Chapter 5: Problem 24
The Jacobson/Karels algorithm sets TimeOut to be 4 mean deviations above the mean. Assume that individual packet round-trip times follow a statistical normal distribution, for which 4 mean deviations are \(\pi\) standard deviations. Using statistical tables, for example, what is the probability that a packet will take more than TimeOut time to arrive?
Short Answer
Expert verified
0.0793
Step by step solution
01
Identify the Given Information
The problem provides that 4 mean deviations are equal to \(\pi\) standard deviations. The distribution of packet round-trip times follows a normal distribution.
02
Calculate the Z-Score
The Z-score corresponds to how many standard deviations an observation is from the mean. Since 4 mean deviations equal to \(\pi\) standard deviations, the Z-score for the time-out value is \(\pi\).
03
Use the Z-Score to Find Probability
Using statistical tables (Z-tables), look up the cumulative probability that a packet will arrive within \(\pi\) standard deviations from the mean. The Z-score of \(\pi\) is approximately 1.41. The cumulative probability for Z = 1.41 is about 0.9207, which reflects the proportion of packets arriving within the time-out.
04
Calculate the Probability of Exceeding TimeOut
The probability of a packet taking more than the timeout time to arrive is the complement of the cumulative probability found in Step 3. It is given by \(\ 1 - 0.9207 = 0.0793 \).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Normal Distribution
A normal distribution, also known as a Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. This distribution is symmetrical, centered around its mean, and characterized by its bell-shaped curve.
Key properties include:
In the context of packet round-trip times, we assume that these times are normally distributed. This assumption allows us to use standard statistical methods to make inferences about the likelihood of different outcomes.
Key properties include:
- The mean, median, and mode of the distribution are equal.
- The curve is symmetrical around the mean.
- Approximately 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.
In the context of packet round-trip times, we assume that these times are normally distributed. This assumption allows us to use standard statistical methods to make inferences about the likelihood of different outcomes.
Z-Score
A Z-score, or standard score, quantifies the number of standard deviations a data point is from the mean of its distribution. Calculating the Z-score is essential for understanding how extreme or typical a particular observation is within a dataset.
The formula for a Z-score is:
\ Z = \frac{(X - \mu)}{\sigma} \
where:
In this exercise, the Z-score for the timeout value is given by \ \pi \, which corresponds approximately to 1.41 standard deviations. This Z-score helps us determine how unusual or usual a packet's round-trip time is compared to the mean.
The formula for a Z-score is:
\ Z = \frac{(X - \mu)}{\sigma} \
where:
- \(X\): the value of the observation
- \(\mu\): the mean of the distribution
- \(\sigma\): the standard deviation of the distribution
In this exercise, the Z-score for the timeout value is given by \ \pi \, which corresponds approximately to 1.41 standard deviations. This Z-score helps us determine how unusual or usual a packet's round-trip time is compared to the mean.
Cumulative Probability
Cumulative probability refers to the probability that a random variable will take on a value less than or equal to a specific value. It is the accumulation of probabilities up to that point and is obtained using the cumulative distribution function (CDF).
To find the cumulative probability for a Z-score, we use Z-tables. These tables provide the proportion of observations that lie below a given Z-score in a standard normal distribution.
In our scenario, we found that the Z-score of 1.41 corresponds to a cumulative probability of 0.9207. This means that 92.07% of packets will arrive within the timeout value set by the Jacobson/Karels algorithm.
To find the cumulative probability for a Z-score, we use Z-tables. These tables provide the proportion of observations that lie below a given Z-score in a standard normal distribution.
In our scenario, we found that the Z-score of 1.41 corresponds to a cumulative probability of 0.9207. This means that 92.07% of packets will arrive within the timeout value set by the Jacobson/Karels algorithm.
Packet Round-Trip Time
Packet round-trip time (RTT) measures the time it takes for a packet to travel from a source to a destination and back. This metric is crucial in network performance assessment and impacts the setting of timeouts in protocols.
Factors affecting RTT include:
In the context of the Jacobson/Karels algorithm, RTT is assumed to be normally distributed, enabling the calculation of probabilities based on standard deviations. Setting timeouts effectively minimizes packet loss and improves network reliability.
Factors affecting RTT include:
- Network congestion and traffic levels.
- Quality of the transmission medium.
- Router and switch performance.
In the context of the Jacobson/Karels algorithm, RTT is assumed to be normally distributed, enabling the calculation of probabilities based on standard deviations. Setting timeouts effectively minimizes packet loss and improves network reliability.
