[Rust Frontend] Extract shared tracing setup logic into vllm-tracing (#50129)

Signed-off-by: Bugen Zhao <i@bugenzhao.com>

rust/src/bench/Cargo.toml

@@ -32,9 +32,9 @@ tokenizers.workspace = true
 tokio.workspace = true
 tokio-stream.workspace = true
 tracing.workspace = true
-tracing-subscriber.workspace = true
 url.workspace = true
 uuid.workspace = true
+vllm-tracing.workspace = true
 
 [lints]
 workspace = true

rust/src/bench/src/main.rs

@@ -19,18 +19,8 @@ struct Cli {
     args: vllm_bench::BenchServeArgs,
 }
 
-// TODO: unify the tracing subscriber used by different binaries.
-fn init_tracing() {
-    let filter = tracing_subscriber::EnvFilter::try_from_default_env()
-        .unwrap_or_else(|_| tracing_subscriber::EnvFilter::new("info"));
-    let _ = tracing_subscriber::fmt()
-        .with_env_filter(filter)
-        .with_writer(std::io::stderr)
-        .try_init();
-}
-
 fn main() -> anyhow::Result<()> {
-    init_tracing();
+    vllm_tracing::init_tracing("Bench");
 
     let cli = Cli::parse();
     vllm_bench::prepare_process();