[router] Introduce router integration tests (#10086)

sgl-router/py_test/integration/load_balancing/test_round_robin.py

@@ -0,0 +1,34 @@
+import collections
+import time
+
+import pytest
+import requests
+
+
+@pytest.mark.integration
+def test_round_robin_distribution(mock_workers, router_manager):
+    procs, urls, ids = mock_workers(n=3)
+
+    rh = router_manager.start_router(worker_urls=urls, policy="round_robin")
+
+    counts = collections.Counter()
+    with requests.Session() as s:
+        for i in range(30):
+            r = s.post(
+                f"{rh.url}/v1/completions",
+                json={
+                    "model": "test-model",
+                    "prompt": f"hello {i}",
+                    "max_tokens": 1,
+                    "stream": False,
+                },
+            )
+            assert r.status_code == 200
+            wid = r.headers.get("X-Worker-Id") or r.json().get("worker_id")
+            assert wid in ids
+            counts[wid] += 1
+
+    # Expect near-even distribution across 3 workers
+    # 30 requests -> ideally 10 each; allow small tolerance ±3
+    for wid in ids:
+        assert 7 <= counts[wid] <= 13, counts