If you don't care about the order of requests then you can just set up a flag to denote if a task is running, and keep rescheduling the other tasks. Something like

      let isProcessing = false;

      async function checkFlagAndRun(task) {
          if (isProcessing) {
              return setTimeout(() => checkFlagAndRun(task), 0);
          }

          isProcessing = true;
          await task();
          isProcessing = false;
      }

should do the trick. You can test it with

      function delayedLog(message, delay) {
          return new Promise(resolve => {
              setTimeout(() => {
                  console.log(message);
                  resolve();
              }, delay);
          });
      }

      function test(name,num) {
          for (let i = 1; i <= num; i++) {
              const delay = Math.floor(Math.random() * 1000 + 1);
              checkFlagAndRun(() => delayedLog(`${name}-${i} waited ${delay} ms`, delay));
          }
      }

      test('t1',20); test('t2',20); test('t3',20); 

BTW, for 4 scheduled tasks, it basically always keeps the order, and I am not sure why. Even if the first task always runs first, the rest 3 should race each other. 5 simultaneously scheduled tasks ruins the order.