body {
font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
line-height: 1.6;
color: #333;
max-width: 1200px;
margin: 0 auto;
padding: 20px;
background-color: #f4f7f6;
}
.calculator-container {
background: #ffffff;
padding: 30px;
border-radius: 12px;
box-shadow: 0 4px 20px rgba(0,0,0,0.1);
margin-bottom: 40px;
}
.calculator-title {
text-align: center;
color: #2c3e50;
margin-bottom: 25px;
font-size: 28px;
font-weight: 700;
}
.input-grid {
display: grid;
grid-template-columns: 1fr 1fr;
gap: 20px;
}
@media (max-width: 768px) {
.input-grid {
grid-template-columns: 1fr;
}
}
.form-group {
margin-bottom: 20px;
}
.form-group label {
display: block;
margin-bottom: 8px;
font-weight: 600;
color: #555;
}
.form-group input {
width: 100%;
padding: 12px;
border: 1px solid #ddd;
border-radius: 6px;
font-size: 16px;
box-sizing: border-box;
transition: border-color 0.3s;
}
.form-group input:focus {
border-color: #3498db;
outline: none;
}
.form-group .help-text {
font-size: 12px;
color: #888;
margin-top: 5px;
}
.btn-container {
text-align: center;
margin-top: 20px;
}
.btn {
background-color: #3498db;
color: white;
border: none;
padding: 14px 30px;
font-size: 16px;
border-radius: 6px;
cursor: pointer;
transition: background-color 0.3s;
margin: 0 10px;
}
.btn:hover {
background-color: #2980b9;
}
.btn-reset {
background-color: #95a5a6;
}
.btn-reset:hover {
background-color: #7f8c8d;
}
#result-section {
margin-top: 30px;
padding: 25px;
background-color: #f8f9fa;
border-radius: 8px;
border-left: 5px solid #3498db;
display: none;
}
.result-item {
display: flex;
justify-content: space-between;
align-items: center;
margin-bottom: 15px;
padding-bottom: 15px;
border-bottom: 1px solid #e9ecef;
}
.result-item:last-child {
border-bottom: none;
margin-bottom: 0;
padding-bottom: 0;
}
.result-label {
font-weight: 600;
color: #555;
}
.result-value {
font-weight: 700;
color: #2c3e50;
font-size: 18px;
}
.article-content {
background: #fff;
padding: 40px;
border-radius: 12px;
box-shadow: 0 2px 10px rgba(0,0,0,0.05);
}
.article-content h2 {
color: #2c3e50;
margin-top: 30px;
}
.article-content h3 {
color: #34495e;
margin-top: 25px;
}
.article-content p {
margin-bottom: 15px;
}
.article-content ul {
margin-bottom: 20px;
padding-left: 20px;
}
.formula-box {
background-color: #e8f4fc;
padding: 15px;
border-radius: 5px;
font-family: monospace;
margin: 20px 0;
border-left: 4px solid #3498db;
}
.error-msg {
color: #e74c3c;
text-align: center;
margin-top: 10px;
display: none;
font-weight: 600;
}
How to Calculate Page Fault Rate
In operating systems using paging for memory management, the Page Fault Rate is a critical metric for understanding system performance. It represents the frequency at which a process tries to access a page of memory that is currently not mapped into physical RAM (Random Access Memory), requiring the operating system to fetch it from secondary storage (like a hard disk or SSD).
The Formula
To calculate the Page Fault Rate ($p$), you need two primary variables: the total number of memory accesses and the number of page faults that occurred during those accesses.
Page Fault Rate (p) = Number of Page Faults / Total Memory Accesses
The result, $p$, is a value between 0 and 1, where 0 means no page faults occurred, and 1 means every single memory access caused a fault.
Calculating Effective Access Time (EAT)
While the rate itself is useful, its true impact is seen in the Effective Access Time (EAT). This metric combines the fast access time of physical memory with the slow penalty of a page fault service.
EAT = (1 – p) × m + p × S
Where:
- p: Page Fault Rate (calculated above)
- m: Memory Access Time (typically in nanoseconds, e.g., 200ns)
- S: Page Fault Service Time (typically in milliseconds, e.g., 8ms). Note: 1ms = 1,000,000ns.
Why is Page Fault Rate Important?
