How to correctly use a fixed size thread pool?

Question

I am quite new to using concurrency in Java, so please forgive if this is a trivial question.
I would like to make use of something like pool=Executors.newFixedThreadPool(n) to automatically use a fixed number of threads to process pieces of work. I understand that I can asynchronously run some Runnable by one of the threads in the threadpool using pool.execute(someRunnable).
My problem is this: I have some fixed amount of N datastructures myDS (which are not reentrant or sharable) that get initialized at program start and which are needed by the runnables to do the work. So, what I really would like to do is that I not only reuse N threads but also N of these datastructures to do the work.
So, lets say I want to have 10 threads, then I would want to create 10 myDS objects once and for all. Each time some work comes in, I want that work to get processed by the next free thread, using the next free datastructure. What I was wondering is if there is something in the library that lets me do the resusing of threads AND datastructures as simply as just reusing a pool of threads. Ideally, each thread would get associated with one datastructure somehow.
Currently I use an approach where I create 10 Runnable worker objects, each with its own copy of myDS. Those worker objects get stored in an ArrayBlockingQueue of size 10. Each time some work comes in, I get the next Runner from the queue, pass it the piece of work and submit it to the thread pool.
The tricky part is how to get the worker object back into the Queue: currently I essentially do queue.put(this) at the very end of each Runnable's run method but I am not sure if that is safe or how to do it safely.
What are the standard patterns and library classes to use for solving this problem correctly?

Answer 1

Thank you for that feedback!
There is one issue that worries me though and I obviously do not understand it enough: as I said I hand back the Runnable to the blocking queue at the end of the Runnable.run method using queue.put(this). This is done via a static method from the main class that creates the threads and runnable objects in a main method. Originally I tried to make that method for putting back the Runnable objects serialized but that inevitably always led to a deadlock or hang condition: the method for putting back the runnable was never actually run. So I ended up doing this without serializing the put action in any way and so far it seems to work ... but is this safe?
To reiterate: I have a static class that creates a thread pool object and a ArrayBlockingQueue queue object of runnable objects. In a loop I use queue.take() to get the next free runnable object, and pass this runnable to pool.execute. Inside the runnable, in method run, i use staticclass.putBack(this) which in turn does queue.put(therunnableigot). Can I trust that this queue.put operation, which can happen from several threads at the same time works without problem without serializing it explicitly? And why would making the staticclass.putBack method serialized cause a hang? I also tried to serialize using the queue object itself instead of the static class, by doing serialize(queue) { queue.put(therunnable) } but that also caused a hang. I have to admit that I do not understand at all why that hang occurred and if I need the serialization here or not.