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/**
* SPDX-License-Identifier: GPL-2.0 only
*
* Copyright (c) 2025 Sandisk Corporation or its affiliates.
*/
#include <stdlib.h>
#include <stdint.h>
#include <math.h>
#include "lib/pow2.h"
#include "fio.h"
#include "file.h"
#include "sprandom.h"
/*
* Model for Estimating Steady-State Data Distribution in SSDs
*
* This model estimates the distribution of valid data across a flash drive
* in a steady state. It is based on the key insight from Desnoyers' research,
* which establishes a relationship between data validity and the physical
* space it occupies.
*
* P. Desnoyers, "Analytic Models of SSD Write Performance,"
* ACM Transactions on Storage,
* vol. 8, no. 2, pp. 1–18, Jun. 2012, doi: 10.1145/2133360.2133364.
*
* The Core Principle
* ==================
*
* The fundamental concept is that for a drive in a steady state, the product
* of a block's validity and the fraction of drive space occupied by such
* blocks is constant.
*
* Key Equation (1): i * f(i) = k
*
* Where:
* - i: The number of valid pages in a block.
* - f(i): The fraction of the drive composed of blocks with 'i' valid pages.
* - k: A constant for the drive.
*
* This implies that for any two validity levels i and j: i * f(i) = j * f(j).
* In other words, regions with lower validity (more invalid data) must
* occupy proportionally more physical space than regions with high validity.
*
*
* Modeling Steps
* ==============
* The model is built by following these steps:
*
* 1. Normalize Validity & Relate to Write Amplification (WA)
* We normalize 'i' into a validity fraction:
*
* valid_frac(i) = i / num_pages_per_region
*
* A greedy garbage collection (GC) algorithm reclaims the block with the
* lowest validity. The validity of this GC block (`valid_frac_gc`) is
* determined by the drive's WA:
*
* valid_frac_gc = 1 - (1 / WA)
*
* 2. Determine Write Amplification (WA) from Over-Provisioning (OP)
* The WA can be calculated from the drive's OP. A simple approximation
* is often sufficient for most cases:
*
* WA ≈ 0.5 / OP + 0.7
*
* Note: The precise formula from Desnoyers uses
* alpha = T/U
* where
* OP = alpha - 1
*
* in the equation:
* alpha
* WA = ----------------------------
* (alpha + W(-alpha*e^-alpha)
*
* with W being the Lambert W function).
*
* 3. Define the Distribution Curve
*
* Using the steady-state principle, we can find the relative size f(i) of a
* region given its validity (`valid_frac_i`) by comparing it to the GC block.
*
* valid_frac(i) * f(i) = valid_frac_gc * f_gc
*
* By defining the base size f_gc = 1, we get a simple relationship:
*
* f(i) = valid_frac_gc / valid_frac(i)
*
* This formula defines a curve where points are spaced equally by validity.
*
* 4. Resample for Equal-Sized Regions
*
* The final step is to make the model practical. We take the curve defined
* above and resample it to get points that are equally spaced by region
* size f(i). This resampling gives the expected validity for each
* equal-sized region of the drive, completing the model.
*/
#define PCT_PRECISION 10000
static inline double *d_alloc(size_t n)
{
return calloc(n, sizeof(double));
}
struct point {
double x;
double y;
};
static inline struct point *p_alloc(size_t n)
{
return calloc(n, sizeof(struct point));
}
static void print_d_array(const char *hdr, double *darray, size_t len)
{
struct buf_output out;
int i;
buf_output_init(&out);
log_buf(&out, "[");
for (i = 0; i < len - 1; i++)
log_buf(&out, "%.2f, ", darray[i]);
log_buf(&out, "%.2f]\n", darray[len - 1]);
if (hdr)
dprint(FD_SPRANDOM, "%s: ", hdr);
dprint(FD_SPRANDOM, "%s", out.buf);
buf_output_free(&out);
}
static void print_d_points(struct point *parray, size_t len)
{
struct buf_output out;
unsigned int i;
buf_output_init(&out);
log_buf(&out, "[");
for (i = 0; i < len - 1; i++)
log_buf(&out, "(%.2f %.2f), ", parray[i].x, parray[i].y);
log_buf(&out, "(%.2f %.2f)]\n", parray[len - 1].x, parray[len - 1].y);
dprint(FD_SPRANDOM, "%s", out.buf);
buf_output_free(&out);
}
/* Comparison function for qsort to sort points by x-value */
static int compare_points(const void *a, const void *b)
{
/* Cast void pointers to struct point pointers */
const struct point *point_a = (const struct point *)a;
const struct point *point_b = (const struct point *)b;
if (point_a->x < point_b->x)
return -1;
if (point_a->x > point_b->x)
return 1;
return 0;
}
/**
* reverse - Reverses the elements of a double array in place.
