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.

The mean increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.

Accepted Answer

Welcome back, everyone. A bank teller can serve a maximum of 3 clients per minute. The average number of clients arriving is 4 per minute. Create a table showing the number of clients waiting at the end of each minute for 15 minutes and answer the following question. What is the number of clients waiting at the end of the 15 15th minute? If the arrivals each minute are 52463527436534 and 5. A says it's 11, B 15, C 19, and D says it would be 20. Now what do we know here that can help us to create this table? What are we given? Well, we know that at the start no clients are waiting. We also know that during each minute all arriving clients join the queue, and the teller can serve up to 3 clients per minute. Any unserved clients then we'll have to wait for the next minute, so we can simulate minute by minute to help us to create this table because we know that at each minute. Then the total to serve, OK, is going to be equal to the arrivals. Plus those waiting from the previous minute. I know the new waiting or the number of persons that will now be waiting, OK? The new waiting number will be equal to the total to serve. OK, minus the number that we served. So now let's use both of these to help us to create this table. So let's break it down minute by minute, OK? And now we're gonna make a note of the arrivals. OK. We're going to make a note of how many were served, and then we're going to make a note of how many were waiting, OK, at the end of the minute. So let's see how we can do that. So let's write on our numbers, our minutes. 123456789, 1011, 12. 1314 15, OK. And in our problem, we were given how many people were arriving each time. So remember we're told that our arrivals are 52 4. 635. OK. At 7 minutes, it's 2, then 7, then 4, then 36, and 5. OK. And then 34 and 5. OK? So we have our minutes and we have our arrivals, OK? Let me just try to keep the line going here, just to keep them separated properly. Now, here, remember, we know that the the teller can serve a maximum of 3 each minute, OK? So, for a first minute, since the teller can serve 3, that means 2 would be waiting. In the second minute, we have 2 arrivals and 2 waiting. So the teller can serve 3 and that would leave 1 waiting. With 1 waiting and 4 new arrivals, the teller can serve 3 and that leaves 2 waiting. Then in minute 4 with 6 arrivals, the teller can serve 3 and that would leave 5 waiting. At minute 5 then again can serve 3 and then 5 would be waiting at minute 6. Then 5 with 5 arrivals, you can serve 3 and that leaves us with 7 waiting. At minute 7 you can serve 3 again and that leaves with 6 waiting. So you notice here the teller is always at maximum capacity in this problem, OK. No, let me just continue this, uh, continue the table here. Then in minute 8 with 7 arriving, then we can serve 3 that leaves us with 10 waiting. In minute 94 arrives, so we can serve 3 that leaves us with 11 waiting, minute 10, 3 have arrived, so 3 served with 11 waiting doesn't change anything. Then minute 11, 6 have arrived, so we can serve 3, leaves us with 14 waiting. Minute 12, 5 have arrived. So again, we can serve 3, that leaves us with 16 waiting. Um, minute 13, 3 have arrived, so we can serve 3, still left with the 16 waiting. And then minute 14, 4 arrived, so we can serve 31 extra person to wait. So no, that's 17. And then finally in minute 15 with 5 arrive, with 5 people arriving, we can serve 3, and that leaves us with 1919 persons waiting. Thus, the number of clients waiting at the end of the 15th minute is 19. If we look back at our answer choices, that means C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Key Concepts

Poisson Distribution

Random Number Generation

Queueing Theory

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