The idea of this is to model the system as a Jackson`s Network. For this purpose we are going to assume that the service and arrivals time are exponential. Additionally, we assume that the capacity of each station is infinity.
With these assumptions in mind, we are going to find the respective rates of arrivals of each station (In the diagram we show the arrivals rate we have calculated). The results are shown in the table :
For a better analysis and understanding of the system, we are going to calculate three performance measures. The first measure we are going to calculate is the utilization of each station of the system. We chose this performance measure because we want to know if the system can get a steady state and we also want to know if any station needs another server for that purpose. Secondly, we are going to calculate the average amount of customers in steady state in the whole system. The idea is to know if the system has the structure necessary for managing the average number of people is in the system. Finally we want to know how much time a customer spends in the whole system. The idea is to know if the system is taking more time than the possible time an average person could spend in this kind of process.
With these results, we can conclude that the station 2 needs 5 more servers to get an steady state, and that the station 3 needs one more server for the same reason. On the contrary, station 1 is already stable. Now, assuming we put these new servers with the purpose of achieve a steady state, we calculate L and W.
For calculate the L for the stations 2 and 3, we used the equations for a system M/M/s/GD/∞/∞ and the results are the following:
As we see, the model shows that the station 3 has 84.57 customers in average in a steady state. This number is quite big but is really close to the reality, because when we were collecting data, we could see the big amount of people in this station. On the other hand we can see that in average, the amount of people in the system is really big and sometimes the system can not manage this number of customers.
Finally we find the average time a customer spends in the system (W):
