During busy periods (9:00 a.m.-11:30 a.m. and 2:00 p.m.-3:45 p.m.), the central appointment office receives 40 calls per hour, on average. Each call takes an average of 3.11 minutes to serve. It is desired that at least 90 percent of calls are received without waiting. What is the minimum number of staff needed during these busy times?

When doctors referred their patients to the Lourdes Hospital in Binghamton, New York, for various services such as X-rays, their office had a tough time getting through to the centralized appointment office of Lourdes. Most of the time, the line was busy. The installation of a call waiting system did not improve the situation, because callers were put on hold for indefinite lengths of time. The poor service had resulted in numerous complaints.

One of the managers was put in charge of finding a solution, and a goal of answering at least 90 percent of calls without delay was set. The hospital was willing to employ more staff to receive calls. The manager studied this queueing problem by collecting data for 21 workdays during which additional staff was used to answer calls and no call received a busy signal or was put on hold. The number of calls per day ranged between 220 and 350, with no day-of-the-week seasonality. Most days, the number of calls was between 250 and 300. The average number of calls arriving during each 15-minute interval peaked at about 10 calls during the 9:00 a.m.-11:30 a.m. and 2:00 p.m.-3:45 p.m. periods. The 944 service duration's during the data collection period had a distribution similar to exponential with a mean of 3.11 minutes. The manager also found out that previously the 6.5 full-time-equivalent employees usually spent half their time doing other tasks and turned off their phones while busy with other tasks. Using the multiple servers queueing model and a service goal of at least 90 percent probability of not having to wait, the manager determined the number of staff required during each 15-minute interval. When the original staffing levels were compared with the model-determined ones, it was discovered that more staff were required earlier in the day and later in the afternoon, and fewer were needed around noon. The problem was solved by rearranging work shifts.