Burrito King (a new fast-food franchise opening up nationwide) has successfully automated burrito production for its drive-up fast-food establishments. The Burro-Master 9000 requires a constant 45 seconds to produce a batch of burritos. It has been estimated that customers will arrive at the drive-up window according to a Poisson distribution at an average of one every 50 seconds. To help determine the amount of space needed for the line at the drive-up window, Burrito King would like to know the following:

What is the average waiting line length (in cars)?

**Note: Do not round intermediate calculations. Round your answer to 2 decimal places.**

What is the average number of cars in the system (both in line and at the window)?

**Note: Do not round intermediate calculations. Round your answer to 2 decimal places.**

What is the average time in the system, in minutes?

**Note: Do not round intermediate calculations. Round your answer to 2 decimal places.**

To solve this problem, we will apply the **queuing theory**, specifically the **M/M/1 queue model**. The system has a single server (the Burro-Master 9000), customers arrive according to a Poisson process, and service times follow an exponential distribution. Here's the relevant dataTo solve this problem, we will apply the **queuing theory**, specifically the **M/M/1 queue model**. The system has a single server (the Burro-Master 9000), customers arrive according to a Poisson process, and service times follow an exponential distribution. Here's the relevant 

- Arrival rate, \(\lambda\) (customers per second): \(\frac{1}{50} \approx 0.02\) customers per second.
- Service rate, \(\mu\) (batches per second): \(\frac{1}{45} \approx 0.0222\) batches per second.

### Step 1: Utilization (\(\rho\))
The utilization factor, \(\rho\), is the fraction of time the system is busy and is given by:

\[
\rho = \frac{\lambda}{\mu} = \frac{0.02}{0.0222}
\]

### Step 2: Average Waiting Line Length (Lq)
The average number of cars in the waiting line is given by the formula for the M/M/1 queue:

\[
L_q = \frac{\rho^2}{1 - \rho}
\]

### Step 3: Average Number of Cars in the System (Ls)
The average number of cars in the system (both in line and at the window) is given by:

\[
L_s = \frac{\rho}{1 - \rho}
\]

### Step 4: Average Time in the System (Ws)
The average time a car spends in the system is related to the average number of cars in the system:

\[
W_s = \frac{1}{\mu - \lambda}
\]

We'll now compute the values for these metrics.Here are the results based on the calculations:

1. The **average waiting line length** is approximately \( 8.10 \) cars.
2. The **average number of cars in the system** (both in line and at the window) is \( 9.00 \) cars.
3. The **average time in the system** is approximately \( 7.50 \) minutes.

Would you like further details or have any questions about these calculations?

Here are 5 related questions to expand your understanding:
1. How does the utilization factor (\(\rho\)) affect the waiting line length?
2. What happens to the system if the service rate decreases?
3. How can we reduce the average waiting time in the system?
4. What assumptions of the M/M/1 model are critical for these calculations?
5. How would the results change if the arrival rate increased to 1 customer every 40 seconds?

**Tip:** Increasing the service rate slightly can dramatically reduce the average waiting line length in queueing systems.

Burrito King (a new fast-food franchise opening up nationwide) has successfully automated burrito production for its drive-up fast-food establishments. The Burro-Master 9000 requires a constant 45 seconds to produce a batch of burritos. It has been estimated that customers will arrive at the drive-up window according to a Poisson distribution at an average of one every 50 seconds. To help determine the amount of space needed for the line at the drive-up window, Burrito King would like to know the following:

What is the average waiting line length (in cars)?

What is the average number of cars in the system (both in line and at the window)?

What is the average time in the system, in minutes?

Discover how to calculate the average waiting line length, the number of cars in the system, and the average time in the system for Burrito King's drive-up service using queuing theory. Learn how to apply the M/M/1 queue model with Poisson arrivals and exponential service times. Suitable for advanced mathematics students.

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