Extensions and algorithm

The market shares for HSR and CR along a transportation corridor can be analyzed by calculating and comparingC^Ii, C^IIi, C_1j^IIi, andC^II_2jⁱ as shown in the preceding sections. On the basis of these analyses, we will discuss and describe consequences of relaxing some important assumptions.

Fig. 12. The market share vs. the standard deviation of passengers’ time value distribution

6.1. Relaxing equivalent spacing assumption

Suppose there are n+1 HSR and CR joint stations (including the origin sta-tion 0) spaced withLi, i = 1, ..., n along the HSR line, and there are mi CR stations (not including joint stations i–1 and i ) in zone i of the CR lines, spaced with l_jⁱ, j = 1, ...,mi‡ 1, then the riding time in an HSR train be-tween two consecutive joint stations, thi, is Li=Vh‡ Vh=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=Vr‡ Vr=b, i=1,..., n, j=1,... mi‡ 1. The values for thi 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-enVh,Vr,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 C^Ii, C^IIi, C_1j^IIi, andC^II_2jⁱ 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 C^Ii,C^IIi,C_1j^IIi and C_2j^IIi:

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 mi

to represent the station at which HSR and CR trains actually stop. Accord-ingly, joint station spacing, Li, i = 1, ..., n, and CR station spacing l_jⁱ, j = 1, ...,mi‡ 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, Li, 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 …n^L_rt^r_lÿ t^h_L† ÿ L …dhÿ dr†=c, where c is a random vari-able and all other varivari-ables are exogenous in the definition of X. Let C1ˆ n^L_rt^r_l ÿ t^h_L and C2ˆ L …dhÿ dr†, then the p.d.f. of X, fX…x†, can be

Substituting fi…X^I† for f_XIi …X^Ii† and X^I for X^Ii in (A-3) will yield the simplified mathematical expression shown in (8). The probability density functions of other random variables such as X^IIi, Y, g^Ii, g^IIi, g^II_1jⁱ, and g^II_2jⁱ 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 n^L_rt^r_lÿ t^h_Lÿ L …dhÿ dr†=c₁> n^L_rt^r_l ÿ t^h_Lÿ L …dhÿ dr†=c₂ˆ X2

…B-1†

Y1ˆ l dr=c₁‡ t^r_l < l dr=c₂‡ t^r_l ˆ Y2 …B-2†

Appendix C

We give here graphical proofs of the following statements:

1. If C^IIi > C_1j^IIi, 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,C^I_1jⁱ and1 ÿ C^II_1jⁱ.

2. If C^IIi < C_1j^IIi, then it is true for C_2j^IIi > C_1j^IIi, i.e. C^IIi < C^II_1jⁱ < C_2j^IIi, then passengers alighting at station j of zone i have all three routing alter-natives available. C^IIi, C^II_2jⁱÿ C^IIi, and 1 ÿ C^II_2jⁱ 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:

Z^c0

(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 gotC^IIi, C_1j^IIi, C^II_2jⁱ 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 C^IIi > C^II_1jⁱ, we can divide the area below X’s distribution curve into three sections, I, II, and III, by using C^IIi and C^II_1jⁱ 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 C^IIi > C_1j^IIi.

Sec-Fig. C.1

Fig. C.2

tion III can not be attributed to the F or B markets with certainty untilC_2j^IIi is jointly compared underC^IIi > C_1j^IIi.

(i). If C_2j^IIi > C^II_1jⁱ, the inference “Section between C_1j^IIi and C_2j^IIi: 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). IfC^II_2jⁱ < C_1j^IIi, 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 ifC^IIi > C^II_1jⁱ, 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,C_1j^IIi and 1 ÿ C_1j^IIi (see Fig. 7).

2. If C^IIi < C_1j^IIi, we divide the area below X’s distribution curve into three Sects. I, II, and III, withC^IIi andC^II_1jⁱ as shown in Fig. C.4.

Fig. C.3

Fig. C.4

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 whenC^IIi < C^II_1jⁱ 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 untilC^II_2jⁱ is jointly compared underC^IIi > C_1j^IIi.

