Discrete Event Dynamic SystemsDiscrete Event Dynamic Systems— Lecture #1 —

by

Y.C.Ho

September, 2003

Tsinghua University, Beijing, CHINA

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Discrete Event Dynamic Systems— An Overview —

• What are DEDS?What are DEDS?

• Models of DEDSModels of DEDS

• Tools for DEDSTools for DEDS

• Future Directions for DEDSFuture Directions for DEDS

TOPICS:

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Resources• Books:

– Y.C. Ho, DEDS Analyzing Performance and Complexity in the Modern World IEEE Press book, 1992

– C. Cassandras & S Lafortune, Discrete Event Systems, Kluwer 1999 (text book for this course)

• WWW Pages: www.hrl.harvard.edu/~ho/CRCD or DEDS with links to Boston University and U. of Mass. Amherst

• Video Tape: IEEE Educational Services Video Tape: “Analyzing Performance and Complexity in the Modern World” 1992

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What are Continuous Variable Dynamic Systems

(CVDS)?

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What are Discrete Event Dynamic Systems (DEDS)?

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An Airport

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More Examples of DEDS

• Manufacturing Automation - a SC Fab

• Communication Network - the Internet

• Military C3I systems

• Traffic - land, sea, air

• Paper processing bureaucracy - insurance co.

••

The pervasive nature of DEDS in modern civilization

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Nature of DEDS

• A set of tasks or jobs: parts to be manufactured, messages to be transmitted, etc

• A set of resources: machines, AGVs, nodal CPUs, communication links and subnetworks, etc

• Routing of job among resources: production plans, virtual circuits, etc

• Scheduling of jobs as they compete for resources: queues and event timing sequences

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A Typical DEDS Trajectory

time

Discrete state

x1

x2

x3

x4

x5

e1 e2 e4 e5 e6e3

Holding time

STATES are piecewise constant HOLDING TIMES are deterministic/random EVENTS triggers state transition TRAJECTORY defined by (state, holding time) sequence

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Comparison with a CVDS Trajectory

time

Discrete state

dx/dt = f(x,u,t)

Hybrid System: can hideeach state CVDS behavior

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Modeling Ingredients

• Discrete States: combinatorial explosion

• Stochastic Effects: unavoidable uncertainty

• Continuous time and performance measure

• Dynamical:

• Hierarchical:

• Computational vs conceptual

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Mathematical Specification

• State Space Approach:– X the state space, a finite set, xX. state: # in queue.– A Event set, finite A.e.g. arrival(a), departure(d).– (x) Enabled event set in x, (x)A, if x≥1, (x)={a,d}; if

x=0, (x)={a}.– f State transition function Xx(x)->X. Could write

down f∈{+1;0;-1}, because these are transitions possible.

• Input/output Approach:– String: sequence of events – Language: all possible event sequences in a DEDS– Operations: defined on strings,e.g., “shuffle”– Score: # of occurrences of event types in a string– Trace: sequence of state, event pair

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Mathematical Specification (contd.)Introduction of “TIME” for quantitative performance analysis purposes

Clock Mechanism (a two dimensional array of numbers)

cn() = the nth lifetime of the event

nthe timeof the nth occurrence of the event cn()

1 2 n-1 n

Event type

time

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Time Evolution of a DEDS

Event enabling

(x)

One event delay

Life timegeneration

Minimum oflifetimes

Statetransition

x

*

cn()

Simulation of a DEDS

Search for next event to occur

New state

Generate lifetime of new event

Place the end of event in future event list

Transition to next state

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Ingredients and ModelsSTATE: not inherent for in/out, not necessarily completeEVENT: fundamental, instantaneous, marks state transitionEVENT FEASIBILITY: basic to controlSTATE TRANSITION: basic to dynamicsTIMING: essential for performance evaluationRANDOMNESS: facts of life

GSMPFSM

(Markov Chains)

QueuingNetwork

Min-MaxAlgebra

Petri NetsLanguage

&Processes

STATEEVENT

FEASIBLEEVENTTIME

TRANSITION

RANDOM-NESS

yesinput

yes

noyes

no/yes

yesyes

yes

yesyes

yes

yesyes

yes

(yes)yes

no

graphicalyes

yes

yesyes

yes

noyes

not really

nono

no

yesyes

yes

yesyes

yes

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Model of DEDS

Logical Algebraic

Untimed

Performance

Finite State Machines &

Petri Nets

Finitely RecursiveProcesses

GeneralizedSemi-Markov

Processes

Min-Max AlgebraTimed

GOAL: Finite representation.Qualitative properties, Quantitative Performance

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Performance Design & Evaluation

• Building Models ~40%

• Validation and analysis ~10%

• Evaluating the model ~25%

• Optimization and tuning ~25%

Emphasis of this course on last two topics!

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Rationale for Performance Evaluation

• Answer “what-if ” questions J(+)=? Sensitivity analysis

• Explore performance surface, J() at (i), i=1, 2, 3, . . .

