Suppose that the log-ons to a computer network follow a...

Suppose that the log-ons to a computer network follow a Poisson process with an average of 3 counts per minute. (a) What is the mean time between counts? (b) What is the standard deviation of the time between counts? (c) Determine $x$ such that the probability that at least one count occurs before time $x$ minutes is $0.95 .$

Suppose that the log-ons to a computer network follow a Poisson process with an average of 3 counts per minute.
(a) What is the mean time between counts?
(b) What is the standard deviation of the time between counts?
(c) Determine $x$ such that the probability that at least one count occurs before time $x$ minutes is $0.95 .$

Key Concepts

Poisson Process

A Poisson process is a stochastic process that models the occurrence of random events over a continuous interval of time or space. It is characterized by the property of independent increments, meaning the number of events occurring in disjoint time intervals are independent, and events occur with a constant average rate. This process is widely used to model random counts like arrivals, phone calls, or log-ons in computer networks.

Exponential Distribution

In a Poisson process, the time intervals between successive events are described by the exponential distribution. This distribution is continuous and models the waiting time until the next event occurs. Its key feature is the memoryless property, meaning that the probability of an event occurring in the future is independent of the past.

Rate Parameter and Moments

The exponential distribution is defined by a rate parameter (?), where the mean (or expected value) of the time between events is given by 1/?. The standard deviation of the inter-arrival times is also 1/?, reflecting the fact that for an exponential distribution, the mean and standard deviation are equal. These properties are essential when analyzing the timing characteristics of a Poisson process.

Cumulative Distribution Function

The cumulative distribution function (CDF) for the exponential distribution provides the probability that the waiting time until the next event is less than or equal to a given time x. Mathematically, it is expressed as F(x) = 1 - e^(-?x). This function is used to determine thresholds or specific time values that correspond to given probability levels, such as finding the time by which there is a 95% chance that at least one event has occurred.

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