task-queue-proof/HOW_TO_MATHS.md

How to Read the Mathematics in This Paper

This guide explains every piece of mathematical notation used in the paper, in the order you will encounter it. No prior math background beyond basic arithmetic is assumed.

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The Basics

Variables and Subscripts

A variable is a letter that stands for a number. In this paper:

• n = the total number of tasks
• p_i = the processing time (how many hours of work) for task i
• C_i = the completion time (when task i is finished) under a given schedule
• q_i = the priority class of task i (1 = Critical, 4 = Low)
• w(q) = the weight assigned to priority class q

The subscript i is just a label. p_1 means "the processing time of task 1," p_2 means "the processing time of task 2," and so on. When we write p_i, we mean "the processing time of task i, for any i."

Lists of Things

p_1, p_2, \ldots, p_n means "a list of processing times, from task 1 up to task n." The \ldots just means "and so on." If there are 5 tasks, this is p_1, p_2, p_3, p_4, p_5.

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Symbols You Will See

Symbol	Name	What it means	Example
=	Equals	Left side is the same as right side	2 + 3 = 5
\ne	Not equal	These two things are different	3 \ne 4
>	Greater than	Left is bigger	5 > 3
<	Less than	Left is smaller	3 < 5
\le	Less than or equal to	Left is smaller or the same	3 \le 5, 3 \le 3
\ge	Greater than or equal to	Left is bigger or the same	5 \ge 3, 5 \ge 5
\approx	Approximately equal	Close but not exact	10/3 \approx 3.33
\cdot	Multiplication	Same as \times	3 \cdot 4 = 12
\perp	Independent of	No relationship between them	p_i \perp q_i means task size and priority are unrelated
\forall	For all	"This is true for every..."	S_i = S_j \; \forall \, i, j means "slowdown is equal for all tasks"
\in	Is a member of	Belongs to a set	q_i \in \{1, 2, 3, 4\} means priority is one of 1, 2, 3, or 4
\blacksquare	End of proof	"The proof is complete"	Appears at the end of every proof

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The Big Sigma: Summation (\sum)

This is the most important symbol in the paper. It means "add up a list of things."

\sum_{i=1}^{n} p_i

Read this as: "Add up p_i for every i from 1 to n."

If there are 3 tasks with processing times p_1 = 2, p_2 = 5, p_3 = 3:

\sum_{i=1}^{3} p_i = p_1 + p_2 + p_3 = 2 + 5 + 3 = 10

The letter under the \sum (i = 1) tells you where to start counting. The number on top (n or 3) tells you where to stop. The expression after the \sum (p_i) tells you what to add up.

Variations You Will See

Summing a product:

\sum_{i=1}^{n} w(q_i) \cdot C_i

Add up w(q_i) \times C_i for each task. If task 1 has weight 8 and completion time 3, and task 2 has weight 1 and completion time 5:

= 8 \cdot 3 + 1 \cdot 5 = 24 + 5 = 29

Summing with a condition:

\sum_{\substack{a \ne b}} p_a \, p_b

Add up p_a \times p_b for every pair of tasks a and b where a and b are different tasks. The condition (a \ne b) filters which terms to include.

Summing up to a variable position:

\sum_{j=1}^{k} p_{\sigma(j)}

Add up processing times for the first k tasks in the schedule. This is how completion time is defined — it is the total work done up to and including position k.

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Fractions

\frac{a}{b}

This means a \div b. The top is the numerator, the bottom is the denominator. When you see:

\bar{C}(\sigma) = \frac{1}{n} \sum_{k=1}^{n} C_{\sigma(k)}

This means: add up all the completion times, then divide by n (the number of tasks). This is an average — the same operation as calculating a mean in everyday life.

Nested Fractions

\bar{C}_w(\sigma) = \frac{\sum_{k=1}^{n} p_{\sigma(k)} \cdot C_{\sigma(k)}}{\sum_{k=1}^{n} p_{\sigma(k)}}

The top (numerator) is: add up (processing time \times completion time) for each task. The bottom (denominator) is: add up all the processing times. This is a weighted average — tasks with more work count more heavily.

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Schedules and Permutations (\sigma)

\sigma (the Greek letter "sigma") represents a schedule — the order in which tasks are done.

If you have 3 tasks and decide to do them in order 2, 3, 1, then:

• \sigma(1) = 2 — the first task you do is task 2
• \sigma(2) = 3 — the second task you do is task 3
• \sigma(3) = 1 — the third task you do is task 1

When you see p_{\sigma(k)}, it means "the processing time of whichever task is in position k of the schedule."

Ordering Symbols

• b \preceq_\sigma a means "task b is scheduled at the same time as or before task $a$" — i.e., b comes first (or they are the same task)
• a \prec_\sigma b means "task a is scheduled strictly before task $b$"

These are just ways of saying "which task comes first in the schedule."

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The Bar: Averages (\bar{C})

A bar over a letter means "average." So:

• \bar{C} = average completion time
• \bar{C}_w = weighted average completion time (the subscript w distinguishes it)
• \bar{p} = average processing time

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The Delta: Change (\Delta)

\Delta (capital delta) means "the change in" or "the difference."

• \Delta_i = C_i - p_i — the delay for task i (how much longer it waited beyond its own processing time)
• \Delta D = w(q_j) \cdot p_i - w(q_i) \cdot p_j — how much the priority-weighted delay cost changes when you swap two tasks

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Subscripts and Superscripts

Subscripts (below) are labels or indices:

• p_i = processing time of task i
• p_{\max} = the largest processing time
• p_{\min} = the smallest processing time
• \bar{C}_{\text{SPT}} = average completion time under the SPT schedule
• \bar{C}_{\text{priority}} = average completion time under priority scheduling

Superscripts (above) indicate context or power:

• p_a^2 = p_a \times p_a (squared)
• \bar{C}^{\text{SPT}} = completion time under SPT (same idea as subscript, just different formatting)

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Set Notation

q_i \in \{1, 2, 3, 4\}

The curly braces \{ \} define a set — a collection of possible values. \in means "is a member of." So this reads: "the priority q_i is one of the values 1, 2, 3, or 4."

