The County Hospital emergency room always has one doctor on duty. In the past, having just a single doctor there has been sufficient. However, because of a growing tendency for emergency cases to use these facilities rather than go to a private doctor, the number of emergency room visits has been steadily increasing. By next year, it is estimated that patients will arrive randomly at a mean rate of two per hour during peak usage hours (the early evening). Therefore, a proposal has been made to assign a second doctor to the emergency room next year during those hours. Hospital management (an HMO) is resisting this proposal, but has asked a management scientist (you) to analyze whether a single doctor will continue to be sufficient next year.
The patients are not treated on a first-come, first-served basis. Rather, the admitting nurse divides the patients into three categories: (1) critical cases, where prompt treatment is vital for survival; (2) serious cases, where early treatment is important to prevent further deterioration; and (3) stable cases, where treatment can be delayed without adverse medical consequences.
Patients are then treated in this order of priority, where those in the same category are normally taken on a first-come, first served basis. A doctor will interrupt treatment of a patient if a new case in a higher priority category arrives. Approximately 10 percent of the patients fall into the first category, 30 percent into the second, and 60 percent into the third. Because the more serious cases will be sent to the hospital for further care after receiving emergency treatment, the average treatment time by a doctor in the emergency room actually does not differ greatly among these categories. For all of them, the treatment time can be approximated by an exponential distribution with a mean of 20 minutes.
Hospital management has established the following guidelines.
The average waiting time in the emergency room before treatment begins should not exceed 2 minutes for critical cases, 15 minutes for serious cases, and 2 hours for stable cases.
a. What kind of queueing model fits this queueing system? E b. Use this model to determine if the management guidelines would be satisfied next year by continuing to have just a single doctor on duty.
c. Use the formula for W q for the M/M/ 1 model to determine if these guidelines would be satisfied if treatment were given on a first-come, first-served basis instead.
d. The mean arrival rate of two patients per hour during peak usage hours next year is only an estimate. Perform sensitivity analysis by repeating part b if this mean arrival rate were to turn out to be 2.25 patients per hour instead.