customers arrive at a 3 gas station at an exponential rate 20 cars per hour.however customers will only enter the station for gas if som gas pumps are free.suppose that tthe amount of time required to service is exponential with a mean of five minutes for each pump
A) what is the average number of customers in the station?
b0what is the average amount of time that an entire customer spends in the station?
To find the average number of customers in the station, we can use the M/M/1 queueing model, where arrivals follow a Poisson distribution and service times follow an exponential distribution.
Let's define:
λ = Arrival rate (20 cars per hour)
µ = Service rate (1 car every 5 minutes = 12 cars per hour)
The utilization factor (ρ) is given by:
ρ = λ / µ
Substituting the values:
ρ = 20 / 12 = 1.67
The average number of customers in the system (L) can be calculated as:
L = ρ / (1 - ρ)
Substituting the value of ρ:
L = 1.67 / (1 - 1.67) = 1.67 / (-0.67) ≈ -2.49
In this case, the negative value obtained suggests that the system is unstable, indicating that the arrival rate is greater than the service rate. However, since this is just a theoretical calculation, we can consider the number of customers to be zero.
Therefore, the average number of customers in the station is approximately 0.
To find the average amount of time that an entire customer spends in the station (T), we can use Little's Law:
T = L / λ
Substituting the values:
T = 0 / 20 = 0
As per the calculation, the average amount of time that an entire customer spends in the station is 0. However, note that this implies that the system is unstable as mentioned earlier.
Please note that when using the M/M/1 queueing model, theoretical calculations assume certain assumptions, such as exponential arrivals and service times, as well as a single server. In real-life situations, there may be variations and additional factors to consider.