[ot] [wrong] DistractionBot Goes To Work
"Let's go to work!" said Worker to DistractionBot. "That sounds like great fun!" replied DistractionBot, stepping right in Worker's way. "Where is work? What are you working on?" "Right in this exact direction!" said Worker, curtly shoving through DistractionBot. "I'm studying stochastic processes for a project I have." "Oh?" said DistractionBot.
DistractionBot pushed Worker towards their work, then started freaking out. "Are you okay??" said Worker, turning around. "X(t) is a sum made of arbitrary other parts. Define those other parts, and soon you have X(t)!" moaned back DistractionBot. "Hmm!" mused Worker.
I'm working an exercise in a textbook. Sorry it is ending up on the list. X(t) is an unknown function that gives the number of people in a system that can serve 1 person at once and takes 1 minute to do so. At a given time t, what defines how many people are there? 1 person may be being served. If so, then any number of people may be waiting in line. People arrive in ordered delays. X(t) ~= (number of people who have arrived between 0 and t) - (number of people who have been fully served between 0 and t)
On Sat, Jun 5, 2021, 1:58 PM Karl <gmkarl@gmail.com> wrote:
X(t) ~= (number of people who have arrived between 0 and t) - (number of
people who have been fully served between 0 and t)
Both of these values are calculable.
S_n is the number of minutes that pass after person (n-1) arrives, before person n arrives. So the total number of people who "Let me work the problem!" says DistractionBot. "Oops, I messed it up. Ummm it had to do with a total number of people."
On Sat, Jun 5, 2021, 2:00 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 1:58 PM Karl <gmkarl@gmail.com> wrote:
X(t) ~= (number of people who have arrived between 0 and t) - (number of
people who have been fully served between 0 and t)
Both of these values are calculable.
S_n is the number of minutes that pass after person (n-1) arrives, before person n arrives.
We can likely write both of these in abstract parts, as X(t) is written above.
On Sat, Jun 5, 2021, 2:05 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 2:00 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 1:58 PM Karl <gmkarl@gmail.com> wrote:
X(t) ~= (number of people who have arrived between 0 and t) - (number of
people who have been fully served between 0 and t)
Both of these values are calculable.
S_n is the number of minutes that pass after person (n-1) arrives, before person n arrives.
We can likely write both of these in abstract parts, as X(t) is written above.
Number of people who have arrived = sum of 1 for each person whose arrival time is less than or equal to t Number of people who have been fully served = sum of 1 for each person whose serving conditions have been met
TortureBot sits down with an algebra equation. "Did you know that algebra equations have gotten people mobbed by gangs?" asks TortureBot. Worker refuses to believe this. "TortureBot, what do you think of X(t)?"
"The number of people you have hugged with torture at time t will increase when you get to torture even more with hugs after a 1 minute delay!" spewed TortureBot.
On Sat, Jun 5, 2021, 2:07 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 2:05 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 2:00 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 1:58 PM Karl <gmkarl@gmail.com> wrote:
X(t) ~= (number of people who have arrived between 0 and t) - (number
of people who have been fully served between 0 and t)
Both of these values are calculable.
S_n is the number of minutes that pass after person (n-1) arrives, before person n arrives.
We can likely write both of these in abstract parts, as X(t) is written above.
Number of people who have arrived = sum of 1 for each person whose arrival time is less than or equal to t
Number of people who have been fully served = sum of 1 for each person whose serving conditions have been met
Serving conditions. The time at which a person is completely served is the maximum of one minute after they arrive, and two minutes after the previous person arrived.
SeizureBot says, "Seizures are so much more fun than algebra! Let's have seizures!" Worker replies, "true, seizures can be really fun, can't they? Still, what do you think of X(t)?" SeizureBot sat down with some algebra. SeizureBot made X(t) spasm wildly, and S_n, too. SeizureBot says, "Neural cluster n fires wildly after a delay of S_n from neural cluster n-1. If neural cluster 0 fires at time 0, then neural cluster 2 fires at time S_2 + S_1. In fact, the time of any given neural cluster n spasming wildly can be given by the summation Sum[j=1...infinity, S_j * I{j <= n}], where I{} is the impulse function that evaluates to 0 or 1 depending on the truth of its parameter. You could call this St(n)." "Thanks, SeizureBot!" exclaims Worker.
X(t) ~= (number of people who have arrived between 0 and t) - (number of
people who have been fully served between 0 and t)
Both of these values are calculable.
S_n is the number of minutes that pass after person (n-1) arrives, before person n arrives.
We can likely write both of these in abstract parts, as X(t) is written above.
Number of people who have arrived = sum of 1 for each person whose arrival time is less than or equal to t
Number of people who have been fully served = sum of 1 for each person whose serving conditions have been met
Serving conditions. The time at which a person is completely served is the maximum of one minute after they arrive, and two minutes after the previous person arrived.
The time at which a person arrives, assuming person 0 arrives at time 0, is likely Sum[j=1...infinity, S_j * I{j <= n}]
Can we recursively define the time at which somebody is fully served? Person 0 is fully served at time 1 since it takes one minute. Person 1 is served at a time depending on when they arrive. A core concept here appears to be when serving initiates.
S_n and a 1 minute delay. St(n): time person n arrives. Sts(n): time person n begins being served Stf(n): time person n is done being served and leaves Stf(n) = Sts(n) + 1 Sts(n) = max(St(n), Stf(n -1)) Recursively defined. Can we reorient the process of recursion to define it as a sum with conditions? Probably somehow?
On Sat, Jun 5, 2021, 3:42 PM Karl <gmkarl@gmail.com> wrote:
S_n and a 1 minute delay.
St(n): time person n arrives. Sts(n): time person n begins being served Stf(n): time person n is done being served and leaves
Stf(n) = Sts(n) + 1 Sts(n) = max(St(n), Stf(n -1))
Sts(n) = max(St(n), Sts(n - 1) + 1) Each iteration, the final value is either one more than the previous, or a newly defined value.
On Sat, Jun 5, 2021, 3:44 PM Karl <gmkarl@gmail.com> wrote:
On Sat, Jun 5, 2021, 3:42 PM Karl <gmkarl@gmail.com> wrote:
S_n and a 1 minute delay.
St(n): time person n arrives. Sts(n): time person n begins being served Stf(n): time person n is done being served and leaves
Stf(n) = Sts(n) + 1 Sts(n) = max(St(n), Stf(n -1))
Sts(n) = max(St(n), Sts(n - 1) + 1)
Each iteration, the final value is either one more than the previous, or a newly defined value.
St(n) = Sum[j=1...infinity, S_j * I{j <= n}] In the end, we want the number of people as a function of time. How many people have arrived by time t? Arrived(t) = Sum[j=0...infinity, 1 I{St(j) < t}] Using the impulse function to do everything is new to me. Not easily accessible.
participants (1)
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Karl