The critical offered load in variants of the symmetric shortest and longest queue systems

We study the ergodicity behavior of three truncated variants of the memoryless two-server symmetric shortest queue system and of two truncated variants of the memoryless two-dimensional symmetric longest queue system. These variants, which can be solved efficiently by the matrix-geometric approach, lead to flexible bounds on some performance measures in the corresponding original system. As a function of the truncating thresholds, we compute the supremum over the offered loads which guarantee ergodicity, and we study the limiting behavior of these suprema