Phone calls arrive at the rate of 48 per hour at the...

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. a. Compute the probability of receiving three calls in a 5-minute interval of time. b. Compute the probability of receiving exactly 10 calls in 15 minutes. c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time? What is the probability that none will be waiting? d. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?

Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.
a. Compute the probability of receiving three calls in a 5-minute interval of time.
b. Compute the probability of receiving exactly 10 calls in 15 minutes.
c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time? What is the probability that none will be waiting?
d. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?

Key Concepts

Expected Value of a Poisson Process

The expected value in a Poisson process is the mean number of events expected to occur within a given time interval and is directly equal to the adjusted rate parameter (?t) for that interval. This value provides a benchmark for predicting how many events, such as incoming calls, are likely to occur during that period.

Time Interval Conversion

Time interval conversion in the context of the Poisson process involves scaling the rate parameter to the appropriate time frame under consideration. Since the rate is given per unit time (like per hour), when analyzing a different interval (such as 5 or 15 minutes), the rate must be multiplied by the fraction of that interval relative to the original unit of time to compute the expected number of events correctly.

Poisson Distribution

The Poisson distribution is the probability distribution that governs the number of events occurring in a fixed interval of time or space when these events happen with a known constant rate and independently of the time since the last event. Its probability mass function, given by P(X=k) = e^(–?t) * (?t)^k / k!, is used to calculate the probability of observing a specific number of events (k) within a specified time period.

Poisson Process

A Poisson process is a statistical model that describes events occurring randomly over time at a constant average rate. In this process, events occur independently of one another, meaning that the occurrence of one event does not affect the probability of another event occurring. This model is particularly useful for analyzing scenarios like phone calls arriving at a reservation desk or other similar random event arrivals.

Rate Parameter (?)

The rate parameter (?) represents the average number of events occurring per unit time in a Poisson process. It is a critical value that defines the behavior of the process and must be adjusted according to the length of the time interval considered. For instance, converting an hourly rate to a per-minute or per-interval rate is accomplished by multiplying ? by the appropriate time fraction.

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