Multiprocessor State

In the following, we define a model of identical multiprocessors, i.e., of processors with multiple cores of identical capabilities. The multiprocessor model is generic in the type of processor state of the cores. That is, it is possible to combine any uniprocessor state (such as the ideal state) with the following generic multiprocessor construction. (In fact, by combining the below multiprocessor model with variable speed processors, it is even possible to obtain a so-called uniform multiprocessor model.) NB: For now, the definition serves only to document how this can be done; it is not actually used anywhere in the library.

Consider any types of jobs...

... and consider any type of per-processor state.

Given a desired number of processors [num_cpus], we define a finite type of integers from [0] to [num_cpus - 1]. The purpose of this definition is to obtain a finite type (i.e., set of values) that can be enumerated in a terminating computation. Syntax hint: the ['I_] before [num_cpus] is ssreflect syntax for the finite set of integers from zero to [num_cpus - 1].

Next, for any given number of processors [num_cpus]...

...we represent the type of the "multiprocessor state" as a function that maps processor IDs (as defined by [processor num_cpus], see above) to the given state on each core.

Based on this notion of multiprocessor state, we say that a given job [j] is currently scheduled on a specific processor [cpu], according to the given multiprocessor state [mps], iff [j] is scheduled in the processor-local state [(mps cpu)].

Similarly, the service received by a given job [j] in a given multiprocessor state [mps] on a given processor of ID [cpu] is exactly the service given by [j] in the processor-local state [(mps cpu)].

Finally, we connect the above definitions with the generic Prosa interface for processor models.

From the instance [multiproc_state], we get the function [service_in]. The service received by a given job [j] in a given multiprocessor state [mps] is given by the sum of the service received across all individual processors of the multiprocessor.