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Regardless of the number of nodes or available resources, SolrCloud begins to have stability problems when the number of collections reaches the low hundreds. With thousands of collections, any little problem or change to the cluster can cause a stability death spiral that may not recover for tens of minutes. Try to keep the number of collections as low as possible. When each collection has many shards, the problem can multiply. These problems are due to how SolrCloud updates cluster state in ZooKeeper in response to cluster changes. Work is underway to try and improve this situation. This problem surfaced in Solr 4.x where the state is kept in a single "clusterstate.json" file. Subsequent Solr versions (5x and above) by default store each collection's data in an individual "state.json" as a child of each collection's znode (e.g. /collections/my_collection/state.json). If you started with a Solr 4x installation, the MIGRATESTATE command will change to the newer, more scalable state. That said, the load on Zookeeper certainly increases as the number of collections (and replicas) increases.
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