[Frontend][Responses API] Fix arrival_time recording for TTFT on initial request (#37498)

Signed-off-by: Andrew Xia <axia@meta.com>

docs/design/metrics.md

@@ -244,6 +244,7 @@ statistics relating to that iteration:
   prefill in this iteration. However, we calculate this interval
   relative to when the request was first received by the frontend
   (`arrival_time`) in order to account for input processing time.
+  Currently `arrival_time` starts when tokenization begins.
 
 For any requests that were completed in a given iteration, we also
 record:

vllm/entrypoints/openai/responses/serving.py

@@ -710,9 +710,11 @@ class OpenAIServingResponses(OpenAIServing):
                 "Only 'auto' tool_choice is supported in response API with Harmony"
             )
 
+        arrival_time = time.time()
         messages = self._construct_input_messages_with_harmony(request, prev_response)
         prompt_token_ids = render_for_completion(messages)
         engine_prompt = token_inputs(prompt_token_ids)
+        engine_prompt["arrival_time"] = arrival_time
 
         # Add cache_salt if provided in the request
         if request.cache_salt is not None: