prometheus: Expose 3 activity metrics

This exposes some of the info from /api/status in the prometheus
metrics endpoint, so they can be scraped over time and queried.

- Server started timestamp, so we can easily see how long servers
  have been running
- Last activity timestamp, so we can see when users were actually
  active
- User activity duration (in seconds), so we can more easily build
  reports of how long users are actually active over time.

All these would be very helpful in JupyterHub environments.

Follow-up to #1470

jupyter_server/prometheus/metrics.py

@@ -58,6 +58,9 @@
     "Jupyter Server Extensiom Version Information",
     ["name", "version", "enabled"],
 )
+LAST_ACTIVITY = Gauge("jupyter_server_last_activity_timestamp_seconds", "Timestamp of last seen activity on this Jupyter Server")
+SERVER_STARTED = Gauge("jupyter_server_started_timestamp_seconds", "Timestamp of when this Jupyter Server was started")
+ACTIVE_DURATION = Gauge("jupyter_server_active_duration_seconds", "Number of seconds this Jupyter Server has been active")
 
 __all__ = [
     "HTTP_REQUEST_DURATION_SECONDS",

jupyter_server/serverapp.py

@@ -110,7 +110,7 @@
     GatewaySessionManager,
 )
 from jupyter_server.log import log_request
-from jupyter_server.prometheus.metrics import SERVER_EXTENSION_INFO, SERVER_INFO
+from jupyter_server.prometheus.metrics import SERVER_EXTENSION_INFO, SERVER_INFO, SERVER_STARTED, LAST_ACTIVITY, ACTIVE_DURATION
 from jupyter_server.services.config import ConfigManager
 from jupyter_server.services.contents.filemanager import (
     AsyncFileContentsManager,
@@ -2708,6 +2708,12 @@ def init_metrics(self) -> None:
                 name=ext.name, version=ext.version, enabled=str(ext.enabled).lower()
             )
 
+        started = self.web_app.settings["started"]
+        SERVER_STARTED.set(started.timestamp())
+
+        LAST_ACTIVITY.set_function(lambda: self.web_app.last_activity().timestamp())
+        ACTIVE_DURATION.set_function(lambda: (self.web_app.last_activity() - self.web_app.settings["started"]).total_seconds())
+
     @catch_config_error
     def initialize(
         self,