Postgres 12

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CREATE TABLE event (
  event_id int PRIMARY KEY
, event_type_id int NOT NULL
, period tstzrange NOT NULL
, quantity int NOT NULL
);

-- improved test data
INSERT INTO event (event_id, event_type_id, period, quantity) VALUES
  (1, 1, '[2016-01-06 09:00:00+00,2016-01-08 17:00:00+00]', 1)  -- range with 3 rows (q=3)
, (2, 1, '[2016-01-07 09:00:00+00,2016-01-07 11:00:00+00]', 1)
, (3, 1, '[2016-01-07 13:00:00+00,2016-01-07 17:00:00+00]', 1)
, (4, 2, '[2016-01-07 12:00:00+00,2016-01-07 17:00:00+00]', 5)  -- differeent type

, (5, 1, '[2016-01-09 09:00:00+00,2016-01-09 11:00:00+00]', 4)  -- range with 1 row  (q=4)
, (6, 1, '[2016-01-09 13:00:00+00,2016-01-09 17:00:00+00]', 2)  -- range with 2 rows (q=5)
, (7, 1, '[2016-01-09 14:00:00+00,2016-01-09 18:00:00+00]', 3);
7 rows affected

-- subquery variant, best performance
SELECT running_sum - lag(running_sum, 1, 0) OVER (ORDER BY p_start) AS sum_quantity
FROM  (
   SELECT lower(period) AS p_start
        ,(sum(quantity)                      OVER w)::int AS running_sum
        , lead(lower(period), 1, 'infinity') OVER w
                        > max(upper(period)) OVER w AS range_end
   FROM   event
   WHERE  event_type_id = 1
   AND    period && '[2016-01-01 0:0+0,2016-01-10 0:0+0]'::tstzrange 
   WINDOW w AS (ORDER BY lower(period))
   ) sub
WHERE  range_end
ORDER  BY 1 DESC
LIMIT  1;


sum_quantity


5


-- same, slower and more verbose
WITH cte1 AS (
   SELECT quantity
        , lower(period) AS p_start
        , upper(period) AS p_end
   FROM   event
   WHERE  event_type_id = 1
   AND    period && '[2016-01-01 0:0+0,2016-01-10 0:0+0]'::tstzrange
   )
,    cte2 AS (
   SELECT (sum(quantity)               OVER w)::int AS running_sum
        , lead(p_start, 1, 'infinity') OVER w               -- next start ..
                          > max(p_end) OVER w AS range_end  -- .. after last end
        , p_start, p_end
   FROM   cte1
   WINDOW w AS (ORDER BY p_start)
   )
SELECT running_sum - lag(running_sum, 1, 0) OVER (ORDER BY p_start) AS sum_quantity
                    -- subtract the previous sum to get the sum of this range
     , p_end::text
FROM   cte2
WHERE  range_end    -- only rows at the end of each range
ORDER  BY 1 DESC    -- biggest sum first
LIMIT  1;           -- only return the winner


sum_quantity
p_end


5
2016-01-09 18:00:00+00