Add monthly active users documentation

changelog.d/13617.doc

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+Document how ["monthly active users"](https://matrix-org.github.io/synapse/latest/usage/administration/monthly_active_users.html) is calculated and used.

docs/SUMMARY.md

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     - [Manhole](manhole.md)
     - [Monitoring](metrics-howto.md)
       - [Reporting Homeserver Usage Statistics](usage/administration/monitoring/reporting_homeserver_usage_statistics.md)
+    - [Monthly Active Users](usage/administration/monthly_active_users.md)
     - [Understanding Synapse Through Grafana Graphs](usage/administration/understanding_synapse_through_grafana_graphs.md)
     - [Useful SQL for Admins](usage/administration/useful_sql_for_admins.md)
     - [Database Maintenance Tools](usage/administration/database_maintenance_tools.md)

docs/usage/administration/monthly_active_users.md

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+# Monthly Active Users
+
+Synapse can be configured to record the number of monthly active users (also referred to as MAU) on a given homeserver.
+For clarity's sake, MAU only tracks local users.
+
+Please note that the metrics recorded by the [Homeserver Usage Stats](../../usage/administration/monitoring/reporting_homeserver_usage_statistics.md)
+are calculated differently. The `monthly_active_users` from the usage stats does not take into account any
+of the rules below, and counts any users who have made a request to the homeserver in the last 30 days.
+
+See the [configuration manual](../../usage/configuration/config_documentation.md#limit_usage_by_mau) for details on how to configure MAU.
+
+## Calculating active users
+
+Individual user activity is measured in active days. If a user performs an action, the exact time of that action is then recorded. When
+calculating the MAU figure, any users with a recorded action in the last 30 days are considered part of the cohort. Days are measured
+as a rolling window from the current system time to 30 days ago.
+
+So for example, if Synapse were to calculate the active users on the 15th July at 13:25, it would include any activity from 15th June 13:25 onwards.
+
+A user is **never** considered active if they are either:
+ - Part of the trial day cohort (described below)
+ - Owned by an application service.
+   - Note: This **only** covers users that are part of an application service `namespaces.users` registration. The namespace
+     must also be marked as `exclusive`.
+
+Otherwise, any request to Synapse will mark the user as active. Please note that registration will not mark a user as active *unless* 
+they register with a 3pid that is included in the config field `mau_limits_reserved_threepids`.
+
+The Prometheus metric for MAU is refreshed every 5 minutes.
+
+Once an hour, Synapse checks to see if any users are inactive (with only activity timestamps later than 30 days). These users
+are removed from the active users cohort. If they then become active, they are immediately restored to the cohort.
+
+It is important to note that **deactivated** users are not immediately removed from the pool of active users, but as these users won't
+perform actions they will eventually be removed from the cohort.
+
+### Trial days
+
+If the config option `mau_trial_days` is set, a user must have been active this many days **after** registration to be active. A user is in the
+trial period if their registration timestamp (also known as the `creation_ts`) is less than `mau_trial_days` old.
+
+As an example, if `mau_trial_days` is set to `3` and a user is active **after** 3 days (72 hours from registration time) then they will be counted as active.
+
+The `mau_appservice_trial_days` config further extends this rule by applying different durations depending on the `appservice_id` of the user.
+Users registered by an application service will be recorded with an `appservice_id` matching the `id` key in the registration file for that service.
+
+
+## Limiting usage of the homeserver when the maximum MAU is reached
+
+If both config options `limit_usage_by_mau` and `max_mau_value` is set, and the current MAU value exceeds the maximum value, the 
+homeserver will begin to block some actions.
+
+Individual users matching **any** of the below criteria never have their actions blocked:
+  - Considered part of the cohort of MAU users.
+  - Considered part of the trial period.
+  - Registered as a `support` user.
+  - Application service users if `track_appservice_user_ips` is NOT set.
+
+Please not that server admins are **not** exempt from blocking.
+
+The following actions are blocked when the MAU limit is exceeded:
+  - Logging in
+  - Sending events
+  - Creating rooms
+  - Syncing
+
+Registration is also blocked for all new signups *unless* the user is registering with a threepid included in the `mau_limits_reserved_threepids`
+config value.
+
+When a request is blocked, the response will have the `errcode` `M_RESOURCE_LIMIT_EXCEEDED`.
+
+## Metrics
+
+Synapse records several different prometheus metrics for MAU.
+
+`synapse_admin_mau:current` records the current MAU figure for native (non-application-service) users.
+
+`synapse_admin_mau:max` records the maximum MAU as dictated by the `max_mau_value` config value.
+
+`synapse_admin_mau_current_mau_by_service` records the current MAU including application service users. The label `app_service` can be used
+to filter by a specific service ID. This *also* includes non-application-service users under `app_service=native` .
+
+`synapse_admin_mau:registered_reserved_users` records the number of users specified in `mau_limits_reserved_threepids` which have
+registered accounts on the homeserver.

docs/usage/configuration/config_documentation.md

@@ -595,6 +595,8 @@ server owner wants to limit to the number of monthly active users. When enabled
 reached the server returns a `ResourceLimitError` with error type `Codes.RESOURCE_LIMIT_EXCEEDED`.
 Defaults to false. If this is enabled, a value for `max_mau_value` must also be set.
 
+See [Monthly Active Users](../administration/monthly_active_users.md) for details on how to configure MAU.
+
 Example configuration:
 ```yaml
 limit_usage_by_mau: true