diff --git a/distributed/active_memory_manager.py b/distributed/active_memory_manager.py
index bb918a24ff..f4df05c181 100644
--- a/distributed/active_memory_manager.py
+++ b/distributed/active_memory_manager.py
@@ -13,6 +13,8 @@
 from collections.abc import Generator
 from typing import TYPE_CHECKING, Any, Literal, NamedTuple, Union
 
+from tornado.ioloop import PeriodicCallback
+
 import dask
 from dask.typing import Key
 from dask.utils import parse_timedelta
@@ -22,7 +24,6 @@
 # https://github.com/agronholm/sphinx-autodoc-typehints#dealing-with-circular-imports
 from distributed import client
 from distributed import scheduler as scheduler_module
-from distributed.compatibility import PeriodicCallback
 from distributed.core import Status
 from distributed.metrics import time
 from distributed.utils import import_term, log_errors
diff --git a/distributed/client.py b/distributed/client.py
index 0d41254a04..bdbc693c29 100644
--- a/distributed/client.py
+++ b/distributed/client.py
@@ -50,7 +50,7 @@
 from packaging.version import parse as parse_version
 from tlz import first, groupby, merge, partition_all, valmap
 from tornado import gen
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask
 from dask._expr import Expr, HLGExpr, LLGExpr
@@ -76,7 +76,6 @@
 from distributed import versions as version_module
 from distributed.batched import BatchedSend
 from distributed.cfexecutor import ClientExecutor
-from distributed.compatibility import PeriodicCallback
 from distributed.core import (
     CommClosedError,
     ConnectionPool,
diff --git a/distributed/compatibility.py b/distributed/compatibility.py
index 3ec4c9212e..dfd3005fb6 100644
--- a/distributed/compatibility.py
+++ b/distributed/compatibility.py
@@ -2,15 +2,12 @@
 
 import asyncio
 import logging
-import random
 import sys
 import warnings
 from collections.abc import Callable, Coroutine
 from typing import Any, TypeVar
 
-import tornado
-
-__all__ = ["logging_names", "PeriodicCallback", "to_thread", "randbytes"]
+__all__ = ["logging_names"]
 
 logging_names: dict[str | int, int | str] = {}
 logging_names.update(logging._levelToName)  # type: ignore
@@ -20,172 +17,6 @@
 MACOS = sys.platform == "darwin"
 WINDOWS = sys.platform == "win32"
 
