Students arrive at the Administrative Services Office at an average of one every 5 minutes, and their requests take, on average, 3 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times. What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line?

Note: Do not round intermediate calculations. Round your answer to 4 decimal places.

To solve this problem, we will use **queuing theory**, specifically the **M/M/1** queue model, where:

- The arrival process follows a Poisson distribution.
- The service times follow an exponential distribution.
- There is one server (Judy Gumshoes) at the counter.

### Step 1: Define the system parameters

1. **Arrival rate (\(\lambda\))**:
   - Students arrive at a rate of one every 5 minutes.
   - \[
   \lambda = \frac{1}{5} \, \text{students per minute}
   \]

2. **Service rate (\(\mu\))**:
   - Each student takes 3 minutes to be served.
   - \[
   \mu = \frac{1}{3} \, \text{students per minute}
   \]

### Step 2: Calculate the system utilization (\(\rho\))

The utilization factor \(\rho\) is the ratio of the arrival rate to the service rate:

\[
\rho = \frac{\lambda}{\mu} = \frac{\frac{1}{5}}{\frac{1}{3}} = \frac{3}{5} = 0.6
\]

### Step 3: Calculate the probability that no students are in the system

The probability that no students are in the system (\(P_0\)) is given by:

\[
P_0 = 1 - \rho = 1 - 0.6 = 0.4
\]

### Step 4: Calculate the probability that there is at least one student in the system

The probability that there is at least one student in the system is \(1 - P_0\):

\[
P(\text{at least one student}) = 1 - 0.4 = 0.6
\]

### Step 5: Calculate the probability that an arriving student will find at least one other student waiting in line

This is the probability that there is **at least one student already in the queue**, which is equivalent to the probability that the number of students in the system is 2 or more. This is calculated by:

\[
P(\text{at least one other waiting}) = 1 - P(\text{no one waiting and no one being served}) = 1 - P_0 = 0.6
\]

Thus, the probability that an arriving student will find at least one other student waiting in line is **0.6**.

### Final Answer:
\[
\boxed{0.6000}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the average number of students in the system (L)?
2. What is the average time a student spends in the system (W)?
3. How would the probabilities change if there were two clerks?
4. What happens to the waiting time if the service time increases to 4 minutes per student?
5. How does the system change if arrivals follow a different distribution?

**Tip**: In an M/M/1 queue, system performance is heavily dependent on the utilization rate \(\rho\).

Students arrive at the Administrative Services Office at an average of one every 5 minutes, and their requests take, on average, 3 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times. What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line?

Explore the probability of an arriving student finding at least one other student waiting in line using queuing theory's M/M/1 model with Poisson arrivals and exponential service times. This example analyzes key concepts like utilization rate, arrival rate, and service rate in a single-server system.

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