## IntroductionEdit

### The Early YearsEdit

The eastern United States was well developed and had been operating railroads for nearly 30 years before major settlements took shape in California during the 1850's. After the discover of gold in 1848, numerous people fled to seek wealth in California during the gold rush. Approximately 200,000 came between 1849 and 1851 with over 100,000 more coming in the next five years.[1] As a direct result of the sudden population growth, California became the 31st state in 1850.

Theodore Judah was the chief engineer for the Sacramento Valley RR and later the Central Pacific RR.[2] Judah was the main proponent in Sacramento for the development of a transcontinental railroad, but the idea did not gain support for several years. The idea of an east to west link had gained interest after the settlement of Oregon in 1846, acquiring territories from Mexico in 1848, and the discovery of gold in California.[3] Settlers traveled to the west coast by traversing the dangerous trails by horseback and wagon trains. Adding a railroad would rapidly speed up travel time and make travel safer as well as reducing the cost of shipping goods long distances.

## MethodologyEdit

The analysis of the life cycle of most modes of transportation follows a traditional logistic curve also known as an S-curve. In the early years, advancements in the technology result in rapid growth that can be modeled as a period of exponential growth. A single innovation can spark several improvements and multiple benefits. As the system expands, the rate of return weakens to a linear relationship. Adding new rail lines results in some benefit, but not as much as implementing the first lines did. Eventually the network grow big enough such that the most profitable lines were are built and new line result in little benefit to the entire system. As the system matures, the extent of the network levels off and begins to decline as a result of consolidation and optimization of the system. Typically, a new technology comes along to make the old technology obsolete and it declines to extinction or to an optimal lower level of service.

The Logistic Formula:

${\displaystyle S(t)=K/\left(1+e^{-b(t-t_{0})}\right)}$

Where:

• S(t) is a measure of the transportation system at a given time [miles of railroad in California]
• t is time [years]
• t0 is the time when S(t) is at the inflection point of the S-curve [years]
• K is the maximum capacity of the transportation system [miles]
• b is a coefficient that affects how quickly a system reaches maturity

The S(t) formula can be transformed to get a the following linear relationship:

${\displaystyle \ln \left({\cfrac {S(t)}{(K-S(t))}}\right)=c+b\times t}$

Where:

• b from the S-curve
• c = -b * t0

There are three unknown parameters (K, n, c) and only two can be fitted, so different K values are chosen and linear regression is done to get the other two using the Least Squares method. Microsoft Excel is used to perform the analysis and obtain the slope and intercept of the best fit line. Since the miles of railroad peaked in 1915 at 8451, the real K value is set at 8460. However, this does not fit the data very well, so Excel's solver is used to obtain the best K, m,& a while minimizing the sum of the square errors. The optimal parameters for the S-curves can be seen in Table 2. The S-curves only model the growth phase and only use the data until 1915 to do the linear regression to fit K, n,& c.

The plot of the data in Figure 1 indicates that the birthing and the maturity phases were very short compared to the time span of the entire system. This indicates that the data may actually follow a two stage linear model, one for the growth phase and one for the decline phase. The peak number of miles occurs in 1915, so the year 1915 serves as the breaking point for the analysis of the growth and decline phases. Again Excel is used to do the linear regression and the optimal parameters can be seen in Table 3.

Data about California's railroads was obtained from numerous annual Statistical Abstracts from the U.S. Census Bureau.[7] Each report gave data for every five or ten years and the individual year data for two or three years before the publication of the report. In order to make a complete data set, data from several reports was merged together and can be seen in Table 1 below. For some of the annual reports, the numbers contradicted each other; thus, for the purposes of this analysis, the most recent value was consider to be more accurate.

## ResultsEdit

### Table 1: Miles of Railroad Track by Year with Comparison to Predicted ValuesEdit

 Year Miles of Track Predicted Miles (k=9152.104) Predicted Miles (k=8460) Predicted Miles (2 stage - Linear) 1860 23 169.0 133.6 -796.4 1865 214 285.9 259.5 24.4 1870 925 479.2 497.0 845.2 1875 1503 791.5 927.1 1666.0 1877 2080 961.7 1176.2 1994.3 1878 2149 1058.4 1320.5 2158.5 1879 2209 1163.5 1479.0 2322.6 1880 2195 1277.3 1652.1 2486.8 1881 2309 1400.3 1840.2 2651.0 1882 2636 1532.9 2043.2 2815.1 1883 2881 1675.3 2261.0 2979.3 1884 2911 1827.8 2492.9 3143.4 1885 3044 1990.4 2738.2 3307.6 1886 3297 2163.3 2995.4 3471.8 1887 3677 2346.2 3263.1 3635.9 1888 4126 2539.0 3539.1 3800.1 1889 4202 2741.2 3821.4 3964.2 1890 4356 2952.4 4107.4 4128.4 1891 4601 3171.8 4394.5 4292.6 1892 4624 3398.5 4680.1 4456.7 1893 4630 3631.6 4961.6 4620.9 1894 4635 3870.0 5236.5 4785.0 1895 4757 4112.3 5502.8 4949.2 1896 4996 4357.3 5758.4 5113.4 1897 5199 4603.5 6001.7 5277.5 1898 5292 4849.6 6231.7 5441.7 1899 5455 5094.2 6447.3 5605.8 1900 5751 5335.7 6648.2 5770.0 1901 5684 5573.0 6834.0 5934.2 1902 5773 5804.8 7005.0 6098.3 1903 5773 6030.0 7161.4 6262.5 1904 5820 6247.6 7303.7 6426.6 1905 6507 6456.8 7432.7 6590.8 1906 6655 6657.0 7549.1 6755.0 1907 6836 6847.6 7653.8 6919.1 1908 7222 7028.3 7747.5 7083.3 1909 7529 7198.9 7831.3 7247.4 1910 7772 7359.2 7905.9 7411.6 1911 7885 7509.4 7972.2 7575.8 1912 8105 7649.6 8031.0 7739.9 1913 8183 7780.0 8083.0 7904.1 1914 8368 7901.0 8129.0 8068.2 1915 8451 8012.8 8169.6 8409.7 1916 8441 8116.0 8205.4 8389.8 1917 8359 8210.9 8236.9 8369.9 1918 8269 8298.0 8264.5 8349.9 1920 8356 8450.9 8310.2 8310.1 1930 8240 8900.5 8421.0 8111.0 1940 7947 9064.8 8449.9 7911.8 1948 7567 9115.0 8456.6 7752.5 1950 7533 9122.2 8457.4 7712.7 1960 7630 9141.9 8459.3 7513.6 1968 7438 9147.8 8459.8 7354.2

