STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |1| STDEVS A Formal Framework for STochastic DEVS Modeling and Simulation Rodrigo Castro * Ernesto Kofman * Gabriel Wainer ** * Universidad Nacional de Rosario ** Carleton University System Dynamics and Signal Processing Lab. Advanced Real-Time Simulation Lab. Argentina Canada Argentina Canada

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |2| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |3| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |4| INTRODUCTION DEVS formalism Developed as a general system theoretic based language. Universal description of discrete event systems. Stochastic models Play a fundamental role in discrete event system theory. Any system involving uncertainties, unpredictable human actions or system failures requires a non–deterministic treatment. Widely adopted stochastic discrete event formalisms: Markov Chains, Queuing Networks, Stochastic Petri Nets...

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |5| PROBLEM STATEMENT Even though most of the DEVS simulation tools have incorporated the use of random functions... DEVS has originally only been formally defined for deterministic systems. Early works on mapping DEVS to stochastic systems are not completely general. The DEVS formal framework has limited extent to a wide family of (generalized) stochastic systems. No previous general DEVS–based formalism for stochastic discrete event systems.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |6| EARLY WORKS Previous efforts on mapping DEVS to stochastic systems behavior have limited scope: “DES models driven by pseudo-random sequences define DEVS models” (Aggarwal, U. of Michigan, 1975) Problem: Not a methodology to describe DEVS stochastic models. “Relationship established between random experiment outcomes and externally observed possible state trajectories of a DEVS simulation” (Melamed, U. of Michigan, 1976) Problem: Limited to models described at the input/output level. “Extended DEVS formalism taking into account internal stochastic behavior at the state transition level” (Joslyn, NASA Goddard Space Flight Center, 1996) Problem: Limited to finite state sets models (not general sets).

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |7| CONTRIBUTION OBJECTIVES Provide an extension of DEVS that establishes a formal framework for modeling and simulation of general stochastic discrete event systems. DRIVERS Rely on the deterministic DEVS Atomic Model definition as a starting point.  Keep the essence of the DEVS model structure, then  derive from it a new stochastic model structure by  introducing the new probabilistic features needed  replacing the way internal dynamics are described.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |8| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |9| STDEVS STRATEGY (What can we do ?) Define the new STochastic DEVS (STDEVS) Atomic Model structure, where internal dynamics incorporate probabilistic components, relying on the general Theory of Probability Spaces: Think of DEVS state transitions as “Random Experiments” Keep the general and arbitrary nature of all the original DEVS sets. Respect the deterministic nature of the original DEVS deterministic functions.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |10| Keep general Keep deterministic Replace with Probability Spaces components Think of DEVS state transitions as “Random Experiments” STDEVS STRATEGY DEVS Atomic Model components: ta 0 → e → ta X Y S DEVS Atomic Model

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |11| STDEVS FORMAL DEFINITION A STDEVS Model (M ST ) has the structure A DEVS Model (M D ) has the structure = = Same Functionality Replaced Components

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |12| STDEVS FORMAL DEFINITION A STDEVS Model (M ST ) has the structure Components obtained from a Probability Space construct. Will have to answer: Given the present model state s Є S, and after the next state transition (internal or external “random experiment”), ¿ What is the probability that the new future state s’ Є S belongs to any given subset of S ? Arbitrary (Finite, Infinite, Continuous, Discrete, Hybrid...) Can't assign probabilities to an individual s (will render always 0 for S continuous !)

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |13| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |14| PROBABILITY SPACES INFORMAL IDEA Arbitrary Subsets of S sp Sample Space : S sp All the possible outcomes (elements) of a random experiment. We have to make sure that S sp is measurable, given S sp is a totally arbitrary set. Takes the role of the DEVS State Space : S Samples s Є S sp States s Є S Analogous Outcomes of a general random experiment. Outcomes of a STDEVS state transition.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |15| PROBABILITY SPACES INFORMAL IDEA F : a Sigma Field of S sp F is a collection of subsets (Not any collection, has special properties) F : a member of F The F members of F can be assigned probabilities, but not the single elements s Є F of them. Sample Space : S sp P(F) : Probability Measures for members F Є F. Now, the structure ( S sp, F, P ) is a Probability Space. It can fully describe random experiments on the arbitrary Sample Space S sp.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |16| PROBABILITY SPACES FORMAL DEFINITION The pair ( S sp, F ) is a Measurable Space if F is a Sigma Field of S sp 1)Build a Measurable Space from the Sample Space. F : a member of F Now this measurable structure ( S sp, F ) can be equipped with Probability Measures. Recall that S sp plays the role of the DEVS State Space. F is a Sigma Field of S sp if it satisfies :

