As per self-regulatory theory people can have two types of regulatory focus: promotion and prevention.
Promotion focus is an eagerness-strategy where if we draw an analogy with SDT, one is more eager to detect signals even though there may be a few false alarms. One is ok with errors of commission. This type of motivation will reflect in seeded responses and is in direct opposition to accuracy that is modeled by the other preventive focus.
In preventive focus, you are more vigilant and on-guard , more focused on avoiding failure or in terms of SDT, ready to miss the signal but not miss out a correct rejection. The more cautious approach will lead to less fast repose but more accurate response bias.
Thus, the way I think speed or response time is to be incorporated in SDT is by pitting it against accuracy and it measuring and indicating the propensity to detect signal even at the cost of false alarms.The more the response time is less for a subject the more eagerness or promotion related focus he has - of detecting signal or achieving success.
Of course this is lay mans integration of response time/ speed with SDT; if you are looking for more mathematical models , perhaps the other answers will work better.