A quick look at the POD suggests that Net::OpenSSH implements timeouts that will close connections after a lack of network activity.

A grep inside turns up this calculation which suggests that the default timeout is 90 (or 30?) seconds:

my $timeout = int((($self->{_timeout} || 90) + 2)/3);
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My guess is that if the remote system is idle, tail -f syslog will produce no output and so the module closes the connection. Maybe you should consider using an explicit timeout.