Minimizing page faults is essential for high-performance computing. Because accessing secondary storage (disk) is orders of magnitude slower than accessing RAM, even a very small page fault rate can drastically degrade performance. This phenomenon is known as Thrashing.
Example Calculation
Imagine a system with the following parameters:
- Total Accesses: 1,000,000
- Page Faults: 50
- RAM Access Time: 200 nanoseconds
- Fault Service Time: 8 milliseconds (8,000,000 nanoseconds)
Step 1: Calculate Rate (p)
$p = 50 / 1,000,000 = 0.00005$
Step 2: Calculate EAT
$EAT = (1 – 0.00005) \times 200 + 0.00005 \times 8,000,000$
$EAT = 199.99 + 400$
$EAT = 599.99 \text{ ns}$
In this example, a tiny fault rate caused the effective access time to triple compared to the raw memory speed (from 200ns to ~600ns).
Factors Influencing Page Fault Rate
- Page Replacement Algorithms: Algorithms like LRU (Least Recently Used) or FIFO (First-In-First-Out) determine which pages to swap out. Efficient algorithms reduce faults.
- Frame Allocation: The number of physical memory frames allocated to a process. More frames generally mean fewer faults.
- Page Size: Larger pages reduce the size of the page table but can lead to internal fragmentation and potentially more unnecessary data loading.
- Locality of Reference: Programs written with good spatial and temporal locality experience fewer faults.
function calculatePageFault() {
// Get input values
var totalAccesses = document.getElementById("totalAccesses").value;
var numFaults = document.getElementById("numFaults").value;
var memAccessTime = document.getElementById("memAccessTime").value;
var faultServiceTime = document.getElementById("faultServiceTime").value;
var errorMsg = document.getElementById("errorMsg");
var resultSection = document.getElementById("result-section");
// Validate inputs
if (totalAccesses === "" || numFaults === "" || memAccessTime === "" || faultServiceTime === "") {
errorMsg.innerText = "Please fill in all fields.";
errorMsg.style.display = "block";
resultSection.style.display = "none";
return;
}
var accesses = parseFloat(totalAccesses);
var faults = parseFloat(numFaults);
var memTimeNs = parseFloat(memAccessTime);
var serviceTimeMs = parseFloat(faultServiceTime);
// Logic check: Faults cannot exceed accesses
if (faults > accesses) {
errorMsg.innerText = "Number of Page Faults cannot be greater than Total Memory Accesses.";
errorMsg.style.display = "block";
resultSection.style.display = "none";
return;
}
if (accesses <= 0 || memTimeNs < 0 || serviceTimeMs < 0 || faults < 0) {
errorMsg.innerText = "Please enter valid positive numbers.";
errorMsg.style.display = "block";
resultSection.style.display = "none";
return;
}
// Hide error message if validation passes
errorMsg.style.display = "none";
// 1. Calculate Page Fault Rate (p)
var rate = faults / accesses;
var percentage = rate * 100;
// 2. Calculate Effective Access Time (EAT)
// Formula: EAT = (1 – p) * memTime + p * serviceTime
// NOTE: Must convert Service Time from ms to ns for consistency
var serviceTimeNs = serviceTimeMs * 1000000;
var eat = ((1 – rate) * memTimeNs) + (rate * serviceTimeNs);
// 3. Calculate Performance Overhead (How much slower is it than raw memory?)
// Overhead % = ((EAT – memTime) / memTime) * 100
var overhead = ((eat – memTimeNs) / memTimeNs) * 100;
// Display Results
document.getElementById("resRate").innerHTML = rate.toExponential(4); // Scientific notation for small numbers
// Format percentage logic
var percentDisplay = percentage 0
? "< 0.0001%"
: percentage.toFixed(6) + "%";
document.getElementById("resPercent").innerHTML = percentDisplay;
document.getElementById("resEAT").innerHTML = eat.toFixed(2) + " ns";
document.getElementById("resOverhead").innerHTML = overhead.toFixed(2) + "%";
resultSection.style.display = "block";
}
function resetCalculator() {
document.getElementById("totalAccesses").value = "";
document.getElementById("numFaults").value = "";
document.getElementById("memAccessTime").value = "200";
document.getElementById("faultServiceTime").value = "8";
document.getElementById("result-section").style.display = "none";
document.getElementById("errorMsg").style.display = "none";
}