Final answer:
The average time a customer will spend from entering the line to leaving the restaurant is 0.5 hours or 30 minutes, considering the arrival rate (λ) is 6 per hour and the service rate (μ) is 8 per hour.
Step-by-step explanation:
The question involves queueing theory, where the arrivals follow a Poisson distribution and service times follow an exponential distribution. Given the average number of arrivals is 6 per hour, we can find the average time between arrivals. Also, with a service rate of 2 every 15 minutes, or 8 per hour, we can determine the average service time as well.
a. The average time between two successive arrivals (arrival rate λ) can be calculated as:
1 hour / 6 arrivals = 0.1667 hours (or 10 minutes).
b. The average service rate (μ) is:
2 services / 15 minutes = 8 services / 1 hour.
Hence, the average service time is:
1 hour / 8 services = 0.125 hours (or 7.5 minutes).
The total average time a customer spends in the system (system time) is the sum of the average service time and the time waiting in line. This is often modeled as 1 / (μ - λ) for M/M/1 queue:
System Time = 1 / (μ - λ) = 1 / (8 - 6) = 1 / 2
= 0.5 hours (or 30 minutes).