Question

A large urban police department's CSI unit laboratory has 20 identical, very accurate and very expensive scanning electron microscopes available for use by its crime scene investigators. Assume crimes are distributed evenly over the day in this city and, on average, an investigator requests a microscope once every four hours with a standard deviation of 4 hours. Also, on average, he or she uses the microscope for 2 days with a standard deviation of 1 day. Finally, assume that CSI agents work around the clock (24 hours a day), until the crime they are working on is solved.

What is the average wait time a CSI analyst has to endure before getting his or her hands on a microscope to solve a super hot crime that the mayor wants solved immediately?

Answer 1

Steps:

1. Determine arrival patterns - arrival time, inter-arrival time (a), standard deviation (SDa), and Coefficient of variation of arrival (CVa). Convert all the times in hours.

2. Determine service patterns – service time (p), standard deviation (SDp), and Coefficient of variation of service time (CVp). Convert all the times in hours.

3. Identify number of servers (m) in system.

4. Determine utilization rate (u) of servers. u = (flow rate) / (Capacity) = p/(a x m)

5. Determine the expected waiting time in queue (Tq) by formula mentioned below.

Inter-arrival time = 1 request every four hours = 4 hours per request

Standard deviation of inter-arrival time = 4 hours

Service time = 2 days per request = 2 x 24 hours/day = 48 hours

Standard deviation of service time = 1 day per request = 1 x 24 hours/day = 24 hours

	Inter arrival rate	Service rate
Mean time	a = 4 hour per customer	p = 48 hours per customer
Standard Deviation	S.D.a = 4 hour	SDp = 24 hours per customer
Coefficient of variance	CVa = SDa/a CVa = 4/4 CVa = 1	CVp = SDp/p CVp = 24/48 CVp = 0.5
Number of servers/window	m = 20
Utilization (U)	u = (flow rate) / (Capacity) = (1/a) / (m/p) = p/(a x m) u = 48/(20 x 4) u = 0.6

The formula for determining average time the customer waits in queue is given as follow:

Tq = 2.4 x 0.1521 x 0.625

Tq = 0.2281 hours

Tq = 13.7 minutes

average time a CSI analyst has wait in line = 0.2281 hours or 13.7 minutes