An airline is planning to open a satellite ticket desk in a new shopping plaza, staffed
by one ticket agent. It is estimated that requests for tickets and information will
average 15 per hour, and requests will have a Poisson distribution. Service time is
assumed to be exponentially distributed. Previous experience with similar satellite
operations suggests that mean service time should average about three minutes per
request. Determine each of the following:
1. System utilization
2. Percentage of time the server (agent) will be idle
3. The expected number of customers waiting to be served
4. The average time customers will spend in the system
5. The probability of zero customers in the system and the probability of four
customers in the system

To analyze the airline ticket desk's performance, we can model it using the **M/M/1 queuing system**, where:

- **M/M/1** represents a single-server queue with exponential inter-arrival and service times.
- \(\lambda\) (arrival rate): Average number of requests per hour.
- \(\mu\) (service rate): Average number of customers served per hour.
  
Given:
- \(\lambda = 15\) customers per hour (arrival rate).
- Average service time \( = 3\) minutes per request.

### 1. System Utilization (\(\rho\))
System utilization (\(\rho\)) is the ratio of the arrival rate (\(\lambda\)) to the service rate (\(\mu\)):

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

First, we need to calculate the service rate \(\mu\).

\[
\mu = \frac{1}{\text{Average Service Time}}
\]

Since the average service time is given in minutes, we need to convert it to hours.

\[
\text{Average Service Time} = \frac{3}{60} \text{ hours}
\]

So,

\[
\mu = \frac{1}{3/60} = 20 \text{ customers per hour}
\]

Now calculate the system utilization:

\[
\rho = \frac{15}{20} = 0.75
\]

### 2. Percentage of Time the Server (Agent) will be Idle
The idle time is \(1 - \rho\):

\[
\text{Idle Time} = 1 - 0.75 = 0.25
\]

So, the percentage of time the server is idle is:

\[
0.25 \times 100 = 25\%
\]

### 3. Expected Number of Customers Waiting to be Served (\(L_q\))
The expected number of customers in the queue (\(L_q\)) can be calculated using:

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

\[
L_q = \frac{0.75^2}{1 - 0.75} = \frac{0.5625}{0.25} = 2.25
\]

### 4. Average Time Customers Will Spend in the System (\(W\))
The average time a customer spends in the system is the sum of time in queue (\(W_q\)) and the service time.

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

\[
W = \frac{1}{20 - 15} = \frac{1}{5} = 0.2 \text{ hours}
\]

This is equivalent to:

\[
0.2 \times 60 = 12 \text{ minutes}
\]

### 5. Probability of Zero and Four Customers in the System
The probability of \(n\) customers in the system is given by:

\[
P_n = (1 - \rho) \rho^n
\]

#### Probability of Zero Customers (\(P_0\))
\[
P_0 = (1 - 0.75) = 0.25
\]

#### Probability of Four Customers (\(P_4\))
\[
P_4 = (1 - 0.75) \times (0.75)^4
\]

\[
P_4 = 0.25 \times (0.3164) = 0.0791
\]

### Summary
1. **System Utilization**: 75%
2. **Percentage of Time the Server will be Idle**: 25%
3. **Expected Number of Customers Waiting**: 2.25
4. **Average Time Customers will Spend in the System**: 12 minutes
5. **Probability of Zero Customers in the System**: 0.25; Probability of Four Customers in the System: 0.0791

Would you like to go through any of these in more detail or ask a follow-up question?

### Related Questions:
1. How does increasing the arrival rate affect system utilization?
2. What happens to the average waiting time if the service rate decreases?
3. How would the results change if we added another agent to the system?
4. Can you calculate the expected number of customers in the system (\(L\))?
5. How would you determine the maximum allowable arrival rate for an 80% utilization?

### Tip:
In queuing systems, increasing the number of servers (agents) or improving service efficiency can significantly reduce wait times and system congestion.

An airline is planning to open a satellite ticket desk in a new shopping plaza, staffed by one ticket agent. It is estimated that requests for tickets and information will average 15 per hour, and requests will have a Poisson distribution. Service time is assumed to be exponentially distributed. Previous experience with similar satellite operations suggests that mean service time should average about three minutes per request. Determine each of the following: 1. System utilization 2. Percentage of time the server (agent) will be idle 3. The expected number of customers waiting to be served 4. The average time customers will spend in the system 5. The probability of zero customers in the system and the probability of four customers in the system

Learn how to calculate system utilization, idle time, customer wait times, and probabilities using the M/M/1 queuing model for an airline's satellite ticket desk. This problem covers essential queuing theory concepts, including Poisson and exponential distributions. Suitable for undergraduate students in Operations Research and Business Analytics.

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