Recently, parity-declustered layouts have been studied as a tool for reducing the time needed to reconstruct a failed disk in a disk array. Construction of such layouts for large disk arrays generally involves the use of a balanced incomplete block design (BIBD), a type of subset system over the set of disks. This research has been somewhat hampered by the dearth of effective, easily implemented constructions of BIBDs on large sets and by inefficiencies in some parity-distribution methods that create layouts that are larger than necessary. We make progress on these problems in several ways. In particular, we v demonstrate new BIBD constructions that generalize some previous constructions and yield simpler BIBDs that are optimally small in some cases, v show how relaxing some of the balance constraints on data layouts leads to constructions of approximately-balanced layouts that greatly increase the number of feasible layouts for large arrays, and v give a new method for distributing parity that produces smaller data layouts, resulting in tight bounds on the size of data layouts derived from BIBDs.Our results use a variety of algebraic, combinatorial, and graphtheoretic techniques, and together greatly increase the number of parity-declustered data layouts that are appropriate for use in large disk arrays. ]

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Abstract. We present an algebraic approach to the model checking of fault-tolerant systems. Fault models and fault-handling mechanisms are modelled using special-purpose process operators. Besides providing for natural models, speciM-purpose operators Mlow systems with large state spaces to be verified using systems with small state spaces. To support this verification technique we show that a kind of simulation relation on processes preserves all process operators in tyft/tyxt format. IntroductionModel checking -in which a system model is automatically checked to see if it satisfies a temporal logic formula -has two serious limitations. The size problem is that the state space of the model can grow exponentially with the state space of its components. Additionally, model checking algorithms for many logics have high time complexity. The generality problem is that model checking tells us only that some instance of a system satisfies a property. We usually want to know that a system works properly over a range of parameter values and initial conditions. These limitations are especially serious in the application of model checking to fault-tolerant systems. By modelling failures of a system component one can increase its possible interactions with other components and thus dramatically increase the state space of the system. Furthermore, one usually wants to show that a fault tolerance mechanism is general purpose; in such cases it is not enough to show that it works for a particular underlying system.Here we present an approach to the model checking of fault-tolerant systems based on process algebra. To model faults and fault-handling mechanisms we define new, special-purpose process operators. A faulty version of a process is modelled by applying a fault operator to it. For example, suppose process P models a system or system component. To obtain a crash-faulty version of P we define a new operator Cr and apply it to P. Similarly, a fault-tolerant or faultdetecting version of a process is defined by applying a process operator to it. This approach has modelling advantages because it is natural to understand faults and fault-handling mechanisms (e.g., triple-modular redundancy [19]) as behavior transformers that are independent of an underlying process or computation.Defining faults and fault-handling mechanisms as process operators also has technical advantages. We show that if a temporal property holds of a faulthandling mechanism applied to a simple underlying process, then it holds of the mechanism applied to more complex underlying processes. In this way both the

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