Before revision:  We consider a limited fork-join model with n servers, where each job consists of k ^{(n)}  ntasks. Upon arrival, each jobchooses k^{(n) under the condition k^{(n)} = o(n^{1/4}) , the steady state response times at any k^{(n) greatly generalizes the asymptotic-independence type of results in the literature where asymptotic independence is shown only for a fixed constant number of queues.
After revision:  We consider a limited fork-join model with n servers, where each job consists of k ^{(n)}  queues under the condition k^{(n)} = o(n^{1/4}) , the steady state response times at any k^{(n) greatly generalizes the asymptotic-independence type of results in the literature where asymptotic independence is shown only for a fixed constant number of queues.