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.

검증된 단계별 안내

Step 1: Understand the problem. The grocery store can process up to 4 customers per minute. If more than 4 customers arrive in a given minute, the excess customers will have to wait. The goal is to create a table showing the number of customers waiting at the end of each minute for 20 minutes.

Step 2: Define the variables. Let the number of arrivals each minute be represented by the given Poisson-distributed random numbers. Let the number of customers processed per minute be fixed at 4. The number of customers waiting at the end of each minute can be calculated as the cumulative excess arrivals minus the cumulative processing capacity.

Step 3: Calculate the excess arrivals for each minute. For each minute, subtract the processing capacity (4 customers) from the number of arrivals. If the result is positive, it represents the number of customers who cannot be processed and will wait. If the result is negative or zero, it means no customers are waiting.

Step 4: Compute the cumulative waiting customers. For each minute, add the excess arrivals from the current minute to the cumulative waiting customers from the previous minute. This will give the total number of customers waiting at the end of each minute.

Step 5: Create the table. Organize the results into a table with two columns: one for the minute (1 through 20) and one for the number of customers waiting at the end of that minute. Use the calculations from Steps 3 and 4 to populate the table.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is commonly used in scenarios where events happen randomly, such as customer arrivals at a store.

Queueing Theory

Queueing theory is the mathematical study of waiting lines or queues. It helps analyze the behavior of queues in various systems, including how many customers arrive, how long they wait, and how many can be served. In this context, it is essential for understanding customer flow and service efficiency at the grocery store's checkout counters.

Random Number Generation

Random number generation is the process of creating a sequence of numbers that lack any predictable pattern. In statistics, it is often used to simulate real-world processes, such as customer arrivals in this scenario. The generated random numbers can represent various outcomes, allowing for analysis and modeling of different situations.