Question

If λ = 6 CUSTOMERS/HOUR and μ = 11 CUSTOMERS/HOUR, in a single server model, find the probability that there will be exactly 1 person in LINE?

1) 0.162
2) 0.297
3) 0.248
4) 0.135

Answer 1

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.