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An Introduction to Matrix Theory for Passenger Trains

August 31st, 2000

Dr. Adrian Herzog, VP Research, URPA

The potential ridership on any given transportation system can be accurately modeled mathematically. We begin by modeling a single route going from location A to location B. Such a non-stop system models non-stop long distance air service and express non-stop rail service. Only two “routes” are possible in such a system, from A to B and from B back to A. One might argue that this represents only one round trip market. However, since the demographics of A and B are often quite different, each direction represents a different market to be exploited.

Let’s expand the system by adding just one more stop, so the route serves locations A, B, and C. Now we find that there are six potential markets between:

  • A and B
  • B and A
  • A and C
  • C and A
  • B and C
  • C and B

Adding just one stop increases the number of potential markets from 2 to 6. From this we can derive a formula for the number of potential markets in this “network” based on the number of stops. In essence, this can be done when we observe that for some number N of potential origin and destination stations there are N-1 stations with which the other N stations in the system can interact. So the number of origin / destination (O/D) pairs is given by:


To illustrate, we can test EQUATION 1 for the cases of TWO STATIONS , THREE STATIONS and FOUR STATIONS

FOR N = 2, MARKETS = 2 x ( 2 -1 )) = 2 x 1 = 2
FOR N = 3, MARKETS = 3 x ( 3 -1 )) = 3 x 2 = 6
FOR N = 4, MARKETS = 4 x ( 4 -1 )) = 4 x 3 = 12

Curiously, analysis of empirical results of real trains in real markets has established that these simple equations are also accurate predictors of actual transaction volume (ridership), within certain ranges of accuracy, especially in larger networks. Why? It can be argued that not all stops are equivalent. This is true, of course: some rural stops are quite small while some urban stations serve very large populations. It is also true, however, that the small rural stops often also have no significant competition from air service, so that while the market may be small, the market share of rail is much larger than it would be in a large metropolitan area where the travelling public has many competitive modes and options available. Large urban areas are by definition large markets, but the market share for rail or any other mode is necessarily small due to all the competition.

Using actual ridership data for several long-distance trains, URPA research staff found that the contribution of RIDERSHIP by a stop IS NOT LINEARLY PROPORTIONAL TO POPULATION. The actual ridership yield of any given market approximates one-third of the cube of the log of the population. This relationship is illustrated in the following table:

Table 1:

Population Log Cube of the Log Ridership Factor









































The data in this table shows that while the ratio of population between two stops may be as much as a factor of 50,000, the actual ridership potential of the largest market is only some 40 times greater than the smallest market. To put it another way, a system made up of only seven of the smallest stations in the network not only is capable of producing but usually will produce the same ridership contribution as the largest single station in the in the network! Given that the smallest communities served by intercity rail services have a population typically of at least 10,000, it only takes four of five such stations to compete with the market potential of a large station on the system and on any long distance route, since on most routes service is offered at a half dozen or more smaller communities for every one large city. Given that most routes have more or less the same mix of large, medium and small station stops, the only factor that really determines the actual market potential of a train system (in a large network) is the number of realistically usable O/D pairing opportunities in the system, and as that number increases, the market potential of the network increases nearly as the square of the numbers of O/D pairs.

To calculate actual ridership projections for any given single O/D station pair, it is also necessary to take into account a number of factors. First, the distance between the two stations; demand for long-distance rail transportation rises rapidly for travel from 100 up to around 900 miles and then decreases slowly for distances greater than 900 miles. In addition, the time of day the stop is served has a strong impact on individual station performance. However, for general application to computing train performance across larger networks of stations, these and other local effects cancel out as there are always station pairs with a short, medium and large separation as well as stations served at various times of the day. For maximum long distance train performance, it is of course advisable that the end points be major terminals and that both the end points and most large intermediate terminals be served at reasonable times of the day. In the case of corridor services across a major urban area, this does not necessarily hold.

Using software based on Matrix Theory, URPA was able to calibrate the effects of population, distance and time of day so that the model was able to predict on and off passenger loadings for all stations along the SOUTHWEST CHIEF route (Chicago-Kansas City-Albuquerque-Los Angeles) with a high degree of accuracy. Using the calibration based on the this route, using actual on-off data, URPA has been able to predict accurately the actual performance of other long-distance services. Calculations for all existing services produce results consistently close to actual ridership data. This gives URPA a high degree of confidence that Matrix Theory-forecasted results for new routes and services are reliable.

