ K† ˆ i X …15†

Random variablesc and X are respectively in left- and right-hand sides of (15), so markets resulting from the comparison between travel via CR alone and via HSR and backtracking CR can’t be analyzed using X distribution alone. LetY ˆ …ldr

c ‡ trl†, and trl < Y < 1, since X ˆ nLrtrl ÿ thLÿ K,

there-fore X is an increasing function ofc and Y is a decreasing function of c. Any passenger who has a specificc value will also have specific X and Y values. The p.d.f. of Y, f (Y), can be derived from the theorem shown in Appendix A. That is: f …Y† ˆ fc  l dr Y ÿ tr l 
l dr …Y ÿ tr l†2 …16†

X and Y are both random variables transformed fromc, though distributed

with different parameters. The proof in Appendix B shows that if c₁> c₂, then it is true forX₁> X₂, andY₁< Y₂. Letc0 stand for ac value, and X' stand for its corresponding X value. The position of X' in the X curve is then the same as that of c0 in the c curve as shown in Fig. 5a and b. On the other hand, the position of Y', the corresponding Y value, is on the op-posite side of the Y curve as shown in Fig. 5 c.

From (15), if the (X, Y ) values of Type II passengers, who alight at sta-tion j in zone i, makes …t_d‡ t0r_wiÿ t_s† ‡ 2…nL_r ÿ j†Y >_{ˆ i X held, they will} then choose a CR train directly to their destinations; otherwise, they will choose to travel via an HSR train and a backtracking CR train. The market boundary along the corridor for these two routing strategies can be

esti-mated by finding the (X, Y ) values that satisfy the equality of (15). Let … X; Y† be …X; Y† values that satisfy the equality condition of (15). Then

…td‡ t0rwiÿ ts† ‡ 2 …nLr ÿ j† Y ˆ i X …17†

From the definitions of X, and Y, we know 

X ˆ …nL_rtr_lÿ th_L† ÿ L…dhÿ dr†
1=c …18†



Y ˆ l dr
1=c ‡ trl …19†

Substitute 1=c ˆ … Y ÿ tr_l†=l d_r derived from (19) into (18), rearrange, and solve the simultaneous equations for X and Y, Y can then be solved for by:

 Y ˆi……nLrtrlÿ thL† ‡ d trl† ÿ …td‡ t0rwiÿ ts† 2 …nL r ÿ j† ‡ i d ˆ Y i j …20†

where d ˆ L…d_hÿ d_r†=l d_r. The value of Y in (20) will change as destina-tion stadestina-tion j in zone i varies, so we use Y_ji to represent Y value in (20). As shown in Fig. 6, for any Type II passengers who alight at station j in zone i, if their Y values are greater than Y_ji, then their X values are smaller than X. That is

…td‡ t0rwiÿ ts† ‡ 2 …nrLÿ j†Y > …td‡ t0rwiÿ ts†

‡ 2 …nL

r ÿ j† Yjiˆ i X > i X …21†

Therefore, these passengers will choose to travel in CR trains directly to their destination stations. On the other hand, passengers whose Y values are smaller than Y_ji will have X values greater than X. That is

…td‡ t0rwiÿ ts† ‡ 2 …nrLÿ j†Y < …td‡ t0rwiÿ ts†

‡ 2 …nL

r ÿ j† Yjiˆ i X < i X …22†

therefore, these passengers will choose to travel via HSR trains and then transfer to CR trains that backtrack to their destination stations. In the same zone i, the value of Y_ji decreases as j decrease as shown in (20). In other words, among all passengers whose destination stations are in zone i, those

alighting at stations closer to the origin station 0, i.e. with smaller j values, are more likely to choose conventional trains, while those alighting at sta-tions closer to joint station i, are more likely to travel in an HSR train to station i, then transfer to a CR train that backtracks to their destination sta-tions. Similarly, in situations where only two routings, i.e. CR alone vs. HSR and backtracking CR, are compared, the proportion of Type II passen-gers who alight at any station j of zone i, i = 1, ..., n and choose CR trains alone,CIIi

1j, can be calculated by:

CIIi 1j ˆ Z1  Yi j f …Y†dY for i ˆ 1; :::; n …23†

Normally, the market for traveling beyond desired destinations in an HSR train and then backtracking via a CR train is competitive only when the to-tal travel distance is long enough and transfer time is short. In reality, sta-tions like this are usually located in large metropolitan areas with strong de-mand, thereby allowing small headway and waiting time. As noted before, passengers alighting at station j in zone 1 are unable to travel via HSR and progressive CR, so the proportion of these passengers, who choose to trav-el via HSR and backtracking CR can be calculated as1 ÿ CII1

1j.

