A method for forecasting time delays added to a scheduled start time and a scheduled end time of a task includes generating a stochastic model of the task and resources affecting the task, the stochastic model includes a reactionary delay component that is a function of previous task end times and a root cause delay component that is an independent random process at a specific time. The method further includes: calculating a probability distribution of time delays added to the scheduled start time as a combination of the reactionary delay component and the root cause delay component using the stochastic model to provide a probability distribution of start times; and calculating a probability distribution of time delays added to the scheduled end time as a combination of the reactionary delay component and the root cause delay component using the stochastic model to provide a probability distribution of end times.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method for forecasting time delays added to a scheduled start time and a scheduled end time of a task, the method comprising: generating a stochastic model of the task and resources affecting the task, the stochastic model comprising a reactionary delay component and a root-cause delay component, the reactionary delay component being a function of previous task end times and the root-cause delay component being an independent random process at a specific time; calculating a probability distribution of time delays added to the scheduled start time as a combination of the reactionary delay component and the root cause delay component using the stochastic model to provide a probability distribution of start times; and calculating a probability distribution of time delays added to the scheduled end time as a combination of the reactionary delay component and the root cause delay component using the stochastic model to provide a probability distribution of end times; wherein the generating, the calculating a probability distribution of time delays added to the scheduled start time, and the calculating a probability distribution of time delays added to the scheduled end time are implemented by a processor. 2. The method according to claim 1, wherein the stochastic model comprises a hidden Markov model (HMM). 3. The method according to claim 2, wherein the HMM comprises: (a) an initial state distribution ρ0(x) to estimate a previous state xt-1 for each resource; (b) a current state xt conditioned on xt-1 obtained using conditional probability distribution p(xt|xt-1) for each resource; (c) additional details ci conditioned on xt-1 obtained using conditional probability distribution q(ct|xt) for each resource; and (d) a departure-related noise-term Yαi,i conditioned on xt and ct obtained using conditional probability distribution r(yt|ct, xt) for each resource. 4. The method according to claim 3, further comprising training the HMM using historical schedule and delay data. 5. The method according to claim 1, wherein task is a transport stopping at a station and the scheduled start time is a scheduled arrival time of the transport at the station and the scheduled end time is a scheduled departure time of the transport at the station. 6. The method according to claim 5, wherein the transport is a train. 7. The method according to claim 5, wherein calculating a probability distribution of time delays added to the scheduled end time comprises performing a convolution of Yαi,i, the scheduled departure time si, and a function of the maximum of arrival times {Aj} of predecessor transports. 8. The method according to claim 5, wherein calculating a probability distribution of time delays added to the scheduled start time comprises performing a convolution of a departure time Di, a scheduled travel time τi, and an arrival-related noise-term Vi. 9. The method according to claim 1, wherein the task is machining a work piece using a machine and a start time is an arrival time of the work piece at the machine and an end time is a departure time of the work piece at the machine after the work piece has been machined. 10. The method according to claim 1, further comprising displaying the probability distribution of start times and the probability distribution of end times to a user using a user interface. 11. The method according to claim 1, further comprising storing the probability distribution of start times and the probability distribution of end times in memory or a storage medium. 12. The method according to claim 1, further comprising identifying a mode or an expectation or another risk measure of the probability distribution of start times as a most likely start time and identifying a mode or an expectation or another risk measure of the probability distribution of end times as a most likely end time. 13. The method according to claim 12, further comprising updating a schedule to include the most likely start time and the most likely end time. 14. The method according to claim 1, wherein the task comprises a plurality of sub-tasks and the method further comprises: generating a stochastic model of each of the sub-tasks and resources affecting the sub-tasks, the stochastic model comprising a reactionary delay component and a root-cause delay component, the reactionary delay component being a function of previous sub-task end times and the root-cause delay component being an independent random process at a specific time; calculating a probability distribution of time delays added to a scheduled start time of each sub-task as a combination of the reactionary delay component and the root cause delay component of each sub-task using the stochastic model of each of the sub-tasks and resources affecting the sub-tasks to provide a probability distribution of start times of the sub-tasks; and calculating a probability distribution of time delays added to a scheduled end time of each sub-task as a combination of the reactionary delay component and the root cause delay component of each sub-task using the stochastic model of each of the sub-tasks and resources affecting the sub-tasks to provide a probability distribution of end times of the sub-tasks.