Speeded and/or simple response tasks may be cognitively modeled by a random walk process that accumulates to threshold. In cases of tasks where mainly one characteristic response is observed, at varying latencies, then random walks involving only positive drifts that each arrive at a single threshold, provide a suitable accumulation modeling account of the data; and advantageously, this accumulation model is exactly described by the shifted Wald (SW) probability density function. We will demonstrate how the SW distribution is thus a noteworthy cognitive model for these tasks, which uniquely possesses simultaneously, high utility as an objective data measurement tool for the response time (RT) distributions. Per each experiment condition, its three parameters can decompose the observed mean RT value, quantify the shape and characteristics of the observed RT distribution, and account for significant differences between distributions with near-identical mean values; regardless of whether one accepts the cognitive interpretation of the random-walk accumulation process. We present the SW model and demonstrate its efficiency and utility on both simulated and real data.