diff --git a/lhotse/cut/mono.py b/lhotse/cut/mono.py
index 75bbe23f7..0fa3f5abd 100644
--- a/lhotse/cut/mono.py
+++ b/lhotse/cut/mono.py
@@ -389,3 +389,13 @@ def from_dict(data: dict) -> "MonoCut":
             recording=recording,
             supervisions=[SupervisionSegment.from_dict(s) for s in supervision_infos],
         )
+
+    def __lt__(self, other):
+        token_length = len(self.supervisions[0].custom['tokens']['text'])
+        other_token_length = len(other.supervisions[0].custom['tokens']['text'])
+        if token_length - other_token_length >= 3:
+            return True
+        elif token_length - other_token_length <= -3:
+            return False
+        else:
+            return self.duration > other.duration
diff --git a/lhotse/dataset/sampling/base.py b/lhotse/dataset/sampling/base.py
index ca43b36fc..6e33e5f5f 100644
--- a/lhotse/dataset/sampling/base.py
+++ b/lhotse/dataset/sampling/base.py
@@ -437,6 +437,7 @@ class TimeConstraint(SamplingConstraint):
     current: Union[int, Seconds] = 0
     num_cuts: int = 0
     longest_seen: Union[int, float] = 0
+    longest_seen_text: Union[int, float] = 0
     quadratic_duration: Optional[Seconds] = None
 
     def __post_init__(self) -> None:
@@ -454,18 +455,23 @@ def add(self, example: Cut) -> None:
         selecting the right property from the input ``cut`` object.
         """
         if self.max_duration is not None:
-            duration = self._maybe_apply_quadratic_correction(example.duration)
-            self.current += duration
+            duration = self._maybe_apply_quadratic_correction(example.duration, quadratic_expand_value=1.5)
+            try:
+                text_duration = self._maybe_apply_quadratic_correction(len(example.supervisions[0].custom['tokens']['text']) * 0.046)
+            except:
+                text_duration = 0
+            self.current += duration + text_duration
             self.longest_seen = max(self.longest_seen, duration)
+            self.longest_seen_text = max(self.longest_seen_text, text_duration)
         self.num_cuts += 1
 
-    def _maybe_apply_quadratic_correction(self, duration: Seconds) -> Seconds:
+    def _maybe_apply_quadratic_correction(self, duration: Seconds, quadratic_expand_value: Seconds = 1.0) -> Seconds:
         if self.quadratic_duration is None:
             return duration
         # For the quadratic complexity case, we add a term that accounts for
         # extra memory occupied by the model. The 1/quadratic_duration term causes
         # the effective duration to be doubled when it's equal to quadratic_duration.
-        return duration + (duration**2) / self.quadratic_duration
+        return duration + ((duration * quadratic_expand_value)**2) / self.quadratic_duration
 
     def exceeded(self) -> bool:
         """Is the constraint exceeded or not."""
@@ -473,7 +479,7 @@ def exceeded(self) -> bool:
             return True
         if self.max_duration is None:
             return False
-        effective_duration = self.num_cuts * self.longest_seen
+        effective_duration = self.num_cuts * (self.longest_seen + self.longest_seen_text)
         return effective_duration > self.max_duration
 
     def close_to_exceeding(self) -> bool:
@@ -487,7 +493,7 @@ def close_to_exceeding(self) -> bool:
             return True
 
         if self.max_duration is not None:
-            effective_duration = (self.num_cuts + 1) * self.longest_seen
+            effective_duration = (self.num_cuts + 1) * (self.longest_seen + self.longest_seen_text)
             return effective_duration > self.max_duration
         return False
 
