# Waiting Line Models – An example

A hospital ward has 30 beds in one section, and the problem centers on the appropriate level of nursing care. The hospital management believes that patients should have immediate response to a call at least 80 percent of the time because of possible emergencies. The mean time between calls is 95 minutes per patient for the 30 patients. The service time is a negative exponential distribution and mean service time is 5 minutes.

The hospital manager wishes to staff the ward to give service so that 80 percent of the time there will be no delay. Nurses are paid \$ 15 per hour, and the cost of idle time at this level of service must be considered. Also, the manager wishes to know how much more patients will have to pay for the 80 percent criterion compared with a 50 percent service level for immediate response, which is the current policy.

The solutions to the problems posed by the hospital manager are developed through a finite waiting line model. The situation requires a finite model because the maximum possible queue is 30 patients waiting for nursing care, and if 1 patient calls for service, there are only 29 patients who could now possibly call for service. Thus, because of the relatively small population, the potential for more arrivals has been changed by the occurrence of an arrival.

In terms of the finite waiting line model for this situation, the mean service time is 5 minutes (µ = 0.2/min or 12/hr), the mean time between calls is 95 minutes per patient (ג = 0.0105 / min or 0.632 / hr) and therefore the service factor is X = ג / ( ג + µ ) = 0.632 / 12.632 = 0.05.

Scanning the finite queuing tables under the population N=30 and X=0.05, we seek data for the probability of a delay of D=.20 because we wish to establish service such that there will be no delay 80 percent of the time. The closest we can come to providing this level of service is with M = 3 nurses and corresponding data of D = .208, F = 0.994, and Lq =0.18. Note that we must select an integer number of servers (nurses).

The cost of this level of service is the cost of employing 3 nurses or 15 x 3 = \$45 per hour or \$ day, assuming day and night care. The average number of calls waiting to be serviced will be Lq=0.18 and the mean waiting time will be

W q = 1 / µ X (1 – F / F)

= 1 / 0.2 x 0.05 (1 – 0.994 / 0.994)

= 0.6 minutes

The waiting time is, of course, negligible which is intended.

The average number of patients being served will be H = FNX = 0.994 x 30 x 0.05 = 1.49, and the average number of nurses idle will be 3 – 1.49 = 1.51. The equivalent value of this idleness is 1.51 x 15 x 24 = \$ 543.60 per day.

Finally, the number of nurses needed to provide immediate service 50 percent of the time is M = 2 from Table (D = .571, F = 0.963 and Lq=1.11). The average waiting time under this policy is Wq=3.84 minutes. The average cost to patients of having the one additional nurse to provide the higher level of service is \$ 15 per hour or \$ 360 per day. Divided among 30 patients, the cost is \$ 12 per patient per day.