* @arr: pointer to the array of doubles to be reversed.
* @size: number of elements in the array.
*/
static void reverse(double arr[], size_t size)
{
size_t left = 0;
size_t right = size - 1;
if (size <= 1)
return;
while (left < right) {
double temp = arr[left];
arr[left] = arr[right];
arr[right] = temp;
left++;
right--;
}
}
/**
* linspace - Generates a linearly spaced array of doubles.
* @start: The starting value of the sequence.
* @end: The ending value of the sequence.
* @num: The number of elements to generate.
*
* Allocates and returns an array of @num doubles, linearly spaced
* between @start and @end (inclusive). If @num is 0, returns NULL.
* If @num is 1, the array contains only @start.
*
* Return: allocated array, or NULL on allocation failure or if @num is 0.
*/
static double *linspace(double start, double end, unsigned int num)
{
double *arr;
unsigned int i;
double step;
if (num == 0)
return NULL;
dprint(FD_SPRANDOM, "linespace start=%0.2f end=%0.2f num=%d\n",
start, end, num);
arr = d_alloc(num);
if (arr == NULL)
return NULL;
if (num == 1) {
arr[0] = start;
return arr;
}
/* Calculate step size */
step = (end - start) / ((double)num - 1.0);
for (i = 0; i < num; i++)
arr[i] = start + (double)i * step;
return arr;
}
/**
* linear_interp - Performs linear interpolation or extrapolation.
* @new_x: The x-value at which to interpolate.
* @x_arr: Array of x-values (must be sorted in strictly increasing order).
* @y_arr: Array of y-values corresponding to x_arr.
* @num: Number of points in x_arr and y_arr.
*
* Returns the interpolated y-value at new_x using linear interpolation
* between the points in x_arr and y_arr. If new_x is outside the range
* of x_arr, returns the nearest endpoint's y-value (extrapolation).
* Handles edge cases for zero or one point, and avoids division by zero
* if two x-values are nearly identical.
*/
static double linear_interp(double new_x, const double *x_arr,
const double *y_arr, unsigned int num)
{
unsigned int i;
double x1, y1, x2, y2;
if (num == 0)
return 0.0;
if (num == 1)
return y_arr[0]; /* If only one point, return its y-value */
/* Handle extrapolation outside the range */
if (new_x <= x_arr[0])
return y_arr[0];
if (new_x >= x_arr[num - 1])
return y_arr[num - 1];
/* Find the interval [x_arr[i], x_arr[i + 1]] that contains new_x */
for (i = 0; i < num - 1; i++) {
if (new_x >= x_arr[i] && new_x <= x_arr[i + 1]) {
x1 = x_arr[i];
y1 = y_arr[i];
x2 = x_arr[i + 1];
y2 = y_arr[i + 1];
/* Avoid division by zero if x values are identical
* Using a small epsilon for float comparison
* Return y1 if x1 and x2 are almost identical
*/
if (fabs(x2 - x1) < 1e-9)
return y1;
return y1 + (y2 - y1) * ((new_x - x1) / (x2 - x1));
}
}
/* Should not reach here if new_x is within bounds
* and x_arr is strictly increasing
*/
return 0.0;
}
/**
* sample_curve_equally_on_x - Resamples a curve at equally spaced x-values.
* @points: array of input points (must have strictly increasing x-values).
* @num: Number of input points.
* @num_resampled: number of points to resample to.
* @resampled_points: An output array of resampled points.
*
* Sorts the input points by x-value, checks for strictly increasing x-values,
* and generates a new set of points with x-values equally spaced between the
* minimum and maximum x of the input. Uses linear interpolation to compute
* corresponding y-values.
* Note: The function allocates memory for the output array.
