There are essentially two results described in this paper. First, it is shown that for any random access machine (RAM) time-constructable function $T(n) \geqq n$, there are languages L such that L can be recognized in time $O(T(n))$ by a RAM (using the unit cost measure), but L cannot be recognized by any deterministic multitape Turing machine in time $O(T(n))$. Secondly, a family of random access stored program machines (RASP’S) are considered. For these RASP’S it is shown that there is an arbitrarily complex (infinitely often) partial recursive function $f(n)$. which has only 0–1 values (whenever defined) such that $f(n)$ can be computed in time $F(n)$ by some RASP, but cannot be computed in time $(1 - \varepsilon )F(n)$, for any $\varepsilon > 0$, by any RASP in this family.