docs: Add Lambda Managed Instance mode documentation (#951)

docs: Add Lambda Managed Instance mode documentation

https://datadoghq.atlassian.net/browse/SVLS-8083

## Overview
Add comprehensive documentation for Lambda Managed Instance support
(v90+):
- Overview of Managed Instance mode and how it differs from standard
Lambda
  - Automatic detection and optimization behavior
- Background continuous flushing architecture with zero per-invocation
overhead
- Key differences comparison table (invocation model, flushing, use
cases)
  - Getting started guide for users

  Also clarifies that custom continuous flush intervals are respected in
  Managed Instance mode (not completely ignored as previously stated).

## Testing 
n/a

Author: litianningdatadog

README.md

@@ -29,6 +29,46 @@ APM Tracing is supported for Python, NodeJS, Go, Java, and .NET runtimes.
 
 We'd love to hear your feedback on the next-generation Lambda Extension. You can open a GitHub issue here using the `version/next` tag, find us on the [Datadog Community Slack](https://chat.datadoghq.com/) in the #serverless channel, or reach out to me directly at [email protected].
 
+## Lambda Managed Instance Support
+
+The Datadog Lambda Extension (v90+) now supports **AWS Lambda Managed Instances**, a deployment mode where Lambda functions run on EC2 instances managed by AWS with multi-concurrent invocation handling. This is ideal for steady-state, high-volume workloads where optimizing costs with predictable capacity is desired.
+
+### What is Lambda Managed Instance Mode?
+
+Lambda Managed Instances run your functions on EC2 instances (managed by AWS) where one execution environment can handle multiple invocations simultaneously, unlike traditional Lambda's one-invocation-per-environment model. This requires setting up a **capacity provider** - a configuration that defines VPC settings, instance requirements, and scaling parameters.
+
+Learn more:
+- [Introducing AWS Lambda Managed Instances: Serverless simplicity with EC2 flexibility](https://aws.amazon.com/blogs/aws/introducing-aws-lambda-managed-instances-serverless-simplicity-with-ec2-flexibility/)
+- [Lambda Managed Instances - AWS Lambda Developer Guide](https://docs.aws.amazon.com/lambda/latest/dg/lambda-managed-instances.html)
+
+### How the Extension Supports Managed Instances
+
+The Datadog Extension automatically detects when running in Managed Instance mode (via the `AWS_LAMBDA_INITIALIZATION_TYPE` environment variable) and optimizes its behavior:
+
+- **Background Flushing**: A dedicated background task continuously flushes observability data at regular intervals (default: 30 seconds), completely independent of invocation processing
+- **Zero Per-Invocation Overhead**: All flush operations are non-blocking and run concurrently with your function invocations
+- **Optimized Event Handling**: The extension processes telemetry events directly from the Lambda Telemetry API without blocking on invocation lifecycle calls
+- **Enhanced Monitoring**: Generates managed-instance-specific metrics and statistics
+
+### Key Differences from Standard Lambda
+
+| Aspect | Managed Instance Mode | Standard (On-Demand) Mode |
+|--------|----------------------|---------------------------|
+| **Invocation Model** | Multi-concurrent invocations per environment | Single invocation per environment |
+| **Data Flushing** | Continuous background flushing (non-blocking) | Configurable flush strategies |
+| **Flush Overhead** | Zero overhead per invocation | Small overhead based on flush strategy |
+| **Configuration** | Always continuous (custom intervals respected) | Flush strategy is configurable |
+| **Use Case** | Steady-state, high-volume workloads | General serverless workloads |
+
+### Getting Started
+
+1. Set up your Lambda function with a [capacity provider](https://docs.aws.amazon.com/lambda/latest/dg/lambda-managed-instances.html)
+2. Install the Datadog Lambda Extension (v90+) as usual
+3. The extension will automatically detect Managed Instance mode and optimize its behavior
+4. Monitor your telemetry in Datadog as normal
+
+No additional configuration is required - the extension handles everything automatically when it detects the Managed Instance environment.
+
 ## Upgrading
 
 To upgrade, update the Datadog Extension version in your Lambda layer configurations or Dockerfile (for Lambda functions deployed as container images). View the latest [releases](https://github.com/DataDog/datadog-lambda-extension/releases) and corresponding changelogs before upgrading.

bottlecap/README.md

@@ -24,7 +24,7 @@ Step 3: Test your change running `./runBottlecap.sh`
 Bottlecap supports several flush strategies that control when and how observability data (metrics, logs, traces) is sent to Datadog. The strategy is configured via the `DD_SERVERLESS_FLUSH_STRATEGY` environment variable.
 
 **Important**: Flush strategies behave differently depending on the Lambda execution mode:
-- **Managed Instance**: Uses continuous background flushing (flush strategies are ignored)
+- **Managed Instance**: Always uses continuous background flushing (only custom continuous intervals are respected)
 - **On-Demand**: Uses configurable flush strategies
 
 ### Managed Instance Mode vs On-Demand Mode
@@ -37,7 +37,7 @@ Lambda Managed Instances run your functions on EC2 instances (managed by AWS) wi
   - A dedicated background task continuously flushes data at regular intervals (default: 30 seconds)
   - All flushes are **non-blocking** and run concurrently with invocation processing
   - Prevents resource buildup by skipping a flush cycle if the previous flush is still in progress
-  - `DD_SERVERLESS_FLUSH_STRATEGY` is **ignored** in this mode
+  - Only `DD_SERVERLESS_FLUSH_STRATEGY=continuously,<ms>` is respected; all other strategies are overridden to continuous with default interval
 - **Shutdown Behavior**:
   - Background flusher waits for pending flushes to complete before shutdown
   - Final flush ensures all remaining data is sent before the execution environment terminates
@@ -164,7 +164,7 @@ Located in `bottlecap/src/bin/bottlecap/main.rs`:
 | **Scaling** | Asynchronous, CPU-based scaling | Reactive scaling with cold starts |
 | **Pricing** | EC2 instance-based | Per-request duration-based |
 | **Flush Trigger** | Background interval timer | Invocation lifecycle + interval |
-| **Strategy Config** | Ignored (always continuous) | Configurable via env var |
+| **Strategy Config** | Always continuous (custom intervals respected) | Configurable via env var |
 | **Main Loop** | Event bus processing | `/next` + event bus processing |
 | **Shutdown Detection** | Separate task monitors `/next` | Main loop receives from `/next` |