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

Distribution-valued heavy-traffic limits for the $G/\mathit{GI}/\infty$ queue