📝 Add async logger lifecycle API docs

docs/api/async-logger-is-running.md

@@ -0,0 +1,76 @@
+---
+name: async-logger-is-running
+group: api
+category: async
+update-time: 20260512
+description: Read whether the async logger worker is currently running and draining the queue.
+key-word:
+    - async
+    - logger
+    - lifecycle
+    - public
+---
+
+## Async-logger-is-running
+
+Read whether the async logger worker is currently active. This helper is useful for runtime diagnostics and shutdown coordination.
+
+### Interface
+
+```moonbit
+pub fn[S] AsyncLogger::is_running(self : AsyncLogger[S]) -> Bool {}
+```
+
+#### input
+
+- `self : AsyncLogger[S]` - Async logger whose worker state should be inspected.
+
+#### output
+
+- `Bool` - Whether the async worker loop is currently running.
+
+### Explanation
+
+Detailed rules explaining key parameters and behaviors
+
+- `run()` sets the running state while the worker loop is active.
+- The flag is cleared when the worker exits normally or after failure handling finishes.
+- A logger may be closed while still running briefly during final drain or shutdown processing.
+- This helper focuses on worker activity rather than queue size or failure details.
+
+### How to Use
+
+Here are some specific examples provided.
+
+#### When Need Worker Activity Diagnostics
+
+When runtime output should show whether the worker loop is active:
+```moonbit
+println(logger.is_running().to_string())
+```
+
+In this example, diagnostics can distinguish an idle configured logger from one with an active worker.
+
+#### When Coordinate Shutdown Waiting
+
+When code should poll worker completion explicitly:
+```moonbit
+while logger.is_running() {
+  @async.pause()
+}
+```
+
+In this example, callers watch the worker lifecycle directly.
+
+### Error Case
+
+e.g.:
+- If `is_running()` is `false`, pending records may still exist if the worker was never started or has already failed.
+
+- If callers need a one-shot lifecycle flow, `shutdown()` is usually better than manual polling.
+
+### Notes
+
+1. Use this helper for worker activity checks, not as a complete health signal.
+
+2. Pair it with `has_failed()` or `state()` when diagnosing stalled pipelines.