91Ó°ÊÓ

An article in Ad Hoc Networks ["Underwater Acoustic Sensor Networks: Target Size Detection and Performance Analysis" (2009, Vol. 7(4), pp. \(803-808\) ) ] discussed an underwater acoustic sensor network to monitor a given area in an ocean. The network does not use cables and does not interfere with shipping activities. The arrival of clusters of signals generated by the same pulse is taken as a Poisson arrival process with a mean of \(\lambda\) per unit time. Suppose that for a specific underwater acoustic sensor network, this Poisson process has a rate of 2.5 arrivals per unit time a. What is the mean time between 2.0 consecutive arrivals? b. What is the probability that there are no arrivals within 0.3 time units? c. What is the probability that the time until the first arrival exceeds 1.0 unit of time? d. Determine the mean arrival rate such that the probability is 0.9 that there are no arrivals in 0.3 time units.

Step by step solution

01

Mean Time Between Consecutive Arrivals

For a Poisson process with a rate of \( \lambda \) arrivals per unit time, the time between arrivals follows an exponential distribution with mean \( \frac{1}{\lambda} \). Given \( \lambda = 2.5 \), the mean time between consecutive arrivals is \( \frac{1}{2.5} = 0.4 \) time units.

02

Probability of No Arrivals in Given Time

The probability of getting no arrivals within \( t \) time units in a Poisson process is given by \( P(X=0) = e^{-\lambda t} \). For \( t = 0.3 \) and \( \lambda = 2.5 \), calculate \[ P(X=0) = e^{-2.5 \times 0.3} = e^{-0.75} \approx 0.4724 \].

03

Probability that Time Until First Arrival Exceeds Given Time

The probability that the time until the first arrival exceeds \( t \) time units is equivalent to no arrivals being present in that time, which is also \( e^{-\lambda t} \). For \( t = 1.0 \) and \( \lambda = 2.5 \), calculate \[ P(T > 1.0) = e^{-2.5 \times 1.0} = e^{-2.5} \approx 0.0821 \].

04

Determine Mean Arrival Rate for Specific Probability

To find the mean arrival rate \( \lambda \) such that the probability of no arrivals in \( t \) time units is 0.9, solve \( e^{-\lambda \times 0.3} = 0.9 \). Take the natural logarithm of both sides: \[ -\lambda \times 0.3 = \ln(0.9) \]. Then, solve for \( \lambda \): \[ \lambda = -\frac{\ln(0.9)}{0.3} \approx 0.352 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Poisson Process

The Poisson Process is a powerful tool in applied statistics used to model random events occurring over time or space. It helps predict the likelihood of a certain number of events happening in a fixed period. This process is defined by a single parameter, the mean arrival rate, denoted by \( \lambda \).
In this context, the arrival of signals in an underwater sensor network is assumed to follow such a process.

• The process is characterized by randomness and independence, meaning the occurrence of an event does not affect the next event.
• It's commonly used in situations involving rare occurrences over continuous time, like signal arrivals in networks.

This helps understand and manage complex systems like sensor networks, where exact predictions are challenging due to various factors affecting arrivals.

Exponential Distribution

The Exponential Distribution is directly linked to the Poisson process and is used to model the time between incoming events. When signals arrive in a Poisson fashion, the time intervals between them follow an exponential distribution.
The core feature of this distribution is its memoryless property. This means that the probability of an event occurring in future time intervals is the same, regardless of when the last event occurred. The mean of this distribution is given by \( \frac{1}{\lambda} \), where \( \lambda \) is the arrival rate.

• For example, with \( \lambda = 2.5 \), the expected time between signal arrivals is 0.4 time units.
• This concept helps predict and optimize system performance by understanding timing between events.

It's particularly useful in situations where we analyze the efficiency or waiting times in queues.

Probability Calculations

Probability Calculations allow us to make quantitative predictions in scenarios modeled by Poisson processes and exponential distributions. We often calculate probabilities for specific scenarios using exponential decay functions.
For instance, the probability of no signal arrivals within a fixed time \( t \) is exponentially dependent on \( \lambda \) and time \( t \).

• This is represented by the formula: \( P(X=0) = e^{-\lambda t} \).
• For \( t = 0.3 \) and \( \lambda = 2.5 \), the probability turns out to be approximately 0.4724.
• Likewise, the chance that the time until the first arrival exceeds 1.0 time unit can also be determined from exponential calculations, yielding 0.0821.

These calculations are essential for designing efficient systems that minimize downtime or maximize responsiveness.

Mean Arrival Rate

The Mean Arrival Rate \( \lambda \) in a Poisson process is crucial as it dictates the average number of events in a given time span. It serves as a crucial parameter to model and predict the behavior of the system.
To decide on an appropriate \( \lambda \), one can set specific criteria such as desired probabilities for events. For example, if we need a probability of 0.9 for no arrivals in 0.3 time units, we adjust \( \lambda \) accordingly using the mathematical relationship between \( \lambda \) and probability:

• Manipulating the formula \( e^{-\lambda t} = 0.9 \) gives us \( \lambda \approx 0.352 \).
• This shows how adjusting arrival rates can meet specific operational goals, making it a valuable tool in practical applications like network design.

Understanding \( \lambda \) helps tailor networks to specific conditions and expected loads.

Sensor Network Analysis

Sensor Network Analysis involves understanding and making the best use of signal data received by a sensor network. By using statistical models like the Poisson process, we can assess and improve the efficiency of sensor networks.
This analysis is crucial for networks in demanding environments, such as underwater acoustic networks, to ensure reliable data transmission and operation.

• Assessing the rate and probability of signal arrivals helps optimize sensor deployment and resource allocation.
• It aids in planning for expected traffic loads and managing unexpected bursts of data.

Overall, applying these statistical principles to sensor networks can enhance their performance, leading to more accurate and timely data collection.