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The questions says:

A printer is used to process print jobs. The interarrival times between jobs are exponentially distributed with a mean of 70 seconds. The time that is required to perform a job is exponentially distributed with a mean of 35 seconds.

a) Assume that 2 printers are used to process the incoming print jobs. Determine the mean time an incoming job has to wait until it can be processed (in seconds).

So I tried to figure this out:

In seconds: $\lambda=70$, $\mu=35$, $\rho=\frac{70}{35}=2$ we have two printers so my $c=2$.

Now to see if there's a steady state: $\frac{\rho}{c}=\frac{2}{2}=1\nless1$, so there is no steady state.

So with the formulas I know I can't calculate the Wq, can someone help me figure this out?

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    $\begingroup$ Don't we actually have $\lambda=1/70$ and $\mu=1/35$ (from which $\rho=1/4$)? $\endgroup$
    – ə̷̶̸͇̘̜́̍͗̂̄︣͟ ♦
    Dec 17, 2021 at 13:09
  • $\begingroup$ @TheSimpliFire This works thanks, why do we have to do this tho? $\endgroup$
    – Bas van Amstel
    Dec 17, 2021 at 13:22
  • $\begingroup$ Why you have to do what exactly? $\endgroup$
    – ə̷̶̸͇̘̜́̍͗̂̄︣͟ ♦
    Dec 17, 2021 at 13:23
  • $\begingroup$ @TheSimpliFire Why it is 1/70 and not just 70. Because 70 is also in seconds or did I miss the definition of lambda? $\endgroup$
    – Bas van Amstel
    Dec 17, 2021 at 13:55
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    $\begingroup$ $\lambda$ is the mean arrival rate. An interarrival time of $70$ seconds means one arrival every $70$ seconds (so $1/70$ per second). If $\lambda=70$ that would mean $70$ arrivals per second. $\endgroup$
    – ə̷̶̸͇̘̜́̍͗̂̄︣͟ ♦
    Dec 17, 2021 at 13:57

1 Answer 1

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Most of the formulas I have come across in queueing theory assume steady state, and I don't know of a purely mathematical approach for all cases. So let me show you a simple simulation approach.

Using Ciw, let us simulate for $10^4$ jobs.

import ciw
import pandas as pd

ciw.seed(2018)


network = ciw.create_network(
    arrival_distributions = [ciw.dists.Exponential(1/70)],
    service_distributions = [ciw.dists.Exponential(1/35)],
    number_of_servers = [2]
    )

simulation = ciw.Simulation(network)

simulation.simulate_until_max_customers(10000)

records = pd.DataFrame(simulation.get_all_records())


print(
    (
        records.waiting_time
        .describe()
        .to_markdown()
        )
    )

And here are the results.

waiting_time
count 10000
mean 2.14293
std 9.50576
min 0
25% 0
50% 0
75% 0
max 161.518

And here is a plot of the arrival time vs waiting time:

enter image description here

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  • $\begingroup$ Would you say what does the waiting time equal to $160$ mean on top of $300000$ arrival_date? $\endgroup$
    – A.Omidi
    Dec 20, 2023 at 10:33
  • $\begingroup$ @A.Omidi Right, I see that spike. It means that one of the jobs waited on the queue for $\approx 160$ units of time and they arrived on the queue at time $\approx 300000$. $\endgroup$
    – Galen
    Dec 20, 2023 at 17:09
  • $\begingroup$ thanks for the clarification. As the arrival rate is more than the service rate, it seems there would be no queue on the server queue. It is a bit different from what you proposed on the graph. Do you check this? $\endgroup$
    – A.Omidi
    Dec 23, 2023 at 8:49
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    $\begingroup$ There are different notions of stochastic dominance if it peaks your interest to get into the mathematics of order relations on statistical populations. $\endgroup$
    – Galen
    Dec 23, 2023 at 16:59
  • 1
    $\begingroup$ thanks for the clarification. $\endgroup$
    – A.Omidi
    Dec 23, 2023 at 19:23

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