An office has 5 copy machines that need to be serviced approximately once an hour either for paper, staples, or toner or repair. Each machine therefore runs approximately 1 hour before needing some attention. The average service time is 7 minutes. Copier downtime in this busy office costs approximately $20 per hour. The cost of attendant is $15 per hour. Using the finite queuing analysis, answer the following questions:.
a. What is the average number of copiers in line?
b. What is the average number of copiers still in operation?
c. What is the average number of copiers being serviced?
d. The firm is considering adding another attendant at the same $15 rate. Should the office do it?

a. Average number of copiers in line is 0.275

b. Average number of copiers still in operation is 4.23

c. Average number being serviced is 0.496

d. No, the office should not do it

Explanation:

N = number of copy machines = 5

U = average time between unit service requirements = 1 hour = 60 minutes

T = average service time = 7 minutes

M = number of servers = 1

Cost of copier downtime = $20

Cost of attendant = $15

(a)  Service factor, X = T / (T+U) = 7 / (60+7) = 0.105

From the Finite Queuing Tables for a Population of N = 5,

For X = 0.105 and M=1, the efficiency factor, F = 0.945

So,  The average number waiting in line, L = N × (1 - F) = 5 × (1 - 0.945) = 0.275

(b) Average number in operation, J = N×F×(1 - X) = 5×0.945×(1 - 0.105) = 4.23

(c) Average number being serviced, H = F×N×X = 0.945×5×0.105 = 0.496

(d)  For M=1

The average number of copier down = N - J = 5 - 4.23 = 0.77

So, cost of downtime per hour = $20×0.77 = $15.4

Also, the cost of the server per hour = $15×M = $15×1 = $15

So, total cost = 15.4 + 15 = $30.4 per hour (i)

For M=2

From the Finite Queuing Tables for a Population of N = 5, with X = 0.105 and M=2, the efficiency factor, F = 0.997

J = N×F×(1 - X) = 5×0.997×(1 - 0.105) = 4.46

The average number of copier down = N - J = 5 - 4.46 = 0.54

So, cost of downtime per hour = $20×0.54 = $10.8

Also, the cost of the server per hour = $15×M = $15×2 = $30

So, total cost = 10.8 + 30 = $40.8 per hour (ii)

Comparing (i) and (ii), we can say that having another attendant is not cost-effective.

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