diff --git a/testsuite/expect/test1.35 b/testsuite/expect/test1.35
index f01045b66b1a6098adc1f84e51eebbf9178a49c8..5ab95693268262bf72191adc96fe3cab68c4746b 100755
--- a/testsuite/expect/test1.35
+++ b/testsuite/expect/test1.35
@@ -57,11 +57,15 @@ if {[test_cray]} {
 # Build input script file
 # Sleep between initiation of job steps to avoid overwhelming slurmd
 #
+# NOTE: Explicity set a small memory limit. Without explicitly setting the step
+#   memory limit, it will use the system default (same as the job) which may
+#   prevent the level of parallelism desired.
+#
 exec $bin_rm -f $file_in $file_out $file_err
 make_bash_script $file_in "
   for ((i = 0; i < $steps_started; i++)); do
     j=`expr $steps_started + 15 - \$i`
-    $srun $bin_sleep \$j &
+    $srun -N1 -n1 --mem-per-cpu=1 $bin_sleep \$j &
     $bin_sleep 1
   done
   $bin_sleep 2
diff --git a/testsuite/expect/test9.8 b/testsuite/expect/test9.8
index 288e821dc3f2dd237f00614cc48b43f19e8d44dc..4a0434e986972312f288da98ab37ddd6453ff381 100755
--- a/testsuite/expect/test9.8
+++ b/testsuite/expect/test9.8
@@ -64,14 +64,17 @@ if {[test_front_end] != 0 || $enable_memory_leak_debug != 0} {
 # NOTE: The initial sleep is so that all of the submissions have time
 #   to occur before contending with a multitude of job step creations.
 #   This is especially important on very slow systems (e.g. AIX).
+# NOTE: Explicity set a small memory limit. Without explicitly setting the step
+#   memory limit, it will use the system default (same as the job) which may
+#   prevent the level of parallelism desired.
 #
 make_bash_script $file_in "
 $bin_sleep 5
 ulimit -u `ulimit -u -H`
 for ((inx=0; inx < $task_cnt; inx++)) ; do
-        $srun -N1 -n1 $bin_sleep $sleep_time &
+        $srun -N1 -n1 --mem-per-cpu=1 $bin_sleep $sleep_time &
 done
-$srun -N1 -n1 $bin_sleep $sleep_time
+$srun -N1 -n1 --mem-per-cpu=1 $bin_sleep $sleep_time
 "
 
 #