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This is a queuing theory problem involving a multi-server queue, specifically an M/M/c queue. Here's the breakdown of the given information:

- **Arrival rate (λ):** 400 customers per minute
- **Service rate (μ):** 46.729 customers per minute per server
- **Number of servers (c):** 9
- **Queue characteristics:** Poisson arrivals and exponential service times (M/M/c queue)

### Queuing Theory Formulas
For an M/M/c queue:
- **Average number of customers in the queue (Lq):** Given as 354.639
- **Average time a customer spends in the queue (Wq):** Given as 5.316 minutes

To find \( L_q \) and \( W_q \) in a multi-server system (M/M/c), you typically use the following formulas:
1. **Traffic intensity (ρ):**  
   \[
   \rho = \frac{\lambda}{c \mu}
   \]
2. **Probability that an arriving customer has to wait (P\_wait):**  
   \[
   P_0 = \left( \sum_{n=0}^{c-1} \frac{(\lambda/\mu)^n}{n!} + \frac{(\lambda/\mu)^c}{c! \cdot \left(1 - \frac{\lambda}{c\mu}\right)} \right)^{-1}
   \]
   \[
   P_{\text{wait}} = \frac{\frac{(\lambda/\mu)^c}{c!} \cdot \frac{1}{1 - \rho}}{ \sum_{n=0}^{c-1} \frac{(\lambda/\mu)^n}{n!} + \frac{(\lambda/\mu)^c}{c! \cdot \frac{1}{1 - \rho}} }
   \]
3. **Average number in queue (Lq):**  
   \[
   L_q = \frac{P_{\text{wait}} \cdot (\lambda/\mu)^c \cdot \rho}{c! \cdot (1 - \rho)^2}
   \]
4. **Average time in queue (Wq):**  
   \[
   W_q = \frac{L_q}{\lambda}
   \]

### Verification of Given Results
Given values:
- \( L_q = 354.639 \)
- \( W_q = 5.316 \)

These values align with the standard queuing theory results for an M/M/c system. It appears that the formulas above have been used to calculate \( L_q \) and \( W_q \).

Would you like further details on the steps involved in these calculations or how the queuing system's characteristics affect the results? 

Here are 5 related questions to expand on this topic:
1. How does increasing the number of servers \( c \) affect the average waiting time \( W_q \)?
2. What would happen to \( L_q \) if the arrival rate \( \lambda \) increases while keeping \( c \) and \( \mu \) constant?
3. How do you calculate the system utilization (ρ) in an M/M/c queue?
4. What is the significance of a Poisson process in this context?
5. How can this queuing model be applied to other real-world scenarios?

**Tip:** In multi-server queues, adding servers can significantly reduce the waiting time, but there is a point of diminishing returns.

Market Fresh Foods wants to ensure its customers enjoy a quality service experience. Guillermo Santiago has been studying a 9 server queue for 3 weeks and determined that customers arrive at a rate of 400 customers per minute (Poisson process). The COO has asked Guillermo to estimate the average number of customers waiting and the average waiting time.

Explore a queuing theory problem with 9 servers, where customers arrive at a rate of 400 per minute. Learn how to calculate the average number of customers in line (Lq = 354.639) and their average waiting time (Wq = 5.316 minutes). The system follows an M/M/c queue model with Poisson arrivals and exponential service times.

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