And I, on the other hand, would contribute a huge "minus one" to posting on this forum (it does seem to violate the tenets of this forum). This forum is not for questions about z/OS system services that you just happen to invoke in assembler language.
If you want the best audience who can answer a question about an interface, then the assembler list is the wrong place. And surely you want the best audience when seeking help. Fortunately for the OP many of the same folks watch both and no one minds much answering from either place. I do agree that it would have been nice to be able to pass a "parameter" that the exit routine could be given. STIMER has this same flaw; it was done better with STIMERM. The suggestion of using a name/token is a good one, available for any authorization. If you're key 0 (or can get into key 0) STCBUSER is "for you" since you're the issuer of the ATTACH. Many applications accomplish the goal by creating a table that has information about each/every/all the tasks that they have attached (identified by TCB address), including whatever data the ETXR needs to do its thing. The ETXR looks through the table to find the TCB address (which is input to the ETXR) and then continues. The ETXR might locate this table based on some address space level name/token or some other protocol that the application used. Peter Relson z/OS Core Technology Design
