In Capacity Planning For 1st Responders, we considered the problem of dimensioning a group so that there is at least one member available when needed. Not all service groups, however, are expected to respond immediately to all customers. Most, from supermarket check stands and airport check-in counters to clinics for non-emergency health care, allow some amount of queueing, giving rise to the question of how long the queues become when the servers get busy.
At one point in his latest book, Andy and Me And The Hospital, Pascal Dennis writes that the average number of patients in an emergency room is inversely proportional to the availability of the doctors. The busier the doctors are, the more dramatic the effect. For example, if they go from being busy 98% of the time to 99%, their availability drop by half from 2% to 1%, and the mean number of patients doubles. Conversely, any improvement in emergency room procedures that, to provide the same service, reduces the doctors’ utilization from 99% to 98%, which cuts the mean number of patients — and their mean waiting time — in half.