Question 1198576: During the registration period, St. Mary’s Univeristy has a technician in its service center to answer
students’ questions. The number of telephone calls arriving at this center follows a Poisson
distribution with an approximate average rate of 10/h. The time required to answer one call follows an
exponential distribution with an average of 4 min. Answer the following questions:
a. What is the average time between incoming calls?
b. What is the average number of calls that the technician can attend in 1h?
c. What is the probability of there being exactly four calls on hold at a given time?
d. What is the probability of the number of calls in the system exceeding 10?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **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.
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