diff --git a/ouroboros-consensus/src/ouroboros-consensus/Ouroboros/Consensus/Util/LeakyBucket.hs b/ouroboros-consensus/src/ouroboros-consensus/Ouroboros/Consensus/Util/LeakyBucket.hs
index 403ca832c5..01ae4a07b9 100644
--- a/ouroboros-consensus/src/ouroboros-consensus/Ouroboros/Consensus/Util/LeakyBucket.hs
+++ b/ouroboros-consensus/src/ouroboros-consensus/Ouroboros/Consensus/Util/LeakyBucket.hs
@@ -219,28 +219,28 @@ runAgainstBucket ::
   (Handlers m -> m a) ->
   m (State m, a)
 runAgainstBucket config action = do
-  runThreadVar <- atomically newEmptyTMVar -- see note [Leaky bucket design].
+  leakingPeriodVersionTMVar <- atomically newEmptyTMVar -- see note [Leaky bucket design].
   tid <- myThreadId
   bucket <- init config
-  withAsync (leak runThreadVar tid bucket) $ \_ -> do
-    atomicallyWithMonotonicTime $ maybeStartThread Nothing runThreadVar bucket
+  withAsync (leak (readTMVar leakingPeriodVersionTMVar) tid bucket) $ \_ -> do
+    atomicallyWithMonotonicTime $ maybeStartThread Nothing leakingPeriodVersionTMVar bucket
     result <-
       action $
         Handlers
           { fill = \r t -> (snd <$>) $ snapshotFill bucket r t,
             setPaused = setPaused bucket,
-            updateConfig = updateConfig runThreadVar bucket
+            updateConfig = updateConfig leakingPeriodVersionTMVar bucket
           }
     state <- atomicallyWithMonotonicTime $ snapshot bucket
     pure (state, result)
   where
-    -- Start the thread (that is, write to its 'runThreadVar') if it is useful.
-    -- Takes a potential old value of the 'runThreadVar' as first argument,
+    -- Start the thread (that is, write to its 'leakingPeriodVersionTMVar') if it is useful.
+    -- Takes a potential old value of the 'leakingPeriodVersionTMVar' as first argument,
     -- which will be increased to help differentiate between restarts.
     maybeStartThread :: Maybe Int -> StrictTMVar m Int -> Bucket m -> Time -> STM m ()
-    maybeStartThread oldRunThread runThreadVar bucket time = do
+    maybeStartThread mLeakingPeriodVersion leakingPeriodVersionTMVar bucket time = do
       State {config = Config {rate}} <- snapshot bucket time
-      when (rate > 0) $ void $ tryPutTMVar runThreadVar $ maybe 0 (+ 1) oldRunThread
+      when (rate > 0) $ void $ tryPutTMVar leakingPeriodVersionTMVar $ maybe 0 (+ 1) mLeakingPeriodVersion
 
     setPaused :: Bucket m -> Bool -> Time -> STM m ()
     setPaused bucket paused time = do
@@ -253,7 +253,7 @@ runAgainstBucket config action = do
       ((Rational, Config m) -> (Rational, Config m)) ->
       Time ->
       STM m ()
-    updateConfig runThreadVar bucket f time = do
+    updateConfig leakingPeriodVersionTMVar bucket f time = do
       State
         { level = oldLevel,
           paused,
@@ -272,9 +272,9 @@ runAgainstBucket config action = do
             configGeneration = oldConfigGeneration + 1,
             config = newConfig
           }
-      -- Ensure that 'runThreadVar' is empty, then maybe start the thread.
-      oldRunThread <- tryTakeTMVar runThreadVar
-      maybeStartThread oldRunThread runThreadVar bucket time
+      -- Ensure that 'leakingPeriodVersionTMVar' is empty, then maybe start the thread.
+      mLeakingPeriodVersion <- tryTakeTMVar leakingPeriodVersionTMVar
+      maybeStartThread mLeakingPeriodVersion leakingPeriodVersionTMVar bucket time
 
