Question

Mid-West Publishing Company publishes college textbooks. The company operates an 800 telephone number whereby potential adopters can ask questions about forthcoming texts, request examination copies of texts, and place orders. Currently, two extension lines are used, with two representatives handling the telephone inquiries. Calls occurring when both extension lines are being used receive a busy signal; no waiting is allowed. Each representative can accommodate an average of 15 calls per hour. The arrival rate is 30 calls per hour.

How many extension lines should be used if the company wants to handle 90% of the calls immediately?

What is the probability that a call will receive a busy signal if your recommendation in part (a) is used? If required, round your answer to four decimal places.

What number of calls, in decimal form, receive a busy signal for the current telephone system with two extension lines? If required, round your answer to four decimal places.

Answer 1

To handle 90% of the calls immediately, the company should use 20 extension lines. The probability of a call receiving a busy signal is 90%. Currently, 27 calls receive a busy signal for the two extension line system.

Step-by-step explanation:

To handle 90% of the calls immediately, the number of extension lines should be adjusted based on the arrival rate and the service rate of the representatives. In this case, the arrival rate is 30 calls per hour and each representative can handle 15 calls per hour. So, to handle 90% of the calls immediately, we need to calculate the number of representatives required.

The formula to calculate the number of representatives is:

Number of representatives = (Arrival rate) / (Service rate × Probability of waiting)

Probability of waiting is equal to 1 minus the desired percentage of immediate service. So, the probability of waiting is 1 - 0.90 = 0.10.

Substituting the values into the formula, we get:

Number of representatives = 30 / (15 × 0.10) = 20

Therefore, the company should use 20 extension lines to handle 90% of the calls immediately.

The probability that a call will receive a busy signal can be calculated using the formula:

Probability of busy signal = (1 - Probability of waiting) × 100%

Substituting the values, we get:

Probability of busy signal = (1 - 0.10) × 100% = 0.90 or 90%

The number of calls that receive a busy signal for the current telephone system can be calculated as:

Number of calls receiving a busy signal = (Arrival rate) × (Probability of busy signal)

Substituting the values, we get:

Number of calls receiving a busy signal = 30 × 0.90 = 27 calls

Answer 2

(a) 27 extension lines

(b) 0.9000

(c) 30.0000

Step-by-step explanation:

Number of representatives=2, number of extension lines=2, average calls each representative can accommodate per hour = 15 calls, arrival rate per hour = 30 calls

(a) 90% of the arrival rate = 0.09 × 30 = 27 calls

To handle 27 calls immediately, 27 extension lines should be used

(b) Probability is given by number of possible outcomes ÷ number of total outcomes

Number of possible outcomes = 27, number of total outcomes = 30

Probability (call will receive busy signal) = 27/30 = 0.9000

(c) For one extension line, numbers of calls to receive busy signal = 30 - 15 = 15.0000 calls

Number of calls to receive busy signal for the current telephone system with two extension lines = 2 × 15.0000 = 30.000 calls