[Backport 2.x] Add a counter to node stat api to track shard going from idle to non-idle (#9622)

[ruai0511]

Description

2.x Backport PR for search idle tracking
Add a counter to node stat api to track shard going from idle to non-idle

Check List

• New functionality includes testing.
  • All tests pass
• New functionality has been documented.
  • New functionality has javadoc added
• Failing checks are inspected and point to the corresponding known issue(s) (See: Troubleshooting Failing Builds)
• Commits are signed per the DCO using --signoff
• Commit changes are listed out in CHANGELOG.md file (See: Changelog)
• Public documentation issue/PR created

By submitting this pull request, I confirm that my contribution is made under the terms of the Apache 2.0 license.
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[github-actions]

Compatibility status:

Checks if related components are compatible with change e9555f3

Incompatible components

Incompatible components: [https://github.com/opensearch-project/security.git]

Skipped components

Compatible components

Compatible components: [https://github.com/opensearch-project/custom-codecs.git, https://github.com/opensearch-project/flow-framework.git, https://github.com/opensearch-project/neural-search.git, https://github.com/opensearch-project/job-scheduler.git, https://github.com/opensearch-project/cross-cluster-replication.git, https://github.com/opensearch-project/observability.git, https://github.com/opensearch-project/security-analytics.git, https://github.com/opensearch-project/geospatial.git, https://github.com/opensearch-project/k-nn.git, https://github.com/opensearch-project/notifications.git, https://github.com/opensearch-project/asynchronous-search.git, https://github.com/opensearch-project/sql.git, https://github.com/opensearch-project/ml-commons.git, https://github.com/opensearch-project/reporting.git, https://github.com/opensearch-project/index-management.git, https://github.com/opensearch-project/performance-analyzer-rca.git, https://github.com/opensearch-project/anomaly-detection.git, https://github.com/opensearch-project/common-utils.git, https://github.com/opensearch-project/alerting.git, https://github.com/opensearch-project/performance-analyzer.git]

[github-actions]

❕ Gradle check result for e9555f3: UNSTABLE

• TEST FAILURES:

      3 org.opensearch.cluster.coordination.AwarenessAttributeDecommissionIT.testConcurrentDecommissionAction

Please review all flaky tests that succeeded after retry and create an issue if one does not already exist to track the flaky failure.

[codecov]

Codecov Report

Attention: Patch coverage is 78.57143% with 6 lines in your changes are missing coverage. Please review.

Project coverage is 71.16%. Comparing base (0dd892c) to head (e9555f3).
Report is 98 commits behind head on 2.x.

Files	Patch %	Lines
...pensearch/index/shard/SearchOperationListener.java	62.50%	3 Missing ⚠️
...org/opensearch/index/search/stats/SearchStats.java	77.77%	0 Missing and 2 partials ⚠️
...in/java/org/opensearch/index/shard/IndexShard.java	75.00%	0 Missing and 1 partial ⚠️

Additional details and impacted files

@@             Coverage Diff              @@
##                2.x   #12969      +/-   ##
============================================
- Coverage     71.28%   71.16%   -0.12%     
- Complexity    60145    60417     +272     
============================================
  Files          4957     4987      +30     
  Lines        282799   284528    +1729     
  Branches      41409    41603     +194     
============================================
+ Hits         201591   202492     +901     
- Misses        64189    64959     +770     
- Partials      17019    17077      +58     

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