• Find optimal parameter settings =optimal

• On-line real time tuning of the system - tracking optimal as the environment changes

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Qualitative Performance Evaluation

• Deadlocks in communication network or databases - mutual waiting for others to release resources, liveliness in PN, forbidden states in FSM

• Failsafe interlock in manufacturing automation - limit switches, automatic shutdown, reachability, controllability

• Stability issues in C3I simulation - for want of a nail, a horse shoe was undone, . . . , war was lost. Numerical stability of sample paths

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Quantitative Performance Evaluation

• Analytical Tools (including Q-network Theory): quick “what-if”, limited state transition possibilities

• Simulation: completely general, easy to visualize and understand, easy to misuse, and time consuming

• Hybrid Tools: Perturbation analysis, likelihood ratio methods, sample path analysis, ordinal optimization

• Hardware Solutions: massively parallel computers

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Example of DEDS control problems

• Access control: allowing task to compete for resources, e.g., telephone busy signal

• Routing control: assigning a task to one of many possible resources, e.g., which route should a packet be routed

• Scheduling control: determine to order to serve several tasks, e.g., which lot of part to be machined first

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Three Common Implementations of a simple queue-server system

A

B

2C

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Three Approaches to a simple control problem -minimum time path from A to B

I. Open Loop Control: Dead Reckoning

II. Feedback Control: Continuous Dead Reckoning - line of sight policy

III. Stochastic Control: l.o.s.policy with statistical correction

VA B A BV

crosswind

A B

A B

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Analogs in Scheduling Theory and Practice

I. Deterministic Batch Solution (Open Loop): due dates for all orders known; minimizes tardiness; mixed integer programming solution; upset by disturbance

II. Heuristic Dispatch Rule (Feedback Control): earliest due date first, longest make span first, longest buffer first, least slack first, etc

III. Smart Dispatch Policies (Stochastic Control): account for stochastic arrival and disturbances; use AI and learning

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Historical Perspective on the Control and Optimization of DEDS and CVDS

History for CVDS:Development of mechanics for

CVDS

Self regulating governor for steam

enginesWWII Servo-mechanism

Modern control theory and practice

<1940 >1940

History of DEDS:

Birth of OR

Emergence of human made systems

Theoretical foundations &

practical success stories

1945 1970’s present

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The AI-OR-CT Intersection

Control Theory

Computational Intelligence

Operations Research

DEDS

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Related DEDS Models

• Hybrid System

• Queuing Networks

• Petri-nets

• Min-Max Algebra

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HYBRID SYSTEM MODELS

Christos G. Cassandras CODES Lab. - Boston University

PLANT

CONTROLLER

REPLACE THE USUAL CONTROL LOOP BYREPLACE THE USUAL CONTROL LOOP BY

PLANT

CONTROLLER

EVENTS

SUPERVISOR

Plant assumed to haveonly time-driven dynamics?(time and event driven)

TIME DRIVEN

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PLANT

EVENT-DRIVENDYNAMICS

TIME-DRIVENDYNAMICS

HYBRID SYSTEM MODELS

Christos G. Cassandras CODES Lab. - Boston University

CONTROLLER

• Plant: time-driven + event-driven dynamics

• Controller affects bothtime-driven + event-driven components

• Control may becontinuous signal and/or discrete event

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HYBRID SYSTEM MODELS

Christos G. Cassandras CODES Lab. - Boston University

CONTINUED

…

xi

Physical State, z

Temporal State, xx1 x2

Switching Time

),,( tuzgz iiii

xi+1 = fi(xi,ui,t)SWITCHING TIMESHAVE THEIR OWN

DYNAMICS!

SWITCHING TIMESHAVE THEIR OWN

DYNAMICS!

Simulation Language -SHIFT

http://www.path.berkeley.edu/shift/

http://www.gigascale.org/shift/

for Lambda-SHIFT (advanced)

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• Pepyne D.L., and Cassandras, C.G., "Modeling, Analysis, and Optimal Control of a Class of Hybrid Systems", J. of Discrete Event Dynamic Systems, Vol. 8, 2, pp. 175-201, 1998.

• Cassandras, C.G., and Pepyne D.L., "Optimal Control of a Class of Hybrid Systems", Proc. of 36th IEEE Conf. Decision and Control, pp. 133-138, December 1997.

• Cassandras, C.G., Pepyne D.L., and Wardi, Y., "Generalized Gradient Algorithms for Hybrid System Models of Manufacturing Systems", Proc. of 37th IEEE Conf. Decision and Control, December 1998.

• Cassandras, C.G., Pepyne D.L., and Wardi, Y., "Optimal Control of Systems with Time-Driven and Event-Driven Dynamics", Proc. of 37th IEEE Conf. Decision and Control, December 1998.

Christos G. Cassandras CODES Lab. - Boston University

SELECTED REFERENCES

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Queuing Networks

• Server are work stations: service duration can be deterministic or random

• Jobs pass from server to server according to routing plan: routing probability matrix

• Performance measure: delay, queue length, through put, etc.

• Closed Form Solutions: product form formula

• Software Solution: QNA and MPX®

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Essence of Q-Network Equations

• Traffics Equations (Global)– traffic mean (linear eqs., continuity of flow)– traffic variance (linear eqs. Approximate)

• Nodal equations (Local)– solution of G/G/m queue

• This is Queuing Network Analysis (QNA)

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MPX® Demo

Separately presented

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Petri-Nets

• Finite graphical representation for possibly infinite state machines

• Incorporates detail timing information explicitly

• Good for small size problems

• Many books and forthcoming special issue of Journal on DEDS, Jan 2000.

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Min-Max Algebra

• Primarily for deterministic and periodic DEDS

• Best application - Analyzing and optimizing complex train schedule

• Experts in France, Netherlands, and China