What is the ultimate schedule for a public bus system?


One problem I have looked at is "what is the ultimate or best schedule scheme for a bus and a bus system?"

What are the functions that a schedule scheme should optimize?
  • Reduce the waiting time between transfers between one bus and another bus.
  • Provide "meets" during off hours so night time travelers are not marooned at a bus stop when a slightly early bus misses a transfer time with a slightly later bus.
  • Organize the rolling stock to carry the most people during rush hours.
  • Arrange for the driver to start and end his driving work at the bus terminal.
Topology provides some very useful ideas for organizing a route network.
  • Think of each bus route of a bus system as a loop.
  • Think of all the bus routes of a bus system as many loops with many loops crossing. Like a pile of rubber bands.
  • The place where s bus route crosses another route is called a node.
  • The length of bus route between nodes is called an edge.
  • As long as there are an even number of bus routes entering and exiting each node, there exists an Euler path.
  • An Euler path can be completely driven by a bus driving over each stretch of road (or edge) .
  • The Euler path is in honor of the famous problem solved by Leonard Euler called the "Seven Bridges of Konigsberg." A Wikipedia entry for Eulerian Path
What timing structure would work best with an Euler path bus network?

  • Suppose we look for a timing structure where there is the smallest wait time when transfering from one bus to a different bus.
  • If two buses on different routes arrive at the same time and wait for passengers to transfer from the other bus this is the shortest wait for all travelers.
  • If the travel time between system nodes is made the same then the most nodes will have the least transfer time in all directions.

Transfer payments and The End of Oil by Paul Roberts

Seen at Pillar Point on a Sunday morning.
What a great way to get from San Francisco to one of the best beach walks in Northern California, with dogs.

Now, I haven't got around to working on the published spreadsheets. And I also have not managed to figure out the really difficult m and b coefficients I am seeing for the San Mateo bus system.

These are the questions that I would like to explore now:

  1. I need to get the spreadsheets and the data entry forms into good condition. Rewrite for clarity, list the validation tests, work out a way so users can print out a graph and the matching assumption values.

  2. What about looking at schemes for equalizing the "perceived personal cost" between cars and bus riding? Lets look at various kinds of transfer payment.
I was reading The End of Oil by Paul Roberts. One of the things I found interesting was how Mr. Roberts describes the chronic problems of desirable solutions that cost more than the present oil based solutions.

The End of Oil, by Paul Roberts, book website: http://the-end-of-oil.com/

I recommend to you The End of Oil as an enjoyably sane and well organized broad view of the "end of oil" and the paths available and the problems in moving American and the world society towards using much less oil.

Getting paid for riding the bus? Getting paid for leaving your car at home on grocery day? How about time specific coupons to direct freight cart traffic onto lightly loaded buses.

Noting the broader view from Wendell Berry

Sunrise from Pillar Point West Shoreline Access

Here is an essay by Wendell Berry, pointing to changes in social philosophy that correspond with a simpler, low energy American society.

Wendell Berry is writing about very big ideas.

This "Put carts on the public bus..." essay is a relatively small project to bring forward a lower energy transportation structure.

If we can solve some of the problems identified from this small project, then perhaps other low energy transportation projects will also become more feasible.


Spreadsheet, chart and form available

Sorry, the bugs are not fully worked out for this posting process. The graph and the spreadsheet are not synchronized and the underlying assumptions are not clearly presented.

Here is a chart showing the relative perceived costs of trips made using an automobile and a bus.

Chart of perceived cost for a given hourly wage, gasoline price, car mileage and bus fare

Here is a form for changing the input values of the perceived cost chart.

Input data form

Total cost of a trip: Car versus Bus

Personal time $5/hr, Gasoline $3.70/gallon,Bus fare $2.00, Vehicle 22.5 mi/gallon, bus and auto coefficients mentioned earlier.

So here is a graph to begin exploring problems with the economic utility of using carts on the public bus as an alternative transportation system to the auto or truck.

The lines show a kind of total cost for making a trip of a given distance.

For the car, the total cost being graphed is $ 5/hr for personal time plus $.33 per mile for gasoline and maintenance.

