ably affected by weather conditions, especially by winter temperatures. The important relation it is intended to show by these charts is the tendency for every train to consume more time on the road, the greater the number of trains on the road per day. This tendency is indicated by the slope of the lines drawn through the middle of the group of points. This relation is brought out in the methematical development of the theory of train operation discussed in the Exhibit.

Simplification of the Train-Hour Diagrams.

Last year a method of plotting actual train-hour and crew-expense diagrams was discussed. The method of obtaining and arranging the data for these charts is tedious and it is felt that some simpler method can be developed which will make it possible to obtain the data with much less effort and at the same time establish mathematical relationships which can be simplified. The scheme briefly is to find the equation of the train-hour diagram and to ascertain if the results obtained by empirical formulas are close enough to the actual to be of practical value. Similarity Between the Typical Train-Hour Diagrams and

Probability Curve.

Train operations have in them the element of luck, that is, every train-after it leaves a terminal-runs a chance of being delayed from one cause or another. Some trains will be delayed more than others and the time which a train consumes on the road depends largely upon various combinations of circumstances. It is, therefore, logical to assume that an equation in the form of the Probability Curve y = ke-(nx)2 will most nearly fit the curve of a train-hour diagram. The origin of coordinates for such a curve is (0,0) and the curve is symmetrical both sides of the y axis, as shown in Fig. 3. The typical train-hour diagram is obtained by using only half of the probability curve and substituting a rectangle for the other half, as indicated.

to

2(2017

FIG. 3-DIAGRAM SHOWING DEVELOPMENT OF THE TYPICAL TRAIN-HOUR DIAGRAM FROM THE PROBABILITY CURVE.

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.

TABLE 2-COMPARATIVE DIMENSIONS OF ACTUAL AND THEORETICAL TRAINHOUR DIAGRAMS ILLUSTRATED IN FIG. 4

FIG. 4 ACTUAL AND THEORETICAL TRAIN-HOUR DIAGRAMS, SECTIONS P-R,

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 conclusions 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

I-Derivation of Theoretical Train-Hour Diagram.

Let the equation of the theoretical train-hour curve be y=ke-(nt)2 where y number of trains taking more than t hours to complete their runs, ka constant and hours referred to a system of coördinates where yk, when to. 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 (y1, t) and (y, fa) are the coördinates of the two points a and b, respectively, then

y=ke (nt) and yake (nt ̧)2

or (nt,)2=loge (*) and (nt.)2 = log. (^)

since t+c.

= n2 (t1 + c)2

This leaves two equations with two unknown quantities n and t1, from which the following values are obtained:

Where to is the abscissa of the origin of the assumed system of coordinates referred to the origin of the train-hour curve and t is the abscissa of the point a referred to the origin of the train-hour curve. Therefore th' = 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:

*To obtain accuracy ye and v should be selected so that the ratios N/y; and N/y1 will be approximately 1.25 and 5.0, respectively.

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

NOTE A-Values of e (nt) for convenience are given on curve sheet Fig. 5 for various values of (nt)2.

A

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 00. If the area under the probability curve between the limits -00 and 00 is taken as unity it can be shown that k =

n

Υπ

For our purposes we will consider the area under the probability curve y = ke (nt)2 between the limits o and 0 equals unity, hence k will equal

On the basis of train-hours the area of the rectangle is Nto or

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

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 =

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.

n = .398.

nt .398 X 2.987 = 1.189.

Ordinate of Curve B corresponding to nt = 1.189 = .909.

=

100 V T 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 2X.398 (1-.909) 800 + 30.38 = 830.38 train-hours paid for on eight-hour basis of pay, time and one-half for overtime.