Matrix Theory and Transcontinental Trains

The general practice in the United States is that trains do not operate from coast to coast. Instead Amtrak tends to operate Western Longhauls and Eastern Longhauls with little or no connectivity, and no marketing at all, between them, either collectively or individually. Let us look at the market potential of actually marketing, for example, the SOUTHWEST CHIEF and CAPITOL LIMITED as a single train operating continuously from Los Angeles to Washington DC via Kansas City and Chicago.

Table 2:

Southwest Limited: 33 Stations = 33 x 32 = 1056 City Pairs
Capitol Limited: 16 Stations = 16 x 15 = 240 City Pairs
Total As Two Trains = 1296 City Pairs
Combined Train 49 Stations = 49 x 48 = 2356 City Pairs

Table 2 illustrates how the proper positioning of these two trains as a single transcontinental service would double the market potential without adding any additional train miles (i.e., incremental operating costs are essentially zero) simply by marketing two trains as a single service . Route potentials exist for about ten such transcontinental service. If operated as ten routes with 20 independent trains, the total number of city pairs would be 12,960, but if integrated into 10 routes with 10 trains the number of city pairs hence market potential would nearly double, to 23,560, on the same number of train miles.

Matrix Theory and Hub-and-Spoke Operations

Many potential U.S. routes are comparable to the CAPITOL LIMITED with approximately 16 stations per route. These can be operated as a series of independent operations each supplying 240 city pairs, or they could be integrated at one or more stations so that complete through car operations can occur among several such routes. Suppose that we begin by integrating four regional long distance trains serving 16 stations each. If operated as four independent trains they would have 960 city pairs (240 per train). But if they operated as an integrated rail system they would generate a system of 64 stations connected by four integrated trains. Such a system would generate 64 x 63 = 4032 city pairs instead of the 960 (on the same number of train miles) and be more than four times as productive as four independent trains, using exactly the same number of train miles, the same level of market penetration, and exactly the same nature and quality of service.

Integrating a network of regional trains can be even more productive than extending a long-distance train from coast to coast! If such regional networks, serving the Northeast, Southeast, Midwest, and Far West are developed and integrated with a series of east west transcontinental trains the benefits become truly staggering. If we take the ten proposed transcontinental trains discussed above, they would have about 490 stations. These stations are not integrated unless we develop regional networks that connect the basic east west transcontinental routes (and the East and West Coast north-south routes) with each other. The transcontinental routes then act as connectors between the regional networks. Suppose we develop four such regional networks each contributing 64 stations. Then the 4 networks would have 256 stations. If we assume that 20% if the regional network stations are common to the transcontinental networks, then they contribute an additional 205 stations (80% of 256). Such a fully integrated national and regional system would nave 695 stations. This is only a modest increase over the current system but if operated as a fully integrated system would provide a network of not 695 but a staggering 482,330 city pairs!

Table 3:

One Western N/S Long-Distance Train = 1,056 City Pairs
One Eastern N/S Long-Distance Train = 240 City Pairs
One Transcontinental Train = 2,356 City Pairs
Four Integrated Regional Trains = 4,032 City Pairs
Ten Independent Transcontinental = 23,560 City Pairs
Four Regional Networks = 16,128 City Pairs
Ten LD and 4 Regions No Connections = 39,688 City Pairs
Ten LD and 4 Regions Fully Connected = 482,330 City Pairs

NOTE the last two systems in Table 3 operate the same number of stations and train miles, but by integrating the system completely the integrated system (i.e., the MATRIX) is 12 times larger.

These results are entirely consistent with the proven non-linearity of flow densities through larger polynodal networks of all types including power distribution grids, the Internet, etc.

The empirical verification of Matrix Theory shows that interconnecting a network into a complex matrix of origin-destination pairs even at constant levels of market penetration drives increases in transaction volume (ridership) exponentially. In the example above, predicted volume increases 12-fold (1200%) with only constant levels of service and market penetration (share) and with no new routes or stations added to the network matrix. But, it is wrong to assume for purposes of business planning that no additional trains, train miles, employees or other cost drivers will be required as the national network becomes integrated. Unless prices are raised to supra-competitive, market-clearing levels to stifle demand, a 12-fold increase in carrying capacity will be required to accommodate expanded demand (still assuming constant levels of market penetration, performance and share), which implies significant incremental capital investment in new cars and locomotives, longer trains and additional frequencies, maintenance bases, more parking facilities at stations, increased employment, etc.

Integration of the network is like starting a chain reaction: it ignites a growth cycle or upward spiral in utility, usage, output, and financial results.

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