Answer:
the length of the drive-way required to accommodate all the arrivals on average is approximately 20 feet.
Step-by-step explanation:
To solve this problem, we can use queuing theory, specifically the M/M/1 queuing model, where "M" denotes a Markovian arrival process and "M" denotes a Markovian service process with a single server.
Given:
Average arrival rate (λ) = 28 customers/hour
Average service time (μ) = 2 minutes/customer
(i) Proportion of time the system will be idle:
The utilization (ρ) of the system can be calculated as the ratio of the arrival rate to the service rate:
ρ = λ / μ
The proportion of time the system will be idle is given by:
P_idle = 1 - ρ
Substituting the values:
ρ = 28 customers/hour / (60 minutes/hour / 2 minutes/customer) = 0.9333
P_idle = 1 - 0.9333 = 0.0667 or 6.67%
Therefore, the proportion of time the system will be idle is approximately 6.67%.
(ii) Average waiting time for a customer:
The average waiting time (Wq) in the queue can be calculated using Little's Law:
Wq = (λ / (μ * (μ - λ)))
Substituting the values:
Wq = (28 customers/hour / (60 minutes/hour / 2 minutes/customer)) / (2 minutes/customer * (2 minutes/customer - 28 customers/hour))
Wq ≈ 0.0381 hours or 2.29 minutes
Therefore, on average, a customer will have to wait approximately 2.29 minutes before reaching the server.
(iii) Length of the drive-way required:
The length of the drive-way required can be calculated by multiplying the average number of customers in the system (L) by the length required per car (20 feet):
L = λ * W
Substituting the values:
L = 28 customers/hour * 0.0381 hours = 1.066 or approximately 1 car