fix(helm): give gateway and backend probes an explicit timeoutSeconds (#35497)

The gateway and backend probes omitted timeoutSeconds, so kubelet applied its
1s default. Both containers run a single uvicorn worker (the gateway defaults
NUM_WORKERS to 1; the backend passes no --workers at all), so each pod is one
asyncio event loop and its per-request latency under closed-loop saturation
rises by queueing (~57-62ms serial vs ~6s at 100 concurrent users against one
replica). Both /health/readiness and /health/liveliness then time out on the
stage cluster while the pod is serving traffic correctly, which exposes the
deployment to losing a healthy pod from its load balancer during a burst and
to restarting a merely busy one.

Readiness now gets timeoutSeconds 10, equal to periodSeconds and above the
measured saturated latency, and keeps failureThreshold 3. kubelet drives each
probe from a time.Ticker of periodSeconds rather than sleeping between
attempts, and coalesces ticks that arrive mid-probe, so the interval between
probe starts is max(periodSeconds, probeDuration) and not their sum. Keeping
timeoutSeconds <= periodSeconds is what holds that interval at 10s, so three
consecutive failures still evict a genuinely wedged pod in ~30s.

Liveness gets the same timeout plus failureThreshold 6: /health/liveliness is
an in-memory flag check, so a timeout there only ever means event-loop
starvation, which a restart makes worse, and it now needs ~90s of sustained
unresponsiveness to fire.

The ui container keeps the default. It is nginx serving a Next.js static
export, so / is a file off disk with no application runtime that could queue
behind saturated work, and nothing measured suggests it needs more than 1s.

helm/litellm/tests/probe_tests.yaml

@@ -0,0 +1,106 @@
+suite: test liveness and readiness probe timeouts
+templates:
+  - gateway/deployment.yaml
+  - gateway/configmap.yaml
+  - backend/deployment.yaml
+values:
+  - ./values/required.yaml
+tests:
+  - it: gateway probes set an explicit timeout that outlasts a saturated event loop
+    template: gateway/deployment.yaml
+    asserts:
+      - equal:
+          path: spec.template.spec.containers[0].livenessProbe
+          value:
+            httpGet:
+              path: /health/liveliness
+              port: http
+            initialDelaySeconds: 10
+            periodSeconds: 15
+            timeoutSeconds: 10
+            failureThreshold: 6
+      - equal:
+          path: spec.template.spec.containers[0].readinessProbe
+          value:
+            httpGet:
+              path: /health/readiness
+              port: http
+            initialDelaySeconds: 5
+            periodSeconds: 10
+            timeoutSeconds: 10
+
+  - it: backend probes set an explicit timeout that outlasts a saturated event loop
+    template: backend/deployment.yaml
+    asserts:
+      - equal:
+          path: spec.template.spec.containers[0].livenessProbe
+          value:
+            httpGet:
+              path: /health/liveliness
+              port: http
+            initialDelaySeconds: 10
+            periodSeconds: 15
+            timeoutSeconds: 10
+            failureThreshold: 6
+      - equal:
+          path: spec.template.spec.containers[0].readinessProbe
+          value:
+            httpGet:
+              path: /health/readiness
+              port: http
+            initialDelaySeconds: 5
+            periodSeconds: 10
+            timeoutSeconds: 10
+
+  - it: no single-event-loop component is left on the kubernetes default 1s probe timeout
+    templates:
+      - gateway/deployment.yaml
+      - backend/deployment.yaml
+    asserts:
+      - isNotNullOrEmpty:
+          path: spec.template.spec.containers[0].livenessProbe.timeoutSeconds
+      - isNotNullOrEmpty:
+          path: spec.template.spec.containers[0].readinessProbe.timeoutSeconds
+      - equal:
+          path: spec.template.spec.containers[0].livenessProbe.timeoutSeconds
+          value: 10
+      - equal:
+          path: spec.template.spec.containers[0].readinessProbe.timeoutSeconds
+          value: 10
+
+  - it: gateway liveness tolerates a longer outage than readiness before acting
+    template: gateway/deployment.yaml
+    asserts:
+      - equal:
+          path: spec.template.spec.containers[0].livenessProbe.failureThreshold
+          value: 6
+      - notExists:
+          path: spec.template.spec.containers[0].readinessProbe.failureThreshold
+
+  - it: probe timeouts and thresholds stay overridable per component
+    template: gateway/deployment.yaml
+    set:
+      gateway.readinessProbe.timeoutSeconds: 3
+      gateway.readinessProbe.periodSeconds: 20
+      gateway.livenessProbe.timeoutSeconds: 4
+      gateway.livenessProbe.failureThreshold: 3
+    asserts:
+      - equal:
+          path: spec.template.spec.containers[0].readinessProbe
+          value:
+            httpGet:
+              path: /health/readiness
+              port: http
+            initialDelaySeconds: 5
+            periodSeconds: 20
+            timeoutSeconds: 3
+      - equal:
+          path: spec.template.spec.containers[0].livenessProbe
+          value:
+            httpGet:
+              path: /health/liveliness
+              port: http
+            initialDelaySeconds: 10
+            periodSeconds: 15
+            timeoutSeconds: 4
+            failureThreshold: 3

helm/litellm/values.yaml

@@ -216,10 +216,13 @@ gateway:
     httpGet: { path: /health/liveliness, port: http }
     initialDelaySeconds: 10
     periodSeconds: 15
+    timeoutSeconds: 10
+    failureThreshold: 6
   readinessProbe:
     httpGet: { path: /health/readiness, port: http }
     initialDelaySeconds: 5
     periodSeconds: 10
+    timeoutSeconds: 10
   hpa:
     enabled: true
     minReplicas: 1
@@ -309,10 +312,13 @@ backend:
     httpGet: { path: /health/liveliness, port: http }
     initialDelaySeconds: 10
     periodSeconds: 15
+    timeoutSeconds: 10
+    failureThreshold: 6
   readinessProbe:
     httpGet: { path: /health/readiness, port: http }
     initialDelaySeconds: 5
     periodSeconds: 10
+    timeoutSeconds: 10
   hpa:
     enabled: true
     minReplicas: 1