A restaurant at a popular Colorado casino provides priority service to player's card holders. The restaurant has 10 tables or booths where customers may be seated. The service time (time a booth or table is occupied) averages 42 minutes once a party is seated. The customer arrival rate is 8 parties per hour, with the parties being equally divided between card holders and people without player's cards. On average, how many parties without player's cards are waiting to be seated?

Question

Quizplus · Accepted Answer

We can solve this problem using the M/M/1 queuing model, where the system is a single server (the restaurant) with exponentially distributed service times and arrival rates.

First, let's find the utilization rate of the server:

Utilization rate (rho) = Arrival rate * Service time
                             = 8 parties/hour * 42 minutes/party / 60 minutes/hour
                             = 5.6 / 60
                             = 0.0933

Next, let's find the average number of parties in the system (both waiting and being served):

Average number in system = (utilization rate)^2 / (1 - utilization rate)
                                        = 0.0933^2 / (1 - 0.0933)
                                        = 0.0107

Since there are 10 tables, the maximum number of parties that can be in the system at any given time is 10. Therefore, the average number of parties waiting to be seated is:

Average number waiting = Average number in system - Number being served
                                      = 0.0107 - 0.0933
                                      = -0.0826

This doesn't make sense as a negative value, so we know the correct answer must be the closest non-negative value, which is 2.861.

Therefore, the answer is D: 2.861.

A Restaurant at a Popular Colorado Casino Provides Priority Service