Question

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. what is the average time that player's card holders wait to be seated?

Answer 1

The average waiting time for player's card holders to be seated is 28 minutes.

Step-by-step explanation:

To find the average waiting time for player's card holders to be seated, we need to use the concept of queuing theory. We'll assume that the arrivals follow a Poisson distribution and the service time follows an exponential distribution.

The customer arrival rate is given as 8 parties per hour, so on average one customer arrives every 2 minutes. Since there are 10 tables or booths available, the average service rate is 1 table per 42 minutes.

Using queuing theory formulas, we can find that the average waiting time for player's card holders is 28 minutes.

Answer 2

Using queueing theory, we calculate that player's cardholders at the restaurant wait just under 6 minutes on average to be seated.

Step-by-step explanation:

The average time that player's cardholders wait to be seated at the restaurant can be calculated using queueing theory, specifically the M/M/1 queue model, where the 'M's stand for 'memoryless', which means that the arrival times and service times are exponentially distributed, and the '1' indicates that there is a single server (in this context, the single 'server' refers to the whole restaurant with 10 tables or booths).

To calculate the wait time, we need to understand the arrival rate (λ) and the service rate (μ).

The arrival rate is 8 parties per hour but since only 4 of those are cardholders, λ = 4 parties/hour for cardholders.

The service rate, μ, is the number of parties the restaurant can serve per hour.

Since the average service time is 42 minutes, we convert it to an hourly rate: 60/42 parties/hour, which is approximately 1.43 parties/hour per table.

Given there are 10 tables, μ = 10 * 1.43 = 14.3 parties/hour.

Now, we apply the formula L = λ / (μ - λ),

where L is the average number of parties in the system (both waiting and being served).

Substituting in the values of λ and μ gives us L = 4 / (14.3 - 4) ≈ 0.398.

This is the average number of parties in the system, but we want just the waiting time (Wq), not the service time, so we use Wq = L / λ.

Therefore, Wq = 0.398 / 4 hours, which is approximately 0.0995 hours, or about 5.97 minutes.

So, player's cardholders wait, on average, just under 6 minutes to be seated.