Assuming one has to go with an image carousel (see this question for reasons why this isn't the best UX to begin with), how long should the carousel wait before beginning and switching?
I recently was giving UX feedback on a website containing a carousel which started almost immediately and switched off the "logo" homepage type image, which was disorienting, so more than 0.25 seconds is clearly necessary.
According the opinion that carousels are evil and one should not use them, regardless of what the customer thinks, the conclusion is: "Longer is better".
The ideal time is "infinity" (i.e. don't use carousel). If you are forced to use some finite time, then make is as longer as possible without getting fired.
In television production, standard "hold" time for on-screen copy is 2x reading speed. In other words, if you have a block of copy on the screen, it should stay on the screen long enough to quickly read it through twice. This is to allow time for orientation, focusing on the text and slow readers.
That being said, a web page is much different than a television screen -- much more content vying for your attention. You will also need to take into account what percentage of the screen is being taken up by the carousel and what else is visible. You might be able to assume that users have learned that carousels normally change content and will focus on it relatively early to see if it contains anything they are interested in.
This is difficult to answer without more info on the carousel content, as others have said. I also agree with the comment on using reading time as a rule of thumb, if text is involved.
One additional answer could be to do a usability study (potentially with eye tracking data if you have access to those tools). You could set all transitions to be manually initiated then ask test participants to "click next" (however that fits into your carousel design) when they feel they have seen and absorbed the contents of the image. Average those times over some number of users, and add something like 50% to account for various reading/viewing speeds and other biases, and use that number.
This could be done pretty quickly and informally in a "hallway test" fashion. All you need is a few people roaming your halls, a stopwatch, and a build of your carousel with automatic transitions disabled. Change up the imagery in this test version if everyone you've got available is already familiar with your carousel, to further help with accounting for bias.
We had a carousel on our sites and we are progressively moving away to a single large image that can be changed by clicking on to all the exposed thumbnails below.
Its early days (2 weeks) yet but we have seen a small increase on time on page, and a small lowering in exits from these pages.
A good workaround (since carousel is evil) is to pause the carousel when the user hovers over it. It is not ideal, but at least people get to see what they are about to click on.
I think this is impossible to answer precisely because there are too many variables to assess when deciding how long someone needs to look at an image or read text: do they have their glasses on; are they trying to read the text on the side of your product which is difficult to read because it's slanted; are they really interested in the image and want to have a long, clear look; etc, etc.
Therefore I think the answer is not to automatically move content.
Give the user controls, such as play and pause, or give them left and right scrolling buttons, or pips that allow them to select the slide they wish to look at.
Give the user clear controls and let them make the decision if and when they want to look at the other slides.
