PERFORMANCE MEASURES - G/G/m SYSTEM

Hello everyone!

For this week, we have to compute the performance measures of the system, without failures and setups and including them.

For this calculation, we use the following relations according to Little’s Law:

ORDERING STATION

Including Failures

As we mentioned previously, the station number two has a failure which is the result of sporadic accidents because of misuse of knifes, shake milk machines or stoves. To calculate the average time between failures is necessary to take into account the rate, 0.00472 accidents/day which means 3.278E-06 accidents per minute, so mf is 305084,75 minutes.  Additionally, the average time of reparation is 30 minutes. In order to compute the performance measures of this station, it is important to calculate the percentage of time in steady state that the station is available to proceed, that we are going to represent with the letter A.

COOKING AND DELIVERING STATION

Model G/G/m including Failures

Now we develop the model with non-planned failures for accidents handling tools, knifes or machines.

The failures occurs with a rate of 0.00472 accidents/day, that means and accident occurs in average every 211.86 days or minutes

We observed a mean time of 5 minutes each time for any kind of accident.

As A tends to 1 the performance measures will be the same for the model without preempt and not preempt stops. The following table resumes the performance measures for this station.

SAUCE STATION

As we said before this station presents a setup that is when the  employees have to refill the sauce bottles, that takes 3 minutes and is every 100 clients, according to this, we proceeded to calculate how long a client is in the row and how many customers are going to be on the station at any time. For the computation of the performance measure, we said that the station has 3 servers.

First we proceeded to calculate the arrival coefficient of variance, the effective time and the variance of the effective time having clear that the refilling time is a setup and not a failure.

Secondly, modeling the station as a G/G/M we computed the time that a client is in the row with the utilization of the station,

Finally, the same variables were calculated without the refilling time to look how this setup affected the quantity and time that customers were in the station.

Model G/G/m including Failures

Using Little’s Law, we compute the expected number of customers in system, in queue and in service, obtaining the following table.

EATING AREA STATION

According to G/G/m systems, we calculate the expected time in queue whit the next formula:

Using Little’s Law, we have the following resume table, for this station.

We hope you enjoyed the entry.

Nataly Patacón

Camila Fonseca

Alejandro Moreno

Freddy Guevara

HAMBURGER AS A G/G/m SYSTEM

Hello everyone!

For this week, we have to model the system based on the real distributions of times, obtained a few weeks ago with Crystal Ball.
We only have the distribution of the external inter-arrival time, we assume that the time of the arrivals for all stations is distributed as the external time. The distributions of the inter-arrival time for each station is Extreme Value, affected by the routing probabilities.
Times, failures and setups are presented for the full period, which means that HAMBURGER is full from 12-2 pm.

The description for each station is based on the next formulation.

ORDERING:

1. The distribution of the inter-arrival time is Extreme Value with the following parameters.
     Mean: 2.2 minutes
     Variance: 20.7 minutes^2

2. The distribution of the service time is Gamma with the following parameters.
     Mean: 1.4 minutes
     Variance: 0.9 minutes^2

3. This station has 2 servers.


     Setups and Failures

This station presents a possible setup is when the billing paper ends. The cashier has to open a drawer and put the new billing paper in the machine. This operation last 2 minutes. This setup occurs every 150 clientes.

The software can fail and the entity has to wait to keep ordering his meal. This failure occurs once a month.  The repairing time is 3 minutes while the cashier restart the machine.


COOKING AND DELIVERING:

1. The distribution of the inter-arrival time is Extreme Value with the following parameters.
     Mean: 2.2 minutes
     Variance: 20.7 minutes^2

2. The distribution of the service time is Lognormal with the following parameters.
     Mean: 6.5 minutes
     Variance: 7.6 minutes^2

3. This station has 3 servers.


According to the possible failures mentioned in previous entries, for the Cooking and Delivering station the possible failure correspond to accidents with tools as knifes or machines. This type of failure is very weird. It can occur with a rate of 0,00472 accidents/day.


     Setups and Failures


We analyze the system when it is full, so, HAMBURGER has strict schedules. Thus, the employees have to go to the restroom and have lunch before or after the full time. That's why this stations has no setups or failures, regarding to employee's physiological needs.



SAUCE STATION:

1. The distribution of the inter-arrival time is Extreme Value with the following parameters.
     Mean: 1.914 minutes
     Variance: 15.6678 minutes^2

2. The distribution of the service time is Beta with the following parameters.
     Mean: 1.0 minutes
     Variance: 0.7 minutes^2

3. This station has infinite servers.


     Setups and Failures


This station presents a setup regarding the sauces. HAMBURGER employees have to refill the sauce bottles every 100 clients. The  refilling time is approximately 3 minutes.

This station has no possible failures, because is a self-server station and the service time depends on the client.


EATING AREA:

1. The distribution of the inter-arrival time is Extreme Value with the following parameters.
     Mean: 2.42 minutes
     Variance: 25.047 minutes^2

2. The distribution of the service time is Lognormal with the following parameters.
     Mean: 21.7 minutes
     Variance: 117.2 minutes^2

3. This station has infinite servers.


     Setups and Failures


This station has no setups or failures because is a self-server station.