For the bus, the total cost being graphed is $5/hr for personal time plus $ 2.00 bus fare.


The graph shows in a nutshell the problem that all kinds of bus travelers face: When you count a cost for the time you spend riding the bus, the "total cost" of the bus ride is greater than the similarly figured "total cost" of a car trip.

  • I'll work on putting up a Google spreadsheet so you can experiment with changing the values of the graph. It is very interesting to see the effects of eight dollar gasoline and a modest improvement in the linear equation coefficients of the bus system.
  • One of the most impressive changes in assumptions is to explore cart transportation schemes that eliminate the cart operator. Carts that can travel on the bus system without an operator are released from the time and distance economics that make individual autos attractive.

Graph for travel time, auto vs. bus, preliminary

Please note the values shown here for the bus are estimates.

For both the auto and bus lines above, a trip of 0 miles does not exist. I simply solved for 0 miles which is also called the "x intercept"

Both lines above are generated using the y=mx+b type of linear equation.

For the auto, the travel time y in minutes for x miles is:

y = travel time in minutes
x = trip distance in great circle miles (or "as the bird flies").

m = 1.41 minutes/mile
b = 12.09 minutes per trip

The automobile coefficients are based on 46 vehicle trips measured with a GPS (global positioning system).

For the bus, I used a set of 6 trips using travel times calculated using the samtrans.org web site.

m = 12.91 minutes/mile
b = -14.96 minutes per trip


The next step is to compute and compare total percieved cost of a given bus trip with a car trip.

The "perceived total cost" depends on a very simple economics generalization that I found in a 1970 vintage economic analysis of the decision to extend the Victoria Subway system in London.

The economists generalized the value of a person's time at 1/3 that person's hourly wage. If you earn $15/hr at work, then the approximate value of your personal time spent doing your daily journeys is $5/hr.

(Footnote to original research paper needed).

So the next step in my freight carts on buses analysis is to compare the total perceived cost of riding a bus with the total perceived cost of driving a car, for a set of realistic distances using travel time data that reflects actual experience.

I am particularly interested in setting up some data input forms so the reader of this blog can change the input values. For instance, just how high does gasoline have to go in price before I switch to riding the bus to work?

I know $4 a gallon gas got me concerned enough to start this blog.

When I reach the point where the cost comparisons have plausible supporting data then it will be possible to explore what economic or physical changes will make freight carts on buses an attractive transportation mode for American society.

Linear equation for travel time by a car

I have finally finished re-creating my linear regression travel time analysis.

Based on measuring 46 car trips in my 1993 Dodge Van, with many trips starting from my El Granada neighborhood, here is a formula for the time required to drive places in a car:

  • Given the distance in miles measured as a bird would fly (great circle miles)

  • Time = 1.41 minutes per mile + 12.09 minutes

Details of the data:
  • The measurement covered 46 vehicle trips.
  • Average distance was 7.90 miles.
  • Average trip time was 23.2 minutes.
  • Average speed for all trips was 17.72 miles per hour.
Another way of looking at the transportation formula is:

1.41 minutes per mile implies 42.55 miles per hour... (take the reciprocal and multiply by 60 to convert miles per minute to miles per hour).

That is pretty reasonable as my neighborhood has mostly 45 and 50 mile per hour speed limits on Highway 1.

The travel time formula is an approximate empirical constant that reflects the performance of a car as a transportation device.

So a car trip consists of "About 12 minutes plus you move toward the destination at 42.55 miles per hour."

Travel time equation - data collection for auto travel

Here is a graph showing travel time and distance for travel by a car.

Each of these data points was marked at the beginning or end of a trip made by driving a car.

What I am going to do with these data points is develop a linear formula that relates distance and the time required to travel that distance for a specific type of transportation.

Example: Given two points D miles apart, how many total minutes is the travel time, by car?

Tc = Kc + Mc X D

Tc -- Time in minutes to drive by car.
Kc -- A constant term of minutes.
Mc -- Number of minutes per mile of D distance between points.
D - Distance to travel in miles.

The travel time formula above is very interesting when you have constant values for automobiles and public buses and other modes of personal travel.