(i). If C^II_2jⁱ < C_1j^IIi, the inference “area between C_2j^IIi and C_1j^IIi 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). IfC^II_2jⁱ > C_1j^IIi, 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 C^IIi < C_1j^IIi, 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, C^IIi,C^II_2jⁱÿ C^IIi, and 1 ÿ C^II_2jⁱ.

Appendix D

1. RevisedC^Ii for Type I passengers. Suppose a Type I passenger alights at joint station i, then, for a given td, the fare conversion time difference be-tween traveling via HSR and via CR can be expressed as:

Fig. C.5

t^h_pÿ t^r_pˆ …L1‡ L2‡ ::: ‡ Li†…dhÿ dr†=c ˆ ‰…dhÿ dr†=cŠXⁱ

kˆ1

Lk (D-1)

and the riding time difference between HSR and CR is

Dt1‡ Dt2‡ ::: ‡ Dti ˆXⁱ

kˆ1

Dtk (D-2)

where Dtk, k = 1, ..., i represents the riding time difference at spacing k be-tween HSR and CR. Dtk, which includes both riding and stop time differ-ences, can be estimated from timetables for both types of trains. (7) thus becomes

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 asg^Ii, that is

2. RevisedC^IIi for HSR and progressive CR vs. CR alone. Suppose a Type II passenger who alights at station j of zone i, for a giventd, the fare con-version time difference between HSR and progressive CR and CR alone can be expressed as:

t^h_p1‡ t^r_p1ÿ t^r_pˆ …L1‡ L2‡ ::: ‡ Liÿ1†…dhÿ dr†=c

ˆ ‰…dhÿ dr†=cŠX^iÿ1

kˆ1

Lk (D-7)

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

Dt1‡ Dt2‡ ::: ‡ Dtiÿ1 ˆX^iÿ1

kˆ1

Dtk (D-8)

whereDtk, 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‡ t^r_w…iÿ1†ÿ ts ˆX^iÿ1

kˆ1

Dtkÿ ‰…dhÿ dr†=cŠX^iÿ1

kˆ1

Lk (D-9)

Similarly, the right hand side of the equality in (D-9) is a random variable.

Assumeg^IIi stands for this variable, and ÿ1 < g^IIi <P_iÿ1

3. Revised C_1j^IIi 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 td, the fare conversion time difference between CR alone and HSR and back-tracking CR,t^h_p2‡ t^r_p2ÿ t^r_p, can be calculated as:

and the riding time difference between the two routings is Xⁱ

kˆ1

Dtkÿ 2X^mi

kˆj

tⁱ_r…k‡1† (D-13)

Therefore, (15) can be rewritten as:

4. RevisedC_2j^IIi for HSR and progressive CR vs. HSR and backtracking CR.

The fare conversion time difference in this situation, t^h_p2‡ t^r_p2ÿ t^h_p1ÿ t^r_p1,

Denote the right hand side of (D-19) as g^II_2jⁱ, and ÿ1 < g^II_1jⁱ < Dtiÿ 2

2. Transform probability density functions and calculate C^Ii (i = 1 to n), C^IIi (i = 2 to n), C^I_1jⁱ (i = 1 to n, j = 1 to mi), C_2j^Ii (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. tds,tds‡ t^r_w…iÿ1†ÿ ts, tds‡ t^0r_wiÿ ts, t^0r_wiÿ t^r_w…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,

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 C^Ii, and the market share for traveling via HSR is 1 ÿ C^Ii.

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_1j^IIi, and the market share for traveling via HSR and backtracking CR is1 ÿ C_1j^IIi:

(2) Zone i, i = 2, ... , n: for i = 2 to n, and for j = 1 ton^L_r ÿ 1;

ifC^IIi > C_1j^IIi, then there are only two markets, the market share for travel-ing via CR alone is C^II_1jⁱ, and the market share for traveling via HSR and backtracking CR is1 ÿ C^II_1jⁱ,

if C^IIi < C_1j^IIi, then there are three markets, the market share for traveling via CR alone is C^IIi, the market share for traveling via HSR and progres-sive CR is C^II_2jⁱÿ C^IIi, and the market share for traveling via HSR and backtracking CR is1 ÿ C^II_2jⁱ.

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