Final answer:
A customer arrives every two minutes on average in a three-server queuing system with an arrival rate λ. It therefore takes on average six minutes for three customers to arrive. Probabilities regarding the times between arrivals can be calculated using the exponential distribution.
Step-by-step explanation:
Understanding Queuing System Parameters
The system in question is a three-server queuing system where customers arrive at a certain rate denoted by λ (lambda). Since it is given that an average of 30 customers arrive per hour, it follows that on average, one customer arrives every two minutes.
Arrival Rate and Customer Flow
The arrival rate λ is key to calculating various probabilities and expectations about the system. For instance, if the store has just opened, it will, on average, take six minutes for three customers to arrive because:
- One customer arrives every two minutes on average.
- Therefore, three customers would take 3 times 2 minutes, which equals six minutes.
Probability of Next Arrival Time
If we are looking to find the probability of a customer arriving in less than a minute after a previous arrival, we need to refer to the exponential distribution because the time spent waiting between events in such systems is often modeled this way. It's useful in determining the likelihood of an event occurring within a certain time frame after a previous event.