4.3. HSR and progressive CR vs. HSR and backtracking CR

Following the analyses described above, letT_H2ÿ T_H1 ˆ 0, then TH2ÿ TH1ˆ …thL‡ t0rwi‡ tsÿ tw…iÿ1†r † ‡ …thp2‡ trp2ÿ thp1ÿ trp1†

ÿ ‰…nL

r ÿ j† ÿ jŠl trl ˆ 0 …24†

Since …th_p2‡ tr_p2ÿ th_p1ÿ tr_p1† ˆ ‰L d_h‡ …nL_r ÿ j†l d_rÿ j l d_rŠ=c ˆ K ‡ 2 …nL_rÿ j† dr=c, therefore (24) can be simplified to:

…t0r wi‡tsÿ trw…iÿ1††‡2 …nLr ÿ j†  l d_cr‡ tr l  ˆ …nL rtrlÿ thLÿ K† ˆ X …25†

The form of (25) is similar to that of (15), so we may solve for Y_j0i (similar to Y_ji in the preceding subsection) by applying the same derivations. Thus,

 Y0i j ˆ ……nL rtrl ÿ thL† ‡ d tlr† ÿ …t0rwiÿ trw…iÿ1†‡ ts† 2 …nL r ÿ j† ‡ d …26†

Similarly, in situations where only two routing strategies, i.e. HSR and pro-gressive CR vs. HSR and backtracking CR, are compared, the proportion of Type II passengers who alight at any station j of zone i, i = 2, ..., n, and

choose to travel via HSR and progressive CR, CIIi 2j, is an integral similar to (23) CIIi 2j ˆ Z1  Y0i j f …Y† dY ; i ˆ 2 ; ::: ; n …27†

Consequently, in these situations, the proportion of Type II passenger who choose to travel via HSR and backtracking CR is1 ÿ CII_2ji, i = 2, ..., n.

4.4. Synthesis

Type II passengers may have three routing alternatives. We have just ana-lyzed their markets by comparing two of them each time as described in the preceding three subsections. CIIi_, CIIi

1j, and CII2ji were derived in these

analyses. Appendix C presents graphical proof and rules for integrating the

results of these separate analyses. IfCIIi > CIIi

1j, passengers alighting at

sta-tion j of zone i are proven to travel via CR alone or via HSR and back-tracking CR, and there is no possibility of traveling via HSR and progres-sive CR. The proportions of passengers chosing these two routes are, re-spectively,CII_1ji and1 ÿ CII_1ji as shown in Fig. 7 a.

As shown in Appendix C, if CIIi < CIIi

1j, then it is true for C2jIIi > CII1ji,

i.e. CIIi < CIIi

1j< CII2ji, then passengers alighting at station j of zone i have

all three routing alternatives available.CIIi_,CIIi

2j ÿ CIIi, and 1 ÿ C2jIIi are,

re-spectively, the proportions of passengers who choose to travel via CR alone, HSR and progressive CR, and HSR and backtracking CR, as shown in Fig. 7 b. Therefore, the markets for Type II passengers can be estimated by calculating CIIi_, CIIi

1j, and C2jIIi, and applying the rules in Appendix C.

Consequently, an overall market analysis for passengers originating at joint station 0 and alighting at different stations along the rail corridor will be completed by combining results of analyses for both Type I and Type II passengers.

Table 1. Values of parameters

Parameter Symbol Value

Number of zones of CR line divided by all HSR stations n 9

Number of CR stations in a zone mi 7

Average spacing between two consecutive HSR stations L 40 km Average spacing between two consecutive CR stations l 5 km

Cruise speed of HSR trains Vh 250 km/h

Cruise speed of CR trains Vr 110 km/h

Rate of acceleration of HSR trains a 1.736 km/min2

Rate of acceleration of HSR trains b 1.44 km/min2 Riding time between two consecutive HSR stations

(including train stopping time)

th

L 14 min

Riding time between two consecutive CR stations

(including train stopping time) t

r

l 6 min

Unit distance fare for HSR trains dh 0.1 $/km Unit distance fare for CR trains dr 0.05 $/km

Average stop time for a station ts 2 min

Waiting time for a progressive CR train at the joint station tr_w 15 min Waiting time for a backtracking CR train at the joint station t0r_w 15 min The mean of passenger’s time value

(assuming normal distribution)

c 0.09 $/min The standard deviation of passenger’s time value

(assuming normal distribution)

5. Example and sensitivity analysis

Along the west corridor of Taiwan, there is an existing CR line. A new HSR project has been proposed in the same corridor to connect the termi-nal stations of Taipei and Kaohsiung. In between, there will be eight inter-mediate stations. The models developed in this research were applied to

