Project 1: Service Time Estimates

Description:

This project aims to simulate service times for oil filter changes by 2 mechanics. The service times follow exponential distributions:

• Mechanic 1 (slow): mean = 12 minutes
• Mechanic 2 (fast): mean = 3 minutes
• You have an 80% chance of being served by the fast mechanic.

Generating Service Times

##  [1]  2.8065419  2.9736822  0.2613884 14.7914727 42.9958515  3.3540987
##  [7]  1.7055927  1.5364692 11.5774357  0.7874212  0.1833506  1.6854978
## [13]  1.3034673  5.6177430  0.4840478 21.1297836  4.0449052  3.0085135
## [19]  4.1293664  0.7242767 49.7944447  3.4666586  0.5676385 10.8404032
## [25]  1.5087962  3.0291367  4.2160368  2.5124400  0.7527660  1.1297600
## [31] 42.0726247 49.4306483  4.9287066  1.8005259  5.8920365  0.2358057
## [37]  0.1691832  5.6857508 10.9808769  2.3874722  6.5090331  2.2878845
## [43]  1.7515729  0.2948972  2.2579344  1.8275674  6.4269146  7.0757824
## [49]  2.4190689  3.6912852

## Estimated mean service time: 4.738109 minutes

## Theoretical mean service time: 4.8

Summary of Simulated Data

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##  0.00051  1.06488  2.56726  4.73811  5.55831 81.93549

## Standard Deviation:

## [1] 6.798332

## Variance:

## [1] 46.21732

Density Histogram

It is important to note that the line for the simulated mean is difficult to see because the theoretical and simulated means are almost the same.

Density with Mixture PDF

Findings

• The distribution is skewed to the right, which is to be expected in an exponential distribution.
• The simulated mean and theoretical mean are very close, which confirms the accuracy of the simulation.
• The peak near 0 shows that most customers were served by the faster mechanic.
• The long right tail indicates that some customers were served by the slower mechanic, pulling the average service time higher.
• The purple mixture PDF line fits the histogram and density curve very closely, validating the theoretical model.
• The overlap between the density curve and the mixture PDF shows that the empirical distribution from simulation aligns well with the mathematical model.