Calculating Call Volume Probabilities in a Call Center Using the Poisson Distribution

In a call center, the number of calls received over a period can be modeled using the Poisson distribution. This distribution is particularly useful in scenarios where events (in this case, call arrivals) occur independently over a fixed time interval with a known average rate, lambda.

The Problem and the Poisson Distribution

Consider a call center where 180 calls are received over a 12-hour period. The probability of receiving between 180 and 200 calls in this time frame can be determined using the Poisson distribution. The Poisson distribution is given by the formula:

[P(X k) frac{lambda^k e^{-lambda}}{k!}]

Modeling the Scenario

In this specific scenario:

The average number of calls lambda 180 calls in 12 hours. (e) is the base of the natural logarithm, approximately equal to 2.71828. is the number of calls.

To find the probability of receiving between 180 and 200 calls, we need to calculate:

[P(180 leq X leq 200) P(X 180) P(X 181) ldots P(X 200)]

While this calculation can be quite tedious, for larger values of lambda, the Poisson distribution can be approximated by a normal distribution. This normal distribution has a mean mu and a standard deviation sigma.

Normal Approximation

For large lambda, the Poisson distribution can be approximated by a normal distribution with:

mu lambda 180 sigma (sqrt{lambda} sqrt{180} approx 13.42)

Standardizing the Values

The z-score formula can be used to standardize the values:

[z frac{X - mu}{sigma}]

For X 180 and X 200, the z-scores are:

z180 (frac{180 - 180}{sqrt{180}} 0) z200 (approx frac{20 - 180}{13.42} approx 1.49)

Using the Z-Table

The z-scores can be looked up in the standard normal distribution table to find the corresponding probabilities:

P(Z leq 0) 0.5 P(Z leq 1.49) approx 0.9319

Final Probability Calculation

The probability of receiving between 180 and 200 calls is:

[P(180 leq X leq 200) approx P(Z leq 1.49) - P(Z leq 0) approx 0.9319 - 0.5 0.4319]

Therefore, the probability of receiving between 180 and 200 calls in a 12-hour period is approximately 0.4319 (or 43.19%).

Conclusion: Using the Poisson distribution and its normal approximation, we can efficiently calculate the probability of receiving a specific range of calls in a call center over a fixed period.