Instructor: Spyros Reveliotis

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IE7201 Production Service Systems
EngineeringFall 2009Closure

• Instructor Spyros Reveliotis

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Course Objectives

• Provide an understanding and appreciation of the
  different resource allocation and coordination
  problems that underlie the operation of
  production and service systems.
• Enhance the student ability to formally
  characterize and study these problems by
  referring them to pertinent analytical
  abstractions and modeling frameworks.
• Develop an appreciation of the inherent
  complexity of these problems and the resulting
  need of simplifying approximations.
• Systematize the notion and role of simulation in
  the considered problem contexts.
• Define a research frontier in the addressed
  areas.

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Course Outline

• 1. Introduction Course Objectives, Context, and
  Outline
• Contemporary organizations and the role of
  Operations Management (OM)
• Corporate strategy and its connection to
  operations
• The organization as a resource allocation system
  (RAS)
• Classification of Production Systems on the basis
  of their workflow structure and control policies
• The underlying RAS management problems and the
  need for understanding the impact of the
  underlying stochasticity
• The basic course structure
• 2. Modeling and Analysis of Production and
  Service Systems as Continuous-Time Markov Chains
• (A brief overview of the key results of the
  theory of Markov Chain and Continuous-Time Markov
  Chains (CT-MC))
• Birth-Death Processes and the M/M/1 Queue
• Transient Analysis
• Steady State Analysis
• Modeling more complex behavior through CT-MCs
• Single station systems with multi-stage
  processing, finite resources and/or blocking
  effects
• Open (Jackson) and Closed (Gordon-Newell)
  Queueing networks
• Gershwins Models for Transfer Line Analysis
• Bucket Brigades

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Course Outline (cont.)

• 3. Accommodating non-Markovian behavior
• Phase-type distributions and their role as
  approximating distributions
• The M/G/1 queue
• The G/M/1 queue
• The G/G/1 queue
• The essence of Factory Physics
• (BCMP networks)
• 4. Performance Control of Production and Service
  systems
• Controlling the event rates of the underlying
  CT-MC model (an informal introduction of the dual
  Linear Programming formulation in standard MDP
  theory)
• A brief introduction of the theory of Markov
  Decision Processes (MDPs) and of Dynamic
  Programming (DP)
• An introduction to Approximate DP
• An introduction to dispatching rules and
  classical scheduling theory
• Buffer-based priority scheduling policies, Meyn
  and Kumars performance bounds and stability
  theory

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Course Outline (cont.)

• 5. Behavioral Control of Production and Service
  Systems
• Behavioral modeling and analysis of Production
  and Service Systems
• Resource allocation deadlock and the need for
  liveness-enforcing supervision (LES)
• Petri nets as a modeling and analysis tool
• A brief introduction to the behavioral control of
  Production and Service Systems

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What (I hope that) we eventually achieved ?

• A solid exposition of
• DTMC theory
• Properties of exponential distributions and
  Poisson processes
• CTMC and Semi-Markov theory
• Classical queueing theory
• (An introduction to reversibility theory and its
  implications)
• Demonstration of the application of the
  aforementioned theory in the modeling and
  performance evaluation of production and service
  systems
• Bucket Brigades
• Factory Physics type of models
• The method of stages for modeling non-Markovian
  behavior through Markovian models
• Other examples presented in class and in reading
  assignments (e.g., excerpt from Gershwin)
• Homework problems
• Establishing some proficiency for working with
  the aforementioned stochastic models
• Indentify / recognize cases accepting exact
  modeling
• Fit more general situations / behaviors to the
  presented queueing models for some approximating
  analysis
• Go back to (semi-)Markov models using the method
  of stages, and maybe some approximating
  techniques on the resulting Markov model to deal
  with the complexity problems arising from the
  curse of dimensionality

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Towards performance control

• In case of a small number of alternatives, use an
  appropriate (descriptive) model to evaluate the
  performance of each alternative.
• For cases where the system performance can be
  explicitly expressed as a function of the
  parameter of interest, formulate and solve an
  appropriate optimization (mathematical
  programming) problem.
• Markov Decision Processes (MDP)
• Transition probabilities out of each state are
  functions of decisions / actions taken at these
  states.
• Selections of decisions to be followed at each
  state (and visit), constitute a policy. A policy
  can also be deterministic or randomized.
• A functional defined over the sequence of visited
  states and performed actions at each state
  characterizes the performance of the policy
  instantiation.
• The expected value of this functional over all
  the possible sample paths generated by the
  considered policy, when starting from a certain
  initial state, defines the policy value at this
  state.
• In an MDP setting we want to identify a policy
  that optimizes the value of each state.
• The state function providing their values under
  an optimal policy is known as the value function
  of the MDP problem.
• Dynamic Programming (DP) is a set of methods that
  tries to compute the value function of a given
  MDP problem by building upon the fundamental
  remark that the value of a (state, action) pair
  at a certain time / period, can be decomposed as
• an immediate value for taking the considered
  action at the given state in the current period,
  plus
• the expected value of the resulting state.
• The above decomposition is the essential content
  of the, so called, Bellmans equation.
• Availability of the value function reduces the
  problem of computing an optimal decision for any
  given state, as a local optimization problem over
  the entire set of actions available at that
  state.
• Classical DP methods are limited by the very
  large size of the underlying state spaces, that
  renders intractable even the enumeration of the
  entire value function! These complexity problems
  are collectively known as the curse of
  dimensionality.
• Approximate DP (ADP) seeks to deal with the curse
  of dimensionality problems by working with an
  approximate compact representation of the value
  function. This approach necessitates the
  development of
• a representational space able to support
  effective compact approximations of the problem
  value function

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Petri nets and behavioral control of RAS
A (generalized stochastic) Petri net modeling the
CQN of Problem 8.15 in Homework 3

• Petri nets offer an unambiguous and compact
  representation of DES structure and behavior.
• Reachability / Coverability analysis is the
  systematic construction of the underlying state
  space through an efficient enumeration of all the
  possible behaviors generated by the system.
• The availability of the reachable state space
  enables an assessment of various qualitative
  properties of the system behavior (like
  boundedness, liveness, deadlock-freedom,
  fairness, etc.)
• For many PN sub-classes, the assessment of many
  such qualitative properties can be performed
  through much more efficient analytical /
  algebraic techniques that exploit the information
  on the system structure and its invariants
  provided by the PN model itself.
• In the context of DES modeling the resource
  allocation taking place in production and service
  systems, an important issue is the maintenance of
  live behavior by preventing the formation of
  deadlock.
• Generalized Stochastic PNs allow the modeling of
  the passage of time by distinguishing between
  instantaneous and timed transitions. Timed
  transitions involve a delay in their firing,
  which is drawn from an exponential distribution.
  Their timed-based dynamics corresponds to a
  semi-Markov process defined on the underlying
  state space.

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Thanks for being in the class and Have a Great
Holiday!