  • You can talk about the personal economic and time choices the different modes of travel provide.
  • You can estimate the time benefits resulting from changes in the transportation system, like "faster buses".

Telling my local community this proposal.

Photo shows a wheelchair chassis carrying 2 pc. of Rubbermaid brand 18 gallon (68 liter) storage container.

The wheelchair with the storage containers is a sample or prototype for carrying stuff on a public bus.

I posted a link to this blog on my local community web site. Thanks for the comment and I'll work to clarify the proposal.


Carrying carts - what are the range of services?

Here is a list of the things that might be transported on the public bus system.

The first 3 items are carried by public bus systems in use today.

The last 4 items are transport challenges of increasing difficulty.
  1. Incidental baggage that depends on hand and arm strength - knapsack, purse, briefcase, shoulder bag, laptop computer in a case, makeup case, cloth grocery bags.
  2. Purpose carriers that use shoulder support - backpacker's pack, laundry bag, two bags of balanced weight with a shoulder strap.
  3. Bicycles carried in a front of bus carrier.
  4. Heavier objects that can be lifted but not held for long - tool boxes, handle grocery bags with canned goods.
  5. Wheeled devices that can be lifted up steps- rolling suitcases.
  6. Wheeled devices operated by a passenger that require ramp or lift to enter the bus: modified shopping carts, modified wheel chairs (seat cushion removed). Contents can include: construction supplies, personal computers, printing, mail, restaurant and office supplies, weekly groceries for a family, automobile repair parts.
  7. Freight containers designed for shipment without an accompanying passenger. These containers are travel on pallets and are boarded and removed based on shipping orders. Contents can include: groceries, business supplies, personal computers and monitors, store purchases, food supplies for restaurants, hotels, schools.

The peak problem: why buses have spare capacity

The load factor of a bus is fraction of seats filled or passengers riding compared to the total number of seats or passenger capacity.

In 1979, the load factor for the Los Angeles public bus system was somewhere below 33%. I am looking for a published load factor with a 2007 to 2008 date.

For the typical public bus system, and for a lot of passenger carrying systems, load factors of 25% and less are quite common. There are a number of reasons for this persistent state of affairs:
  • A passenger car with 1 driver out of a total of 4 or 5 seats is already operating at 25% or 20% load factor.
  • Public buses start out empty in the morning and they gradually fill up as they drive from the suburbs to the city center.
  • There is a big surge of bus patronage each morning. The bus management has to schedule enough buses to reasonably handle the peak of the surge. The peak hour bus fleet continues to run even as the patronage surge declines.
  • The peaks of the twice daily surge are plausibly about a workday plus a lunch hour apart ( or about 8.5 hours).
  • It is a management and labor scheduling challenge for every public bus system to arrange reasonable working hours for bus drivers to match bus service with the demand for it.
As a public bus system critic I see the "below 33% load factor" as a huge service opportunity.
Potentially up to 66% of the gross carrying capacity of the public bus system is available to do useful work.

My daughter runs into the "bus can't do it" problem.

My daughter is having her first exciting month living off campus at a northern California college town.

Her initial scheme for saving money on groceries is to shop at a discount grocery store about a mile away, and perhaps 400 feet lower elevation.

In round numbers, she hopes to buy 4 bags of groceries at $15 each instead of 4 bags at $20 each in the gourmet ghetto.

So the scheme is to save about $20 by going to the discount store once a week. The bus fare is $1.75 each way.

The choices for her are:
  • Travel by bicycle: The trip is a 2 mile (round trip) bicycle ride on heavily traveled side streets.
  • Travel by bus: The bus trip requires walking 2 blocks to the bus stop, waiting, then taking a 1 mile bus ride, then walking 1 block to the store.

I'd say her travel problem puts her on the boundary line of the utility of the public bus system.

Doing this trip by bicycle is a strenuous ride with lots of automobile traffic hazard. Doing this trip by bus will raise problems with arm and hand fatigue. If she doesn't carry the 4 bags of groceries, then she needs a large knapsack and a shoulder bag.

So the point I see as a public bus critic is: the bus does not accommodate the day-to-day need of people to carry 30 to 120 pounds on the bus.

