Before revision:  This limit has been previously determined for SQ(d) and LL(d), two well-known policies that assign an incoming job to a server with either the shortest queue or least work left among d randomly selected servers. However, the derivation of the result for SQ(d) relied on the closed form representation of the mean response time and does not seem to generalize well, moreover the proof for LL(d) is incomplete. In contrast, we present a general result that holds for any policy for which the associated recursion or differential equation satisfies a list of criteria.
After revision:  This limit has a surprisingly simple form for the load balancing algorithms considered. We present a general result that holds for any policy for which the associated recursion or differential equation satisfies a list of criteria.