Final answer:
The probability that there will be exactly 1 person in line with an arrival rate (λ) of 6 customers/hour and a service rate (μ) of 11 customers/hour in a single-server model is approximately 0.248.
Step-by-step explanation:
To find the probability that there will be exactly 1 person in line in a single server model with an arrival rate (λ) of 6 customers/hour and a service rate (μ) of 11 customers/hour, we will use the formula for the steady-state probabilities in a Markovian queue, specifically the M/M/1 queue.
Firstly, calculate the traffic intensity ρ, which is the ratio of the arrival rate to the service rate:
ρ = λ / μ = 6 customers/hour / 11 customers/hour
To find the probability of exactly 1 person in the system (P1), use the formula:
P1 = (1 - ρ) * ρ1
Let's calculate it step by step:
ρ = 6/11 ≈ 0.545
Next, calculate P1:
P1 = (1 - 0.545) * 0.545 ≈ 0.455 * 0.545 ≈ 0.248
Therefore, the probability that there will be exactly 1 person in the line is approximately 0.248.