Advanced architecture processors provide features such as caches and branch prediction that result in improved, but variable, execution time of software. Hard real-time systems require tasks to complete within timing constraints. Consequently, hard real-time systems are typically designed conservatively through the use of tasks? worst-case execution times (WCET) in order to compute deterministic schedules that guarantee task?s execution within giving time constraints. This use of pessimistic execution time assumptions provides real-time guarantees at the cost of decreased performance and resource utilization. In soft real-time systems, however, meeting deadlines is not an absolute requirement (i.e., missing a few deadlines does not severely degrade system performance or cause catastrophic failure). In such systems, a guaranteed minimum probability of completing by the deadline is sufficient. Therefore, there is considerable latitude in such systems for improving resource utilization and performance as compared with hard real-time systems, through the use of more realistic execution time assumptions. Given probability distribution functions (PDFs) representing tasks? execution time requirements, and tasks? communication and precedence requirements, represented as a directed acyclic graph (DAG), this dissertation proposes and investigates algorithms for constructing non-preemptive stochastic schedules. New PDF manipulation operators developed in this dissertation are used to compute tasks? start and completion time PDFs during schedule construction. PDFs of the schedules? completion times are also computed and used to systematically trade the probability of meeting end-to-end deadlines for schedule length and jitter in task completion times. Because of the NP-hard nature of the non-preemptive DAG scheduling problem, the new stochastic scheduling algorithms extend traditional heuristic list scheduling and genetic list scheduling algorithms for DAGs by using PDFs instead of fixed time values for task execution requirements. The stochastic scheduling algorithms also account for delays caused by communication contention, typically ignored in prior DAG scheduling research. Extensive experimental results are used to demonstrate the efficacy of the new algorithms in constructing stochastic schedules. Results also show that through the use of the techniques developed in this dissertation, the probability of meeting deadlines can be usefully traded for performance and jitter in soft real-time systems.