Docs for configuring app servers for production (#3120)

Co-authored-by: José Valim <[email protected]>

Author: hugobarauna

docs/images/shutdown-inactivity.png

@@ -0,0 +1,41 @@
+# Configuring app servers for production
+
+This guide covers recommended configurations for running Livebook app servers in production environments.
+
+## Persisting compiled dependencies
+
+Livebook notebooks can declare Hex dependencies using `Mix.install/2`. When a notebook runs, Mix compiles these dependencies and caches them in a configurable directory. By default, this directory is not persistent in containerized environments, so dependencies must be recompiled every time the container restarts, increasing startup time for deployed apps.
+
+To persist compiled dependencies across container restarts, mount a persistent volume and configure the following environment variable:
+
+```
+MIX_INSTALL_DIR=/data
+```
+
+Where `/data` is the mount point of your persistent volume.
+
+This reduces app startup time after container restarts since dependencies don't need to be recompiled.
+
+## Memory sizing
+
+App servers require memory for the Livebook runtime itself plus each app session. Use this formula as a rule of thumb to calculate the recommended memory:
+
+```text
+Total memory = 1.5 GB (base) + (50 MB × number of app sessions)
+```
+
+The base 1.5 GB accommodates the Livebook web server and the memory required during `Mix.install/2`commands. Each app session requires approximately 50 MB, though this can vary depending on the app. For single-session apps, this equals one session per app. Multi-session apps will use 50 MB per session.
+
+This calculation assumes all apps have active sessions running simultaneously, representing peak memory usage.
+
+**Example:** For 70 single-session apps:
+
+```text
+1.5 GB + (70 × 50 MB) = 1.5 GB + 3.5 GB = 5 GB
+```
+
+> #### Reduce memory usage with inactivity shutdown {: .tip}
+>
+> Configure "Shutdown after inactivity" in the app settings to automatically stop idle sessions. This frees memory when sessions are not in use.
+>
+> ![](images/shutdown-inactivity.png)

docs/teams/app_server_production.md

@@ -266,6 +266,7 @@ defmodule Livebook.MixProject do
       {"docs/teams/deploy_via_cli.md", title: "Deploy Livebook apps via CLI"},
       "docs/teams/deploy_permissions.md",
       {"docs/teams/app_folders.md", title: "Organize apps with folders"},
+      {"docs/teams/app_server_production.md", title: "Production app servers"},
       {"docs/teams/email_domain.md", title: "Email domain auth"},
       {"docs/teams/oidc_sso.md", title: "OIDC SSO"},
       "docs/teams/oidc_groups.md",