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Distributed matrix-block-vector multiplication (Matvec) algorithm is a critical component of many applications, but can be computationally challenging for dense matrices of dimension \(O(10^6\text{--}10^7)\) and blocks of \(O(10\text{--}100)\) vectors. We present performance analysis, implementation, and optimization of our \pname{} library for Matvec under the effect of system variability. Our modeling shows that 1D pipelining Matvec is as efficient as 2D algorithms at small to medium clusters, which are sufficient for these problem sizes. We develop a performance tracing framework and a simulator that reveal pipeline bubbles caused by modest \textasciitilde{}5\% system variability. To tolerate such variability, our \pname{} library, which combines on-the-fly kernel matrix generation and Matvec, integrates four optimizations: inter-process data preloading, unconventional static thread scheduling, cache-aware tiling, and multi-version unrolling. In our benchmarks on \(O(10^5)\) Matvec problems, \pname{} achieves up to 1.85× speedup over COSMA and 17× over ScaLAPACK. For \(O(10^6)\) problems, where COSMA and ScaLAPACK exceed memory capacity, \pname{} maintains linear strong scaling and achieves peak performance of 75\%~FMA~Flop/s. Its static scheduling policy has a 2.27× speedup compared to the conventional work-stealing dynamic scheduler, and is predicted to withstand up to 108\% performance variability under exponential distributed variability simulation.