Why MySQL replication process is idle for so long time?

This depends on the type of queries. If you are doing only select statements, there will be no change in the data.

You can check the status of the slave using:

mysql> SHOW SLAVE STATUS;

If you find these two lines:

Slave_IO_Running: Yes
Slave_SQL_Running: Yes

This means that the replication is working fine. Otherwise, it may be broken due to invalid statement found on the slave machine. Check this link.