4.9 Hospital emergency ward with preemptive priorities The lone doctor in the Puddleduck City Hospital emergency ward encounters two types of patients: emergency and nonemergency patients, who arrive at independent Poisson rates lambda.gif (179
bytes)1, and lambda.gif (179 bytes)2, respectively. Their treatment times are independent and exponentially distributed with parameters mu.gif (189 bytes)1, and mu.gif (189
bytes)2, respectively.
       If an emergency patient arrives during the treatment of a nonemergency patient, the latter's treatment is immediately stopped in favor of the emergency patient. The interrupted treatment is resumed (i.e., from the point at which it was interrupted) when there are no emergency patients present. The treatment of an emergency patient is never interrupted, and within the two groups there is a first-come, firstserved discipline.

a. Find the mean number of emergency patients in the system

b. Show that the mean number of nonemergency patients in the system is

260-1.gif (6960 bytes)

c. Assume that lambda.gif (179
bytes)1/mu.gif (189
bytes)1, = lambda.gif (179 bytes)2/mu.gif (189
bytes)2 < 1/2 Compare the mean waiting times of both types of patient with and without the preemptive priority system.

Hints: The time a nonemergency patient spends in the emergency ward consists of three components:

1 . The time between her arrival and the next time that a nonemergency patient receives attention (T1).

2. The time to complete treatment on all the nonemergency patients in front of her.

3. The time from her first receiving attention until the time that the treatment is completed (completion time, T).

Show that

260-2.gif (3958 bytes)