1. Simulation Examples
STOCHASTIC PROCESSES

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

3. 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

4. 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!)

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

6. 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