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |17| PROBABILITY SPACES FORMAL DEFINITION A Probability Measure P on a Measurable Space ( S sp, F ) is an assignment of a real number P(F) to every member F Є F, such that P obeys the following rules: 2)Build a Probability Space from the Measurable Space. Now, the structure ( S sp, F, P ) is a Probability Space. It can fully describe random experiments on the arbitrary Sample Space S sp.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |18| PROBABILITY SPACES FORMAL DEFINITION 3)Make it more practical ! Sigma Fields F are theoretically essential, but not very useful in practice. Usually we want to pick our own collection of subsets G Є G out of the Event Space S sp that make some practical sense. We are lucky: Any arbitrarily chosen collection G Є G of subsets always generates a minimum Sigma Field F=M(G). → We will use G from now on. The knowledge of P(G) for every G Є G readily defines the function P(F) for every F Є F. → We will use P(G) from now on. Finally: For every G Є G, the function P(G) expresses the probability that the random experiment produces a sample s Є G as the experiment outcome.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |19| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |20| STDEVS THE NEW COMPONENTS Let´s start with the internal transition dynamics: Power Set of S Given a present state s, the collection G int ( s ) contains all the subsets of S that the future state s’ might belong to, with a known probability P int (s,G).

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |21| STDEVS THE NEW COMPONENTS Analogous reasoning for the external transition dynamics: Given a present state s, an elapsed time e, and an input element x, the collection G ext ( q,x ) contains all the subsets of S that the future state s’ might belong to, with a known probability P int (s,G). q=(s,e)

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |22| STDEVS MAIN THEORETICAL PROPERTIES We demonstrated the following properties of STDEVS, analogous to the DEVS main properties: (Formulas and demonstrations in our paper) The STDEVS structure verifies Closure Under Coupling. We can couple STDEVS models in a hierarchical way, encapsulating complex coupled models, and coupling them with other atomic ones. The STDEVS structure is equipped with a Legitimacy Property. We redefined the DEVS Legitimacy Property. Now, it expresses the probability of having an infinite number of transitions in a finite interval of time.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |23| STDEVS THEORETICAL IMPLICATIONS Now, with the new STDEVS formal framework we can: Represent any stochastic system, no matter how complex the stochastic processes driving its dynamics might be. This is true even if the system can not (or it is very difficult, expensive, etc.) be implemented in a practical simulator. This allows to a strong theoretical probabilistic manipulation of the STDEVS structure. (that evolves through a finite number of changes in a finite amount of time)

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |24| STDEVS, DEVS and RND functions MAIN PRACTICAL PROPERTIES We shall call DEVS-RND models to those DEVS models whose transition functions depend on random experiments through any random variable. In practice, probability distributions are typically obtained by some computational manipulation of an Uniform U(0,1) random variable r obtained with a RND() pseudo-random sequence generator in most programming languages. r can be an array of n Uniforms: r ~ U(0,1) n

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |25| STDEVS, DEVS and RND functions MAIN PRACTICAL PROPERTIES We demonstrated the following properties for STDEVS, of strong practical interest: (Formulas and demonstrations in our paper) Theorem 1: A DEVS-RND model always define an equivalent STDEVS model. Corollary 1: A DEVS-RND model depending on n Uniforms: r ~ U(0,1) n in its transition functions always define an equivalent STDEVS model. Corollary 2: A deterministic DEVS model always defines an equivalent STDEVS model. (DEVS is a particular case of STDEVS)

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |26| STDEVS, DEVS and RND functions PRACTICAL IMPLICATIONS Now, with the new STDEVS formal framework (and its properties) we can: Build and couple together any hierarchical system interconnecting DEVS and STDEVS models. (guaranteeing all the desired theoretical properties for doing it) Model most of the practical situations of stochastic behavior in STDEVS without making use of probability spaces. (using the handier DEVS-RND equivalents).

