Suppose that the service rate to a waiting line system is 10 customers per hour (exponentially distributed). Analyze how the average waiting time is expected to change as the arrival rate varies from two to ten customers per hour (exponentially distributed)
To analyze how the average waiting time is expected to change as the arrival rate varies, we can use queuing theory. Queuing theory helps us understand and make predictions about waiting line systems.
In this case, we have a single-server waiting line system with an exponential service rate of 10 customers per hour. The arrival rate is the rate at which customers arrive to the system. We are varying the arrival rate from two to ten customers per hour.
To find the average waiting time, we need to calculate the traffic intensity or utilization of the system. The traffic intensity (ρ) is the ratio of the arrival rate (λ) to the service rate (μ). Mathematically, ρ = λ / μ.
Let's calculate the traffic intensity for each arrival rate:
For an arrival rate of 2 customers per hour:
ρ = 2 / 10
ρ = 0.2
For an arrival rate of 3 customers per hour:
ρ = 3 / 10
ρ = 0.3
For an arrival rate of 4 customers per hour:
ρ = 4 / 10
ρ = 0.4
For an arrival rate of 5 customers per hour:
ρ = 5 / 10
ρ = 0.5
For an arrival rate of 6 customers per hour:
ρ = 6 / 10
ρ = 0.6
For an arrival rate of 7 customers per hour:
ρ = 7 / 10
ρ = 0.7
For an arrival rate of 8 customers per hour:
ρ = 8 / 10
ρ = 0.8
For an arrival rate of 9 customers per hour:
ρ = 9 / 10
ρ = 0.9
For an arrival rate of 10 customers per hour:
ρ = 10 / 10
ρ = 1
Now that we have the traffic intensity for each arrival rate, we can refer to queueing theory formulas to find the average waiting time. In this case, we can use Little's Law, which states that the average waiting time (W) is equal to the average number of customers in the system (L) divided by the arrival rate (λ). Mathematically, W = L / λ.
The average number of customers in the system (L) can be calculated using the formula:
L = (ρ^2) / (1 - ρ)
Let's calculate the average waiting time for each arrival rate using Little's Law and L formula:
For an arrival rate of 2 customers per hour:
L = (0.2^2) / (1 - 0.2)
L = 0.04 / 0.8
L = 0.05
W = L / λ
W = 0.05 / 2
W = 0.025 hours or 1.5 minutes
For an arrival rate of 3 customers per hour:
L = (0.3^2) / (1 - 0.3)
L = 0.09 / 0.7
L = 0.15
W = L / λ
W = 0.15 / 3
W = 0.05 hours or 3 minutes
For an arrival rate of 4 customers per hour:
L = (0.4^2) / (1 - 0.4)
L = 0.16 / 0.6
L = 0.27
W = L / λ
W = 0.27 / 4
W = 0.0675 hours or 4.05 minutes
Continue this process for the remaining arrival rates.
By analyzing these calculations, we can observe how the average waiting time changes with different arrival rates. As the arrival rate increases, the average waiting time also increases. This is because a higher arrival rate leads to more customers in the system, which in turn increases the waiting time for each customer. Conversely, as the arrival rate decreases, the average waiting time decreases as well.
By using queuing theory formulas like Little's Law and analyzing the traffic intensity, we can make predictions about the expected average waiting time in a waiting line system with varying arrival rates.