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