*
* Return: 0 on success, negative error code on failure.
*/
static int sample_curve_equally_on_x(struct point *points, unsigned int num,
unsigned int num_resampled,
struct point **resampled_points)
{
double *x_orig = (double *)0;
double *y_orig = (double *)0;
double *new_x_arr = (double *)0;
struct point *new_points_arr = (struct point *)0;
unsigned int i;
int ret = 0;
if (points == NULL || resampled_points == NULL)
return -EINVAL;
if (num == 0) {
log_err("fio: original points array cannot be empty.\n");
return -EINVAL;
}
if (num_resampled == 0) {
*resampled_points = NULL;
return 0;
}
qsort(points, num, sizeof(struct point), compare_points);
/* Check if x-values are strictly increasing and sort them */
for (i = 0; i < num - 1; i++) {
if (points[i+1].x <= points[i].x) {
log_err("fio: x-values must be strictly increasing.\n");
ret = -EINVAL;
goto cleanup;
}
}
/* 2. Extract x and y into separate arrays for interpolation */
x_orig = d_alloc(num);
y_orig = d_alloc(num);
if (x_orig == NULL || y_orig == NULL) {
log_err("fio: Memory allocation failed for x_orig or y_orig.\n");
ret = -ENOMEM;
goto cleanup;
}
for (i = 0; i < num; i++) {
x_orig[i] = points[i].x;
y_orig[i] = points[i].y;
}
/* 4. Generate new_x values using linspace */
new_x_arr = linspace(x_orig[0], x_orig[num - 1], num_resampled);
if (new_x_arr == NULL) {
ret = -ENOMEM;
goto cleanup;
}
/* 5. Allocate memory for new resampled points */
new_points_arr = p_alloc(num_resampled);
if (new_points_arr == NULL) {
log_err("fio: Memory allocation failed for new_points_arr.\n");
ret = -ENOMEM;
goto cleanup;
}
/* 6. Perform linear interpolation for each new_x to get new_y */
for (i = 0; i < num_resampled; i++) {
new_points_arr[i].x = new_x_arr[i];
new_points_arr[i].y = linear_interp(new_x_arr[i], x_orig, y_orig, num);
}
*resampled_points = new_points_arr;
cleanup:
free(x_orig);
free(y_orig);
free(new_x_arr);
return ret;
}
/**
* compute_waf - Compute the write amplification factor (WAF)
* @over_provisioning: The over-provisioning ratio (0 < over_provisioning < 1)
*
* write amplification approximation equation
*
* 0.5
* WAF = ------------------ + 0.7
* over_provisioning
*
* Return: The computed write amplification factor as a double.
*/
static inline double compute_waf(double over_provisioning)
{
return 0.5 / over_provisioning + 0.7;
}
/**
* compute_gc_validity - validity of the block selected for GC (garbage collector)
*
* @waf: The Write Amplification Factor, must be greater than 1.0.
*
* Return: The computed gavalidity;
*/
static inline double compute_gc_validity(double waf)
{
assert(waf > 1.0); /* Ensure WAF is greater than 1.0 */
return 1.0 - (double)1.0 / waf;
}
/**
* compute_validity_dist - Computes a resampled validity distribution for regions.
* @n_regions: Number of regions to divide the distribution into.
* @over_provisioning: Over-provisioning factor used to calculate WAF and validity.
*
* Calculates the validity distribution across a specified number of regions,
* based on the write amplification factor (WAF) and over-provisioning.
* Steps:
* - Allocates and fills arrays for:
* - validity distribution
* - block ratios
* - accumulated ratios
* - Constructs a set of points representing the curve.
* - Resamples the curve to ensure equal spacing along the x-axis.
* - Reverses the resulting validity distribution before returning.
*
* Note: The function allocates memory for the validity distribution array.
*
* Return: resampled and reversed validity distribution array or NULL on error.
*/
static double *compute_validity_dist(unsigned int n_regions, double over_provisioning)
{
double waf = compute_waf(over_provisioning);
double validity = compute_gc_validity(waf);
double *validity_distribution = NULL;
double *blocks_ratio = NULL;
double *acc_ratio = NULL;
double acc;
unsigned int i;
struct point *points = NULL;
struct point *points_resampled = NULL;
int ret;
if (n_regions == 0) {
log_err("fio: requires at least one region");
goto out;
}
/*
* Use linspace to get equally distributed validity values,
* along the y-axis of the curve we want to generate.