Table 2. Market shares for Type I passengers alighting at station i

Station i td= –10 min td= +10 min

(CR, HSR) (CR, HSR) 1 (0.000021, 0.999979) (0.476267, 0.523733) 2 (0.000259, 0.999741) (0.056067, 0.943933) 3 (0.000625, 0.999375) (0.023429, 0.976571) 4 (0.000978, 0.999022) (0.014958, 0.985042) 5 (0.001281, 0.998719) (0.011400, 0.988600) 6 (0.001535, 0.998465) (0.009504, 0.990496) 7 (0.001747, 0.998253) (0.008345, 0.991655) 8 (0.001926, 0.998074) (0.007568, 0.992432) 9 (0.002077, 0.997923) (0.007014, 0.992986)

Table 3. Market shares for Type II passengers alighting at station j of zone i

Zone i Station j td= –10 min td= +10 min

(CR, HSR+CR–Fa, HSR+CR–Bb) (CR, HSR+CR–F, HSR+CR–B) 1 1*5 (1, 0, 0) (1, 0, 0) 6 (0.970300, 0, 0.029700) (1, 0, 0) 7 (0.998300. 0, 0.001700) (1, 0, 0) 2 1*6 (0.000021, 0.999979, 0) (0.476400, 0.523600, 0) 7 (0.000021, 0.999944, 0.000035) (0.476400, 0.523565, 0.000035) 3 1*6 (0.000260, 0.999740, 0) (0.056060, 0.943940, 0) 7 (0.000260, 0.999705, 0.000035) (0.056060, 0.943905, 0.000035) 4 1*6 (0.000626, 0.999375, 0) (0.023430, 0.976570, 0) 7 (0.000626, 0.999339, 0.000035) (0.023430, 0.976535, 0.000035) 5 1*6 (0.000978, 0.999022, 0) (0.014960, 0.985040, 0) 7 (0.000978, 0.998987, 0.000035) (0.014960, 0.985005, 0.000035) 6 1*6 (0.001281, 0.998719, 0) (0.011400, 0.988600, 0) 7 (0.001281, 0.998684, 0.000035) (0.000060, 0.999905, 0.000035) 7 1*6 (0.001535, 0.998465, 0) (0.009504, 0.990496, 0) 7 (0.001535, 0.998430, 0.000035) (0.009504, 0.990461, 0.000035) 8 1*6 (0.001747, 0.998253, 0) (0.008345, 0.991655, 0) 7 (0.001747, 0.998218, 0.000035) (0.008345, 0.991620, 0.000035) 9 1*6 (0.001926, 0.998074, 0) (0.007568, 0.992432, 0) 7 (0.001926, 0.998039, 0.000035) (0.007568, 0.992397, 0.000035) a

Represents via HSR and progressive CR

b

this corridor by using the data shown in Table 1. Most of the data are based on service characteristics of the existing CR line and planned HSR line (but also slightly modified to fit the model assumptions). The value of time is based on a socioeconomic survey of Taiwan and assumed to be dis-tributed with a normal distribution with a mean of c ˆ 0:09 $/min and a standard deviation of r ˆ0.01 $/min. The model was programmed using Mathematica, and the results are summarized in Tables 2 and 3.

As shown in Table 2, when Type I passengers are confronted with td= 10 min, i.e., the latest CR train is 10 min earlier than the latest HSR

train, except those alighting at joint station 1, most other Type I passengers who alight at stations other than station 1 will choose the HSR train. In other words, CR competes with HSR only in markets with trip lengths of less than 40 km, or approximately the range of commuting distance. Simi-larly, as shown in Table 3, when Type II passengers are confronted with td ˆ 10 min, except for those alighting at CR stations in zones 1 and 2,

most will choose to travel via HSR and progressive CR. Only a few pas-sengers, who alight at the station closest to the next joint station, will choose to travel via HSR and backtracking CR. For t_dˆ ÿ10 min, CR will attract only Type II passengers in zone 1.

Type I and Type II passengers have similar market share patterns along the corridor, i.e., the market for joint station i is similar to that for zone i, so we will perform sensitivity analysis only for Type I passengers. Figs. 8– 12 show how market shares for HSR and CR are affected by changes in waiting time difference,t_d, HSR cruise speed, V_h, HSR unit distance fare, dh, the mean value of time of passengers, c, and the standard deviation of c, r. CR markets exist only for short travel distances when td is positive as

shown in Fig. 8. The improved speed and reduced fares of HSR have the

effect of expanding its market share, and to a lesser extent for longer trips, as shown in Figs. 9 and 10. Fig. 11 shows that CR still can attract a moder-ate to high market share for short to medium travel distances when the average passenger’s value of time is low. Finally, Fig. 12 shows that changes in the standard deviation of distribution of value of time also af-fect the market shares of CR and HSR but at a slower rate.

