Improving performance for each thread using a unique unique identifier

Problem Statement: -

Each thread uses a unique identifier from 1 to 1000, and the program must run for 60 minutes or more. Thus, within 60 minutes, it is possible that the entire identifier will end, so I need to use this identifier again,

I know several ways to do this, one of the ways is the following that I wrote using StackOverflow help, but when I tried to run it, I found that after a few minutes of work this program becomes very slow and it takes a lot of time to print the identifier on the console. And sometimes I get an OutOfMemory error. Is there a better way to solve this problem?

class IdPool {
    private final LinkedList<Integer> availableExistingIds = new LinkedList<Integer>();

    public IdPool() {
        for (int i = 1; i <= 1000; i++) {
            availableExistingIds.add(i);
        }
    }

    public synchronized Integer getExistingId() {
        return availableExistingIds.removeFirst();
    }

    public synchronized void releaseExistingId(Integer id) {
        availableExistingIds.add(id);
    }
}


class ThreadNewTask implements Runnable {
    private IdPool idPool;

    public ThreadNewTask(IdPool idPool) {
        this.idPool = idPool;
    }

    public void run() {
        Integer id = idPool.getExistingId();
        someMethod(id);
        idPool.releaseExistingId(id);
    }

    private void someMethod(Integer id) {
        System.out.println("Task: " +id);
    }
}

public class TestingPool {
    public static void main(String[] args) throws InterruptedException {
        int size = 10;
        int durationOfRun = 60;
        IdPool idPool = new IdPool();   
        // create thread pool with given size
        // create thread pool with given size
    ExecutorService service = new ThreadPoolExecutor(size, size, 500L, TimeUnit.MILLISECONDS, new ArrayBlockingQueue<Runnable>(10), new ThreadPoolExecutor.CallerRunsPolicy()); 


        // queue some tasks
        long startTime = System.currentTimeMillis();
        long endTime = startTime + (durationOfRun * 60 * 1000L);

        // Running it for 60 minutes
        while(System.currentTimeMillis() <= endTime) {
            service.submit(new ThreadNewTask(idPool));
        }

        // wait for termination        
        service.shutdown();
        service.awaitTermination(Long.MAX_VALUE, TimeUnit.DAYS); 
    }
}