@@ -499,6 +505,7 @@ def reset(self) -> None:
         self.current = 0
         self.num_cuts = 0
         self.longest_seen = 0
+        self.longest_seen_text = 0
 
     def measure_length(self, example: Cut) -> float:
         return example.duration
@@ -512,6 +519,7 @@ def load_state_dict(self, state_dict: Dict[str, Any]) -> None:
         self.current = state_dict.pop("current")
         self.num_cuts = state_dict.pop("num_cuts")
         self.longest_seen = state_dict.pop("longest_seen", 0)
+        self.longest_seen_text = state_dict.pop("longest_seen_text", 0)
         self.quadratic_duration = state_dict.pop("quadratic_duration", None)
         # backward compatibility
         state_dict.pop("strict", None)
@@ -537,6 +545,7 @@ def __add__(self, other: "TimeConstraint") -> "TimeConstraint":
             current=self.current + other.current,
             num_cuts=self.num_cuts + other.num_cuts,
             longest_seen=max(self.longest_seen, other.longest_seen),
+            longest_seen_text=max(self.longest_seen_text, other.longest_seen_text),
             quadratic_duration=self.quadratic_duration,
         )
 
diff --git a/lhotse/dataset/sampling/dynamic_bucketing.py b/lhotse/dataset/sampling/dynamic_bucketing.py
index cf4da23f2..69f354529 100644
--- a/lhotse/dataset/sampling/dynamic_bucketing.py
+++ b/lhotse/dataset/sampling/dynamic_bucketing.py
@@ -18,6 +18,7 @@
     Tuple,
     Union,
 )
+import heapq
 
 import numpy as np
 import torch
@@ -458,8 +459,8 @@ def __init__(
                 )
 
         # Init: create empty buckets (note: `num_buckets = len(duration_bins) + 1`).
-        self.buckets: List[Deque[Union[Cut, Tuple[Cut, ...]]]] = [
-            deque() for _ in range(len(duration_bins) + 1)
+        self.buckets: List[List[Union[Cut, Tuple[Cut, ...], Tuple[int, Cut]]]] = [
+            [] for _ in range(len(duration_bins) + 1)
         ]
 
     def __iter__(self) -> Generator[CutSet, None, None]:
@@ -485,6 +486,10 @@ def __iter__(self) -> Generator[CutSet, None, None]:
                     maybe_shuffled = pick_at_random(
                         maybe_shuffled, rng=self.rng, out_indexes_used=indexes_used
                     )
+                else:
+                    maybe_shuffled = pick_at_min_heap(
+                        maybe_shuffled
+                    )
                 # Sample one batch from that bucket and yield it to the caller.
                 batcher = DurationBatcher(
                     maybe_shuffled,
@@ -504,9 +509,8 @@ def __iter__(self) -> Generator[CutSet, None, None]:
                     for idx in indexes_used:
                         del sampling_bucket[idx]
                 else:
-                    # No shuffling, remove first N
-                    for _ in range(batch_size):
-                        sampling_bucket.popleft()
+                    # already pop by heappop
+                    pass
                 # Fetch new cuts and add them to appropriate buckets.
                 self._collect_cuts_in_buckets(batch_size)
         except StopIteration:
@@ -609,7 +613,7 @@ def _collect_cuts_in_buckets(self, n_cuts: int) -> None:
                     cuts[0] if isinstance(cuts, tuple) else cuts
                 )
                 bucket_idx = bisect_right(self.duration_bins, duration)
-                self.buckets[bucket_idx].append(cuts)
+                heapq.heappush(self.buckets[bucket_idx], cuts)
         except StopIteration:
             pass
 
@@ -629,6 +633,12 @@ def pick_at_random(
         out_indexes_used.append(idx)
         yield bucket[idx]
 
+def pick_at_min_heap(
+    bucket: Sequence[Union[Cut, Tuple[Cut, ...]]],
+) -> Generator[Union[Cut, Tuple[Cut, ...]], None, None]:
+    while bucket:
+        yield heapq.heappop(bucket)
+
 
 class BucketsDontHaveEnoughData(Exception):
     pass
diff --git a/lhotse/shar/readers/lazy.py b/lhotse/shar/readers/lazy.py
index c1b0b74a3..3558866d0 100644
--- a/lhotse/shar/readers/lazy.py
+++ b/lhotse/shar/readers/lazy.py
@@ -239,7 +239,7 @@ def __iter__(self):
             map_fns = self._maybe_shuffle_shards(map_fns)
             map_fns = self._maybe_split_for_dataloading(map_fns)
 
