diff --git a/swh/scheduler/backend.py b/swh/scheduler/backend.py
--- a/swh/scheduler/backend.py
+++ b/swh/scheduler/backend.py
@@ -311,14 +311,30 @@
 
     @db_transaction()
     def grab_next_visits(
-        self, visit_type: str, count: int, policy: str, db=None, cur=None,
+        self,
+        visit_type: str,
+        count: int,
+        policy: str,
+        timestamp: Optional[datetime.datetime] = None,
+        db=None,
+        cur=None,
     ) -> List[ListedOrigin]:
         """Get at most the `count` next origins that need to be visited with
         the `visit_type` loader according to the given scheduling `policy`.
 
-        This will mark the origins as "being visited" in the listed_origins
+        This will mark the origins as scheduled in the origin_visit_stats
         table, to avoid scheduling multiple visits to the same origin.
+
+        Arguments:
+          visit_type: type of visits to schedule
+          count: number of visits to schedule
+          policy: the scheduling policy used to select which visits to schedule
+          timestamp: the mocked timestamp at which we're recording that the visits are
+            being scheduled (defaults to the current time)
         """
+        if timestamp is None:
+            timestamp = utcnow()
+
         origin_select_cols = ", ".join(ListedOrigin.select_columns())
 
         query_args: List[Any] = []
@@ -399,7 +415,7 @@
                   url, visit_type, last_scheduled
                 )
               SELECT
-                url, visit_type, now()
+                url, visit_type, %s
               FROM
                 selected_origins
               ON CONFLICT (url, visit_type) DO UPDATE
@@ -413,6 +429,7 @@
             FROM
               selected_origins
         """
+        query_args.append(timestamp)
 
         cur.execute(query, tuple(query_args))
         return [ListedOrigin(**d) for d in cur]
diff --git a/swh/scheduler/interface.py b/swh/scheduler/interface.py
--- a/swh/scheduler/interface.py
+++ b/swh/scheduler/interface.py
@@ -336,13 +336,24 @@
 
     @remote_api_endpoint("origins/grab_next")
     def grab_next_visits(
-        self, visit_type: str, count: int, policy: str
+        self,
+        visit_type: str,
+        count: int,
+        policy: str,
+        timestamp: Optional[datetime.datetime] = None,
     ) -> List[ListedOrigin]:
         """Get at most the `count` next origins that need to be visited with
         the `visit_type` loader according to the given scheduling `policy`.
 
-        This will mark the origins as "being visited" in the listed_origins
+        This will mark the origins as scheduled in the origin_visit_stats
         table, to avoid scheduling multiple visits to the same origin.
+
+        Arguments:
+          visit_type: type of visits to schedule
+          count: number of visits to schedule
+          policy: the scheduling policy used to select which visits to schedule
+          timestamp: the mocked timestamp at which we're recording that the visits are
+            being scheduled (defaults to the current time)
         """
         ...
 
diff --git a/swh/scheduler/simulator/origin_scheduler.py b/swh/scheduler/simulator/origin_scheduler.py
--- a/swh/scheduler/simulator/origin_scheduler.py
+++ b/swh/scheduler/simulator/origin_scheduler.py
@@ -29,7 +29,7 @@
             if remaining < min_batch_size:
                 continue
             next_origins = env.scheduler.grab_next_visits(
-                visit_type, remaining, policy=policy
+                visit_type, remaining, policy=policy, timestamp=env.time
             )
             logger.debug(
                 "%s runner: running %s %s tasks",