Measure-adaptive state-space construction

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Abstract

Measure-adaptive state-space construction is the process of exploiting symmetry in high-level model and performance measure specifications to automatically construct reduced state-space Markov models that support the evaluation of the performance measure. This paper describes a new reward variable specification technique, which combined with recently developed state-space construction techniques will allow us to build tools capable of measure-adaptive state-space construction. That is, these tools will automatically adapt the size of the state space to constraints derived from the system model and the user-specified reward variables. The work described in this paper extends previous work in two directions. First, standard reward variable definitions are extended to allow symmetry in the reward variable to be identified and exploited. Then, symmetric reward variables are further extended to include the set of path-based reward variables described in earlier work. In addition to the theory, several examples are introduced to demonstrate these new techniques.

Introduction

Model-based evaluation of systems is becoming increasingly valuable as performance and dependability requirements become more stringent and systems become more complex. As a result, much research has been focused on developing the proper mathematical and software tools for effective use of modeling in the product life-cycle.

One line of research has explored the use of Markov process models generated from a higher-level formalism, such as stochastic Petri nets, for the purpose of model-based evaluation. Software tools have been developed for the specification of models in a high-level formalism and for the automatic construction of Markov process models from the high-level specification. The Markov process is typically specified in terms of its initial probability vector and its state transition matrix.

The state space of a model specified in a high-level formalism can be very large (>107 states). Research on state-space construction methods has focused on methods for tolerating large state spaces and methods for avoiding large state spaces. Tolerance techniques focus on finding special data structures and efficient algorithms for generating, storing, and computing with the state transition matrix. Examples of this approach include the Kronecker product algorithms of Plateau [1], [2], Buchholz [3], Ciardo and Tilgner [4], [5], Donatelli [6], and Kemper [7], their recent collaboration [8], and the disk-based [9] and “on-the-fly” [10] methods of Deavours and Sanders. Methods for partial exploration of the state space with error bounds for some measures are discussed in [11], [12], [13].

Techniques for avoiding large state spaces typically involve state aggregation, whereby the state space is reduced by partitioning it into equivalence classes and using a single representative from each class in the final state space. Examples of this approach are the papers of Aupperle and Meyer [14], [15], the hierarchical modeling techniques of Buchholz [16], Carrasco’s [17] work with stochastic high-level Petri nets, stochastic well-formed nets [18], performance evaluation process algebra [19], reduced base model construction [20], and the symmetry exploitation algorithm of Somani [21]. These approaches all use symmetry to reduce the state space by aggregating states that correspond to symmetric configurations. Alternatives to these exact methods are the decomposition method of Ciardo and Trivedi [22], which treats nearly independent submodels as independent and uses fixed-point iteration to solve the model, and the bounds on “quasi-lumpable” models described by Franceschinis and Muntz [23], [24]. The most successful largeness-avoidance techniques are able to construct the reduced state space directly, without first constructing the full detailed state space. Also, most largeness-avoidance techniques exploit symmetry in the model to identify equivalent states.

Another problem, which has long been recognized but left unsolved, is the separation of the “performability measure” from the “system model”. Experienced analysts understand that the nature of a model is usually tied directly to the questions one hopes to answer. The performance measure is a formal specification of a question about the system. The system model is a probabilistic specification of the system behavior. Obviously, the choice of modeling methods and the precision with which a model mimics a real system are determined by the accuracy and precision required in the evaluation of the performance measure. This notion is universally accepted. A problem arises when a modeler must artificially add complexity to a model in order to track events or sequences of events that determine the value of the performance measure. We call methods that construct state spaces that are tailored to a particular set of measures “measure-adaptive state-space construction methods”.

In previous work, we developed a new technique for detecting and exploiting symmetry in Markov models [25] for the purpose of constructing compact state spaces. In that work, we applied group theory to the problem of identifying symmetric states in models composed graphically through shared state variables. Composed models yield a “model composition graph”, which is analyzed to find its automorphism group. This group is used to partition the state space into equivalence classes and directly generate a smaller state space. In subsequent work, we developed a new approach to reward variable specification [26] that allows one to define reward measures on sequences of events in the model, as captured in a “path automaton” specification. A “path-based reward variable” can have different reward structures for each state of the path automaton. Another result of this work was a state-space construction procedure for automatically generating a state space from the specification of the system model and the path-based reward variable. This new approach frees the modeler from the need to add additional complexity to a system model in order to support path-based performance measures.

The goal of this paper is to build upon our previous work, presented in [25], [26], to develop specification and model construction techniques for measure-adaptive state-space construction. In this work, we combine our methods for path-based reward variables with our work on detecting and exploiting model symmetry. This is the final step needed to separate the modeling of system structure and behavior from performance measurement. With this new approach, the model of the system need only reflect real system dependencies. Constraints and dependencies created by the nature of the performance measure are dealt with separately in a new symmetric path-based reward variable formalism.

Section snippets

Detecting and exploiting model symmetry

In this section, we briefly summarize the technique developed in [25] for detecting and exploiting model symmetry. Using a simple abstract modeling formalism, we define models and composed models, and discuss how the information embedded in the composed model is used to detect and exploit symmetries for the purpose of state-space reduction. These concepts are the foundation of measure-adaptive state-space construction, since they determine the structural restrictions on state-space reduction.

