Queue simulation of two parallel servers

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!

Answer 1

[Raymond Norris]

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.

[Panagiotis Artemiou]

Thanks for the answer firstly.

Yes you are right on the fisrt section.

Regarding what the code might look like I can describe how I think it should work.

When the first product arrives, it can go to be processed in either machine 1 or machine 2 (which are in parallel in the first station). Lets say it goes in machine 1. Then the second arrival arrives at a specific time. If, at that time, machine 1 is still processing the first product, then it goes in machine 2, if machine 1 has finished processing the first product then it can go to either of the machines.

Generally when a product arrives we should check if the time it arrived is smaller than the finish time of machine 1 and machine 2. If it is then this product will start being processed when one of the 2 machined finishes its previous product. If the arrival time is larger than both finishing times, then this product starts being processed when it arrives simply. If the time it arrives is between the finish time of machine 1 and finish time of machine 2, the it goes in the machine whose finish time is the smaller.

I am not sure if I am easily comprehended. I hope this is helpful. If you want something more specific please let me know.

Thanks again in advance.

[Panagiotis Artemiou]

I can also provide a link to a youtube video of the logic I was describing.

There he explains the logic and simulates this in excel.

(1) Waiting Line Simulation Model with 2 ATM machines - YouTube

[Raymond Norris]

The video lays out the algorithm pretty well. Just need to initialize Machine 1 and Machine 2 properly before entering the for-loop.