Students arrive at the Administrative Services Office at an average of one every 10 minutes, and their requests take, on average, 6 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 percentage of time is Judy idle?

Note: Round your answer to 1 decimal place.



How much time, on average, does a student spend waiting in line?

Note: Do not round intermediate calculations. Round your answer to 1 decimal place.



How long is the (waiting) line, on average?

Note: Round your answer to 2 decimal places.



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.





We can solve this problem using queuing theory, specifically the M/M/1 model. Here's a step-by-step approach for each question:

### Given:
- **Arrival rate (λ)**: 1 student every 10 minutes \( = \frac{1}{10} \text{ students per minute}\).
- **Service rate (μ)**: 1 student processed every 6 minutes \( = \frac{1}{6} \text{ students per minute}\).
- **Judy works for 8 hours per day**.

The M/M/1 queue assumes Poisson arrivals and exponential service times.

### Step 1: Calculate utilization rate (ρ)

The system utilization rate is given by:

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

Substituting the values of \( \lambda \) and \( \mu \):

\[
\rho = \frac{\frac{1}{10}}{\frac{1}{6}} = \frac{6}{10} = 0.6
\]

Thus, Judy's utilization rate (fraction of time she is busy) is 0.6.

---

### 1. **What percentage of time is Judy idle?**

The percentage of time Judy is idle is the complement of her utilization rate:

\[
\text{Idle time} = 1 - \rho = 1 - 0.6 = 0.4
\]

So, Judy is idle **40%** of the time.

### 2. **How much time, on average, does a student spend waiting in line?**

The average waiting time in the queue ( \(W_q\) ) is given by the formula:

\[
W_q = \frac{\rho}{\mu (1 - \rho)}
\]

Substitute the values:

\[
W_q = \frac{0.6}{\frac{1}{6}(1 - 0.6)} = \frac{0.6}{\frac{1}{6} \times 0.4} = \frac{0.6}{0.06667} \approx 9 \text{ minutes}
\]

So, on average, a student spends **9 minutes** waiting in line.

### 3. **How long is the (waiting) line, on average?**

The average number of students in the queue ( \(L_q\) ) is given by:

\[
L_q = \lambda \times W_q
\]

Substituting the values:

\[
L_q = \frac{1}{10} \times 9 = 0.9
\]

Thus, on average, there are **0.90 students** in the queue.

### 4. **What is the probability that an arriving student will find at least one other student waiting in line?**

The probability that there are **no students** in the system (i.e., Judy is idle) is given by \( P_0 = 1 - \rho \), which is 0.4.

So, the probability that **at least one** student is waiting in line is:

\[
P(\text{at least 1 in line}) = 1 - P_0 = 1 - 0.4 = 0.6
\]

Thus, the probability is **0.6000** (or 60%).

---

### Summary:
1. Judy is idle **40.0%** of the time.
2. A student spends, on average, **9.0 minutes** waiting in line.
3. The average length of the waiting line is **0.90 students**.
4. The probability that an arriving student will find at least one other student waiting is **0.6000**.

Would you like to go over the details or ask any follow-up questions?

Here are 5 related questions:
1. How would the results change if the arrival rate doubled?
2. What if the processing time increased to 8 minutes per student?
3. How would a second clerk affect the queue's waiting time?
4. How would the average waiting time change if arrivals occurred every 8 minutes?
5. How does the utilization rate affect the probability of finding students in line?

**Tip:** In queueing systems, the utilization rate (ρ) plays a key role in determining both waiting times and system efficiency.

Students arrive at the Administrative Services Office at an average of one every 10 minutes, and their requests take, on average, 6 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 percentage of time is Judy idle? How much time, on average, does a student spend waiting in line? How long is the (waiting) line, on average? What is the probability that an arriving student will find at least one other student waiting in line?

Solve a queuing theory problem involving Poisson arrivals and exponential service times at an Administrative Services Office. Learn how to calculate utilization rates, average waiting time, average queue length, and the probability of finding other students in line. Ideal for advanced high school or college-level students.

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