Question 1198163: A bank has two tellers working on saving accounts. The first teller handles withdrawals only. The second teller handles deposits only, it has been found the service time distribution for the deposits and withdrawals both are exponential with mean service time 3 minutes per customer. Depositors are found to arrive in a Poisson distribution throughout the day with a mean arrival rate of 16 per hour. Withdrawals also arrive in a Poisson fashion with a mean arrival rate of 14 per hour.
What would be the effect on the average waiting time for depositors and withdrawers if each teller could handle both withdrawals and deposits?
What would be the effect if this could only be by increasing the service time to 3.5 minutes?
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **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!
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