What is the expected number of customers in the queue at checkout, given the service and arrival rates?

To determine the expected number of customers in the queue at checkout, one useful model is the M/M/1 queue system, which assumes a single server with exponential service and arrival distributions.

In an M/M/1 queue, the expected number of customers in the system (including those being served) can be calculated using the formula (L = \frac{\lambda}{\mu - \lambda}), where (L) represents the expected number of customers in the system, (\lambda) is the arrival rate, and (\mu) is the service rate. The expected number of customers specifically in the queue can be calculated using (L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}), where (L_q) is the expected number of customers in the queue.

If the choice represents the expected number of customers in the queue as (1/3), this indicates a specific balance between the arrival and service rates that leads to this expected value.

For the correct answer to hold, we can deduce that the value (1/3) logically arises from a given set of arrival and service rates. For instance, presuming you have calculated or been given certain values for