A family restaurant has one person processing the orders. On average, each custo
ID: 3292954 • Letter: A
Question
A family restaurant has one person processing the orders. On average, each customer that comes to restaurant can be seated at the rate of 6 per minute. According to historical data, customers arrive at the rate of 3 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.
a. What is the average number of customers waiting in line?
b. What is the average time a customer spends in the waiting line?
c. What is the average number of customers in the system?
d. What is a customer's average time in the system?
e. The restaurant estimates that the arrival rate will increase to 10 per minute. Based on the simulation output for a two-server and a three-server system below, how many server would you suggest? Note that cost of waiting per minute is $10/customer and servers are paid $15/hr.
Number of Channels
Arrival Rate
Service Rate
Probability of No Units in System
Average Waiting Time
Average Time in System
Average Number Waiting
Average Number in System
Probability of Waiting
Probability of 11 in System
Number of Channels
2 3Arrival Rate
10 10Service Rate
6 6Probability of No Units in System
.0909 .1727Average Waiting Time
.3788 .0375Average Time in System
.5455 .2041Average Number Waiting
3.7879 .3747Average Number in System
5.4545 2.0414Probability of Waiting
.7576 .2998Probability of 11 in System
.0245 less than .0088Explanation / Answer
a. From information given, lambda=mean arrival rate=3; mu=mean service time=6.
Therefore, the average number of customers waiting in the line is:
Lq=lambda^2/{mu(mu-lambda)}
=3^2/{6(6-3)}
=0.5
b. The average time a customer spends waiting in line (to be served):
Wq=lambda/{mu(mu-lambda)}
=3/{6(6-3)}
=0.1667 minute
c. The average number of customers in the system (customers being serviced and in the waiting line) is:
L=lambda/(mu-lambda)
=3/(6-3)
=1
d. Customers average time in the system (waiting and being served):
W=L/lambda=1/3=0.3333 minute
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