in first sight i think that you should use java script ajax on your a.php for sending data to page b.php but that is not enough so for getting you result a+b=something you need to use something called java script concurrence and those technique allow you to listen for a period of time on the existence of a so i will try to give you the solution o some steps :
Step 1: you've done your ajax sending data
step 2: you need to go check on the java script concurrence you have set interval and set timeout so use one of them but you will need to use set interval cause it will be more efficient
step 3 :use a Php condition called !empty()to check each period of time with the set Interval that a don't equal the default value if it's equal to that default value you should use clear Interval since keeping the execution of set Interval doesn't have any meaning and you will consume of the load of your page and your page will run slower
step 4: you've done with everything and you can do your operation.
i will let a short code for set Interval and you can find them on w3schools with execution example
var myVar;
function myFunction() {
myVar = setInterval(alertFunc, 3000);
}
function alertFunc() {
alert("Hello!");
}
for the clearInterval
var myVar = setInterval(myTimer, 1000);
function myTimer() {
var d = new Date();
var t = d.toLocaleTimeString();
document.getElementById("demo").innerHTML = t;
}
function myStopFunction() {
clearInterval(myVar);
}