1. Lecture 11 – Stochastic Processes
Topics
Definitions
Review of probability
Realization of a stochastic process
Continuous vs. discrete systems
Examples

2. Basic Definitions
Stochastic process: System that changes over time in an uncertain manner
Examples
Automated teller machine (ATM)
Printed circuit board assembly operation
Runway activity at airport
State: Snapshot of the system at some fixed point in time
Transition: Movement from one state to another

3. Elements of Probability Theory
Experiment: Any situation where the outcome is uncertain.
Sample Space, S: All possible outcomes of an experiment (we will call them the “state space”).
Event: Any collection of outcomes (points) in the sample space. A collection of events E1, E2,…,En is said to be mutually exclusive if Ei  Ej =  for all i ≠ j = 1,…,n.
Random Variable (RV): Function or procedure that assigns a real number to each outcome in the sample space.
Cumulative Distribution Function (CDF), F(·): Probability distribution function for the random variable X such that
F(a)  Pr{X ≤ a}

4. Components of Stochastic Model
Time: Either continuous or discrete parameter.
State: Describes the attributes of a system at some point in time.
s = (s1, s2, , sv); for ATM example s = (n)
Convenient to assign a unique nonnegative integer index to each possible value of the state vector. We call this X and require that for each s  X.
For ATM example, X = n.
In general, Xt is a random variable.

5. Model Components (continued)
Activity: Takes some amount of time – duration. Culminates in an event.
For ATM example  service completion.
Transition: Caused by an event and results in movement from one state to another. For ATM example,
# = state, a = arrival, d = departure
Stochastic Process: A collection of random variables {Xt}, where t  T = {0, 1, 2, . . .}.

6. Realization of the Process
Deterministic Process
Time between arrivals
Pr{ ta   } = 0,  < 1 min
= 1,   1 min
Arrivals occur every minute.
Time for servicing customer
Pr{ ts   } = 0,  < 0.75 min
= 1,   0.75 min
Processing takes exactly 0.75 minutes.
Number in system, n
(no transient response)

7. Realization of the Process (continued)
Stochastic Process
Time for servicing a customer
Pr{ ts   } = 0,  < 0.75 min
= 0.6,    1.5 min
= 1,   1.5 min
Number in system, n

8. Markovian Property
Given that the present state is known, the conditional probability of the next state is independent of the states prior to the present state.
Present state at time t is i: Xt = i
Next state at time t + 1 is j: Xt+1 = j
Conditional Probability Statement of Markovian Property:
Pr{Xt+1 = j | X0 = k0, X1 = k1,…, Xt = i } = Pr{Xt+1 = j | Xt = i }
  for t = 0, 1,…, and all possible sequences i, j, k0, k1, , kt–1
Interpretation: Given the present, the past is irrelevant in determining the future.

9. Transitions for Markov Processes
State space: S = {1, 2, , m}
Probability of going from state i to state j in one move: pij
State-transition matrix
Theoretical requirements: 0  pij  1, j pij = 1, i = 1,…,m

10. Discrete-Time Markov Chain
A discrete state space
Markovian property for transitions
One-step transition probabilities, pij, remain constant over time (stationary)
Simple Example
State-transition matrix State-transition diagram 1 2
P =
0.6
0.3
0.1
0.8
0.2

11. (There are other possible bets not included here.)
Game of Craps
Roll 2 dice
Outcomes
Win = 7 or 11
Loose = 2, 3, 12
Point = 4, 5, 6, 8, 9, 10
If point, then roll again.
Win if point
Loose if 7
Otherwise roll again, and so on
(There are other possible bets not included here.)

12. State-Transition Network for Craps

13. Transition Matrix for Game of Craps
Sum 2 3 4 5 6 7 8 9 10 11 12
Prob.
0.028
0.056
0.083
0.111
0.139
0.167
Probability of win = Pr{ 7 or 11 } = = 0.223
Probability of loss = Pr{ 2, 3, 12 } = = 0.112

14. Examples of Stochastic Processes
Single stage assembly process with single worker, no queue
State = 0, worker is idle
State = 1, worker is busy
Multistage assembly process with single worker, no queue
State = 0, worker is idle
State = k, worker is performing operation k = 1, , 5

15. Examples (continued)
Multistage assembly process with single worker and queue
(Assume 3 stages only; i.e., 3 operations)
s = (s1, s2) where
Operations
k = 1, 2, 3

16. Single Stage Process with Two Servers and Queue
s = (s1, s2 , s3) where
i = 1, 2
State-transition network

17. Series System with No Queues
Component
Notation
Definition
State
s = (s1, s2 , s3)
State space
S = { (0,0,0), (1,0,0), ,
(0,1,1), (1,1,1) }
The state space consists of all possible binary vectors of 3 components.
Events
Y = {a, d1, d2 , d3}
a = arrival at operation 1
dj = completion of operation j for j = 1, 2, 3

18. What You Should Know About Stochastic Processes
Definition of a state and an event.
Meaning of realization of a system (stationary vs. transient).
Definition of the state-transition matrix.
How to draw a state-transition network.
Difference between a continuous and discrete-time system.
Common applications with multiple stages and servers.