Sample path large deviation principles (LDP) are established for some pure jump stochastic processes arising from queueing systems, including M/M/1+M, M/M/n+M, and Join-the-Shortest-Queue. The process of evolution suffers from features of infinite dimensional dynamics, vanishing jump rates (near the boundary), and/or discontinuous statistics (at the boundary), which pose challenges on the variational representation and weak convergence approach, in particular the uniqueness argument. Rare events of interest are analyzed by solving the related calculus of variations problems written in terms of the LDP rate functions. If time permits, the Join-the-Shortest-Queue-d (power-of-d) queueing system will be discussed, including conjectures on the LDP and establishment of moderate deviation principles. This is based on several joint works with Rami Atar, Amarjit Budhiraja, Paul Dupuis, Eric Friedlander, and Zhenhua Wang.