Let $(n,l)=1$, there are formulas for the sum of floor and ceiling functions:
$$\sum_{i=1}^{l-1}\left\lfloor \frac{in}{l}\right\rfloor =\frac{1}{2}(n-1)(l-1)$$
and
$$\sum_{i=1}^{l-1}\left\lceil \frac{in}{l}\right\rceil =\frac{1}{2}(n+1)(l-1)$$
For the alternating sum $\displaystyle S(l,n)=\sum_{i=1}^{l-1}(-1)^{i+1}\left\lceil \frac{in}{l}\right\rceil $, is there a way to obtain a formula for $S(l,n)$?
Conjecture: Let $\displaystyle w(j,h)=\sum_{k=1}^{j-1}(-1)^{k+1}\left\lceil\frac{kh}{j}\right\rceil+h-1$, then $\displaystyle \sum_{j=1\ odd}^{h-1}w(j,h)>0$.