The restaurant 20 tables of 6 people. There are 8 to 9 waitress working at anyti

Question

The restaurant 20 tables of 6 people. There are 8 to 9 waitress working at anytime. On a daily basis it serves 400 customers during weekdays and over 1000 customers during weekend. An average of 400 people coming from the restaurant in 3 hours time window of dinner time. It was found out from observation and discussion with manager that each customer spends 55 minutes on average in the restaurant, the queue length is around 36 people on average and the waiting time is around 15 minutes.

Assumed that potential customers will start to balk when they see more than 10 people are already in queuing for the restaurant. Further, it is also assumed that the maximum queue length that a potential customer can tolerate is 40 people. As the capacity of the restaurant when fully occupied is 120 people, we can calculate the probability of 10 people in the queue as the probability when there are 130 people in the system (i.e 120 in the restaurant and 10 or more queuing)

What is the probability that the customers will leave or balk given from the above assumption?
What can be done to reduce queue and avoid customers leaving to go to other competitors?

Can you also find the service rate and arrival rate? please separate it to weekdays and weekdays? thank you

Explanation / Answer

There are 20 tables of 6 people each which means about 120 people can sit in the restaurant. If each customer spends arounf 55 min on average, then, when the first 120 customers arrive, they will leave in the first hour.

Thus in the 3 hour dinner time window a total of 360 customers will come and leave, with a time of 15 min still remaining. The queue length is 36 people and waiting time is 15 min which is also the buffer time equal to 15 min of the 3 hour dinner time remaining.

So the arrival rate thus is approximately 120 people per hour and the service rate = 120 people / 8 waitresses = 15 people per waitress.

mu=120 people/hr

lambda=15 people/waitress

rho=lamba/mu=15/120=0.125

P(Customer will leave)=P(n>=10)=(Rho)^10=(0.125^10)=9.313226e-10

P(n<10)=0.9

1-rho^10=0.9

rho=0.631

mu=lamba/roh=15/0.631=24

serivce rate should increase from 15 people per waitresess to 24 people per waitressess

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