Problem 10-23 (Algo)
You are planning the new layout for the local branch of the Sixth Ninth Bank. You are considering separate cashier windows for the three different classes of service. Each class of service would be separate, with its own cashiers and customers. Oddly enough, each class of service, while different, has exactly the same demand and service times. People for one class of service arrive every 4 minutes and arrival times are exponentially distributed (the standard deviation is equal to the mean). It takes 15 minutes to service each customer, and the standard deviation of the service times is 7 minutes. You assign 5 cashiers to each type of service.

On average, how long will each line be at each of the cashier windows?

Note: Round your intermediate calculations to 4 decimal places and final answer to 3 decimal places.



On average, how long will a customer spend in the bank (assume they enter, go directly to one line, and leave as soon as service is complete)?

We are dealing with a queuing system problem, where we have three separate classes of service, and each class is independent with its own cashiers and customers. The question can be analyzed using queuing theory, specifically the **M/M/c** model, where:

- **M/M/c** refers to a queue where both arrivals and service times follow a **Markov process** (exponentially distributed inter-arrival and service times) with **c** servers (cashiers).

### Step 1: Define the parameters
For each class of service, we have:

- **Arrival rate (\(\lambda\))**: Customers arrive every 4 minutes on average, so the arrival rate is:
  \[
  \lambda = \frac{1}{4} \text{ customers per minute}.
  \]

- **Service rate (\(\mu\))**: Each customer is served in 15 minutes on average, so the service rate is:
  \[
  \mu = \frac{1}{15} \text{ customers per minute}.
  \]

- **Number of servers (\(c\))**: There are 5 cashiers for each class of service, so \(c = 5\).

### Step 2: Calculate traffic intensity (\(\rho\))
The traffic intensity, or utilization factor, is given by:
\[
\rho = \frac{\lambda}{c \mu}.
\]

Substituting the values:

\[
\rho = \frac{\frac{1}{4}}{5 \times \frac{1}{15}} = \frac{\frac{1}{4}}{\frac{5}{15}} = \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4} = 0.75.
\]

This means that, on average, the system is 75% utilized.

### Step 3: Calculate the average number of customers in the system (\(L\))
To calculate the average number of customers in the system, we use the formula for the **M/M/c** system:
\[
L = L_q + \frac{\lambda}{\mu},
\]
where \(L_q\) is the average number of customers in the queue.

The average number of customers in the queue \(L_q\) is given by:
\[
L_q = \frac{P_0 \cdot \left(\frac{\lambda}{\mu}\right)^c \cdot \rho}{c! \cdot (1 - \rho)^2},
\]
where \(P_0\) is the probability that there are no customers in the system. \(P_0\) can be calculated as:
\[
P_0 = \left( \sum_{n=0}^{c-1} \frac{(\lambda / \mu)^n}{n!} + \frac{(\lambda / \mu)^c}{c! \cdot (1 - \rho)} \right)^{-1}.
\]

Now we will calculate \(P_0\) and \(L_q\), and then use them to find the average number of customers in the system.The average number of customers in the system (\(L\)) is approximately **5.135**. This includes both customers being served and those waiting in line.

Out of this, the average number of customers in the queue (\(L_q\)) is approximately **1.385**. This represents the number of customers waiting to be served at any given time.

### Step 4: Time a customer spends in the system (\(W\))
The average time a customer spends in the system, which includes both waiting time and service time, can be calculated using Little's Law:

\[
W = \frac{L}{\lambda}.
\]

Let's calculate the average time a customer spends in the system.The average time a customer spends in the bank, including both waiting and service time, is approximately **20.541 minutes**.

### Summary:
- The average number of customers in the system: **5.135**.
- The average number of customers waiting in line: **1.385**.
- The average time a customer spends in the bank: **20.541 minutes**.

Would you like more details, or do you have any questions? 

Here are 5 related questions to deepen your understanding:
1. What happens to the system if the arrival rate increases?
2. How does increasing the number of cashiers affect the average waiting time?
3. What is the impact of reducing the service time variability on the queue?
4. How would you calculate the system's capacity if the bank wants to reduce waiting times?
5. What are some ways to reduce the traffic intensity in the system?

**Tip:** Understanding the relationship between service rate, arrival rate, and the number of servers is key to optimizing queuing systems.

You are planning the new layout for the local branch of the Sixth Ninth Bank. You are considering separate cashier windows for the three different classes of service. Each class of service would be separate, with its own cashiers and customers. Oddly enough, each class of service, while different, has exactly the same demand and service times. People for one class of service arrive every 4 minutes and arrival times are exponentially distributed (the standard deviation is equal to the mean). It takes 15 minutes to service each customer, and the standard deviation of the service times is 7 minutes. You assign 5 cashiers to each type of service.

On average, how long will each line be at each of the cashier windows?

On average, how long will a customer spend in the bank (assume they enter, go directly to one line, and leave as soon as service is complete)?

Explore a queuing theory problem involving three classes of service at a bank, with exponentially distributed arrival and service times. Learn how to calculate the average queue length and the total time a customer spends in the bank using the M/M/c model. Includes detailed formulas for traffic intensity, number of customers in the queue, and Little's Law. Suitable for undergraduate students.

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