6. Extensions and algorithm

The market shares for HSR and CR along a transportation corridor can be analyzed by calculating and comparingCIi_, CIIi_, CIIi

1j, andCII2ji as shown in

the preceding sections. On the basis of these analyses, we will discuss and describe consequences of relaxing some important assumptions.

6.1. Relaxing equivalent spacing assumption

Suppose there are n+1 HSR and CR joint stations (including the origin sta-tion 0) spaced withL_i, i = 1, ..., n along the HSR line, and there are m_i CR stations (not including joint stations i–1 and i ) in zone i of the CR lines, spaced with l_ji, j = 1, ...,m_i‡ 1, then the riding time in an HSR train be-tween two consecutive joint stations, t_hi, is L_i=V_h‡ V_h=a, i=1,..., n and the riding time in a CR train between two consecutive CR stations of zone

i, ti_rj, is li_j=V_r‡ V_r=b, i=1,..., n, j=1,... m_i‡ 1. The values for t_hi and ti_rj can be obtained from timetables for existing HSR and CR lines. For planned HSR and CR lines, these values can be calculated by utilizing giv-enV_h,V_r,a and b values.

Though it looks complex in the analysis shown in Sects. 3 and 4, there are several critical equations which play key roles in this paper. The critical equations that define market boundaries between HSR and CR are (7), (11), (15), and (25). All four of these equations show a similar physical meaning. That is (waiting and transfer time differences) + (fare conversion time difference)= (Riding time difference) between two routings. In Appen-dix D, we modify these four equations so as to obtain revised CIi_, CIIi_,

CIIi

1j, andCII2ji of relaxing equivalent spacing assumption. Consequently,

fol-lowing the rules in Appendix C and synthesis description of Sect. 4, we can estimate market shares for different routes along the corridor by revised CIi_,CIIi_,CIIi

1j and C2jIIi:

6.2. Relaxing value of time-related assumptions

Traditional methods used in route choice and zonal design problems for transit or rail corridors usually consider an “average” value of time per per-son to estimate the generalized travel cost or the generalized travel time (e.g., Wirasinghe and Seneviratne 1986; Furth 1986; Ghoneim and Wira-singhe 1987; Jansson and Ridderstolpe 1992, etc.). Instead of using an average value of time assumption, this paper treats an individual’s value of time, c, as a random variable, and derives its relationship with trip length, and the resulting route and mode choice of the individual. The p.d.f. of c and otherc transformed variables are further used to accumulate individual route and mode choices so as to estimate markets for different HSR and CR routes. In this paper, we treatc as a random variable in order to recog-nize passengers have differences in the value of time due to variations in income, age, trip purpose, etc. However, passengers may weigh differently between waiting/transfer time and in-vehicle riding time, and between trav-eling in an HSR train and travtrav-eling in a CR train. The theoretical frame-work developed in this paper won’t change due to the incorporation of placing different weights on various time components. So, we may multi-ply the waiting/transfer time components by a relative ratio as compared with the value of time for in-vehicle riding time, and obtained from other empirical literature. The algorithm presented in Appendix E will show how to incorporate this consideration into the computation process.

6.3. Relaxing assumptions that trains make all stops and passengers board at station 0

In reality, HSR and CR trains may not stop at all stations. In these cases, we may just adjust joint station and zonal label n, and CR station label m_i to represent the station at which HSR and CR trains actually stop. Accord-ingly, joint station spacing, L_i, i = 1, ..., n, and CR station spacing l_ji,

j = 1, ...,m_i‡ 1, should also be adjusted. All other procedures may follow

those presented in Sect. 6.1, which describe the revisions for relaxing the equivalent spacing assumption.

The same procedure can also be applied in the analysis of markets for the two rail systems in cases where passengers board at other joint stations. The labels n, L_i, and li_j, are adjusted again, and all other procedures can follow those presented in Sect. 6.1. Furthermore, the model can be applied in analyzing the market of the two rail systems in cases where passengers and trains travel in the opposite direction by using station labels from n+1 to 1.

Finally, we present a simplified revision of an algorithm which shows how to operationalize the theoretical model presented in this paper in Ap-pendix E.

7. Conclusions

This paper develops a new analytical model for exploring how passengers make use of HSR and CR serving the same rail corridor. Passengers are vided into two types according to whether they can take an HSR train di-rectly to their destination stations or not. The route choices for each type of passenger are formulated to depend on the passenger’s departure time, value of time, trip distance, fare and the service characteristics of HSR and CR. Instead of assuming an average value of time for all passengers, this paper treats an individual passenger’s value of time as a random variable, and derives its relationship with the resulting rail route choice and market boundaries. The probability density functions ofc transformed variables are used to estimate market shares for different rail routes along various zones of the rail corridor. Theoretical modeling is operationalized and illustrated through an example. HSR is shown to serve most medium- to long-trip markets and CR is shown to serve commuter trip markets and collection/ distribution markets for HSR. Extension for relaxing some model assump-tions are discussed and a simplified algorithm is presented.

