Use the following queueing network and assume the following distributions for
interarrivals (IAT) and service times (ST) (all in minutes). Determine whether the
system will be stable. Justify your answer.
• IAT ~ exponential(6)
• ST1 ~ triangular(4, 4.5, 5)
• ST2 ~ exponential(4.3)
• ST3 ~ uniform(3,7)

To determine whether the queueing system is stable, we need to analyze the relationship between the **arrival rate** and the **service rates** for each server or service station.

### Key Concepts:

- **Arrival Rate (λ):** This is the rate at which customers arrive at the system, which is the inverse of the mean interarrival time (IAT).
  
- **Service Rate (μ):** This is the rate at which customers are served, which is the inverse of the mean service time (ST) for each server.

For the system to be stable, the utilization factor (ρ) for each server must be less than 1. This means that the service rate must be greater than the arrival rate at each service station (i.e., the server can handle the customer load).

Utilization for each server:
\[
\rho = \frac{\lambda}{\mu}
\]
where \( \lambda \) is the arrival rate and \( \mu \) is the service rate for each service station.

### Step-by-Step Calculation:

#### 1. **Arrival Rate (λ):**
The interarrival times (IAT) follow an exponential distribution with a mean of 6 minutes, so:
\[
\lambda = \frac{1}{\text{Mean of IAT}} = \frac{1}{6} \approx 0.167 \text{ customers per minute}.
\]

#### 2. **Service Rates (μ):**

- **Service Time for ST1 (Triangular(4, 4.5, 5)):**
  
  The mean of a triangular distribution with parameters \(a\), \(b\), and \(c\) is given by:
  \[
  \text{Mean of triangular distribution} = \frac{a + b + c}{3}.
  \]
  Substituting the values:
  \[
  \text{Mean of ST1} = \frac{4 + 4.5 + 5}{3} = \frac{13.5}{3} = 4.5 \text{ minutes}.
  \]
  So the service rate for ST1 is:
  \[
  \mu_1 = \frac{1}{4.5} \approx 0.222 \text{ customers per minute}.
  \]

- **Service Time for ST2 (Exponential(4.3)):**
  
  For an exponential distribution, the mean is the parameter given (4.3 minutes), so:
  \[
  \mu_2 = \frac{1}{4.3} \approx 0.233 \text{ customers per minute}.
  \]

- **Service Time for ST3 (Uniform(3, 7)):**
  
  The mean of a uniform distribution with parameters \(a\) and \(b\) is given by:
  \[
  \text{Mean of uniform distribution} = \frac{a + b}{2}.
  \]
  Substituting the values:
  \[
  \text{Mean of ST3} = \frac{3 + 7}{2} = 5 \text{ minutes}.
  \]
  So the service rate for ST3 is:
  \[
  \mu_3 = \frac{1}{5} = 0.2 \text{ customers per minute}.
  \]

#### 3. **Utilization Factor (ρ):**

Now, let's calculate the utilization factor for each service station:

- **For ST1:**
  \[
  \rho_1 = \frac{\lambda}{\mu_1} = \frac{0.167}{0.222} \approx 0.752.
  \]

- **For ST2:**
  \[
  \rho_2 = \frac{\lambda}{\mu_2} = \frac{0.167}{0.233} \approx 0.717.
  \]

- **For ST3:**
  \[
  \rho_3 = \frac{\lambda}{\mu_3} = \frac{0.167}{0.2} = 0.835.
  \]

### Conclusion:

The utilization factors for all three service stations are less than 1:
- \(\rho_1 \approx 0.752\),
- \(\rho_2 \approx 0.717\),
- \(\rho_3 \approx 0.835\).

Since the utilization factor \(\rho < 1\) for each station, the system will be stable. This means that the service stations can handle the incoming load of customers, and the queues will not grow indefinitely over time.

Would you like further details or have any questions? 

Here are 5 relative questions to deepen your understanding:

1. What happens if the arrival rate increases in this system?
2. How would the system behave if the utilization factor was greater than 1 for any station?
3. What are the implications of different service time distributions on system stability?
4. How can we model this system using simulation to check long-term behavior?
5. How would adding more servers to a station impact the stability?

**Tip:** For stable queueing systems, always ensure that the service rates can handle the customer arrival rates at each station.

Use the following queueing network and assume the following distributions for interarrivals (IAT) and service times (ST) (all in minutes). Determine whether the system will be stable. Justify your answer. • IAT ~ exponential(6) • ST1 ~ triangular(4, 4.5, 5) • ST2 ~ exponential(4.3) • ST3 ~ uniform(3,7)

Analyze the stability of a queueing network with interarrival times following an exponential distribution (mean = 6 minutes) and service times modeled by triangular, exponential, and uniform distributions. Learn how to calculate arrival rates and service rates, and determine the stability condition using utilization factors.

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