Time AND Reliabilty

How The Grinch  Stole Christmas

It came without ribbons, it came without tags.
It came without packages, boxes or bags.
Maybe Christmas doesn't come from a store.
Maybe Christmas (he thought) means a little bit more.

What if congested time in Travel Demand Models was a little bit more

Travel Demand Models that utilize the User Equilibrium assignment method proposed by Sparrow and Dafermos, use the Frank-Wolfe, F-W, algorithm to iteratively solve for this equilibrium. This requires the use of an impedance function.  The purpose of this impedance function is to recompute the impedance of links on paths between F-W iterations.  As implemented in Travel Demand Models, this impedance function is typically called the Volume Delay Function, VDF.  This function expresses impedance in units of time, but this time is more properly a measure of impedance, not observable times. The impedance function must have the volume for which equilibrium is being solved, as one of its independent variables.  The other variables may NOT change in value during F-W iterations. 

VDFs are typically expressed as  a function of volume to capacity ratios, v/c.  The Bureau of Public Roads, BPR, proposed an empirical impedance function which is based on v/c ratios, but the empirical observations were only for v/c ratios less than 1.2 and it was NEVER proposed that this equation be used for conditions with v/c ratios greater than 1.2.  Nonetheless, the BPR equation is the basis of the VDF that is recommended by the Travel Model Improvement Program, and is used for ALL v/c ratios.

Before proposing a theoretical impedance function, it is useful to identify those variables where the impedance will increase with increasing volumes.  The impedance function used in TDMs does not have to include all impedance choice variables, only those choice variables that cause a change in impedance with a change in volume.  If the impedance does not change with a choice variable, that variable  need not be included in the “impedance” function.  More properly what is needed is NOT an impedance function, but a CHANGE in impedance function.  This function need only consider how impedance changes in response to changes in volume.  For example, if the number of traffic control devices on a route is an impedance choice variable, the impedance associated with that choice variable does not change with a change in volumes, and that need not be considered in the UE.

The impedance as a function of volume DOES change as the reliability changes and as the travel time changes.  It is suggested that the “delay” in the BPR is more properly time, and reliability expressed in units of time, compared to the time and reliability with zero volume. If reliability and travel time are functions of volume, then a proposed equation of reliability, expressed in units of time, and a proposed equation of mean travel time, that change with increasing volume, is needed.  The function of reliability is a function of the mean time, but since the mean time is a function of volume, then reliability is, de facto, a function of volume.

It is suggested that the empirical BPR  already must include both mean time, and reliability expressed in units of time. Since the mean travel time is proposed to be a discontinuous function, any impedance function derived using mean travel time thus will also be a discontinuous function.

Azizi and Beagan  showed that route choice, was better correlated with the Planning Index, PI, than with the mean Travel Time Index, TTI.  It is proposed that the impedance function should be a function of PI. It was also shown that PI resulted in fewer assignment, F-W,  iterations before reaching essentially the same equilibrium volumes  It is proposed that the VDF should be PI and that it would include both time and reliability, e.g. be a Volume Delay AND Reliability Function, a VD(R)F.  The proposed theoretical VD(R)F,  based on the equations of travel time and reliability, is proposed to be

If v/c>=1  then TTI = FFS/TS*exp((v/c-1)/γ_T)) else TTI = 1/(γ_L*ln(1-v/c)/FFS+1))

VD(R)F =PI=TTI*(1+0.645*(1-exp(-(TTI-1)/.110171)))

where FFS, TS are link attributes similar to capacity (FFS is already coded on links),  and γ_T and γ_L are functions of FFS and TS, i.e. γ_T=( FFS-TS)/5.8 and γ_L =TS/5.8. The coefficients of reliability, δ and ε,  are shown using their theoretical values which are true for every link. 

This VD(R)F is very well correlated with the existing BPR equation for v/c ratios less than 1.2.  It is proposed that the existing validation of the VDF using the BPR formatted curve be used in this domain.  Figure 1 shows the traditional BPR curve with α =0.5 and β=4, (the original formulation of the BPR has α=0.15 but it also uses the practical capacity in the ratio.  The α =0.5 is the value consistent with the current convention of coding the physical capacity on links), and PI for a Free Flow Speed of 70 MPH, a Transition Speed of 60 MPH, and capacity of 2000 veh./hr./ln . The proposed interim BPR is the validated BPR for v/c less than 1.2,  and a simple queuing equation rotated by an angle θ,,where α=sin(θ), after a transition at v/c=1.2. The queuing equation is also coordinate translated to move the origin from (0,0) to (1.2, 1+α*1.2^β). The congestion portion of the BPR equation, α *(v/c)^β,  is itself coordinate transformed from an origin of  (0,0) to an origin of  (0,1). The proposed interim BPR is

1+α *(v/c)^β                               for v/c ≤ 1.2

and

1+α*(v/c-1.2)+α*1.2^β                for v/c > 1.2

Figure 1 Various Impedance Functions

Traffic Flow

 

Row, Row, RowYour Boat

Row, row, row your boat
Gently down the stream
Merrily merrily, merrily, merrily
Life is but a dream

Do vehicles in traffic behave like water in a stream?

