Efficiency of universal parallel computers

We consider parallel computers (PC's) with fixed communication network and bounded degree. We deal with the following question: How efficiently can one PC, a so-called universal PC, simulate each PC with n processors? This question is asked in [1] where a universal PC with O(n) processors and time loss O(log(n)) is constructed. We improve this result in two ways by construction two universal PC's which many users can efficiently work with at the same time. The first has the same number of processors and the same time loss as that one above. The second has O(O 1 + �) processors for an arbitrary �>0 but only time loss O(loglog(n)). Finally we define three types of simulations the most general of which includes all known simulations. We prove non-linear time-processor tradeoffs for universal PC's associated with the above types.