A fast food restaurant has one server at the counter...

A fast food restaurant has one server at the counter. Customers arrive at the restaurant at an average rate of 10 per hour, according to a Poisson process. The time it takes the server to serve each customer follows an exponential distribution with a mean of 5 minutes. Assume that the system can hold up to 5 customers at a time (i.e., any additional customers will be turned away). a) What is the utilization factor for the server? b) What is the probability that no customers are in the system? c) What is the average number of customers waiting in line? d) What is the average time a customer spends waiting in line?

Transcript

00:01 In this problem, we are having that customers arrive at the drive -up window of a fast food restaurant at the rate of 25 per hour.

00:08 The employee working the window can serve one customer every two minutes.

00:13 So we need to calculate different kind of values.

00:15 Let us see what we can do.

00:17 So first of all, we can write that arrival rate which is assumed as lambda is 24 customers per hour.

00:24 After that, we can write that service rate.

00:27 So service rate is being denoted with m.

00:30 Value.

00:31 So, mu is equals to 60 over 2 that comes out 30.

00:37 So, here we can say that 30 is customers, service rate is 30 customers per hour, similar to that arrival rate.

00:48 Then we will try to calculate the average utilization of the employee.

00:52 So, utilization formula can be written as such and this will be denoted with lambda over mu.

00:59 So this is the average.

01:01 Utilization formula.

01:02 So here if we plug in the value we will have 24 over 30 equals 0 .8.

01:09 So this 0 .8 is the answer for the first part of the problem in which we were asked about the average utilization of the employee.

01:16 Now we will see the second part.

01:19 So in the second part, we can write that it is asking about average number of customers in line.

01:27 So average number of customers in line can be denoted as lq.

01:32 So this will be equals to lambda over mu multiplied lambda over mu minus lambda.

01:40 So these are all formulas.

01:41 So these are well established formulas and we are just plugging the values in it...

Question

Please give Ace some feedback