Question

During tax season, the IRS hires seasonal workers to help answer the questions of taxpayers who call a special 800 telephone number for tax information. Suppose that calls to this line occur at a rate of 60 per hour and follow a Poisson distribution. The IRS workers manning the phone lines can answer an average of 5 calls per hour with the actual service times following an exponential distribution. Assume that 10 IRS workers are available and, when they are all busy, the phone system can keep 5 additional callers on hold.

What is the probability that a caller receives a busy signal?

Answer 1

To find the probability that a caller receives a busy signal at the IRS call center, we use the Poisson distribution for the arrival of calls and the exponential distribution for the service times. Calculating this probability requires knowing the arrival rate, the service rate, and the number of callers the system can handle simultaneously.

Step-by-step explanation:

The scenario described involves the Poisson distribution for the arrival of calls and the exponential distribution for the service times. The probability that a caller receives a busy signal can be determined using the queuing theory. Given that calls are arriving at a rate of 60 per hour (or 1 per minute) and each IRS worker can handle 5 calls per hour (or 1 call every 12 minutes), with 10 workers available, the system is operating with an offered load (or traffic intensity) of λ/cμ, where λ is the arrival rate, c is the number of servers, and μ is the service rate.

Let λ = 60 calls per hour (1 call per minute), c = 10 IRS workers, μ = 5 calls per hour (1 call every 12 minutes).

Offered load (traffic intensity) ρ = λ/cμ = 1 call per minute / (10 workers * 1 call per 12 minutes) = 1/120. The system can handle a total of 15 calls at a time (10 workers + 5 on hold). A busy signal occurs when all 15 lines are occupied. Using Poisson distribution, we can calculate the probability that more than 15 calls are in the system at any time, which gives us the probability of encountering a busy signal.

Let X represent the number of calls in the system. The probability of receiving a busy signal is P(X > 15).

This can be calculated using the Poisson cumulative distribution function:

P(X > 15) = 1 - P(X ≤ 15)

We would compute P(X ≤ 15) using the Poisson CDF with λ = offered load * time period. Since the time period is not specified, let's assume it's 1 hour for simplicity.

Thus, P(X > 15) is a calculation we cannot perform without a time period or without assuming a time period.