STDEVS A Formal Framework for ST ochastic DEVS Modeling and Simulation Rodrigo Castro * Ernesto Kofman * Gabriel Wai

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 http://www.fceia.unr.edu.ar/lsd/http://www.sce.carleton.ca/faculty/wainer/ARS/

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.

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...

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 toa wide family of (generalized) stochastic systems. • No previous general DEVS–based formalism for stochastic discrete event systems.

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 internalstochastic behavior at the state transition level”(Joslyn, NASA Goddard Space Flight Center, 1996) • Problem: Limited to finite state sets models (not general sets).

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 DEVSAtomic Model definition as a starting point. • Keep the essence of the DEVS modelstructure, then • derive from it a new stochastic modelstructureby • introducing the new probabilistic features needed • replacing the way internal dynamicsare described.

STDEVS • STRATEGY (What can we do ?) • Define the newSTochastic DEVS (STDEVS) Atomic Model structure,where internal dynamics incorporate probabilistic components,relying on thegeneralTheory of Probability Spaces:Think ofDEVS 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 deterministicfunctions.

DEVS Atomic Model ta S 0 → e→ ta X Y Keep general Replace with Probability Spaces components Keep deterministic • STDEVS • STRATEGY Think of DEVS state transitions as “Random Experiments” DEVS Atomic Model components:

STDEVS • FORMAL DEFINITION A STDEVS Model (MST) has the structure Replaced Components = = Same Functionality A DEVS Model (MD) has the structure

STDEVS • FORMAL DEFINITION A STDEVS Model (MST) has the structure Can't assign probabilities to an individuals (will render always 0 for S continuous !) Arbitrary (Finite, Infinite, Continuous, Discrete, Hybrid...) 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 Sbelongs to any given subset of S?

PROBABILITY SPACES • INFORMAL IDEA • All the possibleoutcomes (elements) of a random experiment. • We have to make sure that Sspis measurable, given Sspis a totally arbitrary set. Sample Space : Ssp • Takes the roleof the DEVSState Space :S Arbitrary Subsets ofSsp Sampless ЄSsp Statess ЄS Outcomes of a general random experiment. Analogous Outcomes of a STDEVS state transition.

PROBABILITY SPACES • INFORMAL IDEA • Fis a collection of subsets (Not any collection, has special properties) F : a Sigma Field ofSsp P(F):Probability Measures for membersF ЄF. F: a member ofF Sample Space : Ssp • The Fmembers of Fcan be assigned probabilities, but not the single elements s ЄF of them. • Now, the structure (Ssp ,F, P )is a Probability Space. • It can fully describe random experiments on the arbitrary Sample Space Ssp.

PROBABILITY SPACES • FORMAL DEFINITION Build a Measurable Space from the Sample Space. • The pair (Ssp ,F )is a Measurable SpaceifF is a Sigma Field ofSsp • Fis a Sigma Field ofSspifit satisfies : F: a member ofF • Now this measurable structure (Ssp ,F )can be equipped with Probability Measures. • Recall that Ssp plays the role of the DEVS State Space.

PROBABILITY SPACES • FORMAL DEFINITION Build a Probability Space from the Measurable Space. • A Probability Measure P on a Measurable Space (Ssp ,F )is • an assignment of a real number P(F) to every member FЄ F, • such that P obeys the following rules: • Now, the structure (Ssp ,F, P )is a Probability Space. • It can fully describe random experiments on the arbitrary Sample Space Ssp.

PROBABILITY SPACES • FORMAL DEFINITION 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 ЄG out of the Event Space Ssp that make some practical sense. • We are lucky: • Any arbitrarily chosen collection G ЄG 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 ЄG readily defines the function P(F) for every F ЄF. →We will use P(G) from now on. • Finally: For every G ЄG , the function P(G) expresses the probability that the random experiment produces a sample sЄG as the experiment outcome.

STDEVS • THE NEW COMPONENTS • Let´s start with the internal transition dynamics: Power Set of S • Given a present state s , the collection Gint(s)contains all the subsets of Sthat the future state s’ might belong to, with a known probability Pint(s,G).

STDEVS • THE NEW COMPONENTS • Analogous reasoning for the external transition dynamics: q=(s,e) • Given a present state s , an elapsed time e, and an input element x, the collection Gext(q,x) contains all the subsets of S that the future state s’ might belong to, with a known probability Pint(s,G).

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 verifiesClosure 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 • THEORETICAL IMPLICATIONS • Now, with the new STDEVS formal framework we can: • Representany 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, 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 anyrandom variable. • In practice, probability distributions are typically obtained by some computational manipulation of an Uniform U(0,1) random variabler 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, 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, 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 interconnectingDEVS 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).

EXAMPLE • Load Balancer – All components stochastics • A simple illustrative computational system : A generator offering a workload (tasks) to a two-serverscluster, 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 • dr: Departure Rate • bf: Balancing Factor [0 to 1] • sti: Average Service Times • λi, λ’i: Average Traffic Rate • μi: 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-distributedtask workload. • WB will distribute workload according a Uniform distributionbiased by a continuous factor bf .

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 !)

←Equivalent→ STDEVS DEVS-RND No need to be defined Depends on r r ~U(0,1) Continue, real-valued half-open intervals collection Does nothing DETERMINISTIC Practical Inverse Transformation method Explicit stochastic-oriented definition • EXAMPLE • LG: Load (tasks) Generator • Poisson discrete process (dr= λ , with dr = departure rate) ⇒ • Exponentially-distributed inter-departure times σk between task k and task k+1 : where a =dr . • No inputs. Only internal state transitions. State s storages next departure delay.

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 ta(s) or λ(s) functions.

EXAMPLE • WB : Workload Balancer (DEVS-RND components only) Depends on r r ~U(0,1) bf biases each port random selection Depends on r (but doesn't “use” it)

EXAMPLE • Model verification Simulated(Marks) and Theoretical(Curves) results Effective Output Rate and Loss Probabilities vs. Balancing Factor bf Erlang’s Formula for M/M/1/1: LBM Model formulas: Effective Output Rate: Task Loss Probabilities: • Simulation is verified against theoretical expected results.

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 systemtheoretical approach of DEVS. • Makes use of Probability Spaces theory. • STDEVS allows for: • A soundprobabilistic theoretical treatment of generalstochastic 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 !

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 ?

THANK YOU ! More information: rodrigocastro@ieee.org kofman@fceia.unr.edu.ar gwainer@sce.carleton.ca

STDEVS A Formal Framework for ST ochastic DEVS Modeling and Simulation Rodrigo Castro * Ernesto Kofman * Gabriel Wai