Final answer:
Using the M/M/1 queue model, the average queue length when customers arrive at a counter every 10 minutes and the receptionist takes 6 minutes per customer is 1 customer.
Step-by-step explanation:
To determine the average queue length when customers arrive at a reception counter at an average interval of 10 minutes and the receptionist takes an average of 6 minutes for one customer, we need to use the theory of queues, specifically the M/M/1 queue model. Here, the arrival rate, λ (lambda), is 1 customer per 10 minutes (or λ=0.1 customers/minute) and the service rate, μ (mu), is 1 customer per 6 minutes (or μ=1/6 customers/minute). We can apply the formula ρ = λ / μ to find the utilization factor, which is the average number of customers being served at one time.
The utilization factor, ρ, is given by (λ / μ). Therefore, ρ = (0.1 customers/minute) / (1/6 customers/minute) = 0.6. This value represents the proportion of time that the receptionist is busy serving customers.
To calculate the average number of customers in the system (L), we use the formula L = ρ / (1 - ρ). Substituting the value of ρ we have L = 0.6 / (1 - 0.6) = 1.5 customers. Since we are looking for the average queue length, we subtract the customer being served from the total number in the system, leaving us with an average queue length (Lq) of L - 1, which is 0.5 customers. Since customers cannot be split, the closest whole number is 1 customer.
The correct answer to the question is A) 1 customer.