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* Auto-pause threshold calculation utilities
*
* Calculates when to auto-pause based on the student's response times.
* Uses mean + 2*stdDev with clamping between 30s and 5 minutes.
*/
import type { SlotResult } from '@/db/schema/session-plans'
import type { ProgressiveAssistanceTimingConfig } from '@/constants/helpTiming'
// ============================================================================
// Constants
// ============================================================================
/** Default timeout when not enough samples for statistics (5 minutes) */
export const DEFAULT_PAUSE_TIMEOUT_MS = 5 * 60 * 1000
/** Minimum problems needed for statistical calculation */
export const MIN_SAMPLES_FOR_STATISTICS = 5
/** Minimum clamp for the auto-pause threshold (30 seconds) */
export const MIN_PAUSE_THRESHOLD_MS = 30_000
/** Maximum clamp for the auto-pause threshold (5 minutes) */
export const MAX_PAUSE_THRESHOLD_MS = DEFAULT_PAUSE_TIMEOUT_MS
// ============================================================================
// Types
// ============================================================================
/**
* Auto-pause statistics for display and debugging
*/
export interface AutoPauseStats {
/** Mean response time in milliseconds */
meanMs: number
/** Standard deviation of response times in milliseconds */
stdDevMs: number
/** Calculated threshold (mean + 2*stdDev) in milliseconds */
thresholdMs: number
/** Number of samples used to calculate stats */
sampleCount: number
/** Whether statistical calculation was used (vs default timeout) */
usedStatistics: boolean
/** Mean response time per complexity-cost unit (ms/cost), if complexity scaling was used */
meanPerUnitMs?: number
/** Complexity cost of the current problem, if available */
currentProblemCost?: number
/** True when complexity cost data is missing (data quality issue) */
complexityCostMissing?: boolean
}
/**
* Information about why a session was paused
*/
export interface PauseInfo {
/** When the pause occurred */
pausedAt: Date
/** Why the session was paused */
reason: 'manual' | 'auto-timeout' | 'teacher'
/** Auto-pause statistics (only present for auto-timeout) */
autoPauseStats?: AutoPauseStats
/** Teacher's custom message (only present for teacher-initiated pause) */
teacherMessage?: string
}
/**
* Response time statistics
*/
export interface ResponseTimeStats {
/** Mean response time in milliseconds */
mean: number
/** Standard deviation of response times in milliseconds */
stdDev: number
/** Number of samples */
count: number
}
// ============================================================================
// Functions
// ============================================================================
/**
* Calculate mean and standard deviation of response times
*/
export function calculateResponseTimeStats(results: SlotResult[]): ResponseTimeStats {
if (results.length === 0) {
return { mean: 0, stdDev: 0, count: 0 }
}
const times = results.map((r) => r.responseTimeMs)
const count = times.length
const mean = times.reduce((sum, t) => sum + t, 0) / count
if (count < 2) {
return { mean, stdDev: 0, count }
}
const squaredDiffs = times.map((t) => (t - mean) ** 2)
const variance = squaredDiffs.reduce((sum, d) => sum + d, 0) / (count - 1) // Sample std dev
const stdDev = Math.sqrt(variance)
return { mean, stdDev, count }
}
/**
* Calculate the auto-pause threshold and full stats for display.
*
* @returns threshold in ms and stats for debugging/display
*/
export function calculateAutoPauseInfo(results: SlotResult[]): {
threshold: number
stats: AutoPauseStats
} {
const { mean, stdDev, count } = calculateResponseTimeStats(results)
const usedStatistics = count >= MIN_SAMPLES_FOR_STATISTICS
let threshold: number
if (usedStatistics) {
// Use mean + 2 standard deviations
threshold = mean + 2 * stdDev
// Clamp between 30 seconds and 5 minutes
threshold = Math.max(MIN_PAUSE_THRESHOLD_MS, Math.min(threshold, MAX_PAUSE_THRESHOLD_MS))
} else {
threshold = DEFAULT_PAUSE_TIMEOUT_MS
}
return {
threshold,
stats: {
meanMs: mean,
stdDevMs: stdDev,
thresholdMs: threshold,
sampleCount: count,
usedStatistics,
},
}
}
/**
* Get a human-readable explanation of how the auto-pause threshold was calculated.
* Used in the SessionOverview component to explain the timing.
*/
export function getAutoPauseExplanation(stats: AutoPauseStats): string {
if (!stats.usedStatistics) {
return `Default timeout (${formatMs(stats.thresholdMs)}) - need ${MIN_SAMPLES_FOR_STATISTICS}+ problems for statistical calculation`
}
const rawThreshold = stats.meanMs + 2 * stats.stdDevMs
const wasClamped = rawThreshold < MIN_PAUSE_THRESHOLD_MS || rawThreshold > MAX_PAUSE_THRESHOLD_MS
let explanation = `mean (${formatMs(stats.meanMs)}) + 2×stdDev (${formatMs(stats.stdDevMs)}) = ${formatMs(rawThreshold)}`
if (wasClamped) {
explanation += ` → clamped to ${formatMs(stats.thresholdMs)}`
}
return explanation
}
// ============================================================================
// Progressive Assistance Thresholds
// ============================================================================
/**
* Thresholds for progressive assistance escalation.
* All values in milliseconds.
*/
export interface ProgressiveThresholds {
/** When to show encouragement text (based on mean response time) */
encouragementMs: number
/** When to offer help button (based on mean + 1σ) */
helpOfferMs: number
/** When to auto-pause (based on mean + 2σ) */
autoPauseMs: number
}
/**
* Calculate progressive assistance thresholds from historical response times.
