Agentic robots combine a vision-language brain, a tool-using harness, and a vision-language-action (VLA) policy. Serving a fleet of these agents introduces two coupled resource constraints: the brain repeatedly revisits long, growing key-value (KV) cache states, while the VLA executes action chunks under tight control deadlines. We characterize 80 LIBERO-PRO episodes generated by GPT-6-Astra, RPent, and pi0.5 (1,862 brain turns and 1,582 VLA inference chunks), profile both layers on H800 graphics processing units (GPUs), and replay the resulting agent programs on a vLLM serving engine. Brain contexts reach 215k tokens; the median Astra request takes 7.78 s, while the intervening environment-step gap is only 1.0 s. Under a 0.25 s action-chunk deadline, one VLA GPU meets the 1% starvation target for one robot; under a 1.0 s deadline, its measured capacity reaches at least 32. In sustained brain replay, increasing concurrency from 20 to 32 robots reduces throughput from about 2.4 to 0.6 requests/s as repetitive prefill displaces generation. A static active-episode cap restores 2.24-2.25 requests/s and raises completed trace-program throughput from 1.30 to 5.00-5.55 episodes/min. Our measurements expose the workload-dependent shift between action deadlines and brain KV residency, and establish practical principles for capacity planning and admission in sustained agentic robot fleets.