In Exercises 1–7, consider a grocery store that can process a ...

Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

Minitab was used to generate 20 random numbers with a Poisson distribution for . Let the random number represent the number of arrivals at the checkout counter each minute for 20 minutes. 3 3 3 3 5 5 6 7 3 6 3 5 6 3 4 6 2 2 4 1During each of the first four minutes, only three customers arrived. These customers could all be processed, so there were no customers waiting after four minutes.

b. Create a table that shows the number of customers waiting at the end of 1 through 20 minutes.

Accepted Answer

Welcome back, everyone. In this problem, a bank teller can serve up to 5 customers per minute. The number of customers arriving each minute for 10 minutes is 465-73852, 6, and 7 respectively. At the start, no customers are waiting. What is the number of customers waiting at the end of minute 10? A says 2, B4, C5, and D says 6. Now how can we figure out how many customers will be waiting? What do we know? Well, so far we know that for our service capacity, the bank teller can serve up to 5 customers per minute. We also know the number of arrivals for 10 minutes by each minute, and we were told that no customers were waiting initially. So if we're going to figure this out, we can simulate what happens minute by minute to figure out how many we end up with. So let's try to picture what's really going on here, OK? So, let's say here I have our minutes and we're thinking about the first minute. No, in our first minute, notice that 4 customers arrived. So if we imagine that we have a line here. Then in this line, we're gonna have our 4 customers, OK? Then let's imagine that we also have the teller's desk, OK. Or the era for the tiers and remember we are told that the tellers can serve 5. Uh, customers per minute. So let's put these 5 spaces here. And then remember we want to find out how many customers are waiting. So on the side, we can put a little waiting area, OK? So now here we have our little waiting area. Now, in the first minute, since the bank teller serves up to 5 customers per minute, it means that each of these customers would go into 4 of the 5 available spots, OK? And thus, since the 4 of the 5 available spots are filled, then that means there would be no one waiting in the first minute. Now if we go down to the 2nd minute, OK? We know that 6 new customers come, so that's 123456. We know that, again, we have those 5 spaces. These 4 have been served, but no, from the 6, only 5 people can be served. So we can put the 5 people. In a section for the tellers. And now we know that one will be waiting OK? I know we can think about the flow each as we continue to go along to help us figure out how many would be waiting at the end of 10 minutes. So now in the 3rd minute, OK, we know that uh there were 5 arrivals, so 12345, and no. For our spaces, we know 5 were already served, but 1 was waiting. So now these 5 can go into the line with the tellers and we'll still have 1 waiting on the side. Now we go down to minute 4. In minute 47 people are drawing the line, OK. So now we have our 5 spaces. 12345. And now we know that 5 of these people can go to the teller. OK. So we cross off these 5 and notice here that we have 2 remaining. So now, we add those 2 to the one that was already waiting. We have 3 waiting at minute 4. In minute 5, OK, 3 people join. No, we have our spaces, OK? And no, we could put these 3 people with our tellers, cross them out. And now, we have 2 extra spaces. Remember, we had, we had 3 who were waiting before. So we can take 2 of those and put them in our empty sections, and now we'll be left with 1 person waiting. In minute 6 8 people join, that's 12345678. You have our 5 spaces. So we can take 5 to go to the teller. And then the 3 remaining will wait. So we add those 3 to the 1 that was already waiting. Now we have 4 waiting at minute 6. In minute 7, OK, 5 people join the line. So we have those 5 spaces that were filled up. And now we still have our 4 that are still waiting. Now, in minute 8, OK, we know that 2 people joined. So form our 5 spaces, 2 of those people, those 2 people can go to the teller, and then we can take 3 who were previously waiting and put them to the teller, and that leaves us with 1 waiting at minute 8. Now, at minute 9 we have 6 people who join the line, 3456. And no 5 of those people can go to the teller. And the one who is left will have to wait with the other person who was left, so we have 2 waiting. And now we're at minute 1010, 7 people join the line, OK? So now from our 5 spaces. 5 of those people will go to our tellers, and how many are left? 2. So if we add those 2 that are left to the 2 who are already already waiting, it means that at minute 10, 1234 people would be waiting, or 4 customers would be waiting to be served. Therefore, 4 is the correct answer. If we look back at our answer choices, that means B is the correct option. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Poisson Distribution

Queueing Theory

Random Number Generation

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