우선 해당 요구사항을 만족하기 위한 전제조건으로
일련번호나 날짜 등과 같은 정렬용 필드가 추가로 표현되어야만 합니다.
그리고, DB 종류에 따라 구문이 다르니, 질문 시 DB 종류와 버전을 명시해 주세요.
 

-- Oracle --
WITH t AS
(
SELECT 1 idx, 159 id, 'AAA' dept, 10 val FROM dual
UNION ALL SELECT  2, 159, 'AAA',  1 FROM dual
UNION ALL SELECT  3, 159, 'AAA',  2 FROM dual
UNION ALL SELECT  4, 159, 'AAA',  3 FROM dual
UNION ALL SELECT  5, 159, 'AAA',  4 FROM dual
UNION ALL SELECT  6, 161, 'AAA',  5 FROM dual
UNION ALL SELECT  7, 161, 'AAA',  6 FROM dual
UNION ALL SELECT  8, 159, 'AAA', 20 FROM dual
UNION ALL SELECT  9, 159, 'AAA',  1 FROM dual
UNION ALL SELECT 10, 159, 'AAA',  2 FROM dual
UNION ALL SELECT 11, 160, 'AAA',  3 FROM dual
UNION ALL SELECT 12, 159, 'AAA',  1 FROM dual
UNION ALL SELECT 13, 159, 'AAA',  2 FROM dual
UNION ALL SELECT 14, 161, 'AAA',  3 FROM dual
)
SELECT dept
     , seq
     , SUM(val) val
  FROM (SELECT idx, id, dept, val
             , SUM(flag) OVER(PARTITION BY dept ORDER BY idx) seq
          FROM (SELECT idx, id, dept, val
                     , CASE WHEN LAG(id) OVER(PARTITION BY dept ORDER BY idx) <= id
                            THEN 0 ELSE 1 END flag
                  FROM t
                )
        )
 GROUP BY dept, seq
 ORDER BY dept, seq
;