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The sequence number increases whenever a thread acquires the writer lock. You can save the sequence number and pass it to ReaderWriterLock.AnyWritersSince(int) at a later time, if you want to determine whether other threads have acquired the writer lock in the meantime.
You can use WriterSeqNum to improve application performance. For example, a thread might cache the information it obtains while holding a reader lock. After releasing and later reacquiring the lock, the thread can determine whether other threads have written to the resource by calling AnyWritersSince; if not, the cached information can be used. This technique is useful when reading the information protected by the lock is expensive; for example, running a database query.
The caller must be holding a reader lock or a writer lock in order for the sequence number to be useful.