Question

restaurant has 5 tables available. People arrive at times of a Poisson process with rate of 6 per hour. The amount of time for a person to eat has an exponential distribution with a mean of 30 minutes. People who arrive when all 5 tables are full turn around and leave without eating at the restaurant. In the long run, what fraction of time is the restaurant full?

Answer 1

This is a simple question related to finding the probability of the all the servers are full.

For that we first need to find the utilization ratio of the servers.

If the arrival rate is and the service rate is in an environment having m servers ,

The utilization ratio is given as

For our question ,the arrival rate is given as

=6 /hour

and the service rate is

= 1 person/30mins or 2/hour

Thus keeping values we get

The probability of having n customers in a system is given by

where Po is the probability of having no customers in the system.

The probability of having no customer () is given as

Keeping values we get ,

=0.05435

This is the probability of having no customers in the system

Now we know values and hence keeping the values we get

or,

or

Thus there is 11 % probability that the restaurant will be full.

for 11% of the time the restaurant will be full.