Activity: The Tool Crib by Hand
Discrete-Event Modeling Activities
Work in groups of two or three, with pencil and paper. You have about 20 minutes; if your group finishes early, compare your table with another group’s.
The situation. Mechanics arrive at a tool crib to replace worn-out tools, and one attendant serves them first come, first served. Table 1 gives the arrival times and service times of the first 15 mechanics, in minutes.
| Mechanic | Time of arrival | Service time |
|---|---|---|
| 1 | 3 | 4 |
| 2 | 11 | 4 |
| 3 | 13 | 4 |
| 4 | 14 | 3 |
| 5 | 17 | 2 |
| 6 | 19 | 4 |
| 7 | 21 | 3 |
| 8 | 27 | 2 |
| 9 | 32 | 2 |
| 10 | 35 | 4 |
| 11 | 38 | 3 |
| 12 | 45 | 2 |
| 13 | 50 | 3 |
| 14 | 53 | 4 |
| 15 | 55 | 4 |
We want to estimate the average time in the system, the average number of mechanics waiting, and the utilization of the attendant.
1. Define the model
What are the entities, their attributes, the events, and the state variables of this system?
For each event, which state variables can it change?
2. Keep the books
The table below is filled in through time 14. An arrival row (type A) records the mechanic’s arrival time and, because the data are known, the time its service will start and the time it will depart. A departure row (type D) records the mechanic’s time in the system. Every row records the number waiting in line and the attendant’s status after the event (B for busy, I for idle); server idle time is recorded when an idle period ends.
| Event time | Mechanic | Type | Arrival | Start service | Departure | Time in system | Number waiting | Status | Idle time |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | I | |||||||
| 3 | 1 | A | 3 | 3 | 7 | 0 | B | 3 | |
| 7 | 1 | D | 4 | 0 | I | ||||
| 11 | 2 | A | 11 | 11 | 15 | 0 | B | 4 | |
| 13 | 3 | A | 13 | 15 | 19 | 1 | B | ||
| 14 | 4 | A | 14 | 19 | 22 | 2 | B | ||
Complete the next 10 rows of the table.
At one event time, two events happen at once. When? Which did you process first, and would the other order change any of your statistics?
3. Estimate
Use your table through the time of its last row.
Estimate the average time in the system of the mechanics who have departed.
Estimate the average number of mechanics waiting. (Sketch the number waiting as a function of time first; the average is the area under it divided by the elapsed time.)
Estimate the utilization of the attendant.
Why is the average number waiting an average over time, while the average time in the system is an average over mechanics?
If you finish early
Continue the table until the last mechanic departs, and recompute the three estimates. Why do they differ from your estimates in part 3?