Queueing model without steady state

Question

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?

Answer 1

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: