Question 1198163
**1. Current Situation (Separate Queues)**

* **Depositors:**
    * Arrival rate (λ1): 16 customers/hour
    * Service rate (μ1): 20 customers/hour (1 customer every 3 minutes)
    * Utilization factor (ρ1): λ1/μ1 = 16/20 = 0.8

* **Withdrawers:**
    * Arrival rate (λ2): 14 customers/hour
    * Service rate (μ2): 20 customers/hour
    * Utilization factor (ρ2): λ2/μ2 = 14/20 = 0.7

* **Calculate Average Waiting Time (M/M/1 Queue):**
    * For both depositors and withdrawers, we can use the M/M/1 queuing model (Poisson arrivals, exponential service times, single server):
        * Average waiting time in queue (Wq): Wq = (ρ^2) / (μ * (1 - ρ)) 
            * Depositors: Wq1 = (0.8^2) / (20 * (1 - 0.8)) = 0.16 hours = 9.6 minutes
            * Withdrawers: Wq2 = (0.7^2) / (20 * (1 - 0.7)) = 0.082 hours = 4.92 minutes

**2. Effect of Combining Queues and Tellers**

* **Combined Arrival Rate:** λ = λ1 + λ2 = 16 + 14 = 30 customers/hour
* **Combined Service Rate:** μ = μ1 + μ2 = 20 + 20 = 40 customers/hour 
* **Utilization Factor:** ρ = λ/μ = 30/40 = 0.75

* **Calculate Average Waiting Time (M/M/2 Queue):**
    * For M/M/2 queues, the calculations are more complex. We can use queuing tables or software to find the average waiting time. 
    * **Expected Result:** The average waiting time for both depositors and withdrawers will significantly decrease compared to the separate queue scenario. This is because customers can be served by either teller, reducing idle time and improving overall efficiency.

**3. Effect of Increased Service Time (3.5 minutes)**

* **Combined Service Rate:** μ = 60 minutes/hour / 3.5 minutes/customer = 17.14 customers/hour 
* **Utilization Factor:** ρ = λ/μ = 30/17.14 = 1.75 
    * **Note:** This utilization factor is greater than 1, indicating that the system is overloaded.

* **Average Waiting Time:** In an overloaded M/M/2 queue, the waiting times will be significantly longer and potentially unbounded. 

**Conclusion:**

* Combining queues and allowing both tellers to handle both deposits and withdrawals will significantly reduce average waiting times for customers.
* Increasing the service time to 3.5 minutes per customer will significantly increase waiting times due to the high utilization factor and potential system overload.

**Disclaimer:**

* This analysis provides a general understanding of the potential effects. 
* Actual waiting times may vary depending on factors such as customer behavior, queue discipline, and other operational factors not considered in this simplified model. 
* For more accurate predictions, detailed queuing models and simulation techniques may be necessary.

I hope this explanation is helpful!