A Methodology for Formal Expression of Hierarchy in Model Solution zyxw * zyxwv M. Malhotra t Room 2K-314 AT&T Bell Labs Holmdel, NJ 07733 Abstract zyxwvutsr We present a methodology for formal specifica- tion of hierarchy both in model specification and model solution. We allow hierarchy zyxwvuts to exist among diflereni model types used in performance and de- pendability modeling. This oflers a lot of jlexibil- ity and power to the modeler. Our methodology presents a unified view of a variety of modeling techniques such as hierarchical composition, behav- ioral decomposition, iterative hierarchical modeling, reward-based performabilit y modeling, aggregation, etc. This methodology brings the hierarchical mod- eling technique(s , based on which the model is con- structed, to the Are. This results in a better under- standing of the model by the user and it can sim- plify model validation if need be. Such a methodol- ogy would also make the design of modeling toolkits, which allow these modeling techniques, much sim- pler by presenting a conceptually simpler and unijed view of a variety of modeling techniques. The for- mal expression is also expected to assist the modeler in construction of large, complex models. zyxwvut 1 Introduction State-space model types (such as Markov chains) are used to capture various dependencies (such as shared resources) in dependability and performance models. Unfortunately, these model types suffer from the problems of lar eness and stiffness (in case of dependability modelsf for realistic system mod- els. Generation of large Markov models is in it- self a complicated task. To alleviate the problem of model specification and generation, higher level model types such as stochastic Petri nets [l, 21 are used. Stochastic Petri nets provide a more compact representation of a model and can be automatically converted to Markov models. Hence, such higher- level model types do not alleviate the problems of storing and solving the large model. To over- come the problem of largeness, several techniques *This work was supportedin part by the National Science Foundationunder Grant CCR-9108114 and by the Naval Sur- face Warfare Center N60921-92-C-0161. tThis work was done when the author was at Duke University. K.S. Trivedi Dept. of Electrical Engr. Duke University Durham, NC 27706 have been proposed that decompose the Markov chain enerator matrix and yield approximate zyx so- lution 73, 41. Matrix-level decomposition alleviates the problem of largeness in model solution but the problem of large model generation and storage still persists. Therefore the need arises for decomposition ap- proaches that work at model level rather than the matrix level. Examples of such techniques in- clude the flow-equivalent server approximation in- troduced by Chandy et a1 [5], behavioral decompo- sition technique used in the software tool HARP [6], and the approach used by Balbo et a1 [7]. In these approaches, the overall model is not generated but instead smaller models are generated whose solution is combined to yield overall model solution. These are examples of hierarchical composition. It holds a lot of promise since it alleviates all the problems in- cluding large model specification, generation, stor- age, and solution. Depending upon the kind of sys- tem being modeled, this approach could result in an approximate solution or an exact solution. We consider hierarchical composition in its most general form. Our view of hierarchical modeling en- compasses a wide variety of modeling and solution techniques including fixed-point iteration scheme for solving models [8, 9, 10, 111, reward-based per- formability analysis [12], and non-iterative hierar- chical (hierarchical composition or decomposition) models [13, 141. The overall system model con- sists of one or more submodels of possibly different types (this is known as hybrid hierarchical model- ing) which interact with each other in some manner. The model solution is obtained by solving the sub- models and combining submodel solutions in some fashion. Several attempts have been made in the past to formalize the specification of system models. Berson et al [15] proposed an object-oriented paradigm for model specification and generation. Hillston [16] proposed a modular approach to mod- eling which aims to make models more accessible to non-experts. These efforts concentrate on formal- izing model specification and model generation. A more general hierarchical modeling environment is 258 0-81864250-5/93 $03.00 zyxwvutsrqp 0 1993 IEEE