[<< wikibooks] Transportation Deployment Casebook/California Railroads
== Introduction ==


=== The Early Years ===
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.  As a direct result of the sudden population growth, California became the 31st state in 1850.
The first railroad in California was the Sacramento Valley Railroad which operated from 1852-1877.  The 23 mile main line connected the San Fransisco Bay Area and Sacramento.  The track was seven inches wider than standard gauge which limited the compatibility with other railroads that would later be developed.  Since railroad technology had already been developed in the eastern United States, California's railroads could deploy much faster than the early east coast railroads.  As a result, California's railroads had a much shorter birthing phase as they could use the knowledge gained from the earlier railroads.  
Theodore Judah was the chief engineer for the Sacramento Valley RR and later the Central Pacific RR.  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.  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.  
Asa Whitney, a supporter of the idea form New York, had proposed it to congress in 1845.  It took until 1862 for the project to get started when the Pacific Railroad Act of 1862 was passed.  Discrepancies between the north and south had added controversy to the plans and delayed funding until it was passed after the civil war began in 1861 under a Republican Congress.  In 1869, the last "golden" spike was driven completing the first transcontinental railroad from Omaha, Nebraska to Sacramento, California.  By 1893, five lines had been completed connecting the west coast to the rest of the United States.  The Northern Pacific and Great Northern Railroads had built lines connecting Oregon and Washington to Minnesota and the Great Lakes.  The Santa Fe Railroad connected Los Angeles to Atchinson, Kansas, and the Southern Pacific Railroad connected Los Angeles to New Orleans, Louisiana.


=== Steady Growth ===
Once the first transcontinental railway was completed, California's Railroads shifted most of it's construction emphasis to linking the state north to south instead of east to west.  This had substantial benefit, but not as much as linking to the eastern United States, so expansion of the railroad network followed a linear model for the next several decades.  During this time some railroads were already being consolidated.  In 1877, the Sacramento Valley Railroad and the Folsom and Placerville Railroad merged to form the Sacramento and Placerville Railroad.  From 1870 to 1900, larger railroads like the Southern Pacific and the Northern Railway began to take control over numerous smaller railroads.


=== Maturity and Steady Decline ===
With some railroad tracks at capacity and others underutilized, it was time for restructuring and more consolidation.  Over a hundred different railroads had operated in California by 1890 and almost a hundred more started operation between 1890 and 1910.  There were too many railroads to efficiently operate all at once.  The major railroads of Atchison, Topeka and Santa Fe, Southern Pacific, Central Pacific and Union Pacific were consolidating with other smaller railroads.  California was experiencing many of the problems that all railroads in the U.S. faced.  At the start of World War I, railroads were the largest industry in the country and access to eastern ports was congested.  In order to avoid shortages and provide more efficiency, the Interstate Commerce Commission (ICC) was granted more powers.  Later in 1917, the government to control of the largest railroads and put them under the control of the United States Railway Administration (USRA).  Nationalization did not help reduce shipping costs so the Transportation Act of 1920 was passed and returned railroads back to private ownership.  By this time, the miles of railroad track in California had already peaked in 1915.  Track miles were maintained for almost a decade before starting a slow annual decline as a result of the Great Depression and World War II, as well as the rise in airplanes as an alternative mode to the railroads.


== Methodology ==
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:

  
    
      
        S
        (
        t
        )
        =
        K
        
          /
        
        
          (
          
            1
            +
            
              e
              
                −
                b
                (
                t
                −
                
                  t
                  
                    0
                  
                
                )
              
            
          
          )
        
      
    
    {\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:

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

Where:

b from the S-curve
c = -b * t0There 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.  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.


== Results ==


=== Table 1: Miles of Railroad Track by Year with Comparison to Predicted Values ===


=== Table 2: S-curve Model Parameters ===
S(t) = K/[1+exp(-b(t-t0)]


=== Table 3: Linear Model Parameters ===
S(t) = a + m*t


=== Figure 1:Life Cycle for the Miles of Railroad Track in California ===


=== Figure 2: S-Curve Models for the Miles of Railroad Track in California ===


=== Figure 3: Linear Model for the Miles of Railroad Track in California ===


=== Figure 4: Linear Regression to determine b and t0 when k=8460 ===


=== Figure 5: Linear Regression to Determine b and t0 when k=9152.104 ===


== Analysis ==
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.


== Conclusion ==
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.


== References ==