This topic was rejected by a national publication, in part because one of the reviewers disagreed that people in traffic behave  like drops of water in a stream with laminar and turbulent flow. Oh, yeah?

Polynomial equations of speed/time have been proposed and are widely used. These equations describe the behavior of speed/time in two domains.  They require a Free Flow Speed, FFS, the speed at zero volume; the capacity, which separates the uncongested and congested domains; and either an explicit or implicit Transition Speed, TS. The two domains are:

·       Uncongested/ laminar flow, where the volume is less than the capacity, and

·       Congested/ turbulent flow, where the volume is greater than the capacity.

The polynomial equations adequately explain the speed/time in the uncongested domain.  Their use in the congested domain is always not necessary for Traffic Simulation models, but is necessary for Travel Demand Models.  The use of these polynomial equations creates some inconsistencies when they are used in the congested domain.  Polynomial  equations are smooth, i.e. have no discontinuities.  But the use of these equations creates a discontinuity at a v/c of 2.0, and requires the allowance of negative or imaginary, not real and positive, capacities. It is proposed that exponential equations of speed/time which are different in the uncongested and congested domains may be preferable. They will have a discontinuity at the transition between the two domains, but create no additional discontinuities and require only positive capacities.  These equations are:

Uncongested domain, v/c ≤1.

Vol/Cap = 1-exp (-1/γ_L *(FFS-Speed)))

Speed = f(Vol/Cap) = FFS*(γ_L/FFS*ln(1-Vol/Cap)+1)

where γ_L is a range ( transient) coefficient of the laminar equation.  If 99.97%, 3 Sigma, of the transition occurs during the range which is the absolute value of (FFS-TS), it has the value of

γ_L = -|FFS -TS|/ln(1-0.9997) = (FFS-TS)/5.8

This is a horizontal mirroring of the traditional exponential association of speed and a translation of the origin from (0,0) to an origin of (FFS,0), restated with v/c as a function of speed.

Congested domain, v/c >1

Vol/Cap = (1-γ_T*ln(1+(Speed-TS)/TS))

Speed=f(Vol/Cap) = TS*((1-exp((-(Vol/Cap-1))/γ_T ))+1)

where γ_T is a range ( transient) coefficient of the turbulent equation.  If 99.97%, 3 Sigma, of the transition occurs during the absolute value of the range (TS-0), it has the value of

γ_T = -|TS-0|/ln(1-0.997) =  TS/5.8

This is the traditional exponential association of transition and a translation of the origin from (0,0) to an origin of (TS,1), restated with v/c as a function of speed.

 

Reliability

 

Call Me Irresponsible

Call me irresponsible
Call me unreliable
Throw in undependable, too

So what is reliability?

My Blog, my rules.  This is longer and more technical than my typical postings, but so it goes.

The extra time added in order to be reliable is proposed to be

δ*(1-exp (-(TTI-1)/ϵ)))

This is an exponential association,  also known as a “limits to growth” or transitional, equation.  If the travel times are normally distributed, then 1+δ is the (one tailed) Z-Score of the percentile of reliability.  If reliability is 95%, then δ=.645

The value of ε is such that 99.99994%, 5 Sigma, of the transition must take place between TTI =1 and TTI=1+δ. The theoretical value for a 95% reliability is ε = 0.102815.

TTI, the Travel Time Index, is the Mean Time divided by the Free Flow Time, FFT. PI, the Planning Index is the Mean Time plus the Extra Time Added in order to be 95% reliable, divided by the Free Flow Time.  The Planning Index and the Travel Time Index is recorded for various metropolitan area and reported in the FHWA’s Urban Congestion Reports.  The Extra Time Added to be 95% reliable divided by the Mean Time is the Buffer Index, BI.  The relationship between the Planning Index, the Buffer Index, and the Travel Time Index is thus:

PI = TTI*(1+BI).

A non-linear regression of the 156 reported values in the FHWA's Urban Congestion Reports for the Third Quarter of the Federal Fiscal Year, the period including April, May, and June, for 2017, 2018 and 2019, yields the following result.  