*
* Uses the same statistical approach as auto-pause but with three escalation levels:
* - Encouragement: mean response time (clamped)
* - Help offer: mean + 1σ (clamped)
* - Auto-pause: mean + 2σ (clamped) — same formula as existing auto-pause
*/
export function calculateProgressiveThresholds(
results: SlotResult[],
timing: ProgressiveAssistanceTimingConfig
): ProgressiveThresholds {
const { mean, stdDev, count } = calculateResponseTimeStats(results)
const hasSufficientData = count >= MIN_SAMPLES_FOR_STATISTICS
if (!hasSufficientData) {
return {
encouragementMs: timing.defaultEncouragementMs,
helpOfferMs: timing.defaultHelpOfferMs,
autoPauseMs: DEFAULT_PAUSE_TIMEOUT_MS,
}
}
return {
encouragementMs: clamp(mean, timing.minEncouragementMs, timing.maxEncouragementMs),
helpOfferMs: clamp(mean + stdDev, timing.minHelpOfferMs, timing.maxHelpOfferMs),
autoPauseMs: clamp(mean + 2 * stdDev, timing.minAutoPauseMs, timing.maxAutoPauseMs),
}
}
function clamp(value: number, min: number, max: number): number {
return Math.max(min, Math.min(value, max))
}
// ============================================================================
// Complexity-Scaled Thresholds
// ============================================================================
/**
* Normalized response time statistics (per complexity-cost unit)
*/
export interface NormalizedResponseTimeStats {
/** Mean response time per unit of complexity cost (ms/cost) */
meanPerUnit: number
/** Standard deviation per unit of complexity cost (ms/cost) */
stdDevPerUnit: number
/** Number of results with valid complexity costs used */
count: number
/** Number of results skipped due to missing or zero complexity cost */
skippedCount: number
}
/**
* Extract the totalComplexityCost from a problem's generation trace.
* Returns the cost if present and > 0, otherwise null.
*
* Note: basic skills (directAddition, directSubtraction, etc.) have base
* complexity 0, so their totalComplexityCost is legitimately 0/undefined.
* This is NOT a data quality issue — use `hasGenerationTrace` to distinguish.
*/
export function getComplexityCost(
problem: { generationTrace?: { totalComplexityCost?: number } } | null | undefined
): number | null {
const cost = problem?.generationTrace?.totalComplexityCost
if (cost != null && cost > 0) return cost
return null
}
/**
* Calculate normalized response time statistics by dividing each response time
* by its problem's complexity cost. Results without valid complexity cost are skipped.
*/
export function calculateNormalizedResponseTimeStats(
results: SlotResult[]
): NormalizedResponseTimeStats {
const normalized: number[] = []
let skippedCount = 0
for (const r of results) {
const cost = getComplexityCost(r.problem)
if (cost === null) {
skippedCount++
continue
}
normalized.push(r.responseTimeMs / cost)
}
if (normalized.length === 0) {
return { meanPerUnit: 0, stdDevPerUnit: 0, count: 0, skippedCount }
}
const count = normalized.length
const meanPerUnit = normalized.reduce((sum, v) => sum + v, 0) / count
if (count < 2) {
return { meanPerUnit, stdDevPerUnit: 0, count, skippedCount }
}
const squaredDiffs = normalized.map((v) => (v - meanPerUnit) ** 2)
const variance = squaredDiffs.reduce((sum, d) => sum + d, 0) / (count - 1)
const stdDevPerUnit = Math.sqrt(variance)
return { meanPerUnit, stdDevPerUnit, count, skippedCount }
}
/**
* Calculate progressive assistance thresholds scaled by the current problem's
* complexity cost. Harder problems get proportionally more time.
*
* When currentProblemCost is null (missing trace/data), returns flat 5-minute defaults.
* When fewer than MIN_SAMPLES_FOR_STATISTICS results have valid costs, returns timing defaults.
*/
export function calculateComplexityScaledThresholds(
results: SlotResult[],
timing: ProgressiveAssistanceTimingConfig,
currentProblemCost: number | null
): ProgressiveThresholds {
// Missing cost → flat defaults (data quality issue — no generation trace)
if (currentProblemCost === null) {
return {
encouragementMs: DEFAULT_PAUSE_TIMEOUT_MS,
helpOfferMs: DEFAULT_PAUSE_TIMEOUT_MS,
autoPauseMs: DEFAULT_PAUSE_TIMEOUT_MS,
}
}
const { meanPerUnit, stdDevPerUnit, count } = calculateNormalizedResponseTimeStats(results)
// Not enough data with valid costs → timing defaults (same as flat path)
if (count < MIN_SAMPLES_FOR_STATISTICS) {
return {
encouragementMs: timing.defaultEncouragementMs,
helpOfferMs: timing.defaultHelpOfferMs,
autoPauseMs: DEFAULT_PAUSE_TIMEOUT_MS,
}
}
// Scale normalized stats back up by current problem's cost
return {
encouragementMs: clamp(
meanPerUnit * currentProblemCost,
timing.minEncouragementMs,
timing.maxEncouragementMs
),
helpOfferMs: clamp(
(meanPerUnit + stdDevPerUnit) * currentProblemCost,
timing.minHelpOfferMs,
timing.maxHelpOfferMs
),
autoPauseMs: clamp(
(meanPerUnit + 2 * stdDevPerUnit) * currentProblemCost,
timing.minAutoPauseMs,
timing.maxAutoPauseMs
),
}
}
/**
* Format milliseconds as a human-readable time string
*/
export function formatMs(ms: number): string {
if (ms >= 60_000) {
const minutes = ms / 60_000
return `${minutes.toFixed(1)}m`
}
const seconds = ms / 1000
return `${seconds.toFixed(1)}s`
}
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