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A new salon with a single barber operates near MIT. The salon provides two kinds of...

A new salon with a single barber operates near MIT. The salon provides two kinds of services: Quick Barbering (QB) which takes exactly 9 minutes and Normal Barbering (NB) which takes exactly 15 minutes. On average, the customers who request QB service arrive at a rate of 1 per hour, and the customers who request NB service arrive at a rate of 2 per hour. The two arrival streams follow independent Poisson processes. Because the two arrival streams are independent, the overall arrival process is also a Poisson process, with an average arrival rate of 3 customers per hour, and the probability that a new arrival wants for a QB service is 1/(1+2)=1/3. Assume that the customers are served on a first-in-first-out basis and the salon has unlimited capacity to store waiting customers. What is the expected time spent in the system for a customer who wants for a QB service (including waiting and service time)?

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Answer #1

Given:

Time taken for QB Service 9 mins

Time taken for NB Service 15 mins

Probability that new arrival wants QB service is 1/3.

Customer are served on first in first out basis

Arrival of new customer per hour is 1 for QB Service and 2 for QB Service

Total time taken to service one time arrival is 39 mins (9+15+15)

Therefore probability of waiting time spent by new arrival for QB service = 39/3=13 mins

The expected time spent in the system for a customer who wants for a QB Service is 28 mins (13 mins +15 mins)

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