*/
validity_distribution = linspace(1.0, validity, n_regions);
blocks_ratio = d_alloc(n_regions);
if (blocks_ratio == NULL) {
log_err("fio: memory allocation failed for linspace.\n");
goto out;
}
for (i = 0; i < n_regions; i++)
blocks_ratio[i] = 1.0 / validity_distribution[i];
acc_ratio = d_alloc(n_regions);
if (acc_ratio == NULL) {
log_err("fio: memory allocation failed for linspace_c.\n");
goto out;
}
acc = 0.0;
for (i = 0; i < n_regions; i++) {
acc_ratio[i] = acc + blocks_ratio[i];
acc = acc_ratio[i];
}
print_d_array("validity_distribution", validity_distribution, n_regions);
print_d_array("blocks ratio", blocks_ratio, n_regions);
print_d_array("accumulated ratio:", acc_ratio, n_regions);
points = p_alloc(n_regions);
for (i = 0; i < n_regions; i++) {
points[i].x = acc_ratio[i];
points[i].y = validity_distribution[i];
}
print_d_points(points, n_regions);
/*
* Use linspace again to get uniformly distributed x-values,
* and then interpolate the curve to find the validity at those
* uniformly distributed x-values.
*/
ret = sample_curve_equally_on_x(points, n_regions, n_regions,
&points_resampled);
if (ret == 0) {
print_d_points(points_resampled, n_regions);
} else {
log_err("fio: failed to resample curve. Error code: %d\n", ret);
free(validity_distribution);
validity_distribution = NULL;
goto out;
}
for (i = 0; i < n_regions; i++)
validity_distribution[i] = points_resampled[i].y;
print_d_array("validity resampled", validity_distribution, n_regions);
out:
free(points);
free(points_resampled);
free(blocks_ratio);
free(acc_ratio);
reverse(validity_distribution, n_regions);
return validity_distribution;
}
/**
* Calculate the physical size based on logical size and over-provisioning
*
* @over_provisioning: over provisioning factor (e.g. 0.2 for 20%)
* @logical_sz: Logical size in bytes
* @align_bs: Block size for alignment in bytes
*
* return: Physical size in bytes, including over-provisioning and aligned to align_bs
*/
static uint64_t sprandom_physical_size(double over_provisioning, uint64_t logical_sz,
uint64_t align_bs)
{
uint64_t size;
size = logical_sz + ceil((double)logical_sz * over_provisioning);
return (size + (align_bs - 1)) & ~(align_bs - 1);
}
/**
* estimate_inv_capacity - Estimates the invalid capacity of a region.
* @region_cnt: number of offsets in the region.
* @validity: invalidation ration in the regions (between 0 and 1).
*
* Calculates the expected number of invalidion in regions, adding a margin
* of 6 standard deviations to account for statistical variation.
*
* Returns: Estimated invalid capacity
*/
static uint64_t estimate_inv_capacity(uint64_t region_cnt, double validity)
{
double sigma = sqrt((double)region_cnt * validity * (1.0 - validity));
return (uint64_t)ceil(region_cnt * (1.0 - validity) + 6.0 * sigma);
}
/**
* sprandom_setup - Initialize and configure sprandom_info structure.
* @spr_info: Pointer to sprandom_info structure to be initialized.
* @logical_size: Logical size of the storage region.
* @align_bs: Alignment block size.
*
* Calculates physical size and region parameters based on logical size,
* alignment, and over-provisioning. Allocates and initializes validity
* distribution and invalid percentage arrays for regions. Precomputes
* invalid buffer capacity and allocates buffer. Sets up region size,
* write counts, and resets region/phase counters.
*
* Returns 0 on success, enagative value on failure.
