A couple of thoughts have come to my mind recently with respect to the coronavirus.
As more citizens become infected with a virus, fewer citizens are available to become infected. This is identical in principle to the arrival rate in a finite population waiting line system. Consider, Example D7 from your Heizer/Render/Munson textbook. There are 5 printers that each break down at the rate of .05 per hour. Thus, if all five computers are working, the system arrival rate is 5*.05=.25 while if all 5 are broken down the system arrival rate is 0. Over time, the arrival rate changes depending on the number of printers that are working and we can compute the weighted average arrival rate, which we term the effective arrival rate. The Excel worksheet for this example, available on MyOMLab, computes the effective arrival rate as .218 printers per hour. This effective arrival rate is similar to the effective reproductive number that epidemiologists use for viruses.
| Data | Results | ||||
| Arrival rate (l) per customer | 0.05 | Average server utilization(r) | 0.436048 | ||
| Service rate (m) | 0.5 | Average number of customers in the queue(Lq) | 0.203474 | ||
| Number of servers | 1 | Average number of customers in the system(Ls) | 0.639522 | ||
| Population size (N) | 5 | Average waiting time in the queue(Wq) | 0.933264 | ||
| Average time in the system(Ws) | 2.933264 | ||||
| Probability (% of time) system is empty (P0) | 0.563952 | ||||
| Effective arrival rate | 0.218024 | ||||
An interesting graphic related to the virus spread is at this Washington Post web site.
Observation: I recently had the opportunity to attend a concert at the Amalie Arena in Tampa. At intermission, the men’s room had a long line. This is not unusual. However, the line was not for the urinals or stalls but rather for the sinks. This was unusual. The design of the bathrooms was clearly for normal use rather than for a situation like the one we currently have with increased demand for handwashing. I was wondering what an arena might do to handle the increased sink demand.