Example of how private firms block the transit district

My local public bus system was sued by a private transportation firm for "unfair competition". The result is, the public bus system restricts the amount of luggage people can carry on the public bus that goes to the airport.

Here is the story it appears in the SamTrans.org website.

"Luggage Policy on Route KX

SamTrans is unable to allow excess luggage to be carried on Route KX between San Francisco and the San Francisco International Airport in either direction.

SamTrans Carry-on Policy States:

"If a passenger can carry the item(s) on his/her lap or store it under or next to him/her on a seat unneeded and not hurt anyone else, and not interfere with others, the item is probably safe to transport." The final decision will be made by the bus operator, who is responsible for the safe transport of everyone on the vehicle.

Alternatives & Information:

If you are unable to ride Route KX, Route 292 can take you to The City or the airport.

SamTrans wants your trip to be as pleasant as possible. If you have any routing or fare questions, please call 1-800-660-4287.


Since SamTrans receives some federal money to help fund its operation, a private bus operator cited "unfair competition." Under an agreement with the Federal Transit Administration, SamTrans agreed not to transport passengers with luggage between The City and the airport on its express route.

If the item fits into the following definition, SamTrans will gladly allow the passenger to board with the item. "

source: http://samtrans.org/luggage.html

How does a transit district define itself?

Here is how my local San Mateo County transit district defines itself.

The omission... from my viewpoint as a transit critic is there is no commitment to allow the bus rider to carry or board with any substantial load.

"About the District

San Mateo County Transit District provides SamTrans bus service throughout San Mateo County and into parts of San Francisco and Palo Alto.

The transit district also offers Redi-Wheels paratransit service for persons with disabilities who are unable to ride SamTrans.

  • Mission:
    To supply the public with a high-quality, safe and efficient transportation system that should enhance quality of life by increasing access and mobility, reducing congestion, improving the environment and promoting economic vitality.
  • Vision:
    San Mateo County Transit District is a leader in providing mobility alternatives that are relevant and desirable.
  • Values:
    Integrity, Customer focused, Respect, Quality, Teamwork, Leadership, Accountability."
Quoted from: http://samtrans.org/info_about_the_district.html

Walk, wait and ride times explain the bus system network

The structure of a public bus network is reflected in the time a bus traveler spends walking to, waiting for and riding the bus.

If you want to "improve" the time performance of a public bus system, you can decide which time relationship you want to improve.

  • To reduce the walking tine, you add bus routes so the distance is less to walk.
  • To reduce the waiting time, you add buses so the interval between buses is less.
  • To reduce riding time, you add faster buses or provide high speed lane options.
There is a problem with globally "improving" a bus system by adding more buses and routes: The operating costs can go up much faster than patronage arising simply out of shorter trip times.

At the time of my 1979 bus study, the public bus system spent 3 times more than it earned in
farebox revenue.

You can say, conventional American public bus systems operate in a "cost and revenue saddle". The total time of a bus trip can be 2x to 3x the same trip made in a private car. There simply isn't the funding or resources to drive down bus travel times by the brute direct provision of more routes, more buses and faster buses.

(Unfortunately, I have not been able to find my 1979 linear regression travel time analysis data and formulae.) I am still looking. The data was on HP-41 magnetic cards. I hope to find at least the notes and calculation work sheets.

Saving the appearances: Determining a linear equation for travel time on a bus

i wanted to develop some observation based mathematics for the bus system. The purpose was to enable modeling of the bus system and evaluating proposed improvements or changes.

So I began with a "Travel time record". I generated one of these records every time I went on a bus trip. I did several versions of this form as the bus travel experience and what I could do with the data became clear.

To develop a linear travel time equation I collected these data items:

    • Start time, start location of the bus trip, measured from the door.
    • Walk time (time when I arrived at the bus stop).
    • Waiting time (time when the bus arrived, ending the wait)
    • Riding time (time when I got off the bus)
    • ... additional walk, wait and ride times for a 2 bus trip
    • Walk time (time when I walked to the destination and touched the door).
    • Destination location.
With just a few bus rides recorded, I began to see a travel time relationship emerge. For a typical bus trip: **

      • 30% of the trip time is spent walking.
      • 30% of the trip time is spent waiting.
      • 40% of the trip time is spent riding the bus.
Before even beginning a linear regression analysis, the basic characteristics of the public bus system as a network of paths, buses and walking people are beginning to emerge.

