The owner of a (soon to be out of business) restaurant find that an average of 2.3 customers arrive to eat every hour. Find the probability that in any randomly
selected hour, the following number of customers arrive.
a. 0 b. 2 c. 4

SteGiven 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 ?

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.p-by-step explanation: