You can realize it, using lock file:

if(is_file(__DIR__.'/work.lock'))
{
    die('Script already run.');
}
else
{
    file_put_contents(__DIR__.'/work.lock', '');
    // YOUR CODE
    unlink(__DIR__.'/work.lock');
}

Too bad I didn't see this before it was accepted..

I have written a class to do just this. ( using file locking ) and PID, process ID checking, on both windows and Linux.

https://github.com/ArtisticPhoenix/MISC/blob/master/ProcLock.php

I think your are really overdoing it with all the processes and background checks. If you run a PHP script without a session, then you are already essentially running it in the background. Because it will not block any other request from the user. So make sure you don't call session_start();

Then the next step would be to run it even when the user cancels the request, which is a basic function in PHP. ignore_user_abort

Last check is to make sure it's only runs once, which can be easily done with creating a file, since PHP doesnt have an easy application scope.

Combined:

<?php
// Ignore user aborts and allow the script
// to run forever
ignore_user_abort(true);
set_time_limit(0);

$checkfile = "./runningtask.tmp";

//LOCK_EX basicaly uses flock() to prevents racecondition in regards to a regular file open.
if(file_put_contents($checkfile, "running", LOCK_EX)===false) {
    exit();
}

function Cleanup() {
  global $checkfile;
  unlink($checkfile);
}


/*
actual code for task.php    
*/


//run cleanup when your done, make sure you also call it if you exit the code anywhere else
Cleanup();
?>

In your javascript you can now call the task.php directly and cancel the request when the connection to the server has been established.

<script>
function Request(url){
  if (window.XMLHttpRequest) { // Mozilla, Safari, ...
      httpRequest = new XMLHttpRequest();
  } else if (window.ActiveXObject) { // IE
      httpRequest = new ActiveXObject("Microsoft.XMLHTTP");
  } else{
      return false;
  }
  httpRequest.onreadystatechange = function(){
      if (httpRequest.readyState == 1) {
        //task started, exit
        httpRequest.abort();
      }
  };
  httpRequest.open('GET', url, true);
  httpRequest.send(null);
}

//call Request("task.php"); whenever you want.
</script>

Bonus points: You can have the actual code for task.php write occasional updates to $checkfile to have a sense of what is going on. Then you can have another ajax file read the content and show the status to the user.

Lets make the whole process from B to D simple

Step B-D:

$rslt =array(); // output from first exec
$output = array(); // output of task.php execution

//Check if any process by the name 'task.php' is running
exec("ps -auxf | grep 'task.php' | grep -v 'grep'",$rslt);

if(count($rslt)==0) // if none,
  exec('php task.php',$output); // run the task,

Explanation:

ps -auxf        --> gets all running processes with details 
grep 'task.php' --> filter the process by 'task.php' keyword
grep -v 'grep'  --> filters the grep process out

NB:

1. Its also advisable to put the same check in task.php file.

2. If task.php is executed directly via httpd (webserver), it will only be displayed as a httpd process and cannot be identified by 'ps' command

3. It wouldn't work under load-balanced environment. [Edited: 17Jul17]

You can get an exclusive lock on the script itself for the duration of the script running

Any other attempts to run it will end as soon as the lock() function is invoked.

//try to set a global exclusive lock on the file invoking this function and die if not successful
function lock(){
  $file = isset($_SERVER['SCRIPT_FILENAME'])?
    realpath($_SERVER['SCRIPT_FILENAME']):
    (isset($_SERVER['PHP_SELF'])?realpath($_SERVER['PHP_SELF']):false);
  if($file && file_exists($file)){
    //global handle stays alive for the duration if this script running
    global $$file;
    if(!isset($$file)){$$file = fopen($file,'r');}
    if(!flock($$file, LOCK_EX|LOCK_NB)){
        echo 'This script is already running.'."\n";
        die;
    }
  }
}

Test

Run this in one shell and try to run it in another while it is waiting for input.

lock();

//this will pause execution until an you press enter
echo '...continue? [enter]';
$handle = fopen("php://stdin","r");
$line = fgets($handle);
fclose($handle);