Stochastic Processes

1. Stochastic Processes Definition: A stochastic process is defined to be an indexed collection of random variables {Xt}, where the index t runs through a given set T. Often T is taken to be the set of non-negative integers, and Xt represents a measurable characteristic of interest at time t. (example: Xt – size of queue between machine 3 and 4 in Goldratt’s game.)

2. Stochastic Processes A stochastic process often has the following structure:The current status of the system can fall into any one of M + 1 mutually exclusive categories called states. For notational convenience, these states are labeled 0,1,…..,M. The random variable represents the state of the system at time t, so its only possible values are 0,1,….,M. The system is observed at particular points of time, labeled t = 0,1,2,…. Thus, the stochastic process {Xt} = {X0 , X1 , X2 , …} provides a mathematical representation of how the states of the physical system evolves over time.

3. Examples of Stochastic Processes Scheduling Radar Warning Receivers Developing Optimal Geometries/Routing for Sweeping Minefields

4. Scheduling Radar Warning Receivers PROBLEM STATEMENT • Develop an RWR scheduler that minimizes the time to detect multiple threats across multiple frequency bands. Stochastic Nature – What is the initial angle/time offset for each of the threats?

5. Developing Optimal Geometries/Routing for Sweeping Minefields • Develop a route(s) for MCM assets to search a specified area in the minimum time. Size and shape of search area may change over time Previously searched cell may become “unvisited” with probability. Stochastic Nature: Arrival time is variable Cell search time variable Asset may become inactive