Make starting pushers faster during start up

We start all pushers on start up and immediately start a background
process to fetch push to send. This makes start up incredibly painful
when dealing with many pushers.

Instead, let's do a quick fast DB check to see if there *may* be push to
send and only start the background processes for those pushers. We also
stagger starting up and doing those checks so that we don't try and
handle all pushers at once.

synapse/storage/event_push_actions.py

@@ -386,6 +386,36 @@ class EventPushActionsWorkerStore(SQLBaseStore):
         # Now return the first `limit`
         defer.returnValue(notifs[:limit])
 
+    def get_if_maybe_push_in_range_for_user(self, user_id, min_stream_ordering):
+        """A fast check to see if there might be something to push for the
+        user since the given stream ordering. May return false positives.
+
+        Useful to know whether to bother starting a pusher on start up or not.
+
+        Args:
+            user_id (str)
+            min_stream_ordering (int)
+
+        Returns:
+            Deferred[bool]: True if there may be push to process, False if
+            there definitely isn't.
+        """
+
+        def _get_if_maybe_push_in_range_for_user_txn(txn):
+            sql = """
+                SELECT 1 FROM event_push_actions
+                WHERE user_id = ? AND stream_ordering > ?
+                LIMIT 1
+            """
+
+            txn.execute(sql, (user_id, min_stream_ordering,))
+            return bool(txn.fetchone())
+
+        return self.runInteraction(
+            "get_if_maybe_push_in_range_for_user",
+            _get_if_maybe_push_in_range_for_user_txn,
+        )
+
     def add_push_actions_to_staging(self, event_id, user_id_actions):
         """Add the push actions for the event to the push action staging area.