The new model may have several advantages over other simpler modal choice or route choice models. First, the model integrates rail passengers’ mode choice and route choice into a framework, thus should help to better describe the actual phenomenon. Second, the model introduces new con-cepts such as time zones with the same waiting time differences, and the probability density functions of c transformation variables and apply these concepts as a way to aggregate individual choices of passengers with

differ-ent departure times and values of time. Though, the theoretical derivation of the model is technical and long, the application of the model is simple as shown in parameters listed in Table 1 and algorithm in Appendix E. Only estimation for statistical distribution of value of time requires addi-tional efforts, while other parameter values are easy to collect. Therefore, this approach has the advantage of application simplicity as compared to the calibration process and aggregation problem in conventional logit mod-el. Third, the model has potential application beyond the HSR and CR amined here and could be worth examining further. The results of the ex-ample illustrated in the paper could be valid only for the west corridor in Taiwan, and the findings are exploratory. Estimates for statistical distribu-tion of value of time and other input values should be based on actual data in future application of the model in other corridors of interest.

Appendix A

Theorem. Let X be a random variable of the continuous type having p.d.f. f (x). Let A be the one-dimension space where f(x)>0. Consider the ran-dom variable Y = u (X), where y = u (x) defines a one-to-one transformation that maps the set A onto the set B. Let the inverse of y=u(x) be denoted by x = w (y), and let the derivative dx/dy = w'(y) be continuous and not van-ish for all points y inB. Then the p.d.f. of the random variable Y=u(X) is given by

g …y† ˆ f ‰w…y†Š jw0_{y†j y 2 B ;}

ˆ o ; elsewhere :

It is given thatX ˆ …nL_rtr_lÿ th_L† ÿ L …d_hÿ d_r†=c, where c is a random vari-able and all other varivari-ables are exogenous in the definition of X. Let C1ˆ nLrtrl ÿ tLh and C2ˆ L …dhÿ dr†, then the p.d.f. of X, fX…x†, can be

calculated as: * c ˆ ÿC2=XIÿ C1; dc ˆ C2 …XI _{ÿ C1†}2dx …A-1† ) fX…X† ˆ fc  ÿ_{X ÿ C}C2 1 
C2 …X ÿ C1†2 …A-2† Let XIi ˆ iX ˆ iC

1ÿ iC2=c, then the p.d.f. of XIi, fXIi…XIi†, can be

ob-tained directly from (A-2): fXIi …XIi† ˆ fc  ÿ_XIii C_{ÿ i C}2 1 
i C2 …XIiÿ i C₁†2 …A-3†

Substituting f_i…XI† for f_XIi …XIi† and XI for XIi in (A-3) will yield the

simplified mathematical expression shown in (8). The probability density functions of other random variables such as XIIi_{, Y,} gIi_, gIIi_, gIIi

1j, and gII2ji

can be obtained in the same way as shown above.

Appendix B

Supposec₁ andc₂ represent the specific time values of two individuals and their respective (X, Y ) values are (X1, Y1) and (X2, Y2). Ifc₁> c₂, then the following conditions must hold by the definitions of X and Y:

X1ˆ nLrtrlÿ thLÿ L …dhÿ dr†=c1> nLrtrl ÿ thLÿ L …dhÿ dr†=c2ˆ X2

…B-1† Y1ˆ l dr=c1‡ trl < l dr=c2‡ trl ˆ Y2 …B-2†

Appendix C

We give here graphical proofs of the following statements: 1. If CIIi > CIIi

1j, passengers alighting at station j of zone i travel only

via CR alone or via HSR and backtracking CR with no possibility of trav-eling via HSR and progressive CR. The proportions of passengers choosing these two routes are, respectively,CIi

1j and1 ÿ CII1ji.

2. If CIIi < CIIi

1j, then it is true for C2jIIi > C1jIIi, i.e. CIIi < CII1ji < C2jIIi,

then passengers alighting at station j of zone i have all three routing alter-natives available. CIIi_, CIIi

2j ÿ CIIi, and 1 ÿ CII2ji are, respectively, the

portions of passengers who choose to travel via CR alone, HSR and pro-gressive CR, and HSR and backtracking CR.

