Given Information:
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 = $16
Cost of customer waiting time = $25
Required Information:
Use one consultant with an average service time of 8 minutes/customer or use two consultants, each with average service time of 10 minutes/customer ?
Answer:
it is recommended to use one consultant with an average service time of 8 minutes/customer.
Step-by-step explanation:
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*0.442 + $16
Total cost = $27.05
Multi consultant queuing model:
Calculations for two consultants
Average no. of customers waiting for service = Lq = ((λ/μ)^k(λμ)/(k -1)(kμ - λ))*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*6)/(2-1)(2*6 - 2.3)*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*0.5264 + ($16)*2 (since there are 2 consultants now)
Total cost = $45.16
Conclusion:
Therefore, it is recommended to use one consultant with an average service time of 8 minutes per customer since the total cost is less than the other option.