Symmetric reward structures

In this section, we define a symmetric reward structure and give sufficient conditions for constructing reduced state-space Markov processes that support the specified reward structure.

Definition 3

Given a composed model (Σ,I,κ,C) with state mapping set M and event set E, a symmetric reward structure (C,RR) is a pair of functions

  • C:E →R, the impulse reward function;

  • R:M →R, the rate reward function;

and a group ΓR defined on the composed model such that for all γΓR, C(eγ)=C(e) for all eE, and Rγ)=R(μ)

Reward variable specification

In this section, we introduce a reward variable formalism that will allow us to detect and exploit symmetry in the variable definition. Symmetry in the variable definition ultimately is combined with structural symmetry in the model to derive the symmetry group used to reduce the state space.

The basis for this new approach to reward variable specification is the observation that many performance measures may be written as functions that are invariant under permutations of some or all of their

Example reward structure specifications

We use the composed model in Fig. 3 to demonstrate the specification of compound reward structures. This example was used in [25] to demonstrate state-space reduction. For our purpose here, the details of the model are not important. We only need the structure and information on which instances are from the same model. In Fig. 3a, the boxes marked “C” correspond to instances of a computation node, those marked “R” correspond to routing nodes in the interconnection network, and those marked “IO”

Symmetric path-based reward structures

In this section, we extend symmetric reward structures to paths in composed models. Building on our work in [26], we define path automata for composed models and then introduce “symmetric path-based reward structures”. Composed models require an extension of the theory in [26] to multiple interconnected models and automorphisms.

When a path automaton is defined on a composed model that has a nontrivial automorphism group, care must be taken in defining the state transition function. Unless the

Example state spaces for symmetric reward variables

In this section, we introduce an example system and three symmetric reward variables that demonstrate the use of this new technique. We show how the size of the state space changes according to the specified reward variables. The toroidal mesh shown in Fig. 4 serves as the basis for the examples in this section. We will consider dependability measures for this system, and begin with a very simple model composition graph, shown in Fig. 5. In Fig. 5, the CPUs are independent, so the model

Example state space for symmetric path-based reward variable

In this section, we describe an example system and a symmetric path-based reward variable, and demonstrate the construction of the state space that supports the variable. To make the presentation of the example clear, complete, and understandable, we use a small example.

Consider a cluster of two servers where each server may be in one of the three states. The first state is perfect working order, the second state is partially degraded, and the last state is failed. Now suppose that this cluster

Conclusion

We have presented new techniques for specifying performance, dependability, and performability measures, and automatically constructing state spaces tailored to model and measure symmetry. First, we introduced symmetric reward structures; then, we extended that concept to symmetric path-based reward structures. We demonstrated the application of these new reward structures through several examples.

By developing these new techniques, we have separated the specification of the system model from

W. Douglas Obal II is a Performance Engineer at Hewlett-Packard Storage Organization. He received a B.S. in electrical engineering from Pennsylvania State University in 1988, an M.S. in electrical engineering from The University of Arizona in 1993, and a Ph.D. in electrical engineering from The University of Arizona in 1998. From 1996 to 1998, he was a Visiting Scholar at the University of Illinois at Urbana-Champaign. His research interests include model-based performance/dependability

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    W. Douglas Obal II is a Performance Engineer at Hewlett-Packard Storage Organization. He received a B.S. in electrical engineering from Pennsylvania State University in 1988, an M.S. in electrical engineering from The University of Arizona in 1993, and a Ph.D. in electrical engineering from The University of Arizona in 1998. From 1996 to 1998, he was a Visiting Scholar at the University of Illinois at Urbana-Champaign. His research interests include model-based performance/dependability evaluation, performance of storage systems, and dependable computing.

    William H. Sanders received his B.S.E. in computer engineering (1983), his M.S.E. in computer, information, and control engineering (1985), and his Ph.D. in computer science and engineering (1988) from the University of Michigan. He is currently a Professor in the Department of Electrical and Computer Engineering and the Coordinated Science Laboratory at the University of Illinois and is a Vice-Chair of IFIP Working Group 10.4 on Dependable Computing. In addition, he serves on the editorial board of the IEEE Transactions on Reliability. He is a Fellow of the IEEE and a member of the IEEE Computer, Communications, and Reliability Societies, as well as the ACM, Sigma Xi, and Eta Kappa Nu.

    Dr. Sanders’s research interests include performance/dependability evaluation, dependable computing, and reliable distributed systems. He has published more than 75 technical papers in these areas. He was co-Program Chair of the 29th International Symposium on Fault-tolerant Computing (FTCS-29), was program co-Chair of the Sixth IFIP Working Conference on Dependable Computing for Critical Applications, and has served on the program committees of numerous conferences and workshops. He is the developer of two tools for assessing the performability of systems represented as stochastic activity networks: METASAN and UltraSAN. UltraSAN has been distributed widely to industry and academia, and licensed to more than 185 universities, several companies, and NASA for evaluating the performance, dependability, and performability of complex distributed systems.

    This material is based upon work supported by DARPA/ITO under Contract No. DABT63-96-C-0069. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of DARPA/ITO. This work was completed while Dr. Obal was a Ph.D. student in the Electrical and Computer Engineering Department of the University of Arizona and a Visiting Scholar at the Coordinated Science Laboratory of the University of Illinois at Urbana-Champaign.

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