From Appendix B, we know that when c₁> c₂, it is true that X₁> X₂, Y1< Y2. Thus, for a specificc value, c0, with its corresponding (X, Y) as

(X', Y'), the following equality must hold: Zc0 0 fy…c†dc ˆ ZX0 0 fX…X†dX ˆ Z1 Y0 fY…Y†dY …C-1†

(C-1) implies that the position ofc0 in c’s distribution curve is the same as that of X' in X’s distribution curve from the left, and the same as that of Y' in Y’s distribution curve from the right. This implies that as long as we get the market boundary of station j in zone i derived by random variable Y, we get the market boundary of station j in zone i for X distribution graph simultaneously. That means, the positions where one is located in both

dis-tribution curves of X and Y could be found, and the areas defined by a mar-ket boundary in X' and Y' distribution curves stand for the same group of passengers. Therefore, we can transform all the distribution graphs origin-ally expressed in the phase of Y’s distribution into their respective X distri-bution graphs. The following illustrations will further be used to prove the two statements mentioned above.

All figures below are only for illustration purposes. Letters C, F, and B are used to stand for CR alone, HSR and progressive CR, and HSR and backtracking CR, respectively. Since we have gotCIIi_, CIIi

1j, CII2ji by means

of the derivations shown in Sects. 3 and 4, therefore their market bound-aries can be all expressed in the X-based distribution curve as shown in Fig. C.1.

1. If CIIi > CIIi

1j, we can divide the area below X’s distribution curve

into three sections, I, II, and III, by using CIIi _and CIIi

1j as shown in Fig.

C.2.

From Fig. C.2 (a), the attributed market of these three sections can be expressed respectively by “I: C > F”, which stands for passengers with X value located in area I of X distribution curve in Fig. C.2 (a) will choose CR rather than HSR and progressive CR, and “II: C > F” and “III: F > C” represent in the same way as that of “I: C >F”. Similarly, in Fig. C.2 (b) we have “I: C >B”, “II: B > C”, and “III: B > C”. Therefore, we can make an in-ference as follows:

“I: C > F and B”; “II: B > C > F”; “III: F and B > C” (C-2) (C-2) shows that Sect I and II under the X distribution belong to the mar-kets for CR alone, and HSR and backtracking CR when CIIi > CIIi

1j.

Sec-Fig. C.1

tion III can not be attributed to the F or B markets with certainty untilCIIi

2j

is jointly compared underCIIi > CIIi

1j.

(i). If CIIi

2j > CII1ji, the inference “Section between C1jIIi and C2jIIi: F > B”

could be made from Fig. C.3 (a), and this is in contradiction with the infer-ence “II: B > C > F” in (C-2).

(ii). IfCIIi

2j < C1jIIi, then we have market segments from Fig. C.3 (b) as

fol-lows:

“The shade of I: F > B”; “The non-shade of I: B > F”;

“II: B >F”; “III: B > F” (C-3)

From synthesizing the inferences of (C-2) and (C-3), we have “I:

C > F > B”; “II: B > C > F”; and “III: B > F > C”, that is, area I is attributed to

the CR alone market, and area II and III are attributed to the HSR and backtracking CR markets. Thus we have proven that ifCIIi > CIIi

1j,

passen-gers alighting at station j of zone i travel only via CR alone or via HSR and backtracking CR, and have no possibility of traveling via HSR and progressive CR. The proportions of passengers choosing these two routes are, respectively,CIIi

1j and 1 ÿ C1jIIi (see Fig. 7).

2. If CIIi < CIIi

1j, we divide the area below X’s distribution curve into

three Sects. I, II, and III, withCIIi _andCIIi

1j as shown in Fig. C.4.

Fig. C.3

From Fig. C.4 (a), the attributed markets of the three sections are expressed as “I: C > F”; “II: F > C”; “III: B > F”; and those from Fig. C.4 (b) are “I:

C > B”; “II: C > B”; “III: B > C”. Thus, we can make an inference as

fol-lows:

“I: C > F and B”; “II: F > C > B”; “III: F and B > C” (C-4) Thus, the areas I and II under the X distribution curve whenCIIi < CIIi

1j are

inferred to belong to the CR alone market, and the HSR and backtracking CR market, respectively. Area III can not be attributed to the F or B mar-ket with certainty untilCIIi

2j is jointly compared underCIIi > C1jIIi.

(i). If CIIi

2j < C1jIIi, the inference “area between C2jIIi and C1jIIi of II: B > F”

can be made from Fig. C.5 (a), and this is in contradiction with the infer-ence “III: F > C > B” in (C-4).

(ii). IfCII_2ji > C_1jIIi, then we have market segments from Fig. C.5 (b) as fol-lows:

“I: F > B”; “II: F > B”; “The non-shade of III: F > B”;

“shade of III: B > F” (C-5)

From synthesizing the inferences of (C-4) and (C-5), we have “I:

C > F > B”; “II: F > C > B”; “The shade of III: F > C > B”, and “The non-shade of III: B > F > C”. This implies that if CIIi < CIIi

1j, passengers

alight-ing at station j of zone i will travel via CR alone, via HSR and progressive CR, and via HSR and backtracking CR. The proportions of passengers choosing these three routes are, respectively, CIIi_,CIIi

2j ÿ CIIi, and 1 ÿ CII2ji.

