During the early morning hours, customers arrive at a branch post office at the average rate of 45 per hour (exponential interarrival times), while clerks can handle transactions on an average of 4 minutes each (exponential).
If clerk cost is $30 per hour and customer waiting time represents a cost of $20 per hour, how many clerks can be justified on a cost basis?
A) 2.
B) 3.
C) 4.
D) 5.
E) 6.

Question

Quizplus · Accepted Answer

We need to use the M/M/c queuing model where M denotes exponential arrival and service times and c denotes the number of servers (clerks).

Given, arrival rate (λ) = 45 per hour, service rate (μ) = 15 customers per hour (since each transaction takes 4 minutes or 1/15 hour), and cost of clerk (Cs) = $30 per hour and cost of customer waiting (Cw) = $20 per hour.

The formula for the expected total cost is given by:
Total Cost = (Cs * c) + (Cw * expected time spent in queue)

The expected time spent in the queue is given by:
Expected time spent in queue = ((ρ^c) / c!(c-ρ))^(-1) * ((1-(λ/μ))/(1-((λ/μ)^(c+1))))

where
ρ = λ/μ = (45/15) = 3
c = number of servers (unknown)
(! represents factorial)

Now, we can calculate the total cost for different values of c and choose the value that gives the lowest total cost.

For c = 2:
ρ^c / c!(c-ρ) = (3^2)/(2!(2-3)) = 4.5
1 - (λ/μ)^(c+1) = 1 - (3/4)^3 = 37/64
Expected time spent in queue = (4.5)^(-1) * (37/64 - 1) = 0.47 hours
Total cost = (30 * 2) + (20 * 0.47) = $77.40

For c = 3:
ρ^c / c!(c-ρ) = (3^3)/(3!(3-3)) = 1.5
1 - (λ/μ)^(c+1) = 1 - (3/4)^4 = 79/256
Expected time spent in queue = (1.5)^(-1) * (79/256 - 1) = 0.26 hours
Total cost = (30 * 3) + (20 * 0.26) = $98.00

For c = 4:
ρ^c / c!(c-ρ) = (3^4)/(4!(4-3)) = 4.5
1 - (λ/μ)^(c+1) = 1 - (3/4)^5 = 2601/1024
Expected time spent in queue = (4.5)^(-1) * (2601/1024 - 1) = 0.10 hours
Total cost = (30 * 4) + (20 * 0.10) = $132.00

For c = 5:
ρ^c / c!(c-ρ) = (3^5)/(5!(5-3)) = 1.69
1 - (λ/μ)^(c+1) = 1 - (3/4)^6 = 4782965/4096
Expected time spent in queue = (1.69)^(-1) * (4782965/4096 - 1) = 0.03 hours
Total cost = (30 * 5) + (20 * 0.03) = $150.00

For c = 6:
ρ^c / c!(c-ρ) = (3^6)/(6!(6-3)) = 1.69
1 - (λ/μ)^(c+1) = 1 - (3/4)^7 = 10460353203/16384
Expected time spent in queue = (1.69)^(-1) * (10460353203/16384 - 1) = 0.01 hours
Total cost = (30 * 6) + (20 * 0.01) = $180.00

Thus, the lowest total cost is for c = 4 clerks, which gives a total cost of $132. Therefore, the best choice is C) 4.

During the Early Morning Hours, Customers Arrive at a Branch