4. a) In a supermarket, the average arrival rate of customers is 10 every 30 minutes, following Poisson process. The average time taken by a cashier to list and calculate the customer’s purchase is two and a half minutes following exponential distribution. What is the probability that the queue length exceeds six? What is the expected time spent by a customer in the system?
To solve this problem, we can use the M/M/1 queuing model, where the arrival process follows a Poisson distribution and the service time follows an exponential distribution.
a) Finding the probability that the queue length exceeds six:
First, we need to calculate the arrival rate (λ) and the service rate (μ).
Given: The average arrival rate of customers is 10 every 30 minutes.
The arrival rate can be calculated as λ = (Number of arrivals) / (Time) = 10 / 30 = 1/3 customers per minute.
Given: The average time taken by a cashier to process a customer's purchase is two and a half minutes.
The service rate can be calculated as μ = 1 / (Time taken for service) = 1 / 2.5 = 2/5 customers per minute.
Next, we need to calculate the traffic intensity (ρ), which is the ratio of the arrival rate to the service rate.
ρ = λ / μ = (1/3) / (2/5) = 5/6
Using the traffic intensity, we can now calculate the probability that the queue length exceeds six using the Erlang C formula. However, since the queue length is not mentioned, we will assume it to be infinite (as is common in M/M/1 models).
The probability that the queue length exceeds six (P(Q > 6)) can be calculated as:
P(Q > 6) = (1 - ρ) * ρ^6 / (1 - ρ^7)
P(Q > 6) = (1 - 5/6) * (5/6)^6 / (1 - (5/6)^7) = (1/6) * (5/6)^6 / (1 - (5/6)^7)
Calculating this expression will give us the probability that the queue length exceeds six.
b) Finding the expected time spent by a customer in the system:
The expected time spent by a customer in the system (W) is the sum of the time spent waiting in the queue (Wq) and the time spent being serviced (Ws).
The expected time spent in the queue (Wq) can be calculated as:
Wq = (ρ^2) / (1 - ρ) * (1 / μ)
The expected time spent being serviced (Ws) can be calculated as:
Ws = 1 / (μ - λ)
Finally, the expected time spent by a customer in the system (W) can be calculated as:
W = Wq + Ws
Using the values of ρ, λ, and μ calculated earlier, we can plug them into the formulas to find W.
Please note that if you provide specific numerical values for λ and μ, I can provide the exact probability and expected time spent by a customer in the system.