A small diner has one employee and a counter with seating for 8 customers.The diner does not package food for take out.Customers arrive at the diner at the rate of 20 per hour Poisson distributed).Service times are exponentially distributed and average 24 per hour.Customers that arrive when all seats are taken do not enter the diner.What is thehe probability that the diner is full and an arriving customer does not enter?
A)0.8333
B)0.1667
C)0.2067
D)0.0481

Question

Quizplus · Accepted Answer

Let's first find the utilization factor (p) of the diner:
p = arrival rate/service rate = 20/24 = 5/6

The diner has 8 seats, so the maximum number of customers it can hold is 8. If there are already 8 customers in the diner, a new arriving customer will not enter. Therefore, we are looking for the probability that there are 8 customers in the diner at a given time.

Since the arrival rate is Poisson distributed and the service time is exponentially distributed, we can model the system as an M/M/1/8 queue. In this case, we can use the Erlang-C formula to calculate the probability that there are 8 customers in the system:

P(8) = ((p^8)/8!)/(∑(p^i)/i!), i=0 to 8

Plugging in the values, we get:

P(8) = ((5/6)^8/40320)/(∑((5/6)^i/i!), i=0 to 8)
P(8) = 0.0529

Therefore, the probability that the diner is full and an arriving customer does not enter is 0.0529, which is closest to option D (0.0481).

A Small Diner Has One Employee and a Counter with Seating