diff --git a/python/ray/tests/test_advanced_2.py b/python/ray/tests/test_advanced_2.py
index e3a7128a6..81f9b1514 100644
--- a/python/ray/tests/test_advanced_2.py
+++ b/python/ray/tests/test_advanced_2.py
@@ -443,19 +443,17 @@ def test_multiple_raylets(ray_start_cluster):
 
 def test_custom_resources(ray_start_cluster):
     cluster = ray_start_cluster
-    cluster.add_node(num_cpus=3, resources={"CustomResource": 0})
+    cluster.add_node(num_cpus=1, resources={"CustomResource": 0})
     custom_resource_node = cluster.add_node(
-        num_cpus=3, resources={"CustomResource": 1})
+        num_cpus=1, resources={"CustomResource": 1})
     ray.init(address=cluster.address)
 
     @ray.remote
     def f():
-        time.sleep(0.001)
         return ray.worker.global_worker.node.unique_id
 
     @ray.remote(resources={"CustomResource": 1})
     def g():
-        time.sleep(0.001)
         return ray.worker.global_worker.node.unique_id
 
     @ray.remote(resources={"CustomResource": 1})
@@ -463,9 +461,6 @@ def test_custom_resources(ray_start_cluster):
         ray.get([f.remote() for _ in range(5)])
         return ray.worker.global_worker.node.unique_id
 
-    # The f tasks should be scheduled on both raylets.
-    assert len(set(ray.get([f.remote() for _ in range(500)]))) == 2
-
     # The g tasks should be scheduled only on the second raylet.
     raylet_ids = set(ray.get([g.remote() for _ in range(50)]))
     assert len(raylet_ids) == 1
diff --git a/python/ray/tests/test_advanced_3.py b/python/ray/tests/test_advanced_3.py
index 089dd1671..aae6bc4aa 100644
--- a/python/ray/tests/test_advanced_3.py
+++ b/python/ray/tests/test_advanced_3.py
@@ -2,11 +2,9 @@
 import glob
 import logging
 import os
-import shutil
 import json
 import sys
 import socket
-import tempfile
 import time
 
 import numpy as np
@@ -416,24 +414,6 @@ def test_ray_stack(ray_start_2_cpus):
                         "'ray stack'")
 
 
-def test_pandas_parquet_serialization():
-    # Only test this if pandas is installed
-    pytest.importorskip("pandas")
-
-    import pandas as pd
-    import pyarrow as pa
-    import pyarrow.parquet as pq
-
-    tempdir = tempfile.mkdtemp()
-    filename = os.path.join(tempdir, "parquet-test")
-    pd.DataFrame({"col1": [0, 1], "col2": [0, 1]}).to_parquet(filename)
-    with open(os.path.join(tempdir, "parquet-compression"), "wb") as f:
-        table = pa.Table.from_arrays([pa.array([1, 2, 3])], ["hello"])
-        pq.write_table(table, f, compression="lz4")
-    # Clean up
-    shutil.rmtree(tempdir)
-
-
 def test_socket_dir_not_existing(shutdown_only):
     if sys.platform != "win32":
         random_name = ray.ObjectRef.from_random().hex()
diff --git a/python/ray/tests/test_dynres.py b/python/ray/tests/test_dynres.py
index dda0872f8..1470f8919 100644
--- a/python/ray/tests/test_dynres.py
+++ b/python/ray/tests/test_dynres.py
@@ -286,14 +286,15 @@ def test_dynamic_res_deletion_scheduler_consistency(ray_start_cluster):
     target_node_id = node_ids[1]
     ray.get(set_res.remote(res_name, res_capacity, target_node_id))
 
-    def check_resources():
-        return ray.cluster_resources().get(res_name, None) == res_capacity
-
-    wait_for_condition(check_resources)
+    wait_for_condition(
+        lambda: ray.cluster_resources().get(res_name, None) == res_capacity)
 
     # Delete the resource
     ray.get(delete_res.remote(res_name, target_node_id))
 
+    wait_for_condition(
+        lambda: ray.cluster_resources().get(res_name, None) is None)
+
     # Define a task which requires this resource. This should not run
     @ray.remote(resources={res_name: res_capacity})
     def test_func():
diff --git a/python/ray/tests/test_output.py b/python/ray/tests/test_output.py
index 29cfecfcb..5b0e986bf 100644
--- a/python/ray/tests/test_output.py
+++ b/python/ray/tests/test_output.py
@@ -6,11 +6,12 @@ import ray
 
 
 def test_output():
+    # Use subprocess to execute the __main__ below.
     outputs = subprocess.check_output(
         [sys.executable, __file__, "_ray_instance"],
         stderr=subprocess.STDOUT).decode()
     lines = outputs.split("\n")
-    assert len(lines) == 3
+    assert len(lines) == 3, lines
     logging_header = r"\d{4}-\d{2}-\d{2} \d{2}:\d{2}:\d{2},\d{3}\sINFO\s"
     assert re.match(
         logging_header + r"resource_spec.py:\d+ -- Starting Ray with [0-9\.]+ "
@@ -24,7 +25,7 @@ def test_output():
 
 if __name__ == "__main__":
     if len(sys.argv) > 1 and sys.argv[1] == "_ray_instance":
-        ray.init()
+        ray.init(num_cpus=1)
         ray.shutdown()
     else:
         sys.exit(pytest.main(["-v", __file__]))