discrete event dynamic systems discrete event dynamic systems — lecture #1 — by y.c.ho...
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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!
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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