Simulation Examples STOCHASTIC PROCESSES Simulation Types Static Simulation

Simulation Examples STOCHASTIC PROCESSES

Simulation Types Static Simulation Dynamic Simulation Estimation of the mean of a random variable Estimation of a performance measure from a random process Expected Profit Average Queue Length Average Inventory Level Shortage Probability

Stochastic Processes A stochastic process {Xt , t ∈ T} is a sequence of random variables (when T is discrete, e. g. T={ 0, 1, 2, …}) a “random function” (when T is continuous, e. g. T=[0, ∞)) Index set T often represents time Xt is a discrete or a continuous random variable

Examples Inventory level of cars at the beginning of each day (discrete state, discrete time) Queue length at any time (discrete state, continuous time) Water level in a dam ◦ at the beginning of each day (continuous state, discrete time) ◦ at any time of the day (continuous state, continuous time) Queue length observed by arriving customers (index set denotes the customer index, not time!)

Random Variable vs. Stochastic Process ω X(ω) Sample Space X(ω’) ω' ω Real Numbers Xt(ω) Sample Paths Sample Space ω' Xt(ω’) Index

Simulation: A Stochastic Experiment Dynamic Simulation: A Sample Path Analysis Objective: Analyze a finite number of sample paths to determine an expected performance of a stochastic process over all sample paths (there may be infinite number of them) For ergodic systems, observing a single sample path to time infinity one can determine the steady state expected performance over all sample paths