Appendix D

1. RevisedCIi _{for Type I passengers. Suppose a Type I passenger alights at}

joint station i, then, for a given t_d, the fare conversion time difference be-tween traveling via HSR and via CR can be expressed as:

th

pÿ trpˆ …L1‡ L2‡ ::: ‡ Li†…dhÿ dr†=c ˆ ‰…dhÿ dr†=cŠ

Xi kˆ1

Lk (D-1)

and the riding time difference between HSR and CR is

Dt1‡ Dt2‡ ::: ‡ Dti ˆ

Xi kˆ1

Dtk (D-2)

where Dt_k, k = 1, ..., i represents the riding time difference at spacing k be-tween HSR and CR. Dt_k, which includes both riding and stop time differ-ences, can be estimated from timetables for both types of trains. (7) thus becomes td ˆ Xi kˆ1 Dtkÿ ‰…dhÿ dr†=cŠ Xi kˆ1 Lk (D-3)

Considering the market of interest as a whole, then, similarly, the right hand side of (D-3) is a random variable transformed from c. Denote this variables asgIi_{, that is} gIi _ˆX i kˆ1 Dtkÿ ‰…dhÿ dr†=cŠ Xi kˆ1 Lk; ÿ1 < gI < Xi kˆ1 Dtk (D-4)

Then, from Appendix B, the p.d.f. ofgIi_,f

i (gI), is fi…gI† ˆ fc  ÿC2 gI_{ÿ C1} 
C2 …g2_{ÿ C1†}2 ; C1ˆ Xi kˆ1 Dtk; C2ˆ …dhÿ dr† Xi kˆ1 Lk (D-5)

andCIi _{can be revised as:}

CIi _ˆ

Ztd

ÿ1

fi…gI† dgI (D-6)

2. RevisedCIIi _{for HSR and progressive CR vs. CR alone. Suppose a Type}

II passenger who alights at station j of zone i, for a givent_d, the fare con-version time difference between HSR and progressive CR and CR alone can be expressed as:

th p1‡ trp1ÿ trpˆ …L1‡ L2‡ ::: ‡ Liÿ1†…dhÿ dr†=c ˆ ‰…dhÿ dr†=cŠ Xiÿ1 kˆ1 Lk (D-7)

and the riding time difference between HSR and progressive CR and CR alone is

Dt1‡ Dt2‡ ::: ‡ Dtiÿ1 ˆ

Xiÿ1 kˆ1

Dtk (D-8)

whereDt_k, k = 1, ... , i–1 represents the riding time difference at spacing k be-tween HSR and progressive CR and CR alone. (11) then can be rewritten as:

td‡ trw…iÿ1†ÿ ts ˆ Xiÿ1 kˆ1 Dtkÿ ‰…dhÿ dr†=cŠ Xiÿ1 kˆ1 Lk (D-9)

Similarly, the right hand side of the equality in (D-9) is a random variable. AssumegIIi _{stands for this variable, and} ÿ1 < gIIi <Piÿ1

kˆ1Dtk, then the

p.d.f. ofgIIi _{can be derived as:}

fi…gII† ˆ fc  ÿC2 gII_{ÿ C1} 
C2 …gII_{ÿ C1†}2 (D-10)

whereC₁ˆPiÿ1_kˆ1Dt_k,C₂ˆ …d_hÿ d_r†Piÿ1_kˆ1L_k, andCIIi _{can be revised as:}

CIIi ˆ Z td‡tt_w…iÿ1†ÿts ÿ1 f…iÿ1†…gII† dgII (D-11) 3. Revised CIIi

1j for CR alone vs. HSR and backtracking CR. Suppose a

Type II passenger who alights at station j of zone i, then, for a given t_d, the fare conversion time difference between CR alone and HSR and back-tracking CR,th_p2‡ tr_p2ÿ tr_p, can be calculated as:

…dhÿ dr†=c
Xi kˆ1 Lk‡ 2 dr=c
Xmi kˆ1 li …k‡1† ˆ  …dhÿ dr†
Xi kˆ1 Lk‡ 2 dr
Xmi kˆ1 li …k‡1†  =c (D-12)

and the riding time difference between the two routings is Xi kˆ1 Dtkÿ 2 Xmi kˆj ti_r…k‡1† (D-13)

Therefore, (15) can be rewritten as: …td‡ t0rwiÿ ts† ˆ Xi kˆ1 Dtkÿ 2 Xmi kˆj ti r…k‡1† ÿ  …dhÿ dr† Xi kˆ1 Lk‡ 2 dr
Xmi kˆ1 li …k‡1†  =c (D-14)

