Hello,
More of my philosophy about the mathematical modeling of the Jackson network and about mathematics and more of my thoughts..
I am a white arab from Morocco, and i think i am smart since i have also invented many scalable algorithms and algorithms..
I think i am highly smart since I have passed two certified IQ tests and i have scored "above" 115 IQ, and i mean that it is "above", so I have to explain more my mathematical modeling of the Jackson network below, so notice that i have solved the
Jackson network with my mathematical modeling so that to show how you can solve it in general, i mean my mathematical modeling methodology can be applied so that to solve the Jackson network problems more easily, so you have to read my HTML tutorial
below, inside my zip file below, that is like a small paper about the Jackson network problem(s), but i have also created the following software project called PDQ that permits to solve the Jackson network by modeling it with the software, and here it
is:
PDQ for Delphi and Freepascal is here..
This is a port by Amine Moulay Ramdane of PDQ version 6.2.0 to Delphi on Windows and to Freepascal on both Windows and Linux, i have also provided you with two demos, one queuing MM1 demo, and another Jackson network demo. Also i have provided you with
my html tutorial on how to solve analytically the Jackson network problem provided to you as a PDQ demo.
PDQ is an analytic queueing-circuit analyzer made freely available under MIT/X11 license from www.perfdynamics.com
You can download it from my website:
https://sites.google.com/site/scalable68/pdq-for-delphi-and-freepascal
Also about mathematical modeling, i invite you to look at my
following project of my mathematical modeling of a Jackson network problem:
Jackson network problem: (The mathematical modeling by Amine Moulay Ramdane)
Description:
From:
http://www.perfdynamics.com/Tools/PDQpython.html
http://simpy.sourceforge.net/examples/Jackson%20network.htm
Messages arrive randomly at rate 0.5 per unit time at a communication network with 3 nodes (computers). Each computer can queue messages and service-times are exponential with mean m[i] at node i. These values are given in the column headed m[i] in the
table below. On service at a node a message transfers to one of the other nodes or leaves the system. The probability of a transfer from node i to node j is p[i,j]. Node 4 means "outside the system" so a message will transfer out of the system from node
i with probability p[i,4].
Your task is to estimate the average time taken for jobs going through the system and the average number of jobs in the system.
You can download it from my website here:
https://sites.google.com/site/scalable68/jackson-network-problem
I think i am highly smart since I have passed two certified IQ tests and i have scored "above" 115 IQ, and i mean that it is "above", so I have just provided you with my interesting tutorial about Markov chains in mathematics, read it below, and now i
will invite you to look at my following small software project that implements the following interesting mathematical theorem:
Multiple linear regression that uses SIMD SSE2 instructions and that implements the following mathematical theorem:
If A is an m x n rank n matrix, then the least-squares solutions to the system A*vector(X) = vector(C) are the solutions to the system:
A*vector(X)=
A*inverse(transpose(A)*A))*transpose(A)*vector(C).
This system has the following unique solution:
vector(X) = inverse(transpose(A)*A)*transpose(A)*vector(C)
I have updated my multiple linear regression program to support SEE (Standard Error of the Estimate), the "SEE" is important in quality control, the interpretation of SEE is similar to standard deviation's "68-95-99 rule", that is, 68% of the time the
predicted score will fall plus-or-minus 1 x SEE of the actual score based on the regression equation, 95% of the time the predicted score will fall plus-or-minus 2 x SEE , and 99% of the time the predicted score will fall plus-or-minus 3 x SEE.
So my Multiple linear regression program does show the multiple regression equation that approximate the statistical data and it calculates and shows the Coefficient of determination R^2 and it calculates and shows the Standard Error of the Estimate.
So you can download it from my website here:
https://sites.google.com/site/scalable68/multiple-linear-regression
More of my philosophy about Logic and about the Gödel's incompleteness theorems and about Markov chains and more of my thoughts..
I am a white arab from Morocco, and i think i am smart since i have also invented many scalable algorithms and algorithms..
I have searched more on internet the much more precise and correct Gödel's First incompleteness theorem, and here it is:
"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F"
And in mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence "must" be true or false, and it
cannot be both.
So that means that we know that the statement is true or false but it can not be proven true or false, so we then logically infer that we can not prove the consistency of the system , so the statement can be that it is like an axiom in mathematics that
is true but that we can not prove by such logical inference or deduction, so then the system remains really useful even if it's incomplete by Gödel's incompleteness theorems, so i think that Gödel's incompleteness theorems are not so problematic.
More of my philosophy about the truth table of the logical implication and more of my thoughts..
I think i am highly smart since I have passed two certified IQ tests and i have scored "above" 115 IQ, and i mean that it is "above", and now
i will ask a philosophical question of:
What is a logical implication in mathematics ?
So i think i have to discover patterns with my fluid intelligence
in the following truth table of the logical implication:
p q p -> q
0 0 1
0 1 1
1 0 0
1 1 1
Note that p and q are logical variables and the symbol -> is the logical implication.
And here are the patterns that i am discovering with my fluid intelligence that permit to understand the logical implication in mathematics:
So notice in the above truth table of the logical implication
that p equal 0 can imply both q equal 0 and q equal 1, so for
example it can model the following cases in reality:
If it doesn't rain , so it can be that you can take or not your umbrella, so the pattern is that you can take your umbrella since it can be that another logical variable can be that it can rain in the future, so you have to take your umbrella, so as you
notice that it permits to model cases of the reality ,
and it is the same for the case in the above truth table of the implication of if p equal 1, it implies that q equal 0 , since the implication is not causation, but p equal 1 means for example that it rains in the present, so even if there is another
logical variable that says that it will not rain in the future, so you have to take your umbrella, and it is why in the above truth table
p equal 1 implies q equal 1 is false, so then of course i say that the truth table of the implication permits to model the case of causation, and it is why it is working.