PI = TTI*(1+δ*(1-exp(-(TTI-1)/ε))) 

with a coefficient of determination, r², of 0.964,

where:

δ          =0.602 +/-  0.001; and

ε           =0.113 +/-  0.007

One of the data points, which is an outlier to the regression, is for San Juan, Puerto Rico.  If the data for San Juan is removed, then the regression becomes

PI = TTI*(1+δ*(1-exp(-(TTI-1)/ε))) 

with a coefficient of determination, r^2, of 0.934,

where:

δ          =          0.621 +/-  0.008; and

ε          =          0.121 +/-  0.005.

Republicans III

 

Let’s Call The Whole Thing Off

So if I go for scallops and you go for lobsters,
So all right no contest we'll order lobster
For we know we need each other so we
Better call the calling off off,
Let's call the whole thing off.

There are differences, but the voters need both Democrats and Republicans.

The voters of the Commonwealth of Massachusetts have developed a novel approach to government.  The Massachusetts state Legislature has been controlled by the Democrats since 1959.  During this period, the Executive Branch has often been controlled by Republicans.  The current Governor is Republican Charles Baker.  I was appointed to a minor executive position by then Republican Governor William Weld.  Mitt Romney, the current Senator from Utah was once the Governor of Massachusetts. The Federal Transportation Building in Cambridge, MA bears the name of the late Republican Governor John Volpe.

This is because there a difference between justice and the law.  The voters of Massachusetts appear comfortable that the law should reflect those views advanced by the Democrats. However, the voters apparently also realize that the government will be comprised of corruptible humans and that power corrupts and absolute power corrupts absolutely.  A way to limit power and corruption is to promote small government.  When faced with the desire to have the government enact laws, the voters have said that they prefer Democrats.  When it comes to administer justice under those laws, they often prefer Republicans.  At least those Republicans who are not fixated  on advancing their own interests, excluding others from society, and believe in falsehoods.  A Republican party that represents society’s interests, seeks to include all individuals, and believes in the truth, but administers justice and enforces the law equitably is desired.

The current national Republican Party does not serve that role in Massachusetts.  That role is too important to be abandoned.

Judge Ketanji Brown Jackson

 

I Second That Emotion

Oh, but if you feel like lovin' me
If you got the notion,
I second that emotion.
So, if you feel like giving me a lifetime of devotion
I second that emotion.

 

All in favor of the motion to appoint Judge Ketanji Brown Jackson to the U.S. Supreme Court, say aye.

The first federal black judge in Alabama , U. W. Clemon, is claimed to oppose Judge Ketanji Brown Jackson who is being considered as President Biden’s nominee for the Supreme Court.  "She refused to approve the settlement because in her view there were no common factual questions," Clemon wrote. Clemon is named as a counsel at the firm that argued the losing side of the Ross v. Lockheed settlement case.

You mean Judge Jackson based her decision on the facts instead of approving the progressive settlement with which I bet she emotionally approved.  If this is not a ringing endorsement for why Judge Jackson should be approved, I don’t know what is. A Judge who renders opinions on the facts instead of her emotions. What a concept!

Pornography in Schools

 

For What It’s Worth

Paranoia strikes deep
Into your life it will creep
It starts when you're always afraid
Step out of line, the men come and take you away

If it wasn’t so scary it would be laughable.

Senator Ted “Cancun” Cruz, has said that "left-wing educators" are putting "explicit pornography in front of kids." However he would not give examples of what he means by pornography. 

The late Justice Potter Stewart famously said that “"I know it when I see it” in the SCOTUS decision  on obscenity in Jacobellis v. Ohio in 1964.  How do we know that it is explicit pornography, if we can’t see it.  It might indeed be pornography, but we won’t know until we see it.  Until we can see it, I would suggest Senator Cruz shut up and stop trying to scare people. 

Intellectual Property

 

All Across the Universe

Images of broken light which dance before me like a million eyes
They call me on and on across the universe
Thoughts meander like a restless wind inside a letterbox 
They tumble blindly as they make their way across the universe.

If a universe is created by a work of art, who owns that universe?

A role of the sovereign is to protect the property of his subjects, whether that property is physical or intellectual.  Economists distinguish property, goods, by the characteristics of being rival and/or exclusive.  Goods are rival if a price is charged.  (and do NOT confuse price with value.  Price is a way of allocating resources.   A good can be unpriced but have a value, e.g. fresh air. “The best things in life are free”.).  A good is exclusive if my using the good prevents you from using that same good (e.g. if I sit in a seat then you can’t sit in that same seat. However, if I see a movie, you can also see that same movie.  The seat in the movie theater is exclusive, but the movie on the screen is not).   Physical property is almost always rival and exclusive.  Intellectual property is typically non-exclusive. To ensure that intellectual property is created, the sovereign should encourage that a price can be collected for this property, i.e. make sure it is rival.   If there is no price, then the creator has no incentive to create that intellectual property.

