If I have to observe a queue at an electronics shop and I need to perform at least 10 observations, how can I determine what the mean rate of arrival is? or the mean rate of service?
Like lets say that I choose to observe 10 people between 9 AM and 12 PM
Person 1 entered at 9:10 -- Left at 9:15
Person 2 entered at 9:15 -- Left at 9:20
Person 3 entered at 9:40 -- Left at 9:50
Person 4 entered at 9:59 -- Left at 10:20
Person 5 entered at 10:30 -- Left at 10:45
Person 6 entered at 10:45 -- Left at 11:00
Person 7 entered at 11:10 -- Left at 11:25
Person 8 entered at 11:20 -- Left at 11:30
Person 9 entered at 11:30 -- Left at 11:45
Person 10 entered at 11:35 -- Left at 11:50
How can I determine from those observations what the mean arrival rate is and the mean service rate? Lets say its for hours as well as for every 3 hours.
Mean arrival rate can be estimated as total number of arrivals divided by total time spent observing the system, if you assume the arrival rate is constant. If you think the arrival rate varies during the day (for instance, busy at lunch and dinner, slow between meals), then you need to estimate the arrival rate over multiple time intervals (for instance, count arrivals in each 30 minute window and double to get a per-hour rate).
Service rate is trickier. The difference between when a customer enters and when they leave combines waiting time (if there is a line) and service time. One way to estimate mean service rate would be to count the number of customers exiting (after service) per unit time while there is a continuous line (always someone waiting). Another way would be to record when each customer starts and ends service, take the difference to get their time in service, and average that. If you assume Markovian service times, the mean service rate is the reciprocal of the mean service time.