# Library prosa.model.processor.multiprocessor

# Multiprocessor State

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).

Definition multiproc_scheduled_on

(j : Job) (mps : multiprocessor_state) (cpu : processor num_cpus)

:= scheduled_in j (mps cpu).

(j : Job) (mps : multiprocessor_state) (cpu : processor num_cpus)

:= scheduled_in j (mps cpu).

Similarly, the supply produced by a given multiprocessor state
mps on a given core cpu is exactly the supply provided by the
core-local state (mps cpu).

Definition multiproc_supply_on

(mps : multiprocessor_state) (cpu : processor num_cpus)

:= supply_in (mps cpu).

(mps : multiprocessor_state) (cpu : processor num_cpus)

:= supply_in (mps cpu).

Next, 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).

Definition multiproc_service_on

(j : Job) (mps : multiprocessor_state) (cpu : processor num_cpus)

:= service_in j (mps cpu).

(j : Job) (mps : multiprocessor_state) (cpu : processor num_cpus)

:= service_in j (mps cpu).

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

#[local] Program Instance multiproc_state : ProcessorState Job :=

{|

State := multiprocessor_state;

scheduled_on := multiproc_scheduled_on;

supply_on := multiproc_supply_on;

service_on := multiproc_service_on

|}.

Next Obligation.

by move ⇒ ? ? ?; apply: leq_sum ⇒ c _; exact: service_on_le_supply_on.

Qed.

Next Obligation. by move⇒ ? ? ?; apply: service_in_implies_scheduled_in. Qed.

{|

State := multiprocessor_state;

scheduled_on := multiproc_scheduled_on;

supply_on := multiproc_supply_on;

service_on := multiproc_service_on

|}.

Next Obligation.

by move ⇒ ? ? ?; apply: leq_sum ⇒ c _; exact: service_on_le_supply_on.

Qed.

Next Obligation. by move⇒ ? ? ?; apply: service_in_implies_scheduled_in. Qed.

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.