docs: Add documentation for stdio server shutdown behavior

Documents how the stdio server detects parent process termination
via stdin EOF and shuts down gracefully to prevent orphan processes.

Author: Developer

docs/stdio_server_shutdown.md

@@ -0,0 +1,37 @@
+# Stdio Server Shutdown Behavior
+
+## Overview
+
+When using the stdio transport, the MCP server monitors stdin for EOF (End of File)
+to detect when the parent process has terminated. This ensures the server shuts down
+gracefully instead of becoming an orphan process.
+
+## How It Works
+
+1. The server reads from stdin in a loop
+2. When stdin is closed (EOF), the server detects this condition
+3. The server signals shutdown by closing the read stream
+4. All resources are cleaned up properly
+
+## Parent Process Death
+
+If the parent process (MCP client) dies unexpectedly:
+- The server's stdin will be closed by the operating system
+- The server detects EOF and initiates graceful shutdown
+- No orphan processes remain
+
+## Configuration
+
+No additional configuration is required. This behavior is automatic when using
+the stdio transport.
+
+## Example
+
+```python
+from mcp.server.stdio import stdio_server
+
+async def run_server():
+    async with stdio_server() as (read_stream, write_stream):
+        # Server will automatically shut down when stdin closes
+        await server.run(read_stream, write_stream, init_options)
+```