As we do not have these in India, Pedestrian push buttons on roads for stopping traffic were a new thing. It looked like a way to give more power to people waiting to cross the road.

Though, I was wondering as to how they work. Time it takes for lights to turn red after pressing push-button is always different.

Few observations I made were:

Looking at the web it seems I was not the only one curious to find this out.
According to Cecil Adams, push-to-walk buttons only work when there is nobody around to push them. To put it more charitably, the buttons generally are wired to insert a "walk" phase into an otherwise walkless signal cycle, not speed an existing "walk" phase up.

But do they really work at all? The New York Times archives claim that New York city deactivated most of the pedestrian buttons long ago with the emergence of computer-controlled traffic signals, even as an unwitting public continued to push on, according to city Department of Transportation officials. More than 2,500 of the 3,250 walk buttons that still exist
function essentially as mechanical placebos. Any benefit from them is only imagined.

Sometimes, it is interesting to watch people getting impatient and pushing these
buttons again and again, in the hope of getting the "walk" sign. According to a random sample collected in a road, 35.7% of walk light users pushed the button more than once. 26.2% pushed it more than twice. The average number of pushes was 2.2.