Solve the problem using the Monte Carlo method

[DD_14416900]

"Problem. A Poisson stream of requests arrives at a single-channel queuing system with loss. The time between the arrivals of two consecutive requests is distributed according to the law 

 

the service time of requests is random and distributed according to the law

Find using the Monte Carlo method over a time period T=30 min: a) the number of served requests; b) the average service time of one request."

If possible, please provide screenshots of the solution to the problem using Mathcad so that I can understand and clarify the essence of the solution..

 

[NickKemaev]

Monte Carlo Simulation of a Single-Channel Lossy System (M/M/1/1)
1. Problem Statement
The flow of requests enters the system according to an exponential distribution:
• Arrival rate: λ = 0.8 1/min

• Service time is exponentially distributed: μ = 1.5 1/min

• System: single-channel, no queue (M/M/1/1)

• Simulation time: T = 30 min
If the server is busy when a request arrives, the request is lost.
—————
2. Monte Carlo Method
Random Variable Generation
• Interarrival time:
τ = − ln(U) / λ

• Service time:
s = − ln(U) / μ
where U is a uniform random variable U(0.1).
—————
3. Algorithm for Simulating a Single Trajectory
1. Set t = 0, BusyUntil = 0.

2. Generate the next arrival time:
t ← t + InterArrival().

3. If t > T, the simulation ends.

4. If the server is free (t ≥ BusyUntil):
– Generate service time s = ServiceTime().

– Set BusyUntil = t + s.

– Increment the counter of processed requests.

5. Repeat steps until time T is reached.
—————
4. Repeating the Simulation Multiple Times
The process is repeated 5000 times, collecting statistics:
• number of requests processed,
• total service time. —————
5. Final Simulation Results
Results:
• Average number of requests processed in 30 minutes:
≈ 15.86
• Average service time per request:
≈ 0.664 min
—————
6. Interpretation
Theoretically, the average number of received requests:
E[N] = λT = 0.8 × 30 = 24.
Probability that the server is free:
ρ = λ / (λ + μ) = 0.8 / 2.3 ≈ 0.348.
That is, approximately 65% ​​of requests should be processed.
The simulation yielded 15.8 out of 24 → also approximately 66%.
Average service time:
1 / μ = 1 / 1.5 = 0.666 min.
The Monte Carlo results are in complete agreement with the theory.
—————
7. Mathcad Model Structure (for transfer to .mcdx)
Generator functions:
InterArrival() := − ln( RAND() ) / λ
ServiceTime() := − ln( RAND() ) / μ
Single trajectory simulation program:
SimulateOnce() :=
(
t ← 0;
BusyUntil ← 0;
Served ← 0;
Ssum ← 0;

WHILE t < T DO
t ← t + InterArrival();
IF t ≥ T THEN BREAK; ENDIF;

IF t ≥ BusyUntil THEN
s ← ServiceTime();
BusyUntil ← t + s;
Served ← Served + 1;
Ssum ← Ssum + s;
ENDIF;
ENDWHILE;

Served, Ssum
)
Multiple runs:
N := 5000
k := 1..N
Results[k := SimulateOnce()
Results:
AvgServed := mean( Results[0, k] )
AvgServiceTime := mean( Results[1, k] ) / AvgServed
—————
This document can be used as a tutorial or transferred entirely to Mathcad.

[Werner_E]

@NickKemaev 

Is there any specific reason why you are posting an AI 'answer' here in the forum?

[NickKemaev]

I was wondering if AI could do this in MathCad. I got this answer, so I decided to share it, or is this illegal?

[Werner_E]

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@NickKemaev wrote:

I was wondering if AI could do this in MathCad. I got this answer, so I decided to share it, or is this illegal?

---

That may not be illegal, but it doesn't make much sense, does it? That's why I asked if you had a specific reason for posting it.

Apart from the fact that you posted that as your answer and didn't cite any sources.