### Table 2: S-curve Model ParametersEdit

S(t) = K/[1+exp(-b(t-t0)]

 Parameter S-curve 1 S-curve 2 k 9152.104201 8460 b 0.107701516 0.1358 t0 1896.888412 1890.427099

### Table 3: Linear Model ParametersEdit

S(t) = a + m*t

 Parameter Growth Decline a 164.16 -19.91 m -306134 46545

## AnalysisEdit

In Figure 1, the life cycle phases can be seen. There is no distinct point at which the birthing ends and steady growth begins, but the transition year is found to be 1875. Similarly, the growth period ends rather abruptly with a little or no transition period. The year decided to be the beginning of maturity is 1911. A slow steady decline is deemed to begin in 1920 and continue through WWII. The timing of maturity occurs at approximately the same time the entire United States railroads mature and is contributed by the start of WWI and failed nationalization of larger railroads at the time. The continual decline in the 30's and 40's is mainly contributed to the Great Depression and WWII.

In Figure 2, the two best fitting S-curves are illustrated with the data. Since the transition between phases is rather abrupt, the curves do not fit particularly well. Even though the S-curves were fitted using only the growth data, the location of t0 and the tight transitions cause the S-curves to predict capacity is not reached until approximately 1930, 15 years later than reality. Figures 4 and 5 illustrate that the transformed data has an weak linear relationship as the r2 values are 0.84 for K=8460 and 0.91 for K=9152.104.

Since the S-curves were not very accurate at predicting the annual track miles, a two stage linear model was analyzed and can be seen in Figure 3. The fit for the growth phase had an r2 value of 0.99 and the decline phase had an r2 value of 0.91. Only the first year of the birthing phase does not get modeled accurately with a linear relationship and the r2 values are significantly better than the S-curves.

## ConclusionEdit

The deployment of California's railroads was greatly influenced by the historical events of its time. The gold rush created the demand for more efficient transportation modes. The divisions between the north and south delayed the start of construction on the first transcontinental railroad. World War I, shipping congestion, and experimental nationalization of the larger railroads started the maturity phase. The rise of the airplane as an alternative to railroads, the Great Depression, and World War II caused the miles of railroad track to slowly decline in the 30's and 40's.

Since railroad technology was already being improved in the rest of the country, California's railroads experienced a short birthing phase. Railroads were able to deploy technology that was already proven to be successful elsewhere. The relatively unexpected level of track miles in 1915 due to other issue going on in the U.S. could have prematurely ended the growth phase. A short birthing phase and transition periods can explain why a linear model fits the data better. If California's railroads had developed independently from the rest of the country, perhaps an S-curve would fit the data better then and linear model. Like any technology, the success or failure of California's railroad was directly dependent on the alternative modes and the external factor the entire country was facing during the time period.

## ReferencesEdit

1. Lewis, Robert. "Photographing the California Gold Rush." History Today 52.3 (2002): 11-17. Web. [1].
2. a b c Fickewirth, Alvin A. California Railroads: an Encyclopedia of Cable Car, Common Carrier, Horsecar, Industrial, Interurban, Logging, Monorail, Motor Road, Shortlines, Streetcar, Switching and Terminal Railroads in California. San Marino: Golden West, 1992. Print.
3. a b "Transcontinental Railroad." Columbia Electronic Encyclopedia. 6th ed. 1 Oct. 2011. Web. [2].
4. "Asa Whitney." Columbia Electronic Encyclopedia. 6th ed. 1 Oct. 2011. Web. [3].
5. United States. Pacific Railroad Act of 1862. Washington, D.C.: Govt. Print. Off., 1862. Academic Search Premier. Web. [4].
6. Garrison, William L., and David M. Levinson. The Transportation Experience. New York: Oxford UP, 2006. Print.
7. United States. U.S. Census Bureau. U.S. Department of Commerce. Statistical Abstracts. 1881 1887 1891 1896 1900 1904 1908 1911 1913 1914 1916 1917 1920 1950 1970. Web. [5].