help splitting a file into multiple files in bash

I have a file that logs multiple sessions. What I would like to do is split this file inclusive of the lines that include "starting session" and "shutting down" and ignore the data before and after the beginning of the first session and the end of the last session. The output files can be called whatever... session.1 session.2 ...

I am using bash.

blah blah
blah blah starting session blah blah
blah1 blah1 blah1
blah2 blah2 blah2
blah3 blah3 blah3
blah balh shutting down blah blah
blah3 blah5 starting session blah3 1blah
blah103 blah1 blah1
blah3 blah3 blah3
blah balh shutting down blah blah
blah blah

So, if I ran it against the bit above, I'd have

file1 (session.1):

blah blah starting session blah blah
blah1 blah1 blah1
blah2 blah2 blah2
blah3 blah3 blah3
blah balh shutting down blah blah

file2 (session.2):

blah3 blah5 starting session blah3 1blah
blah103 blah1 blah1
blah3 blah3 blah3
blah balh shutting down blah blah

Thanks,
Eric

Something like this?

awk '
/starting session/{f=1;c++}
{print > "session."i}
/shutting down/{f=0}
' file

Yeah. Cool. Thanks!