this is for single queue single server utilization rate Po is probability of ide

Question

this is for single queue single server
utilization rate
Po is probability of ideal system
Wq waiting time in queue
Ws waiting time in the system
Lq number of customers in the queue
Ls number of customers in the system
then the guided questions thank you

A Japanese Restaurant in Manila is studying how to improve its customer service by looking at the queue and applying M/M/1 theory to solve the problem The restaurant provides 20 tables of 6 people. There are 8 to 9 waiters or waitresses working at any one time. On a daily basis, it serves over 400 custo weekdays, and over 1000 customers during weekends. There are on average 40 people are coming to 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 mers during 0 Calculate the theoretical estimates using M/M/1 for the following: Utilization Rate, PO Wa, WS/W, and La, LS/L It is assumed that potential customers will start to balk when they see more than 10 people are already 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 (ie. 120 in the restaurant and 10 or more queuing) What is the probability that customers will leave or balk given the above assumptions?

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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