 -- | Initialise a bucket given a configuration. The bucket starts full at the
 -- time where one calls 'init'.
@@ -299,7 +299,7 @@ init config@Config {capacity} = do
 -- ~~~~~~~~~~~~~~~~~~~~~~~~~~
 --
 -- The leaky bucket works by running the given action against a thread that
--- makes the bucket leak. Since that would be extremely inefficient to actually
+-- makes the bucket leak. Since it would be inefficient to actually
 -- remove tokens one by one from the bucket, the 'leak' thread instead looks at
 -- the current state of the bucket, computes how much time it would take for the
 -- bucket to empty, and then wait that amount of time. Once the wait is over, it
@@ -317,22 +317,21 @@ init config@Config {capacity} = do
 -- for the action to lower the waiting time by changing the bucket configuration
 -- to one where the rate is higher.
 --
--- We fix both those issues with one mechanism, the @runThreadVar@. It is an
--- MVar containing an integer that tells the thread whether it should be
--- running. An empty MVar means that the thread should not be running, for
--- instance if the rate is null. A full MVar (no matter what the integer is)
--- means that the thread should be running. When recursing, the thread blocks
--- until the MVar is full, and only then proceeds as described above.
+-- We fix both those issues with one mechanism, the @leakingPeriodVersionSTM@.
+-- It is a computation returning an integer that identifies a version of the
+-- configuration that controls the leaking period. If the computation blocks,
+-- it means that no configuration has been determined yet.
+-- The leak thread first waits until @leakingPeriodVersionSTM@ yields a
+-- value, and only then proceeds as described above.
 -- Additionally, while waiting for the bucket to empty, the thread monitors
--- changes to the MVar, indicating either that the thread should stop running or
--- that the configuration changed as that it might have to wait less long. The
--- change in configuration is detected by changes in the integer.
+-- for changes to the version of the leaking period, indicating either that the
+-- thread should pause running if the @leakingPeriodVersionSTM@ starts blocking
+-- again or that the configuration changed as that it might have to wait less
+-- long.
 --
--- Note that we call \“start\”/\“stop\” running the action of filling/emptying the
--- MVar. This is not to mistaken for the thread actually being spawned/killed.
 
--- | Monadic action that calls 'threadDelay' until the bucket is empty, then
--- runs the 'onEmpty' action and terminates. See note [Leaky bucket design].
+-- | Neverending computation that runs 'onEmpty' whenever the bucket becomes
+-- empty. See note [Leaky bucket design].
 leak ::
   ( MonadDelay m,
     MonadCatch m,
@@ -340,18 +339,20 @@ leak ::
     MonadAsync m,
     MonadTimer m
   ) =>
-  -- | A variable indicating whether the thread should run (when it is filled)
-  -- or not (otherwise). The integer it carries only helps in differentiating
-  -- between starts and restarts. 'leak' does not modify this variable.
-  StrictTMVar m Int ->
+  -- | A computation indicating the version of the configuration affecting the
+  -- leaking period. Whenever the configuration changes, the returned integer
+  -- must be incremented. While no configuration is available, the computation
+  -- should block. Blocking is allowed at any time, and it will cause the
+  -- leaking to pause.
+  STM m Int ->
   -- | The 'ThreadId' of the action's thread, which is used to throw exceptions
   -- at it.
   ThreadId m ->
   Bucket m ->
   m ()
-leak runThreadVar actionThreadId bucket = forever $ do
-      -- Block until we are allowed to run. Do not modify the TMVar.
-      oldRunThread <- atomically $ readTMVar runThreadVar
+leak leakingPeriodVersionSTM actionThreadId bucket = forever $ do
+      -- Block until we are allowed to run.
+      leakingPeriodVersion <- atomically leakingPeriodVersionSTM
       -- NOTE: It is tempting to group this @atomically@ and
       -- @atomicallyWithMonotonicTime@ into one; however, because the former is
       -- blocking, the latter could get a _very_ inaccurate time, which we
@@ -377,7 +378,7 @@ leak runThreadVar actionThreadId bucket = forever $ do
             atomically $
               (check =<< TVar.readTVar varTimeout)
                 `orElse`
-              (void $ blockUntilChanged id (Just oldRunThread) $ tryReadTMVar runThreadVar)
+              (void $ blockUntilChanged id leakingPeriodVersion leakingPeriodVersionSTM)
 
 -- | Take a snapshot of the bucket, that is compute its state at the current
 -- time.