During the weeknight evenings customers arrive at a video...

During the weeknight evenings, customers arrive at a video and DVD rental outlet at the average rate of 1.5 customers per minute. The checkout clerk can serve an average of two customers per minute. Assume Poisson arrivals and exponential service times. a. What is the probability that no customers are in the system? b. What is the average number of customers waiting for service? c. What is the average time a customer waits for service to begin? d. What is the probability that an arriving customer will have to wait for service? e. Do the operating characteristics indicate that the one-clerk checkout system provides an acceptable level of service?

Transcript

00:01 Hello there, i see we have a question where we're dealing with customers that arrive at a video and dvd rental outlet.

00:14 The average rate of the customers that frequent to this particular rental shop is at a rate of 1 .5 customers per minute.

00:27 Customers per minute, okay.

00:36 And how the checkout clerk can serve an average of two customers per minute.

00:43 Two customers per minute.

00:51 So they want to know what is the probability that no customers are in the system.

00:57 So the possibility, the probability for that, how we're going to calculate this.

01:05 Given that the average arrival rate for a customer is at 1 .5 per, you know, customer per minute.

01:23 And the average rate of the service at two minutes is, i mean, average rate for the employee, the clerk to serve to the customer is two customers per minute.

01:37 We can calculate the probability by doing, by using this equation.

01:45 So p not is equal to one minus our mu.

01:50 And this will be for our customers per minute.

01:54 So that's what we're going to use that for.

01:56 I mean, not mu because our mu is what we're going to divide it by.

02:02 That's our mu.

02:04 We're going to use this symbol to represent for, we're going to use this symbol to represent for our customer per minute, divide it by two.

02:15 And when we do that, we're going to minus that by one.

02:19 So there'll be one minus 1 .5 over two.

02:25 And when you calculate that, you should get, so by calculating that, you should end up with 0 .75.

02:36 Oh, i forgot you got to minus it by one.

02:39 I jumped ahead myself.

02:42 I mean, 0 .25.

02:46 I jumped ahead.

02:47 I forgot to minus it by one.

02:49 So that's your answer for your first step.

02:52 So now we're going to go to the second step where they want to know what's, what is the average number of customers waiting to be served? so let me write point a up here.

03:06 So that way we know which is which.

03:09 I'm going to write point b.

03:11 So given that our average rate, so we know like how much the clerk can serve, they can serve two customers per minute.

03:30 We can calculate this as, we're going to use this equation to represent, oh wait, let me make sure i'm writing this right.

03:42 Okay, i meant to write this as a cube, my bad.

03:45 So we're going to use this symbol and we're going to use this equation to calculate for our average number of customers waiting for service.

04:02 So then we're going to plug in all the numbers...