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In this connection it is interesting to note how closely the theoretical values in Table 2 agree with the values obtained in actual operation. Data for plotting the actual train-hour diagrams were obtained from train dispatcher's sheets for two months for Division E, Fig. 1. The division was subdivided into three sections, P-R, R-S and S-V, and actual trainhour diagrams plotted as shown. Two points (as indicated) on each one of these curves were used to derive the equation of the probability curve and compute the theoretical values. (See Exhibit A for explanation of methods employed.)

The total train-hours from the actual records are given in the following table, together with the theoretical train-hours calculated from the equations of the curves. In no case is the difference between the area of the actual and theoretical train-hour diagrams more than 2 per cent, which is much closer than would ordinarily be expected from empirical formulas.



P-R (31.3 Miles) R-S(48.7 Miles) S-V (30.7 Miles)

Actual Theo. Actual Theo. Actual Theo. N.

125 125 136 136 135 135 to.

1.250 1.207 2.000 2.638 1.000 1.736

.7955 Area (train-hours).. 281.0 278.5 589.7 599.3 378.5 384.9 Ave. hrs. road time. 2.248 2.227 4.335 4.405 2.803 2.850

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R-S, S-V, Division E, Fig. 1.


CONCLUSIONS 1. The Committee has not been able to analyze sufficient data to show how close it is possible to approach the theoretical limit of track capacity in actual practice. Two essentially single track divisions with very few passenger trains have been partially analyzed, but the concluisions which can be drawn from the analyses of these two cases cannot be taken as final. Before making final conclusions it is important to know if there are any other divisions which are doing better with equal facilities and also to secure opinions from those operating these divisions regarding their estimates of how soon additional track facilities will be required.

These features deserve more study and it is suggested that those familiar with actual operating conditions on sections of track which are known to be operating nearly to capacity apply some of the principles here discussed to their conditions and favor the Committee with their conclusions.

2. It has been found that train operations can be represented by a mathematical law, the development of which is given in Exhibit A of this report.

3. The application of this law to different sets of observations makes it possible to compare several months' operation of a given division on a more equal basis. Likewise operations of different divisions which are more or less similar can be compared on more nearly the same basis.

4. By such comparisons it is felt that the effect of extreme weather conditions, greater track facilities, characteristics of motive power, character of commodities and supervisory methods on the average time on the road can be more accurately determined.

5. Exhibit A of the report is most important, because it is arranged for reference and shows how closely actual operations can be forecasted by applications of the mathematical theory.

Exhibit A

1-Derivation of Theoretical Train-Hour Diagram.

Let the equation of the theoretical train-hour curve be y=ke-(nt): where y = number of trains taking more than t hours to complete their runs, k = a constant and t= hours referred to a system of coördinates where y = k, when 1 = 0. See Fig. 3. Then if k is known the coördinates of two other points (a and b) determine the curve. The number of trains considered determines k, hence if (V1, t.) and (y:, tz) are the coördinates of the two points a and b, respectively, then Vi = ke-(nt,) and y:=ke-(nt.)?

? = log

= n(t, to):

or (nt.)?=loge () and (nt.)>= 109. (:)

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since t2 = ttc.

This leaves two equations with two unknown quantities n and tı, from which the following values are obtained :

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Where to is the abscissa of the origin of the assumed system of coordinates referred to the origin of the train-hour curve and ti' is the abscissa of the point a referred to the origin of the train-hour curve. Therefore ti' = hours on the road and is a known quantity:

If N= number of trains considered, is substituted for k, then the calculations are best arranged in table form as given below:




Example Item

Fig. 3 1 Total number of trains considered....

N 100 2 *Number of trains over tå hours (8.2)


20 3 *Number of trains over ti' hours (6.2).

J'i 4 Time interval between tz' and ti'.


2 5 *Ratio item (1)/item (2) = N/y2.

5. 6 *Ratio item (1)/item (3) = N/y.

1.25 7 Loge item (5)

1,60944 8 Loge item (6).

