Reserved or On-Demand Servers?

A cloud provider charges either a high hourly rate with no commitment, or a monthly fee plus a cheaper hourly rate. Each bill is a line in the hours used. The break-even is where the lines cross, one linear equation, and which option wins is a linear inequality.

How it works

Linear equations and inequalities

In one sentence

A linear equation in one variable, $a + mx = b + nx$ , has the unique solution $x = (b - a)/(m - n)$ when $m \ne n$ . A linear inequality is solved the same way, except that multiplying or dividing both sides by a negative number reverses the inequality sign.

General formula

{\htmlData{sym=0}{C_1}} = {\htmlData{sym=2}{a}} + {\htmlData{sym=3}{m}}{\htmlData{sym=6}{x}}, \qquad {\htmlData{sym=1}{C_2}} = {\htmlData{sym=4}{b}} + {\htmlData{sym=5}{n}}{\htmlData{sym=6}{x}}, \qquad {\htmlData{sym=2}{a}} + {\htmlData{sym=3}{m}}{\htmlData{sym=6}{x}} = {\htmlData{sym=4}{b}} + {\htmlData{sym=5}{n}}{\htmlData{sym=6}{x}} \iff {\htmlData{sym=6}{x}} = \frac{{\htmlData{sym=4}{b}} - {\htmlData{sym=2}{a}}}{{\htmlData{sym=3}{m}} - {\htmlData{sym=5}{n}}}

The reserved plan starts behind (a bigger fee) but gains on on-demand with every hour; the break-even is where it catches up.

What the symbols mean

$C_1$
On-demand bill
The month's cost with no commitment.
$C_2$
Reserved bill
The month's cost with the reserved plan.
$a$
On-demand fee
The fixed monthly charge for on-demand (usually $0).
$m$
On-demand rate
Dollars per server-hour without commitment.
$b$
Reserved fee
The fixed monthly charge for reserving the server.
$n$
Reserved rate
Dollars per server-hour once reserved.
$x$
Hours used
Server-hours this month (a month has about 730).

Each bill is a fixed monthly fee plus a price per hour: $C_1 = a + mx$ and $C_2 = b + nx$ . Each is a line whose slope is the hourly price.
The bills are equal where the lines cross. Set $a + mx = b + nx$ , collect the $x$ terms on one side and divide: $x = \frac{b - a}{m - n}$ .
On-demand is cheaper when $a + mx < b + nx$ . Solving this inequality is the same algebra, except that dividing by a negative number flips the inequality, so watch which slope is bigger.
If the hourly prices are equal the lines are parallel and never cross, and the option with the lower fee always wins.

Learn the math

Linear EquationsAlgebra · Lesson 8, Solving Equations and Inequalities Applications of Linear EquationsAlgebra · Lesson 9, Solving Equations and Inequalities Linear InequalitiesAlgebra · Lesson 16, Solving Equations and Inequalities