Background PHP Process with shell_exec() As Part of WordPress Plugin

I'm trying to run a background PHP process with shell_exec() as a part of WordPress plugin.

This is the sample.

/* Plugin Name: Sample Background Process */

add_action('init', 'Sample_Background_Process');    
function Sample_Background_Process() {
    $file = __DIR__ . '/log.html';
    $current = date('l jS \of F Y h:i:s A') . ': accessed<br />';
    file_put_contents($file, $current, FILE_APPEND);
}

It creates a log in the plugin folder and whenever a page gets access it appends some text to it.

Then, I added this line, shell_exec('php "' . ABSPATH . '/index.php" > /dev/null 2>/dev/null &');. So a background process should access the page and run the function which appends text to the log file. But it doesn't seem to be doing it. I expect one page load produces two lines in the log file. But it only adds one line, which means a background process is not run or WordPress does not do anything if it is called like this.

add_action('init', 'Sample_Background_Process');    
function Sample_Background_Process() {
    $file = __DIR__ . '/log.html';
    $current = date('l jS \of F Y h:i:s A') . ': accessed<br />';
    file_put_contents($file, $current, FILE_APPEND);
    shell_exec('php "' . ABSPATH . '/index.php" > /dev/null 2>/dev/null &');
}

What am I doing wrong?

The path, php in shell_exec() is altered by the path to php.exe in my actual script since I'm testing on Windows server. It runs fine in other PHP scripts.

Thanks.