**The travel ratios presented here are subject to editing. I am hunting around in my garage for the original data. 5-28-2008

From "Carry carts" to our shared mythology

Here are two tools for making the public bus service do more. Consider the consequences if a person could board the bus with a shopping cart.

The consequence: An entire class of domestic living activities would be possible without owning a car or hiring a taxi.

Now in thinking about why the bus systems I have studied don't carry shopping carts I finally conclude is because Americans have a mythology about individual automobiles as our primary mode of travel. The public bus system occupies a little corner of that mythology.

By mythology, I mean we have a set of shared values and ideas. For instance, here is a statement that we broadly share:

There is one mind common to all individual men. Every man is an inlet to the same and to all of the same. He that is once admitted to the right of reason is made a freeman of the whole estate. What Plato has thought, he may think ; what a saint has felt, he may feel ; what at any time has befallen any man, he can understand. Who hath access to this universal mind is a party to all that is or can be done, for this is the only and sovereign agent.

Ralph Waldo Emerson,
Essays: History, paragraph 1.

Our shared mythology is continually present and interacting with our perception of what is reasonable, what is possible in the review and development of the public bus system in the new world moving toward a low carbon emission society.

Busbook conclusion: Change bus rules of operation to allow efficent operation.

I studied the public bus system of the Los Angeles metropolitan area in 1979. Within the framework of the rules of operation extended to it, I found the Southern California Rapid Transit District was providing "transportation for those who have no other choice" in a fair manner. The District had a "service standard" to provide service and access. The District put buses on the road and did the best it could with fare income and government support.

Part of the study was I bought a 1 month bus pass and I resolved to not drive my car at any time during the month. On the last three days of the "no car" month, a neighbor offered me $100 to install a gas water heater for him. I was unemployed at the time, staying temporarily in my Dad's house, and I needed the money bad.

I had two toolboxes of plumbing tools in a storage locker 15 miles away. I had previously been in business as "Lee McKusick Water Conservation and Faucet Repair" during the 1973 water crisis. The tool boxes weighed 48 lb. and 52 lb. I decided it was not practical to walk 4 blocks, some of it up hill with this load.

So, in violation of my study discipline, I drove my 1965 Volkswagen sedan (with a bald front tire and problematic liability insurance), got my tools and made my $100.

I had just lived through a practical demonstration of the failure of the public bus system. The service rules are rigged to provide "transportation for people and their incidental baggage only."

So the conclusion of my Busbook research is: The public bus system is caught in a saddle of restrictions on what it can do with it's resources. The restrictions condemn the system to operation at a low load factor and low usability. Low load factor means the overall fraction of seats filled for the entire bus service day is below 1/3. Low usability means most people choose a car or get some other kind of ride.

The affirmation is: The mythology and legal restrictions on the public bus system need to change. The mythology is a belief that a public bus system should not be allowed to provide services offered by the private sector.

The legal restrictions that must be reduced are restrictive clauses in Federal transportation equipment bonds and suits brought by commercial transportation providers that find a bus service steals patrons from their licensed transportation service.

Advocacy: Change public bus service idea

This blog is a rewrite of my 1979 research project "Busbook - A study of the Los Angeles public bus system".

The current (year 2008) global warming and carbon dioxide crisis reminds me of my experience with the 1979 energy crisis.

My 1979 study and my 2008 experience leave me in the uncomfortable position of advocating a substantial social change in the services offered by the public bus system in America.

The public bus system should add services of carrying carts and containers of 30 to 150 pounds with the service purpose of allowing more individuals, families, and small businesses to function sometimes without the use of an automobile or truck.

The public bus systems I have studied all show a "load factor" of 33% or less. That means about 2/3 of the bus capacity is unused.

The public bus must be wedged out of it's assumption that the public bus is for "those who have no other transportation choice" and service is limited to "carry passengers and their incidental baggage only."