20. L. Winston Martin (as allegory) has an excellent system for handling his regular patients who come in just for allergy injections. Patients arrive for an injection and fill out a nurse slip, which is then placed in an open slot that passes into another room staffed by up to three nurses. The specific injections for a patient are prepared, and the patient is called through a speaker system into the room to receive the injection. At certain times during the day, patient load drops and only one nurse is needed to administer the injections.

Let's focus on the simpler case of the two-nursely, when there is one nurse. Also, assume that patients arrive in a Poisson fashion and the service rate of the nurse is exponentially distributed. During this slower period, patients arrive with an interarrival time of approximately three minutes. It takes the nurse an average of two minutes to prepare a patient's serum and administer the injection.
a. What is the average number you would expect to see in Dr. Martin's facilities?
b. How long would it take for a patient to arrive, get an injection, and leave?
c. What is the probability that there will be three or more patients on the premises?
d. What is the utilization of the nurse?
e. Assume three nurses are available. Each takes an average of two minutes to prepare a patient's serum and administer the injection. What is the average total time of a patient in the system?

Question

20. L. Winston Martin (as allegory) has an excellent system for handling his regular patients who come in just for allergy injections. Patients arrive for an injection and fill out a nurse slip, which is then placed in an open slot that passes into another room staffed by up to three nurses. The specific injections for a patient are prepared, and the patient is called through a speaker system into the room to receive the injection. At certain times during the day, patient load drops and only one nurse is needed to administer the injections.

Let's focus on the simpler case of the two-nursely, when there is one nurse. Also, assume that patients arrive in a Poisson fashion and the service rate of the nurse is exponentially distributed. During this slower period, patients arrive with an interarrival time of approximately three minutes. It takes the nurse an average of two minutes to prepare a patient's serum and administer the injection.
a. What is the average number you would expect to see in Dr. Martin's facilities?
b. How long would it take for a patient to arrive, get an injection, and leave?
c. What is the probability that there will be three or more patients on the premises?
d. What is the utilization of the nurse?
e. Assume three nurses are available. Each takes an average of two minutes to prepare a patient's serum and administer the injection. What is the average total time of a patient in the system?

StudyXY · Accepted Answer

I'll solve this queueing theory problem step by step using the M/M/1 and M/M/3 queueing models. : Calculate system utilization $\rho$ : Use $L_{s} = \frac{\rho}{1-\rho}$ formula : Use $W_{s} = \frac{1}{\mu - \lambda}$ formula : Use Poisson distribution probability : Calculate probabilities : Calculate final probability : Recalculate parameters : Use more complex queueing formulas for M/M/3 system a. 2 patients b. 6 minutes c. 0.323 (32.3%) d. 66.7% e. Requires advanced queueing theory calculations

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I'll solve this queueing theory problem step by step using the M/M/ 1 and M/M/ 3 queueing models.

Step 2
: Calculate system utilization \rho

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