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |27| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |28| EXAMPLE Load Balancer – All components stochastics A simple illustrative computational system : A generator offering a workload (tasks) to a two-servers cluster, with an adjustable balancer biased with a balancing factor. LBM : Load Balancer Model LG : Load Generator CL : Cluster (Coupled) WB : Workload Balancer S1 : Server 1 S2 : Server 2 d r : Departure Rate b f : Balancing Factor [0 to 1] s ti : Average Service Times λ i, λ ’ i : Average Traffic Rate μ i : Average Service Rate Rates in [Tasks/second] S1,S2 are M/M/1/1 queues (simplest). No buffer capacity: overflowing tasks are dropped. Service Times are Exponentially-distributed. LG generates Poisson-distributed task workload. WB will distribute workload according a Uniform distribution biased by a continuous factor b f.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |29| EXAMPLE LBM: Load Balancer Model We want to model this system relying on the STDEVS formal framework. Then, execute the model in a DEVS simulator and verify it against analytical results. All the stochastic descriptions of this model are simple ones: can be readily modeled with a DEVS-RND approach. We will show only LG with both STDEVS and DEVS-RND descriptions, for illustrative purposes. For the rest of the components we will forget about the STDEVS description, and make use of Theorem1/Corollary1: Concentrate only on the DEVS- RND description (much easier !)

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |30| EXAMPLE LG: Load (tasks) Generator Poisson discrete process ( d r = λ, with d r = departure rate) ⇒ Exponentially-distributed inter-departure times σ k between task k and task k+1 : where a = d r. No inputs. Only internal state transitions. State s storages next departure delay. STDEVS ← Equivalent → Explicit stochastic-oriented definition No need to be defined Continue, real-valued half-open intervals collection Practical Inverse Transformation method DEVS-RND Depends on r r ~U(0,1) Does nothing DETERMINISTIC

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |31| EXAMPLE WB, S1, S2 For these components we follow an identical technique as with LG to get the DEVS- RND models: Make the transition functions depend on a random variable r =U(0,1): and Define them using the Inverse Transformation Method that uses r and yields the desired stochastic properties with an algorithmically programmable formula. Note that: There is no stochastic description at the t a (s) or λ(s) functions.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |32| EXAMPLE WB : Workload Balancer (DEVS-RND components only) Depends on r r ~U(0,1) Depends on r (but doesn't “use” it) b f biases each port random selection

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |33| EXAMPLE Model verification Simulated (Marks) and Theoretical (Curves) results Effective Output Rate and Loss Probabilities vs. Balancing Factor b f Erlang’s Formula for M/M/1/1: LBM Model formulas: Effective Output Rate: Task Loss Probabilities:  Simulation is verified against theoretical expected results.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |34| AGENDA Introduction. Problem Statement. Early Works. Contribution Objectives & Drivers. STDEVS Strategy. Formal Definition. Probability Spaces Informal Idea. Formal Definition. STDEVS The New Components. Theoretical & Practical Properties and Implications. Example Conclusions. Next Steps.

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |35| CONCLUSIONS We presented STDEVS, a novel formalism for describing stochastic discrete event systems. STDEVS provides: A formal framework for modeling and simulation of generalized stochastic discrete event systems. Shares the system theoretical approach of DEVS. Makes use of Probability Spaces theory. STDEVS allows for: A sound probabilistic theoretical treatment of general stochastic DEVS. From its dynamics, not from its external behavior. An easy practical way of implementation in simulators. Not ‘very’different from what we were doing so far !

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |36| NEXT STEPS We are developing STDEVS–based libraries for simulation tools PowerDEVS and CD++ Research area: Control Theory techniques applied to Admission Control in data networks. QUESTIONS ?

STDEVS – STochastic DEVS Universidad de Rosario | Carleton University HPCS 2008 | SpringSim08 April 15, Ottawa City. Castro R., Kofman E., Wainer G. |37| THANK YOU ! More information: