Summing activity intervals without any obvious column to group by

[Carey Tilden <carey.tilden@xxxxxxxxx>]

Apologies for the awkward title.  I haven't quite thought of the right way to describe my problem, which may be why I've had a hard time figuring out how to solve it.  I have a list of program start/stop times, and I want to know how long each run takes to complete.  The thing that's really tripping me up is there are gaps in the sequence.  I've figured out how to collapse the results down to a single row per attempt, but I can't quite figure out how to further collapse down each full run to its own row.  It'd be easy if I had a session_id or something to group on, but I don't.  All I have are the start/stop times.

Here's some sample data.  Hopefully this clarifies what I'm talking about:

    drop table if exists program_runs;
    
    create temporary table program_runs (
        id serial,
        time_stamp timestamptz,
        action text
    );
    
    insert into program_runs (time_stamp, action) values
        ('2012-01-01 10:00:00 PST', 'started'), ('2012-01-01 10:10:00 PST', 'stopped early'),
        ('2012-01-01 10:20:00 PST', 'started'), ('2012-01-01 10:30:00 PST', 'stopped early'),
        ('2012-01-01 10:40:00 PST', 'started'), ('2012-01-01 10:47:00 PST', 'completed'),
        ('2012-01-01 10:50:00 PST', 'started'), ('2012-01-01 11:00:00 PST', 'stopped early'),
        ('2012-01-01 11:10:00 PST', 'started'), ('2012-01-01 11:13:00 PST', 'completed'),
        ('2012-01-01 11:20:00 PST', 'started'), ('2012-01-01 11:30:00 PST', 'stopped early'),
        ('2012-01-01 11:40:00 PST', 'started'), ('2012-01-01 11:50:00 PST', 'stopped early'),
        ('2012-01-01 12:00:00 PST', 'started'), ('2012-01-01 12:10:00 PST', 'stopped early'),
        ('2012-01-01 12:20:00 PST', 'started'), ('2012-01-01 12:29:00 PST', 'completed');
    
    select
        this_time_stamp as starting_time_stamp,
        next_time_stamp - this_time_stamp as time_elapsed,
        next_action as closing_action
    from (
        select
            time_stamp as this_time_stamp, lead(time_stamp) over (order by id) as next_time_stamp,
            action as this_action, lead(action) over (order by id) as next_action,
            id as this_id, lead(id) over (order by id) as next_id
        from program_runs
    ) q
    where this_action = 'started';

Note that each run has a pair of entries in the table.  The first is always "started", but the second may be either "stopped early" or "completed".  The final results I'd like to see are:

      starting_time_stamp   | total_time_elapsed 
    ------------------------+--------------------
     2012-01-01 10:00:00-08 | 00:27:00
     2012-01-01 10:50:00-08 | 00:13:00
     2012-01-01 11:20:00-08 | 00:39:00

Hope that's enough detail.  Any ideas or suggestions gladly accepted!

Regards,
Carey