Question 1198576
**a. Average Time Between Incoming Calls**

* **Arrival Rate (λ):** 10 calls/hour 
* **Average Time Between Arrivals:** 1 / λ = 1 / 10 calls/hour = 0.1 hours/call 
* **Convert to minutes:** 0.1 hours/call * 60 minutes/hour = 6 minutes/call

**Therefore, the average time between incoming calls is 6 minutes.**

**b. Average Number of Calls Attended in 1 Hour**

* **Service Time per Call:** 4 minutes
* **Calls Attended per Hour:** 60 minutes/hour / 4 minutes/call = 15 calls/hour

**Therefore, the technician can attend an average of 15 calls per hour.**

**c. Probability of Exactly Four Calls on Hold**

* To calculate this, we need to use queuing theory. 
* **Assumptions:**
    * This is an M/M/1 queue (single-server queue with Poisson arrivals and exponential service times).
    * The arrival rate (λ) is 10 calls/hour.
    * The service rate (μ) is 15 calls/hour.
* **Traffic Intensity (ρ):** ρ = λ / μ = 10 calls/hour / 15 calls/hour = 2/3 
* **Probability of n Calls in the System (P_n):**
    * P_n = (1 - ρ) * ρ^n 
    * For exactly four calls on hold (n = 4): 
        * P_4 = (1 - 2/3) * (2/3)^4 = (1/3) * (16/81) = 16/243 
        * P_4 ≈ 0.0658

**Therefore, the probability of there being exactly four calls on hold at a given time is approximately 0.0658.**

**d. Probability of More Than 10 Calls in the System**

* To calculate this, we need to use the formula for the probability of n calls in the system for an M/M/1 queue:
    * P_n = (1 - ρ) * ρ^n 

* The probability of more than 10 calls in the system is:
    * P(n > 10) = 1 - P(n ≤ 10) 
    * P(n > 10) = 1 - (P_0 + P_1 + P_2 + ... + P_10)

* Calculate P_n for n = 0 to 10 using the formula above. 
* Sum these probabilities to find P(n ≤ 10).
* Subtract P(n ≤ 10) from 1 to find P(n > 10).

**Note:**

* Calculating P(n > 10) manually can be tedious. You can use software tools like Excel, R, or specialized queuing theory software to calculate this probability more efficiently.

This analysis provides a basic framework for understanding the call center's performance. 

**Disclaimer:** 
* This analysis assumes a steady-state condition, where the arrival rate and service rate remain constant over time. 
* Real-world scenarios may involve variations in arrival rates and service times.
* This analysis does not consider factors such as call abandonment, priority queues, or multiple servers.

I hope this helps! Let me know if you have any further questions.