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BUSAN 305

Assignment 3

May 1, 2023

PART A

In problems 1 through 6 below you must find all five KPIs: W_q, W, L_q, L and r. That is,

W_q, mean waiting time in the queue

W, mean waiting time in the system

L_q, long-term average number of customers in the queue

L, long-term average number of customers in the system

r, server utilisation

1. Consider the M/M/3 queue with mean interarrival time 1.5 minutes and mean service time 3 minutes at each of the three servers.

2. Suppose now that the service times of the queue in the previous problem are uniformly distributed between 1.5 and 4.5  

3. Further, suppose now that the service times of the queue in the previous problem are triangularly distributed between 1.5 and 4.5 with mode 3.

In each problem above (1, 2, and 3) we would like to see the effect of increasing the arrival rate in small steps.

4. Re-evaluate the five KPIs for an inter-arrival rate that is 10% higher.

5. Re-evaluate the five KPIs again for an interarrival rate that is 20% higher (than the interarrival rate in problem 1)

6. Proceed re-evaluating the five KPIs again for interarrival rates 30% and 50% higher (than the interarrival rate in problem 1)

If, in some case, you run into problems (like system instability), how many more servers would you need to bring the system to stability?

Patients arrive to an urgent-care facility. Every patient must first sign in, possibly after waiting in a queue. After signing in, patients will either go to registration or go to trauma room, if they are seriously ill. They probably must wait in respective queues. Patients who have visited the Exam Rooms either exit the system or go to a Treatment Room and then exit. Patient going through Trauma then go to Treatment and the exit. The figure below depicts the story just presented.

7. Assume patients’ interarrival times are exponentially distributed with mean 6 minutes. The number of servers at each station is shown with a white circle. The split probabilities are also shown. The service times are exponentially distributed, and their means are as shown in the table.

a. For each station find the “traffic intensity” or “station utilization”  r_station.

b. Will this facility be able to handle the external patient load? Why?

c. If you could add a single server to the system, what station would you choose to place the server at? Why?

PART B

Use the Excel file named Data_Assignment 3.xls, which you will find on Canvas-->Modules --> Week 8.

Getting the data ready

You will see 56 points of data collected from a process (first column of data, column B).

Fill in the shifted data on columns C, D, E, F and G for respective time lags of size 2, 3, 4 and 5.

Notice that you must use the blue-shaded cells to get the data ready. That is because every time correlation function is calculated for two data samples, they must be of the same size.

Scatterplot of Service-Time Data

Draw the following scatterplots:

· Lag-1 Scatterplot of Service-Time Data

· Lag-2 Scatterplot of Service-Time Data

· Lag-3 Scatterplot of Service-Time Data

each for any two data sets with a lag of value 1, 2 and 3, respectively. (You have several choices for each; can you see why?)

Building the autocorrelation matrix:

We will build an autocorrelation matrix, that is, a matrix with results that estimate the autocorrelation value of the data when lagged by 1, 2, 3, 4 and 5 time periods.

Notice the section of the matrix (on the bottom right of the Excel worksheet) below the main diagonal. Fill in each entry with the correlation between the corresponding data columns on the left of the worksheet.

Fill in all entries on the lower entries below the main diagonal.

Were the data sampled from statistically independent sources?

Look at the values you just calculated for the autocorrelation matrix. Can you conclude anything about the statistical independence of the data set values?

Compare the values obtained on each diagonal of cells of the same colour. Are they similar?

By using the simple arithmetic of the values on each diagonal, do a correlation plot. (Y-axis is the average of correlations found for each lag, and X-axis is the value of the lag, that is, 0, 1, 2, 3,  …)