.22314 9 Difference (item (7) -item (8))

1.38630 10 Product item (7) X item (8).

.3595 11 Square root item (10)..

.5995 12 Item (8) + item (11).

.82264 13 Item (4) X item (12).

1.64528 14 Item (13)/item (9)..

ti 1.187 15 ti' of item (3) — item (14)

to 5.013 16 Item (8) divided by item (14)”.

no .1584 17 Square root of item (16)...

.398 *To obtain accuracy yz and vs should be selected so that the ratios N/y: and N/yı will be approximately 1.25 and 5.0, respectively.

To plot the curve arrange calculations as follows and plot values in outside columns :

t (nt) =

Ne-(nt)? tott

Item (16) Xt Note A 5.013 5.013 0


100.0 5.513


96.0 6.013


87.0 7.013


52.8 8.013



24.0 9.013



8.0 10.013



2.0 11.013








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Note A-Values of r-(91t)? for convenience are given on curve sheet Fig. 5 for various values of (nt)?.



Economics of Railway Operation

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II–Area of the Theoretical Train-Hour Diagram.

The area of the train-hour diagram is the sum of two areas, one the area of a rectangle having the dimensions k and to and the other, one-half the area under the probability curve y=ke (nt)? between the limits -00 and 0. If the area under the probability curve between the limits 00 and 00 is taken as unity it can be shown that k =

V For our purposes we will consider the area under the probability curve y=ke-(nt)? between the limits o and of equals unity, hence k will equal 2n

2n or the area of the train-hour diagram (to +1) referred to some arbitrary unit of area. On the basis of train-hours the area of the rectangle is Nto or

2nto N Vπ Nto

=1 уп

21 or the area of the train-hour diagram in train-hours is

Nto +


2n III–Area of the Theoretical Crew-Expense Diagram.

If the crews are paid on the basis of 100 miles or eight hours then all crews will be paid for a minimum of eight hours (if the run is not over 100 miles), and those on the road over eight hours will make overtime. Usually some of the trains get over the road in less time than the limit set for overtime to begin. If t" represents the time when overtime begins, then the area of the crew-expense diagram is equal to the area of a rectangle having the dimensions N and t" plus that part of the area of the train-hour diagram outside of the rectangle. Or mathematically Area of crew-expense diagram

nt NV

2n Nt" +

e-(nt)? dt 2n


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curve sheet,


2n The values of the integral

e-(nt)dt are given on

VT Fig. 5, for various values of nt.

See also discussion Crew-Expense Diagrams in last year's report, Volume 22, page 751. EXAMPLE: Find the area of the crew-expense diagram Fig. 3

t" = 8.

to = 5.013. t" - to = t = 2.987.

= 398.

nt = .398 X 2.987 1.189. Ordinate of Curve B corresponding to nt= 1.189 = .909.

100 V7 8 X 100+

(1 – 909) = 800 + 20.25 = 820.25 train-hours paid

2 X .398 for on eight-hour basis of pay, straight time for overtime.

100 V 8 X 100 + 1.5

800 + 30.38 830.38 train-hours paid for on eight-hour basis of pay, time and one-half for overtime.



2X.398 (1 —.909)

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Example To Plot Train Hour Diagram: In the equation Y-e-Int)

assume n = 398. Then for t-3,(nt)2= 1.4256, from curve A' ratio corresponding toint)2-1.4256 is.24. Plot this value for t-3. Other points on the train hour curve found in the same manner.

To Find Partial Area of the Train Hour Diagram:
Ist - Calculate the area of the rectangular portion.
2nd - Total area under train hour curve
3rd - Partial area between 0 and any point t= 3 is found as
follows assuming n =.398, nt-.398x3=1.194. From curve'B'the
ratio corresponding to nt= 1.194 (on top line) is. 91. That is, the
area under the curve between 0 and t-3 is 91% of the total

. If overtime occurs after t-3, the overtime area equals total area minus partial area.

91 VTT

area or 2n

Fig. 5.

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