Denote the right hand side of (D-14) as gII_1ji, and ÿ1 < gII_1ji <Pi_kˆ1Dt_k ÿ2Pmi kˆjtir…k‡1†, then the p.d.f. ofgII1ji,fij…gII1†, is fij…gII1† ˆ fc  ÿC2 gII 1 ÿ C1 
C2 …gII 1 ÿ C1†2 (D-15) where C₁ˆPi_kˆ1Dt_kÿ 2Pmi kˆjtir…k‡1†, C2ˆ  …dhÿ dr†
Pikˆ1Lk‡ 2dr
Pmi kˆjl…k‡1†i 

, and the revisedCIIi

1j can be expressed as:

CIIi 1j ˆ Z td‡t0rwiÿts ÿ1 fij…gII1† dgII1 (D-16)

4. RevisedC_2jIIi for HSR and progressive CR vs. HSR and backtracking CR.

The fare conversion time difference in this situation, th_p2‡ tr_p2ÿ th_p1ÿ tr_p1, can be expressed as:

Li…dhÿ dr†=c ‡ ‰2 dr=cŠ Xmi kˆ1 li …k‡1†ˆ  Li…dhÿ dr† ‡ 2 dr
Xmi kˆ1 li …k‡1†  =d (D-17) and the riding time difference is

Dtiÿ 2

Xmi

kˆj

ti

r…k‡1† (D-18)

Therefore, (25) can be rewritten as: …t0r wiÿ trw…iÿ1†‡ ts† ˆ Dtiÿ 2 Xmi kˆj ti r…‡1† ÿ  Li…dhÿ dr† ‡ 2 dr
Xmi kˆ1 li …k‡1†  =c (D-19)

Denote the right hand side of (D-19) as gIIi 2j, and ÿ1 < gII1ji < Dtiÿ 2 P_mi kˆjtir…k‡1†, then the p.d.f. ofgII2ji,fij…gII2†, is fij…gII2† ˆ fc  _ÿC 2 gII 2 ÿ C1 
C2 …gII 2 ÿ C1†2 (D-20) where C₁ˆ Dt_iÿ 2P_kˆjmi ti_r…k‡1†, C₂ˆL_k…d_hÿ d_r† ‡ 2 d_r
Pm_kˆji l_k‡1†i , andCII_2ji can be revised as:

CIIi 2j ˆ Z t0r wiÿtrw…iÿ1†‡ts ÿ1 fij…gII2† dgII2 (D-21) Appendix E Algorithm:

1. Collect data including tr_w…iÿ1†, t0r_wi, t_ds, Dt_k, ti_rj), n, m_i, V_h, V_r, a, b, dh, d_r, t_s, and f_c…c†. The first five variables may be estimated from timeta-bles for HSR and CR trains.

2. Transform probability density functions and calculate CIi _{(i = 1 to n),}

CIIi _{(i = 2 to n),} CIi

1j (i = 1 to n, j = 1 to mi), C2jIi (i = 2 to n, j = 1 to mi) by

(D-2) to (D-13). The waiting/transfer time components in the upper bounds of the integrals (D-6), (D-11), (D-16), and (D-21), i.e. t_ds,t_ds‡ tr_w…iÿ1†ÿ t_s, tds‡ t0rwiÿ ts, t0rwiÿ trw…iÿ1†‡ ts, may be multiplied by p, the ratio of

wait-ing/transfer time value to the riding time value, so as to account for differ-ent weights placed by passengers on various time compondiffer-ents. For in-stance, CIIi ˆ Z p
…tds‡tr_w…iÿ1††ÿts ÿ1 f…iÿ1†…gII† dgII: (E-1)

3. Estimate market shares

Type I passengers: passengers departing from joint station 0 and alighting at joint station i, i = 1, ..., n. There are two markets, the market share for traveling via CR is CIi_{, and the market share for traveling via HSR is}

1 ÿ CIi_.

Type II passengers: passengers departing from joint station 0 and alighting at CR station j of zone i.

(1) Zone 1: there are only two competitive markets, the market share for traveling via CR alone is C_1jIIi, and the market share for traveling via HSR and backtracking CR is1 ÿ CIIi

1j:

ifCIIi > CIIi

1j, then there are only two markets, the market share for

travel-ing via CR alone is CIIi

1j, and the market share for traveling via HSR and

backtracking CR is1 ÿ CIIi

1j,

if CIIi < CIIi

1j, then there are three markets, the market share for traveling

via CR alone is CIIi_{, the market share for traveling via HSR and}

progres-sive CR is CIIi

2j ÿ CIIi, and the market share for traveling via HSR and

backtracking CR is1 ÿ CIIi

2j.

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