This paper investigates the model of probabilistic program with delays (PPD) that consists of a few program blocks. Performing each block has an additional time-consumption—waiting to be executed—besides the running time. We interpret the operational semantics of PPD by Markov automata with a cost structure on transitions. Our goal is to measure those individual execution paths of a PPD that terminates within a given time bound, and to compute the minimum termination probability, i.e. the termination probability under a demonic scheduler that resolves the nondeterminism inherited from probabilistic programs. When running time plus waiting time is bounded, the demonic scheduler can be determined by comparison between a class of well-formed real numbers. The method is extended to parametric PPDs. When only the running time is bounded, the demonic scheduler can be determined by real root isolation over a class of well-formed real functions under Schanuel's conjecture. Finally we give the complexity upper bounds of the proposed methods.

本文研究了由几个程序块组成的带延迟概率程序模型（PPD）。除了运行时间之外，执行每个块还有一个额外的时间消耗（等待执行）。我们用马尔可夫自动机解释PPD的操作语义，并带有转换的成本结构。我们的目标是测量在给定时间范围内终止的PPD的各个执行路径，并计算最小终止概率，即在恶魔调度程序下的终止概率，该概率解决了从概率程序继承的不确定性。当运行时间加上等待时间是有界的时，可以通过一类格式良好的实数之间的比较来确定恶魔调度程序。该方法扩展到参数化PPD。当只有运行时间受限时，恶魔调度器可以通过在Schanuel的猜想下，通过一类格式良好的实函数的实根隔离来确定。最后，我们给出了所提出方法的复杂度上限。