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Throughput extremization is an important facet of performance modeling for low-power wide-area network (LP-WAN) wireless networks (e.g., LoRaWAN) as it provides insight into the best and worst case behavior of the network. Our previous work on throughput extremization established lower and upper bounds on throughput for random access channel assignment over a collision erasure channel in which the lower bound is expressed in terms of the number of radios and sum load on each channel. In this paper the lower bound is further characterized by identifying two local minimizers (a load balanced assignment and an imbalanced assignment) where the decision variables are the number of radios assigned to each channel and the total load on each channel. A primary focus is to characterize how macro-parameters of the optimization, i.e., the total number of radios, their total load, and the minimum load per radio, determine the regions under which each of the local minimizers is in fact the global minimizer.