  We study the $G/\mathit{GI}/\infty$ queue in heavy-traffic using tempered
distribution-valued processes which track the age and residual service time of
each customer in the system. In both cases, we use the continuous mapping
theorem together with functional central limit theorem results in order to
obtain fluid and diffusion limits for these processes in the space of tempered
distribution-valued processes. We find that our diffusion limits are tempered
distribution-valued Ornstein-Uhlenbeck processes.