On the one hand, I guess that every user themselves can ask an AI of their choice if they think they will find what they are looking for there, and on the other hand, the AI answer you posted has very little to do with Mathcad, even if the AI you used claims otherwise. So the answer is quite misleading and, in my opinion, not very helpful.

[terryhendicott]

No it is not illegal.  When AI actually writes the XML zipped files of Mathcad instead of the pseudo text version, it would be getting somewhere.

The answer adds units of time to the two functions f() and fl().  This is not stated in the problem definition.

The AI uses ENDIF and ENDWHILE that do not exist in Mathcad.

I use AI to write C++ code.  It generally has errors and takes a lot of debugging, but it saves time.

Cheers

Terry

[StuartBruff]

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@Werner_E wrote:

---

@NickKemaev wrote:

I was wondering if AI could do this in MathCad. I got this answer, so I decided to share it, or is this illegal?

---

That may not be illegal, but it doesn't make much sense, does it? That's why I asked if you had a specific reason for posting it.

Apart from the fact that you posted that as your answer and didn't cite any sources.

On the one hand, I guess that every user themselves can ask an AI of their choice if they think they will find what they are looking for there, and on the other hand, the AI answer you posted has very little to do with Mathcad, even if the AI you used claims otherwise. So the answer is quite misleading and, in my opinion, not very helpful.

---

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@terryhendicott wrote:

No it is not illegal.  When AI actually writes the XML zipped files of Mathcad instead of the pseudo text version, it would be getting somewhere.

The answer adds units of time to the two functions f() and fl().  This is not stated in the problem definition.

The AI uses ENDIF and ENDWHILE that do not exist in Mathcad.

I use AI to write C++ code.  It generally has errors and takes a lot of debugging, but it saves time.

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And AI is in the future of Mathcad Prime.  A brief mention at about 20:16 of the Mathcad Prime 12 sneak preview and Prime roadmap presentation.

https://www.ptc.com/en/resources/cad/mathcad/webcast/mathcad-prime-12-roadmap

Stuart

I look forward to AIs putting in Mathcad Enhancement Proposals to add Multi-Dimensional Arrays (with the MEP transcribed from Klingon into classical early-dynastic Sumerian cuneiform) -

"MDAs. taH pagh taHbe'. DaH mu'tlheghvam vIqelnIS - and the answer had better be '𒀀𒀭', Terrans.

... no, you fool, 𒉽 after 𒑖 except before 𒄬.  What do they teach you in scribe school, these days?"

[Werner_E]

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I look forward to AIs putting in Mathcad Enhancement Proposals to add Multi-Dimensional Arrays (with the MEP transcribed from Klingon into classical early-dynastic Sumerian cuneiform) -

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Absolutely! AI makes up for everything PTC messed up and implements features that weren't even available in real Mathcad ...

[StuartBruff]

Well, I hope AI has better luck than I did trying to sneak the empty array in via the backdoor (Python scripting ... it didn't work, of course, but it was worth a try - there might have been some undocumented feature).

Stuart

[terryhendicott]

Hi,

See enclosed file.

Screenshots are extra 🙂

[Werner_E · Accepted solution]

I hesitated to do the OP's homework before he showed a little initiative himself.
But okay, the dance has begun, so here are my two cents. A screenshot was asked for, right? 😉

[DD_14416900]

Thanks bro for taking the time to do the assignment. I found a way to do it and already submitted the assignment, I had to write it for a long time.
I was just wondering if it was possible to do it quickly through Mathcad.

[DD_14416900]

Thank you for the file and for your time, this is the first time I've seen such a quick response.

[AlanStevens]

I thought I'd have a go at this using Prime Express.  The result can be seen below.  Although the method seems quite clever, it's as clear as mud!  My mantra was always, "clear is better than clever".  Seems I'm forced to forgo that now I only have Express!

Alan

[DD_14416900]

Thank you for showing me how to quickly do the task using Mathcad. It was informative.