The Constitution recognizes this.  Article One Sections 8 reads, ”To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries.”  Among the first laws passed by Congress were the establishment of copyrights for arts and patents for science.  ( A digression.  I long for the day when Arts and Sciences were one and the same.  I have a Bachelor of Science, which fortunately my college calls an Sc.B., and not the unfortunate initials which would otherwise distinguish it from a B.A.)

The key terms in the Constitution are “promote” and “limited time”.  The law has acknowledged that when a work is for hire, the inventor or author can be a corporation that employs that “work for hire”.  While a human author and inventor has a limited lifetime, a corporation  can live for much longer.  However the “limited times” applies for Writings and Discoveries whether the authors and inventors are humans or are corporations. 

The sovereign additionally can offer an opinion on the trustworthiness of goods that are being sold. In order to do this the sovereign can help enforce a trademark. When Congress passed the laws covering patents, it also extended it to cover Trademarks.  A Trademark is a separate issue that does not promote the arts or sciences, but addresses the Trust that can be placed in a good.  A Trademark confirms who produced the good. The  sovereign can also agree that the Trademark identifies the source of the good.  It is not a writing or a discovery. Unlike these, it does not exist for a limited time, but exists as long as it is enforced and used.  For example, the Lowenbrau Beer lion Trademark dates to 1383. 

The United States Patent and Trademark Office is an agency in the U.S. Department of Commerce that issues patents to inventors and businesses for their inventions, and trademark registration for product and intellectual property identification. The United States Copyright Office, a part of the Library of Congress, is the official U.S. government body that maintains records of copyright registration in the United States including a Copyright Catalog.

In the United States, a patent is valid for 20 years, if it is renewed during that period.  In the United States, a Trademark is valid potentially forever.  In the United States, copyrighted works published after 1923, but before 1978 are protected for 95 years from the date of publication. If the work was created, but not published, before 1978, the copyright lasts for the life of the author plus 70 years.

The fear is that Intellectual Property is fairly easy to copy and the benefit will accrue to the copy of the  Intellectual Property, especially since it is non-exclusive. If it is copied without a price being paid, there is no incentive for an artist to create that Intellectual Property.  The constitutional protections specifically mention authors and their writings.  At time of the drafting of the Constitution, the only works of art which could be easily copied were writings by authors.  Technology has changed since that time.  Many works of art can now be copied or photographed.  The creators of works of art now include photographers, movie directors, musical artists, etc. in addition to authors.  To promote the arts, protections are extended to their works. 

While 95 years is cited as the “limited time” for older works of art, it mentions the life of the author plus seventy years for works published after 1978.  If the “artist” is a corporation,  then its lifetime is not a reasonable limit.  Should the lifetime be the life of the “work for hire” artist employed by the corporation?  If the copyright is intended to cover the artist’s life plus the lifetime of his heirs, who are the heirs of a “work for hire”? It would appear that 95 years is long enough to cover the remainder of an artist’s life, plus the life of their heirs, if any.

But in addition to a work of art, the best storytellers create a universe in which their work of art is set.  That universe can be Disney, Marvel, the World of Harry Potter, etc.  That universe would not exist if the storyteller had not created it.  It would seem that the universe of the storyteller, whether it is the Star Trek, or Star Wars, that universe should also be protected.  The Intellectual Property which is that universe should be protected for as long as the work of art that was copyrighted. 

The problem is that while 95 years seems like a very long time, 95 years from the date that the Disney Universe was created with the first copyrighted appearance of Mickey Mouse in “Steamboat Willie” is 2023.  Ninety-five years from the  date that the Superman-DC universe was created with the first issue of Action Comics will be in 2033.   The ownership of those created universes is an immediate issue. 

It is suggested that these universes are a Trademark which could endure forever.  The trademark for Lowenbrau does not mean that there is no other beer, only that beer can not call itself Lowenbrau beer if it is not.  Characters in a fictional universe are already protected for 95 years from the date they first appeared in that universe.  Portions of the universe can be sold or licensed, as Sony Pictures has licensed Spider-man from Marvel.  That license presumably enters into the public domain 95 years after the first appearance of Spider-man in 1962. 

Will Disney seek to extend its copyright of Mickey Mouse for longer than 2023? Probably.  But just as Bayer has endured and sells Bayer Aspirin long after the patent for aspirin entered into the public domain, and there is still a market for coats with the Burberry plaid even though a plaid cannot be copyrighted, there will be value in a Trademark for Disneyland, long after Mickey Mouse has entered into the public domain.  The Disney Universe is a Trademark. If the House of Gucci has endured, then the House of Mouse will endure.