Pi is the constant 3.14 (etc…)<\/li>\n<\/ul>\nSimple example:<\/p>\n
If you’re building a website that will run a business application, and you’re expecting 20,000 sessions per day, randomly distributed throughout the day, between 8 AM and 5 PM, then we can calculate the peak as follows:<\/p>\n
T is the time in hours, or 5 PM – 8 AM + 1 hour \u00a0(10)<\/p>\n
p = ( 4 * 20,000 ) \/ ( 3.1416 * 10 )<\/p>\n
p = 80,000 \/ 31.416<\/p>\n
p = 2,547 hits during the peak hour<\/strong><\/p>\nOr, between 42 and 43 sessions\u00a0per minute.<\/p>\n
If you want more information on how or why this works, read on…<\/p>\n
<\/p>\n
<\/span>Distribution Curve<\/span><\/h2>\nA distribution curve is a way to generate, or guess, the points of a histogram based on certain known values – for example, knowing the total number of data points allows a distribution curve to plot how those data points are distributed among multiple slices or buckets, according to a formula.<\/p>\n
<\/p>\n
<\/span>Histogram<\/span><\/h3>\n“Slow down!\u00a0 What in the world is a histogram???”<\/em><\/p>\nFair enough!<\/p>\n
A histogram counts the number of data points that fall within each slice of a specific range of values.<\/strong><\/p>\nFor example, if you go to the grocery store, there might be 10 checkout lanes.\u00a0 If you plot the lane number across the bottom (X-axis) of a graph, and then count the number of people in each lane, plotting that number as the Y coordinate corresponding to that lane’s X coordinate, you have a histogram!<\/p>\n
In this example, you have 10 lanes (or buckets), and you’ve counted the number of people in each lane.<\/p>\n
When performing capacity planning, histograms are often used to count the number of hits for each hour of the day – e.g. the buckets are 0 (12:00 AM) to 23 (11:00 PM), and a hit gets counted in to one of the 24 buckets based on the time of day in which the “hit” (request) is submitted to the server.<\/p>\n
This type of graph helps answer the questions:<\/p>\n
\n- What time of day is my system being used most often?<\/li>\n
- When can I safely perform system maintenance, disrupting the fewest customers?<\/li>\n
- What is my peak resource demand (utilization)?<\/li>\n<\/ul>\n
Creating a usage histogram requires detailed log analysis, and there are plenty of commercial and open source analysis tools that can create one.<\/p>\n
The problem with this approach is that you need usage data to create the histogram!\u00a0 What if you’re building a new service, and there is no data available?<\/p>\n
A distribution curve can help you project what the histogram will look like<\/strong>.<\/p>\n <\/p>\n
<\/span>Distribution Patterns<\/span><\/h3>\nThere are various types of distribution patterns.<\/p>\n
<\/p>\n
<\/span>Bell Curve<\/span><\/h4>\nStatisticians discuss a bell-shaped distribution curve. \u00a0This curve appears when analyzing how data points relate to the average of all data points within the set.<\/p>\n
This sounds really complicated, but let’s walk through it…<\/p>\n
Let’s say that you ask 100 people their age. \u00a0Let’s say that the minimum age is 10, and the maximum age is 80. \u00a0Statistically, the average is going to be about 35. \u00a0If you plot a histogram based on the distance (or difference) between each data point (person’s age) and 35, most data points will cluster at the 0 mark (right at 35) and will drop off in an inverse curve to the left and right of 0 (the data point labeled “35”). \u00a0At the outer edges, there will be a few data points in the low age range to the left, and a few data point in the high age range to the right.<\/p>\n
![\"Bell](\"https:\/\/justinparrtech.com\/JustinParr-Tech\/wp-content\/uploads\/Bell-Curve_Annotated.png\")
<\/p>\n
This shape is known as the bell-shaped curve, where 90% of all data points fall within the center 50% of the entire range of data values.<\/p>\n
This is also the basis of the famous 80\/20 rule, where 80% of data points within the center 20% of the range of data values, while the remaining 20% consume the remaining 80% of the range.<\/p>\n
<\/p>\n
<\/span>Linear Distribution<\/span><\/h4>\nA linear distribution is literally a “line”.<\/p>\n
The formula for a line is:<\/p>\n
y=m * x + b<\/strong><\/p>\n\n- X is the independent variable<\/li>\n
- M is the slope of the line (y\/x)<\/li>\n
- B is the Y intercept – the value of Y, where x=0.<\/li>\n<\/ul>\n
Linear distributions could be flat, meaning every Y value is the same. \u00a0What this means in the real world is that every time bucket has the same number of hits, or that traffic is constant around the clock. \u00a0This never happens. \u00a0Every application has peak and off-peak utilization.<\/p>\n
Linear distributions could be a fixed slope, meaning that traffic progressively increases or decreases at a constant rate (or maybe a combination of the two) until the cycle resets at a specific clock time. \u00a0Again, this never happens.<\/p>\n
Linear distribution is easy to calculate, but not very useful.<\/p>\n
<\/p>\n
<\/span>Cosine Curve<\/span><\/h4>\nA cosine curve is based on the cosine formula for a right-triangle:<\/p>\n
COS = Opposite \/ Hypotenuse<\/p>\n
To find \u00a0the height for a given x coordinate, you must supply an angle.<\/p>\n
In degrees, the angle is specified in a range of 0 (north) through 180 (south) to 360 (or back to 0).<\/p>\n
In radians, the angle is specified in a range of 0 (north) through pi (south) to 2 * pi (or back to 0).<\/p>\n
By pinning the angle to the x coordinate, you can generate a curve.<\/p>\n
Using the time buckets example, if we map 360 degrees to 24 time buckets, that’s 15 degrees per hour.<\/p>\n
Given some peak magnitude, p, the height of the curve\u00a0at a specific time t is:<\/p>\n
y = COS (t * 15) * p<\/p>\n
This produces a familiar shape…<\/p>\n
![\"Cosine_Curve\"](\"https:\/\/justinparrtech.com\/JustinParr-Tech\/wp-content\/uploads\/Cosine_Curve-1024x512.png\")
<\/p>\n
In fact, the bell curve can be approximated using a cosine curve.<\/p>\n
Most real-world usage patterns can be approximated using a cosine curve.<\/p>\n
<\/p>\n
<\/span>Elliptical Distribution<\/span><\/h4>\nAn ellipse is a stretched-out circle.<\/p>\n
Using half of an ellipse, you can generate a very good approximation of real-world usage, using the much simpler formulas that describe a circle.<\/p>\n
<\/p>\n
<\/p>\n
<\/span>Generating an Elliptical Distribution<\/span><\/h2>\nGiven a total daily volume of server hits, V, we can quickly approximate the peak volume, p, as well as the volume y, at any given time, t.<\/p>\n
<\/p>\n
<\/span>Calculate Peak Volume<\/span><\/h3>\nThe area of a circle is described by the following formula:<\/p>\n
A = pi * r^2<\/p>\n