-
-def to_thread(*args, **kwargs):
-    warnings.warn(
-        "to_thread is deprecated and will be removed in a future release; use asyncio.to_thread instead.",
-        DeprecationWarning,
-        stacklevel=2,
-    )
-    return asyncio.to_thread(*args, **kwargs)
-
-
-def randbytes(*args, **kwargs):
-    warnings.warn(
-        "randbytes is deprecated and will be removed in a future release; use random.randbytes instead.",
-        DeprecationWarning,
-        stacklevel=2,
-    )
-    return random.randbytes(*args, **kwargs)
-
-
-if tornado.version_info >= (6, 2, 0, 0):
-    from tornado.ioloop import PeriodicCallback
-else:
-    # Backport from https://github.com/tornadoweb/tornado/blob/a4f08a31a348445094d1efa17880ed5472db9f7d/tornado/ioloop.py#L838-L962
-    # License https://github.com/tornadoweb/tornado/blob/v6.2.0/LICENSE
-    # Includes minor modifications to source code to pass linting
-
-    # This backport ensures that async callbacks are not overlapping if a run
-    # takes longer than the interval
-    import datetime
-    import math
-    from collections.abc import Awaitable
-    from inspect import isawaitable
-
-    from tornado.ioloop import IOLoop
-    from tornado.log import app_log
-
-    class PeriodicCallback:  # type: ignore[no-redef]
-        """Schedules the given callback to be called periodically.
-
-        The callback is called every ``callback_time`` milliseconds when
-        ``callback_time`` is a float. Note that the timeout is given in
-        milliseconds, while most other time-related functions in Tornado use
-        seconds. ``callback_time`` may alternatively be given as a
-        `datetime.timedelta` object.
-
-        If ``jitter`` is specified, each callback time will be randomly selected
-        within a window of ``jitter * callback_time`` milliseconds.
-        Jitter can be used to reduce alignment of events with similar periods.
-        A jitter of 0.1 means allowing a 10% variation in callback time.
-        The window is centered on ``callback_time`` so the total number of calls
-        within a given interval should not be significantly affected by adding
-        jitter.
-
-        If the callback runs for longer than ``callback_time`` milliseconds,
-        subsequent invocations will be skipped to get back on schedule.
-
-        `start` must be called after the `PeriodicCallback` is created.
-
-        .. versionchanged:: 5.0
-        The ``io_loop`` argument (deprecated since version 4.1) has been removed.
-
-        .. versionchanged:: 5.1
-        The ``jitter`` argument is added.
-
-        .. versionchanged:: 6.2
-        If the ``callback`` argument is a coroutine, and a callback runs for
-        longer than ``callback_time``, subsequent invocations will be skipped.
-        Previously this was only true for regular functions, not coroutines,
-        which were "fire-and-forget" for `PeriodicCallback`.
-
-        The ``callback_time`` argument now accepts `datetime.timedelta` objects,
-        in addition to the previous numeric milliseconds.
-        """
-
-        def __init__(
-            self,
-            callback: Callable[[], Awaitable | None],
-            callback_time: datetime.timedelta | float,
-            jitter: float = 0,
-        ) -> None:
-            self.callback = callback
-            if isinstance(callback_time, datetime.timedelta):
-                self.callback_time = callback_time / datetime.timedelta(milliseconds=1)
-            else:
-                if callback_time <= 0:
-                    raise ValueError(
-                        "Periodic callback must have a positive callback_time"
-                    )
-                self.callback_time = callback_time
-            self.jitter = jitter
-            self._running = False
-            self._timeout = None  # type: object
-
-        def start(self) -> None:
-            """Starts the timer."""
-            # Looking up the IOLoop here allows to first instantiate the
-            # PeriodicCallback in another thread, then start it using
-            # IOLoop.add_callback().
-            self.io_loop = IOLoop.current()
-            self._running = True
-            self._next_timeout = self.io_loop.time()
-            self._schedule_next()
-
-        def stop(self) -> None:
-            """Stops the timer."""
-            self._running = False
-            if self._timeout is not None:
-                self.io_loop.remove_timeout(self._timeout)
-                self._timeout = None
-
-        def is_running(self) -> bool:
-            """Returns ``True`` if this `.PeriodicCallback` has been started.
-
-            .. versionadded:: 4.1
-            """
-            return self._running
-
-        async def _run(self) -> None:
-            if not self._running:
-                return
-            try:
-                val = self.callback()
-                if val is not None and isawaitable(val):
-                    await val
-            except Exception:
-                app_log.error("Exception in callback %r", self.callback, exc_info=True)
-            finally:
-                self._schedule_next()
-
-        def _schedule_next(self) -> None:
-            if self._running:
-                self._update_next(self.io_loop.time())
-                self._timeout = self.io_loop.add_timeout(self._next_timeout, self._run)
-
-        def _update_next(self, current_time: float) -> None:
-            callback_time_sec = self.callback_time / 1000.0
-            if self.jitter:
-                # apply jitter fraction
-                callback_time_sec *= 1 + (self.jitter * (random.random() - 0.5))
-            if self._next_timeout <= current_time:
-                # The period should be measured from the start of one call
-                # to the start of the next. If one call takes too long,
-                # skip cycles to get back to a multiple of the original
-                # schedule.
-                self._next_timeout += (
-                    math.floor((current_time - self._next_timeout) / callback_time_sec)
-                    + 1
-                ) * callback_time_sec
-            else:
-                # If the clock moved backwards, ensure we advance the next
-                # timeout instead of recomputing the same value again.
-                # This may result in long gaps between callbacks if the
-                # clock jumps backwards by a lot, but the far more common
-                # scenario is a small NTP adjustment that should just be
-                # ignored.
-                #
-                # Note that on some systems if time.time() runs slower
-                # than time.monotonic() (most common on windows), we
-                # effectively experience a small backwards time jump on
-                # every iteration because PeriodicCallback uses
-                # time.time() while asyncio schedules callbacks using
-                # time.monotonic().
-                # https://github.com/tornadoweb/tornado/issues/2333
-                self._next_timeout += callback_time_sec
-
-
 _T = TypeVar("_T")
 
