The optimization process and recommendations:

1. Avoid Calling Functions With Indexed Columns (query line: 29): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `stamp` is indexed, the index won’t be used as it’s wrapped with the function `date_part`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
2. Avoid Calling Functions With Indexed Columns (query line: 30): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `stamp` is indexed, the index won’t be used as it’s wrapped with the function `date_part`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
3. Avoid Calling Functions With Indexed Columns (query line: 13): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `calltime` is indexed, the index won’t be used as it’s wrapped with the function `date_part`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
4. Avoid Calling Functions With Indexed Columns (query line: 14): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `calltime` is indexed, the index won’t be used as it’s wrapped with the function `date_part`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
5. Avoid Calling Functions With Indexed Columns (query line: 16): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `calltime` is indexed, the index won’t be used as it’s wrapped with the function `date_trunc`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
6. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

CREATE INDEX mytable_idx_calltime ON "mytable" ("calltime");

The optimized query:

WITH cal AS (SELECT
        generate_series('2011-02-02 00:00:00'::timestamp,
        '2012-04-01 05:00:00'::timestamp,
        '1 hour'::interval) AS stamp), qqq AS (SELECT
        date_trunc('hour',
        mytable.calltime) AS stamp,
        count(*) AS zcount 
    FROM
        mytable 
    WHERE
        mytable.calltime >= '07-13-2011 00:21:09' 
        AND mytable.calltime <= '07-31-2011 21:11:21' 
        AND date_part('hour', mytable.calltime) >= 0 
        AND date_part('hour', mytable.calltime) <= 21 
    GROUP BY
        date_trunc('hour',
        mytable.calltime)) SELECT
        cal.stamp,
        COALESCE(qqq.zcount,
        0) AS zcount 
    FROM
        cal 
    LEFT JOIN
        qqq 
            ON cal.stamp = qqq.stamp 
    WHERE
        cal.stamp >= '07-13-2011 00:00:00' 
        AND cal.stamp <= '07-31-2011 21:11:21' 
        AND date_part('hour', cal.stamp) >= 0 
        AND date_part('hour', cal.stamp) <= 21 
    ORDER BY
        cal.stamp ASC