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The Piecewise Function (Cases)

w(q) = \begin{cases} 8 & q = 1 \text{ (Critical)} \\ 4 & q = 2 \text{ (High)} \\ 2 & q = 3 \text{ (Medium)} \\ 1 & q = 4 \text{ (Low)} \end{cases}

This defines a function that gives different outputs depending on the input. Read it as: "If q is 1, then w is 8. If q is 2, then w is 4." And so on. It is just a lookup table written in math notation.

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Limits and Convergence

\lim_{T \to \infty} \frac{W(T)}{T} = \mu

Read as: "As T gets larger and larger (approaching infinity), the value of W(T) / T gets closer and closer to \mu." In this paper, it means: over a long enough time horizon, the throughput settles to a fixed rate \mu regardless of scheduling order.

• \to means "approaches" or "goes toward"
• \infty means infinity (without bound)

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Mutual Information (I)

I(\sigma_{\text{SPT}}) = 0 \quad \text{when } p_i \perp q_i

I here measures how much knowing one thing tells you about another. I = 0 means "knowing a task's position in the SPT schedule tells you absolutely nothing about its priority." The schedule and the priority system are statistically independent — they contain no information about each other.

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Functions

f(\bar{C}) means "some function of \bar{C}." A function takes an input and produces an output. When the paper writes:

U_{\text{client}} = f\!\left(\bar{C}(\sigma)\right), \quad f' < 0

It means: client satisfaction (U) depends on the average completion time, and f' < 0 means the function is decreasing — when the average goes up, satisfaction goes down. The ' (prime) notation refers to the derivative, which indicates the direction of change.

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Correlation (\text{Corr})

\text{Corr}(p_i, q_i) \approx 0

Correlation measures whether two quantities move together. A correlation of 0 means they are unrelated — knowing the size of a task tells you nothing about its priority. A positive correlation means they increase together; a negative correlation means one goes up when the other goes down.

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Variance (\text{Var}) and Standard Deviation (\sigma_p)

Variance measures how spread out a set of numbers is. If all tasks take about the same time, variance is low. If some are 15 minutes and others are 40 hours, variance is high.

The coefficient of variation CV = \sigma_p / \bar{p} normalizes the spread by dividing by the average. Here \sigma_p is the standard deviation (the square root of variance) of the processing times, and \bar{p} is the mean. A CV above 1 means the spread is wider than the average — the data is highly variable.

(Note: \sigma_p uses the same Greek letter as schedule \sigma, but they are different things. Context tells you which is which — \sigma_p with a subscript p always means standard deviation of processing times; \sigma alone or \sigma(k) always means a schedule.)

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How to Read the Proofs

The Exchange Argument

The most common proof technique in this paper works like this:

1. Assume you have any schedule.
2. Find two adjacent tasks that are in the "wrong" order.
3. Swap them and show that the metric improves (or stays the same).
4. Conclude: since every "wrong" pair can be improved by swapping, the best schedule has no "wrong" pairs — i.e., everything is in the "right" order.

This is like proving that a sorted list is optimal by showing that any out-of-order pair can be fixed by swapping, and each swap makes things better.

Reading a Proof Step by Step

Take the simplest proof in the paper (Theorem 1):

"Consider any schedule in which two adjacent tasks i, j satisfy p_i > p_j with task i scheduled immediately before task j."

Translation: Pick any two neighboring tasks where the bigger one comes first.

"Let t be the start time of task i."

Translation: Call the clock time when we start working on task i by the name t. We don't need to know what t actually is.

"The change in the sum of completion times is: $(2p_i + p_j) - (p_i + 2p_j) = p_i - p_j > 0$"

Translation: When the bigger task comes first, the total completion time is p_i - p_j hours more than if the smaller task came first. Since p_i > p_j, this difference is positive — swapping them makes it better.

"Therefore SPT uniquely minimizes \bar{C}(\sigma)." \blacksquare

Translation: Since every swap of big-before-small improves the metric, the best possible schedule is all tasks sorted from smallest to largest. Proof complete.

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Quick Reference Card

When you see...	It means...
\sum_{i=1}^{n}	Add up for each task from 1 to n
\frac{a}{b}	a divided by b
\bar{X}	The average of X
\Delta X	The change in X
\sigma	A schedule (the order tasks are done)
\sigma(k)	Which task is in position k
p_i	How long task i takes
C_i	When task i finishes
q_i	Priority class of task i
w(q_i)	How much priority class q_i is weighted
\le, \ge	Less/greater than or equal to
\ne	Not equal to
\in	Is a member of
\forall	For all
\perp	Independent of
\blacksquare	End of proof

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A Note on Reading Math

Mathematical notation is a compression format. The formula:

\bar{C}(\sigma) = \frac{1}{n} \sum_{k=1}^{n} C_{\sigma(k)}

says in 15 characters what takes a full sentence in English: "The average completion time under schedule \sigma equals the sum of all individual completion times divided by the number of tasks."

If a formula looks intimidating, break it apart:

1. Find the main operation (usually = splitting left from right)
2. Identify the pieces on each side
3. Replace the symbols with words using the tables above
4. Read it as a sentence

Every formula in this paper can be understood this way. The math is not doing anything more complicated than arithmetic — it is just doing it for a general case rather than specific numbers.

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This guide accompanies the paper "Unweighted Average Completion Time Is Not a Fair Metric for Task Scheduling" (2026-03-28).