 if sys.version_info >= (3, 12):
@@ -265,3 +96,17 @@ def _cancel_all_tasks(loop: asyncio.AbstractEventLoop) -> None:
                         "task": task,
                     }
                 )
+
+
+def __getattr__(name: str) -> Any:
+    if name == "PeriodicCallback":
+        warnings.warn(
+            "distributed.compatibility.PeriodicCallback is deprecated and will be removed "
+            "in a future release; use tornado.ioloop.PeriodicCallback instead.",
+            DeprecationWarning,
+            stacklevel=2,
+        )
+        from tornado.ioloop import PeriodicCallback
+
+        return PeriodicCallback
+    raise AttributeError(f"module {__name__!r} has no attribute {name!r}")
diff --git a/distributed/core.py b/distributed/core.py
index dc5f7733fe..e9cc387486 100644
--- a/distributed/core.py
+++ b/distributed/core.py
@@ -29,7 +29,7 @@
 
 import tblib
 from tlz import merge
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask
 from dask.utils import parse_timedelta
@@ -46,7 +46,6 @@
     unparse_host_port,
 )
 from distributed.comm.core import Listener
-from distributed.compatibility import PeriodicCallback
 from distributed.counter import Counter
 from distributed.diskutils import WorkDir, WorkSpace
 from distributed.metrics import context_meter, time
diff --git a/distributed/deploy/adaptive.py b/distributed/deploy/adaptive.py
index 406928b1b0..0b16553945 100644
--- a/distributed/deploy/adaptive.py
+++ b/distributed/deploy/adaptive.py
@@ -6,12 +6,11 @@
 from inspect import isawaitable
 from typing import TYPE_CHECKING, Any, Literal, cast
 
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask.config
 from dask.utils import parse_timedelta
 
-from distributed.compatibility import PeriodicCallback
 from distributed.core import Status
 from distributed.deploy.adaptive_core import AdaptiveCore
 from distributed.protocol import pickle
diff --git a/distributed/deploy/cluster.py b/distributed/deploy/cluster.py
index 7b68f67114..fe283faaa3 100644
--- a/distributed/deploy/cluster.py
+++ b/distributed/deploy/cluster.py
@@ -9,13 +9,12 @@
 from typing import Any
 
 from packaging.version import parse as parse_version
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask.config
 from dask.utils import _deprecated, format_bytes, parse_timedelta, typename
 from dask.widgets import get_template
 
-from distributed.compatibility import PeriodicCallback
 from distributed.core import Status
 from distributed.deploy.adaptive import Adaptive
 from distributed.exceptions import WorkerStartTimeoutError
diff --git a/distributed/diagnostics/memory_sampler.py b/distributed/diagnostics/memory_sampler.py
index 9ee5cc7726..98e6913eb3 100644
--- a/distributed/diagnostics/memory_sampler.py
+++ b/distributed/diagnostics/memory_sampler.py
@@ -6,7 +6,7 @@
 from datetime import datetime
 from typing import TYPE_CHECKING, Any
 
-from distributed.compatibility import PeriodicCallback
+from tornado.ioloop import PeriodicCallback
 
 if TYPE_CHECKING:
     # Optional runtime dependencies
diff --git a/distributed/scheduler.py b/distributed/scheduler.py
index b7c5aabacb..45f93600cf 100644
--- a/distributed/scheduler.py
+++ b/distributed/scheduler.py
@@ -59,7 +59,7 @@
     take,
     valmap,
 )
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask
 from dask._expr import LLGExpr
@@ -96,7 +96,6 @@
     unparse_host_port,
 )
 from distributed.comm.addressing import addresses_from_user_args
-from distributed.compatibility import PeriodicCallback
 from distributed.core import (
     ErrorMessage,
     OKMessage,
diff --git a/distributed/semaphore.py b/distributed/semaphore.py
index 1a53a03758..3ede6d8b61 100644
--- a/distributed/semaphore.py
+++ b/distributed/semaphore.py
@@ -7,10 +7,11 @@
 from asyncio import TimeoutError
 from collections import defaultdict, deque
 
+from tornado.ioloop import PeriodicCallback
+
 import dask
 from dask.utils import parse_timedelta
 