*/
static int sprandom_setup(struct sprandom_info *spr_info, uint64_t logical_size,
uint64_t align_bs)
{
double over_provisioning = spr_info->over_provisioning;
uint64_t physical_size;
uint64_t region_sz;
uint64_t region_write_count;
double *validity_dist;
size_t invalid_capacity;
size_t total_alloc = 0;
char bytes2str_buf[40];
int i;
physical_size = sprandom_physical_size(over_provisioning,
logical_size, align_bs);
validity_dist = compute_validity_dist(spr_info->num_regions,
spr_info->over_provisioning);
if (!validity_dist)
return -ENOMEM;
/* Initialize validity_distribution */
print_d_array("validity resampled:", validity_dist, spr_info->num_regions);
/* Precompute invalidity percentage array */
spr_info->invalid_pct = calloc(spr_info->num_regions,
sizeof(spr_info->invalid_pct[0]));
if (!spr_info->invalid_pct)
goto err;
total_alloc += spr_info->num_regions * sizeof(spr_info->invalid_pct[0]);
for (i = 0; i < spr_info->num_regions; i++) {
double inv = (1.0 - validity_dist[i]) * (double)PCT_PRECISION;
spr_info->invalid_pct[i] = (int)round(inv);
}
region_sz = physical_size / spr_info->num_regions;
region_write_count = region_sz / align_bs;
invalid_capacity = estimate_inv_capacity(region_write_count,
validity_dist[0]);
spr_info->invalid_capacity = invalid_capacity;
spr_info->invalid_buf = pcb_alloc(invalid_capacity);
total_alloc += invalid_capacity * sizeof(uint64_t);
spr_info->region_sz = region_sz;
spr_info->invalid_count[0] = 0;
spr_info->invalid_count[1] = 0;
spr_info->curr_phase = 0;
spr_info->current_region = 0;
spr_info->region_write_count = region_write_count;
spr_info->writes_remaining = region_write_count;
/* Display overall allocation */
dprint(FD_SPRANDOM, "Summary:\n");
dprint(FD_SPRANDOM, " logical_size: %"PRIu64": %s\n",
logical_size,
bytes2str_simple(bytes2str_buf, sizeof(bytes2str_buf), logical_size));
dprint(FD_SPRANDOM, " physical_size: %"PRIu64": %s\n",
physical_size,
bytes2str_simple(bytes2str_buf, sizeof(bytes2str_buf), physical_size));
dprint(FD_SPRANDOM, " op: %02f\n", spr_info->over_provisioning);
dprint(FD_SPRANDOM, " region_size: %"PRIu64"\n", region_sz);
dprint(FD_SPRANDOM, " num_regions: %u\n", spr_info->num_regions);
dprint(FD_SPRANDOM, " region_write_count: %"PRIu64"\n", region_write_count);
dprint(FD_SPRANDOM, " invalid_capacity: %zu\n", invalid_capacity);
dprint(FD_SPRANDOM, " dynamic memory: %zu: %s\n",
total_alloc,
bytes2str_simple(bytes2str_buf, sizeof(bytes2str_buf), total_alloc));
free(validity_dist);
return 0;
err:
free(validity_dist);
free(spr_info->invalid_pct);
return -ENOMEM;
}
/**
* sprandom_add_with_probability - Adds an offset to the invalid buffer with
* a probability.
*
* @info: sprandom_info structure containing random state and buffers.
* @offset: The offset value to potentially add to the invalid buffer.
* @phase: The current phase index for invalid count tracking.
*
* Generates a random value and, based on the current region's invalid percentage,
* decides whether to add the offset to the invalid buffer.
* If the buffer is full, ogs an error and asserts failure.
*/
static void sprandom_add_with_probability(struct sprandom_info *info,
uint64_t offset, unsigned int phase)
{
int v = rand_between(info->rand_state, 0, PCT_PRECISION);
if (v <= info->invalid_pct[info->current_region]) {
if (pcb_space_available(info->invalid_buf)) {
pcb_push_staged(info->invalid_buf, offset);
info->invalid_count[phase]++;
} else {
dprint(FD_SPRANDOM, "pcb buffer would be overriten\n");
assert(false);
}
}
}
static void dprint_invalidation(const struct sprandom_info *info)
{
uint32_t phase = info->curr_phase;
double inv = 0;
double inv_act; /* actually invalidation percentage */
inv_act = (double)info->invalid_count[phase] / (double)info->region_write_count;
if (info->current_region > 0)
inv = (double)info->invalid_pct[info->current_region - 1] / PCT_PRECISION;
dprint(FD_SPRANDOM, "Invalidation[%d] %"PRIu64" %zu %.04f %.04f\n",
info->current_region,
info->region_write_count,
info->invalid_count[phase],
inv, inv_act);
}
/**
* sprandom_get_next_offset - Generate the next write offset for a region,
* managing invalidation, and region transitions.