-        for shard, cut_map_fn in zip(shards, map_fns):
+        for shard_idx, (shard, cut_map_fn) in enumerate(zip(shards, map_fns)):
             # Iterate over cuts for the current shard
             cuts = LazyManifestIterator(shard["cuts"])
 
@@ -249,33 +249,59 @@ def __iter__(self):
             }
 
             # Open every tarfile/jsonl so it's ready for streaming
-            field_iters = {
-                field: TarIterator(path)
-                if extension_contains(".tar", path)
-                else _jsonl_tar_adaptor(LazyJsonlIterator(path), field=field)
-                for field, path in field_paths.items()
-            }
+            field_iters = {}
+            for field, path in field_paths.items():
+                if path is None:
+                    pass
+                elif extension_contains(".tar", path):
+                    field_iters[field] = TarIterator(path)
+                else:
+                    for item, pseudo_path in _jsonl_tar_adaptor(LazyJsonlIterator(path), field=field):
+                        field_iters[field] = TarIterator(pseudo_path)
+            # field_iters = {
+            #     field: TarIterator(path)
+            #     if extension_contains(".tar", path)
+            #     else _jsonl_tar_adaptor(LazyJsonlIterator(path), field=field)
+            #     for field, path in field_paths.items()
+            # }
 
             # *field_data contains all fields for a single cut (recording, features, array, etc.)
-            for cut, *field_data in zip(cuts, *field_iters.values()):
-                for (field, (maybe_manifest, data_path)) in zip(
+            try:
+                for cut in self.generate_shard_cuts(cuts, field_iters):
+                    cut.shard_origin = shard["cuts"]
+                    cut.shar_epoch = self.epoch
+                    if cut_map_fn is not None:
+                        cut = cut_map_fn(cut)
+                    yield cut
+            except OSError as e:
+                print("WARNING(LazySharIterator): OSError: {}".format(e))
+            if shard_idx % 100 == 0:
+                print("INFO(LazySharIterator): EPOCH({}) Shard {}/{} done".format(self.epoch, shard_idx+1, len(shards)))
+
+        for cut in self.generate_customized_cuts():
+            yield cut
+
+        self.epoch += 1
+
+
+    def generate_customized_cuts(self):
+        for cut in []:
+            yield cut
+
+
+    def generate_shard_cuts(self, cuts, field_iters):
+        for cut, *field_data in zip(cuts, *field_iters.values()):
+            for (field, (maybe_manifest, data_path)) in zip(
                     field_iters.keys(),
                     field_data,
-                ):
-                    if maybe_manifest is None:
-                        continue  # No value available for the current field for this cut.
-                    assert (
+            ):
+                if maybe_manifest is None:
+                    continue  # No value available for the current field for this cut.
+                assert (
                         str(data_path.parent / data_path.stem) == cut.id
-                    ), f"Mismatched IDs: cut ID is '{cut.id}' but found data with name '{data_path}' fsor field {field}"
-                    setattr(cut, field, maybe_manifest)
-
-                cut.shard_origin = shard["cuts"]
-                cut.shar_epoch = self.epoch
-                if cut_map_fn is not None:
-                    cut = cut_map_fn(cut)
-                yield cut
-
-        self.epoch += 1
+                ), f"Mismatched IDs: cut ID is '{cut.id}' but found data with name '{data_path}' fsor field {field}"
+                setattr(cut, field, maybe_manifest)
+            yield cut
 
     def __len__(self) -> int:
         if self._len is None:
diff --git a/lhotse/shar/readers/tar.py b/lhotse/shar/readers/tar.py
index a6c2ecf86..49ae7060c 100644
--- a/lhotse/shar/readers/tar.py
+++ b/lhotse/shar/readers/tar.py
@@ -56,7 +56,11 @@ def iterate_tarfile_pairwise(
         if len(result) == 2:
             yield tuple(result)
             result = []
-        result.append(parse_tarinfo(tarinfo, tar_file))
+        try:
+            result.append(parse_tarinfo(tarinfo, tar_file))
+        except:
+            print("Failed to parse tar info of tar file {}".format(tar_file))
+            return
 
     if len(result) == 2:
         yield tuple(result)