# Queue simulation of two parallel servers

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Panagiotis Artemiou
on 24 May 2023

Commented: Raymond Norris
on 24 May 2023

I've got a problem where I need to simulate o production line which consists of 3 stations having 1 machine each. Each machine serving time follows a specific distribution. The products pass one station after the other in series, and the code I wrote is the below.

hours_day = 8;

days_year = 220;

arrival_rate = 4; %per hour

arrival_rate_m = 4/60; %per minute

minutes_year = 60 * hours_day * days_year;

for y=1:10000

inter_arrival_times(y) = exprnd(1/arrival_rate_m);

arrival_times(y) = sum(inter_arrival_times(1:end)); % generate the arrival times of products for a year

if arrival_times(y)>minutes_year

break

end

end

m1_start = arrival_times(1); % first product arriving at station 1

for i=1:length(arrival_times)-1

m1_serve_time(i) = unifrnd(m1_min,m1_max); % machine in station 1 serve time

m1_finish(i) = m1_start(i) + m1_serve_time(i); % machine in station 1 finish processing

m1_start(i+1) = max(arrival_times(i+1),m1_finish(i)); % machine in station 1 start processing of next

end

m2_start(1) = m1_finish(1); % first product arriving at station 2 (the first to finish from station 1)

for j=1:length(arrival_times)-2

m2_serve_time(j) = unifrnd(m2_min,m2_max); % machine in station 2 serve time

m2_finish(j) = m2_start(j) + m2_serve_time(j); % machine in station 2 finish processing

m2_start(j+1) = max(m1_finish(j+1),m2_finish(j)); % machine in station 2 start processing of next

end

m3_start(1) = m2_finish(1); % first product arriving at station 3 (the first to finish from station 2)

for k=1:length(arrival_times)-3

m3_serve_time(k) = normrnd(m3_mean,m3_std); % machine in station 3 serve time

m3_finish(k) = m3_start(k) + m3_serve_time(k); % machine in station 3 finish processing

m3_start(k+1) = max(m2_finish(k+1),m3_finish(k)); % machine in station 3 start processing of next

end

This is like an MM1 simulation but with 3 "servers" in series.

Now I want to simulate what would happen if we add a second machine in station 1 that has the same servning characteristics as the other one (same distribution and parameters), and so they can process products in parallel.

Is it possible to do that in the way I developed the code until now, and if so how can this be done?

Any helpful idea is really welcome.

Thank you in advance!

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### Accepted Answer

Raymond Norris
on 24 May 2023

If I'm reading this right, I think this code

for y=1:10000

inter_arrival_times(y) = exprnd(1/arrival_rate_m);

arrival_times(y) = sum(inter_arrival_times(1:end)); % generate the arrival times of products for a year

if arrival_times(y)>minutes_year

break

end

end

can be rewritten as follows (and should save a bit of time).

% Get all of the random numbers upfront

inter_arrival_times = exprnd(1/arrival_rate_m,1,10000);

% Calculate the cumulative sumation at each point

arrival_times = cumsum(inter_arrival_times);

% Trim arrival times that exceed the number of minutes in a year

arrival_times(arrival_times>minutes_year) = [];

Regarding how to add a second machine to station 1, can you sketch out what the code might look like? I'm gathering it's not as simple as adding a fourth for-loop (i.e., m1b_serve_time(i) = ...) Does the code then need to lie within for 1st for-loop?

Also, do you have the Parallel Computing Toolbox? If so, all 3 of your for-loops could be run in parallel.

##### 3 Comments

Raymond Norris
on 24 May 2023

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