More of my philosophy about objective truth and subjective truth and more of my thoughts..
Today i will use my fluid intelligence so that to explain more
the way of logic, and i will discover patterns with my fluid intelligence so that to explain the way of logic, so i will start by asking the following philosophical question:
What is objective truth and what is subjective truth ?
So for example when we look at the the following equality: a + a = 2*a,
so it is objective truth, since it can be made an acceptable general truth, so then i can say that objective truth is a truth that can be made an acceptable general truth, so then subjective truth is a truth that can not be made acceptable general truth,
like saying that Jeff Bezos is the best human among humans is a subjective truth. So i can say that we are in mathematics also using the rules of logic so that to logically prove that a theorem or the like is truth or not, so notice the following truth
table of the logical implication:
p q p -> q
0 0 1
0 1 1
1 0 0
1 1 1
Note that p and q are logical variables and the symbol -> is the logical implication.
The above truth table of the logical implication permits us
to logically infer a rule in mathematics that is so important in logic and it is the following:
(p implies q) is equivalent to ((not p) or q)
And of course we are using this rule in logical proofs since
we are modeling with all the logical truth table of the
logical implication and this includes the case of the causation in it,
so it is why it is working.
And i think that the above rule is the most important rule that permits
in mathematics to prove like the following kind of logical proofs:
(p -> q) is equivalent to ((not(q) -> not(p))
Note: the symbol -> means implies and p and q are logical
variables.
or
(not(p) -> 0) is equivalent to p
And for fuzzy logic, here is the generalized form(that includes fuzzy logic) for the three operators AND,OR,NOT:
x AND y is equivalent to min(x,y)
x OR y is equivalent to max(x,y)
NOT(x) is equivalent to (1 - x)
More of my philosophy about eigenvectors and eigenvalues and Markov chains in mathematics and more of my thoughts..
I am a white arab from Morocco, and i think i am smart since i have also invented many scalable algorithms and algorithms..
I think i am highly smart since I have passed two certified IQ tests and i have scored "above" 115 IQ, and i mean that it is "above" 115 IQ, and as you know, i have just invented my philosophy and the new ideas of my philosophy and posted them here and
i have just invented many proverbs and posted them here and i have invented many poems of Love and poems and posted them here... , so now i will quickly create an interesting mathematical tutorial about eigenvectors and Markov chains in mathematics, and
notice carefully my way of creating it, and here it is:
Say v is an eigenvector of a matrix A with eigenvalue λ. Then Av=λv.
So there is an important thing to know in mathematics and it is that mathematical eigenvectors show in phenomenons that exhibit a "stable" behavior with time, and in Markov chains we search also for
an eigenvector where the system will stabilize, so we have to solve:
A*vector(v) = I*vector(v) [1]
The identity matrix I has 1 as its only eigenvalue. All vectors are associated eigenvectors since
I*v = v = (1)v
for all v.
A is the transition matrix
and v a vector.
So since the Eigenvalue is 1, that means we have to solve the
following system of equation that will give you the eigenvector
where the system will "stabilize" its behavior:
(A - I)*vector(x)= vector(0)
I is the identity matrix.
So now here is my tutorial about Markov chains in mathematics:
In mathematics, many Markov chains automatically find their own way to an equilibrium distribution as the chain wanders through time. This happens for many Markov chains, but not all. I will talk about the conditions required for the chain to find its
way to an equilibrium distribution.
So if in mathematics we give a Markov chain on a finite state space and asks if it converges to an equilibrium distribution as t goes to infinity. An equilibrium distribution will always exist for a finite state space. But you need to check whether the
chain is irreducible and aperiodic. If so, it will converge to equilibrium. If the chain is irreducible but periodic, it cannot converge to an equilibrium distribution that is independent of start state. If the chain is reducible, it may or may not
converge.
So i will give an example:
Suppose that for a course you are currently taking there are two volumes on the market and represent them by A and B. Suppose further that the probability that a teacher using volume A keeps the same volume
next year is 0.4 and the probability that it will change for volume B
is 0.6. Furthermore the probability that a professor using B this
year changes to next year for A is 0.2 and the probability that it
again uses volume B is 0.8. We notice that the transition matrix is:
| 0.4 0.6 |
| |
| 0.2 0.8 |
So the interesting question for any businessman is whether his
market share will stabilize over time. In other words, does it exist
a probability vector (t1, t2) such that:
(t1, t2) * (transition matrix above) = (t1, t2) [2]
and notice that the mathematical system [2] looks like the mathematical system [1] above.
So notice that the transition matrix above is irreducible and aperiodic,
so it will converge to an equilibrium distribution that is (t1, t2) that
i will mathematically find, so the system of equations of [2] above is:
0.4 * t1 + 0.2 * t2 = t1
0.6 * t1 + 0.8 * t2 = t2
this gives:
-0.6 * t1 + 0.2 * t2 = 0
0.6 * t1 - 0.2 * t2 = 0
But we know that (t1, t2) is a probability vector, so we have:
t1 + t2 = 1
So we have to solve the following system of equations:
t1 + t2 = 1
0.6 * t1 - 0.2 * t2 = 0
So i have just solved it with R, and this gives the vector:
(0.25 , 0.75)
Which means that in the long term, volume A will grab 25% of the market
while volume B will grab 75% of the market unless the advertising campaign does change the transition probabilities.
Thank you,
Amine Moulay Ramdane.
--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)