Approximate MVA algorithms for separable queueing networks are based upon an iterative solution of a set of modified MVA formulas. Although each iteration has a computational time requirement of O(MK²) or less, many iterations are typically needed for convergence to a solution. (M denotes the number of queues and K the number of closed chains or customer classes.) We present some faster approximate solution algorithms that are noniterative. They are suitable for the analysis and design of communication networks which may require tens to hundreds, perhaps thousands, of closed chains to model flow-controlled virtual channels. The basis of our method is the distribution of a chain's population proportional to loads to get initial estimates of mean queue lengths. This is the same basis used in the derivation of proportional upper bounds for single-chain networks; for a multichain network, such a proportional distribution leads to approximations rather than upper bounds of chain throughputs. Nevertheless, these approximate solutions provide chain throughputs, mean end-to-end delays, and server utilizations that are sufficiently accurate for the analysis and design of communication networks and possibly other distributed systems with a large number of customer classes. Three PAM algorithms of increasing accuracy are presented. Two of them have time and space requirements of O(MK). The third algorithm has a time requirement of O(MK²) and a space requirement of O(MK).