We used M/G/C Queuing Theory, where arrivals are modeled by a Poisson process, service times have a general distribution, and there are c servers.
We set C=35 to simulate just one circulator, and ran 1000 simulations to solve for these several outcomes:
- Utilization Factor (U) ~ Traffic Intensity
- Probability that there are n students in the system (Pn)
- Total Service Time (S)
- Total Wait Time (W)
- Average Line Length
- Average system size (number of students in the system)
- Average total time in the system (T)
After running 1000 simulations with C = 35, we set C = 70 to represent adding another circulator to the system.
The average arrival rate (A) is 0.74 students/minute, which is calculated from our data, where we found the average number of students arriving at both the Lee circ stop and the Clock tower circ stop. Since we are using M/G/C Queuing Theory, we had to use this average arrival rate and model it by a Poisson process.
To find the average service rate, we first considered the service from the South 40 (Lee and Clocktower) to Sam Fox, which we set as our “full” service route. The service rate we found for this was about 1/7.14 minutes. We then considered the service from the South 40 to Mallinckrodt as the “short” service route, which is the route that about 62.17% of students take, which is the majority. The service rate we found for the “short” route was about 1/4.73 minutes. To decide which service rate to use, we used a random number generator and compared that value to the 62.17% of students who take the “short” route. From this comparison, our code uses either the “full” service rate or the “short” service rate.