Is it doing TLS?
It's likely running out of entropy. Debian package Exim linked against GNU's TLS library, rather than the more common OpenSSL. It'll suck /dev/random dry very quickly.
Lousy under-the-gun hack:
root@shangrila:/dev# rm random && mknod -m 664 /dev/random c 1 9
root@shangrila:/dev# ls -l /dev/*random
crw-rw-r-- 1 root root 1, 9 Aug 7 17:39 /dev/random
crw-rw-rw- 1 root root 1, 9 Aug 7 13:42 /dev/urandom
root@shangrila:/dev#
This is bad for all kinds of reasons... (644 is what the manual says set it to, hmm.). Better ways exist.
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