Question

Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.3 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. Agan's management would like to evaluate two alternatives:
Use one consultant with an average service time of 8 minutes per customer.

Expand to two consultants, each of whom has an average service time of 10 minutes per customer.

If the consultants are paid $18 per hour and the customer waiting time is valued at $25 per hour for waiting time prior to service, should Agan expand to the two-consultant system?

No/Yes

What is the total cost for each scenario? Round your answers to the nearest cent.

The total cost for the first scenario where there is one consultant with an average service time of 8 minutes per customer is $ ______ .

The total cost for the second scenario where there are two consultants with an average service time of 10 minutes per customer is $ ______ .

Answer 1

• It is recommended to use one consultant with an average service time of 8 minutes/customer.

• The total cost for the first scenario where there is one consultant with an average service time of 8 minutes per customer is $ 29.05

• The total cost for the second scenario where there are two consultants with an average service time of 10 minutes per customer is $ 49.16

Explanation:

Given,

Average customer arrival = λ =2.3 customers/hour

Average consultant time1 = μ  = 8 minutes/customer

Average consultant time2 = μ  = 10 minutes/customer

Cost of consultants service = $18

Cost of customer waiting time = $25

Convert the average consultant time to hours/customer

Average consultant time1 = μ  = 60/8 = 7.5 hours/customer

Average consultant time2 = μ  = 60/10 = 6 hours/customer

• Single consultant queuing model:

Calculations for one consultant

Average no. of customers waiting for service = Lq = λ²/μ(μ-λ)

Average no. of customers waiting for service = Lq = (2.3)²/7.5(7.5 - 2.5)

Average no. of customers waiting for service = Lq = 0.1356

Average no. of customers in the system = L = Lq + λ/μ

Average no. of customers in the system = L = 0.1356 + 2.3/7.5

Average no. of customers in the system = L = 0.442

Total cost = (customer waiting time cost)*L + consultant service cost

Total cost = $ 25 $\times$ 0.442 + $ 18

Total cost = $ 29.05

• Multi consultant queuing model:

Calculations for two consultants

Average no. of customers waiting for service,

             Lq = ((λ/μ)^k(λμ)/(k -1)(kμ - λ)) $\times$ P₀

Where k = 2 is the number of consultants and P₀ is the probability that all of the k consultants are idle. The value of P₀  can be found in the tables with λ/μ = 2.3/6 = 0.38 and k = 2,  P₀ ≅ 0.685

Average no. of customers waiting for service,

            Lq = (2.3/6)²(2.3 $\times$ 6) / (2-1)(2 $\times$ 6 - 2.3) $\times$ 0.685

Average no. of customers waiting for service = Lq = 0.1431

Average no. of customers in the system = L = Lq + λ/μ

Average no. of customers in the system = L = 0.1431 + 2.3/6

Average no. of customers in the system = L = 0.5264

Total cost = (customer waiting time cost)*L + consultant service cost

Total cost = $25 $\times$ 0.5264 + ($18) $\times$ 2 (since there are 2 consultants now)

Total cost = $ 49.16