-from distributed.compatibility import PeriodicCallback
 from distributed.metrics import time
 from distributed.utils import Deadline, SyncMethodMixin, log_errors, wait_for
 from distributed.utils_comm import retry_operation
diff --git a/distributed/stealing.py b/distributed/stealing.py
index 63b4a31f50..cc6f1e6d65 100644
--- a/distributed/stealing.py
+++ b/distributed/stealing.py
@@ -10,12 +10,12 @@
 from typing import TYPE_CHECKING, Any, ClassVar, TypedDict, cast
 
 from tlz import topk
+from tornado.ioloop import PeriodicCallback
 
 import dask
 from dask.typing import Key
 from dask.utils import parse_timedelta
 
-from distributed.compatibility import PeriodicCallback
 from distributed.core import CommClosedError
 from distributed.diagnostics.plugin import SchedulerPlugin
 from distributed.utils import log_errors, recursive_to_dict
diff --git a/distributed/tests/test_compatibility.py b/distributed/tests/test_compatibility.py
index b6368ee098..b5b9e4c5b1 100644
--- a/distributed/tests/test_compatibility.py
+++ b/distributed/tests/test_compatibility.py
@@ -1,36 +1,9 @@
-from __future__ import annotations
-
-import random
-from collections import Counter
-
 import pytest
 
-from distributed.compatibility import randbytes
-
-
-def test_randbytes():
-    with pytest.warns(
-        DeprecationWarning,
-        match=r"randbytes is deprecated and will be removed in a future release; "
-        r"use random\.randbytes instead\.",
-    ):
-        x = randbytes(256_000)
-    assert isinstance(x, bytes)
-    assert len(x) == 256_000
-    c = Counter(x)
-    for i in range(256):
-        assert 800 < c[i] < 1200, (i, c[i])
 
+def test_deprecated_periodic_callback():
+    from tornado.ioloop import PeriodicCallback
 
-def test_randbytes_seed():
-    state = random.getstate()
-    try:
-        random.seed(0)
-        with pytest.warns(
-            DeprecationWarning,
-            match=r"randbytes is deprecated and will be removed in a future release; "
-            r"use random\.randbytes instead\.",
-        ):
-            assert randbytes(8) == b"\xcd\x07,\xd8\xbeo\x9fb"
-    finally:
-        random.setstate(state)
+    with pytest.warns(DeprecationWarning):
+        from distributed.compatibility import PeriodicCallback as compat
+    assert compat is PeriodicCallback
diff --git a/distributed/tests/test_scheduler.py b/distributed/tests/test_scheduler.py
index e2f714e1c6..b6047c80b8 100644
--- a/distributed/tests/test_scheduler.py
+++ b/distributed/tests/test_scheduler.py
@@ -20,7 +20,7 @@
 import psutil
 import pytest
 from tlz import concat, first, merge
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask
 from dask import bag, delayed
@@ -41,7 +41,7 @@
     wait,
 )
 from distributed.comm.addressing import parse_host_port
-from distributed.compatibility import LINUX, MACOS, WINDOWS, PeriodicCallback
+from distributed.compatibility import LINUX, MACOS, WINDOWS
 from distributed.core import ConnectionPool, Status, clean_exception, connect, rpc
 from distributed.metrics import time
 from distributed.protocol import serialize
diff --git a/distributed/worker.py b/distributed/worker.py
index 79f24b22da..9a1c1ed3e8 100644
--- a/distributed/worker.py
+++ b/distributed/worker.py
@@ -42,7 +42,7 @@
 )
 
 from tlz import keymap, pluck
-from tornado.ioloop import IOLoop
+from tornado.ioloop import IOLoop, PeriodicCallback
 
 import dask
 from dask._task_spec import GraphNode
@@ -64,7 +64,6 @@
 from distributed.comm import Comm, connect, get_address_host, parse_address
 from distributed.comm import resolve_address as comm_resolve_address
 from distributed.comm.addressing import address_from_user_args
-from distributed.compatibility import PeriodicCallback
 from distributed.core import (
     ConnectionPool,
     ErrorMessage,
diff --git a/distributed/worker_memory.py b/distributed/worker_memory.py
index 4f0c75d499..b92b005c20 100644
--- a/distributed/worker_memory.py
+++ b/distributed/worker_memory.py
@@ -32,6 +32,7 @@
 from typing import TYPE_CHECKING, Any, Literal, cast
 
 import psutil
+from tornado.ioloop import PeriodicCallback
 
 import dask.config
 from dask.system import CPU_COUNT
@@ -39,7 +40,7 @@
 from dask.utils import format_bytes, parse_bytes, parse_timedelta
 
 from distributed import system
-from distributed.compatibility import WINDOWS, PeriodicCallback
+from distributed.compatibility import WINDOWS
 from distributed.core import Status
 from distributed.gc import ThrottledGC
 from distributed.metrics import context_meter, monotonic