Question

Problem description:

In the Capital Health System, the Nuclear Medicine Department and the Heart Station work together to provide the service of nuclear stress testing. Nuclear stress testing evaluates the patient for coronary artery disease through a combination of myocardial perfusion imaging scans and treadmill or pharmacologic stress testing. Nuclear Medicine’s role in nuclear stress testing is to prepare all patients (both inpatients and outpatients) for undergoing the procedure, to prepare all radioactive doses utilized for imaging, to inject all patients with radioactive doses, and to image all patients undergoing nuclear stress testing.

Currently, Nuclear Medicine utilizes two gamma cameras for nuclear stress imaging. The department operates 24 hours a day, 7 days a week. On any given day, bottlenecks can occur on the gamma cameras due to high volumes of inpatients and/or ER chest pain patients. Inpatients and ER patients are unlimited and variable on any given day for both Nuclear Medicine gamma cameras. Arrival and service rates were collected on all patients served during regular operating hours for a one-week period. Patient arrival rate per gamma camera is estimated at 2.9 per hour. Patient service rate per gamma camera is three per hour.

Cost:

Nuclear Medicine busy server: $500

Nuclear Medicine idle server: $200

Nuclear Medicine patient waiting: $1500

Nuclear Medicine patient being served: $1200

perform a queuing analysis for the stress test imaging process in Nuclear Medicine to determine the optimal server capacity.

Answer 1

This is M/M/s queue system with following parameters

Arrival rate = 2.9 per hour

Service rate = 3 per hour

Number of servers (gamma cameras), s = 2

Analysis of cost per hour is done in the following spreadsheet to determine the optimal server capacity

FORMULAS:

Formula:  copy these formulas down the rows

B12 =($B$4/$B$5)^A12/FACT(A12)

C12 =SUM(B11:$B$11)

D12 =+B12/(1-$B$4/(A12*$B$5))

E12 =1/(C12+D12)

F12 =$B$4/($B$5*A12)

G12 =+E12*B12*F12/(1-F12)^2

H12 =G12+$B$4/$B$5

I12 =G12/$B$4

J12 =I12+1/$B$5

K12 =(F12*$I$4+(1-F12)*$I$5)*A12+G12*$I$6+(H12-G12)*$I$7

Total cost per hour is the lowest for 3 servers .

Therefore, optimal server capacity = 3 gamma cameras should be installed.