feat: first response time support gemini and claude

relay/channel/gemini/relay-gemini.go

@@ -11,6 +11,7 @@ import (
 	relaycommon "one-api/relay/common"
 	"one-api/service"
 	"strings"
+	"time"
 
 	"github.com/gin-gonic/gin"
 )
@@ -160,7 +161,7 @@ func streamResponseGeminiChat2OpenAI(geminiResponse *GeminiChatResponse) *dto.Ch
 	return &response
 }
 
-func geminiChatStreamHandler(c *gin.Context, resp *http.Response) (*dto.OpenAIErrorWithStatusCode, string) {
+func geminiChatStreamHandler(c *gin.Context, resp *http.Response, info *relaycommon.RelayInfo) (*dto.OpenAIErrorWithStatusCode, string) {
 	responseText := ""
 	dataChan := make(chan string)
 	stopChan := make(chan bool)
@@ -190,10 +191,15 @@ func geminiChatStreamHandler(c *gin.Context, resp *http.Response) (*dto.OpenAIEr
 		}
 		stopChan <- true
 	}()
+	isFirst := true
 	service.SetEventStreamHeaders(c)
 	c.Stream(func(w io.Writer) bool {
 		select {
 		case data := <-dataChan:
+			if isFirst {
+				isFirst = false
+				info.FirstResponseTime = time.Now()
+			}
 			// this is used to prevent annoying \ related format bug
 			data = fmt.Sprintf("{\"content\": \"%s\"}", data)
 			type dummyStruct struct {