*
* @info: sprandom_info structure containing state and configuration.
* @f: fio file associated with the ssd device.
* @b: block offset to store the next write offset.
*
* Generates offsets to write a region and saves a fraction of the offsets
* in a two phase circular buffer.
* When transitioning to the next region (phase is flipped),it first writes
* all saved offsets to achieve the desired fraction of invalid blocks in the
* previous region. The remainder of the current region is then filled with
* new offsets.
*
* Returns:
* 0 if a valid offset is found and stored in @b,
* 1 if no more offsets are available (end of regions or LFSR exhausted).
*/
int sprandom_get_next_offset(struct sprandom_info *info, struct fio_file *f, uint64_t *b)
{
uint64_t offset = 0;
uint32_t phase = info->curr_phase;
/* replay invalidation */
if (pcb_pop(info->invalid_buf, &offset)) {
sprandom_add_with_probability(info, offset, phase ^ 1);
dprint(FD_SPRANDOM, "Write %"PRIu64" over %d\n",
offset, info->current_region);
goto out;
}
/* Move to next region */
if (info->writes_remaining == 0) {
if (info->current_region >= info->num_regions) {
dprint(FD_SPRANDOM, "End: Last Region %d cur%d\n",
info->current_region, info->num_regions);
return 1;
}
dprint_invalidation(info);
info->invalid_count[phase] = 0;
info->current_region++;
phase ^= 1;
info->writes_remaining = info->region_write_count -
info->invalid_count[phase];
info->curr_phase = phase;
pcb_commit(info->invalid_buf);
}
/* Fetch new offset */
if (lfsr_next(&f->lfsr, &offset)) {
dprint(FD_SPRANDOM, "End: LFSR exhausted %d [%zu] [%zu]\n",
info->current_region,
info->invalid_count[phase],
info->invalid_count[phase ^ 1]);
dprint_invalidation(info);
return 1;
}
if (info->writes_remaining > 0)
info->writes_remaining--;
sprandom_add_with_probability(info, offset, phase ^ 1);
dprint(FD_SPRANDOM, "Write %"PRIu64" lfsr %d\n", offset, info->current_region);
out:
*b = offset;
return 0;
}
/**
* sprandom_init - initialize sprandom info
* @td: fio thread data
* @f: fio file associated with the ssd device.
*
* Sets up the sprandom_info structure for the given file according:
* region count, over-provisioning, and file/device size.
*
* Return: 0 on success, negative error code on failure.
*/
int sprandom_init(struct thread_data *td, struct fio_file *f)
{
struct sprandom_info *info = NULL;
double over_provisioning;
uint64_t logical_size;
uint64_t align_bs = td->o.bs[DDIR_WRITE];
int ret;
if (!td->o.sprandom)
return 0;
if (!is_power_of_2(align_bs)) {
log_err("fio: sprandom: bs [%"PRIu64"] should be power of 2",
align_bs);
return -EINVAL;
}
info = calloc(1, sizeof(*info));
if (!info)
return -ENOMEM;
logical_size = min(f->real_file_size, f->io_size);
over_provisioning = td->o.spr_over_provisioning.u.f;
info->num_regions = td->o.spr_num_regions;
info->over_provisioning = over_provisioning;
td->o.io_size = sprandom_physical_size(over_provisioning,
logical_size, align_bs);
info->rand_state = &td->sprandom_state;
ret = sprandom_setup(info, logical_size, align_bs);
if (ret)
goto err;
f->spr_info = info;
return 0;
err:
free(info);
return ret;
}
/**
* sprandom_free - Frees resources associated with a sprandom_info structure.
* @info: Pointer to the sprandom_info structure to be freed.
*
* Releases memory allocated for validity_dist, invalid_buf, and the spr_info
* structure itself. Does nothing if @spr_info is NULL.
*/
void sprandom_free(struct sprandom_info *info)
{
if (!info)
return;
free(info->invalid_pct);
free(info->invalid_buf);
free(info);
}
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