ME.6.MC:4.3 - Construct the connection, including shared resources

For sequential functions, form (g\circ f), meaning first (f), then (g). Check that every relevant output of (f) is an allowed input of (g), including its meaning and conditions. With partial functions, determine the inputs on which the composite is defined. MATH.17 supplies the operation collections and closure argument.

Associativity permits regrouping a fixed sequence. Reordering needs a different property: the two orders must agree in the observations the receiver uses. If the operations return information as well as changing state, retain both in that comparison.

For overlapping work, represent the shared state or resource once. Connecting two models that each contain a private copy of the same bench or performer doubles the modeled capacity. Describe when each operation can begin, what it reads, what it changes or occupies, and when it releases a resource.

For example, let operation (i) have start time (s_i), fixed duration (d_i), and resource demand (r_{ik}) on resource (k). For nonpreemptive work, a precedence (i) before (j) requires

[ s_j\geq s_i+d_i. ]

If the available capacity is (c_k(t)), then at every time (t),

[ \sum_{i:\ s_i\leq t<s_i+d_i}r_{ik}\leq c_k(t). ]

Add input-availability and deadline conditions where they matter. These inequalities describe one timing model; interruptions, setup, rework or uncertain durations require the corresponding extension when they can change the answer.

Use CMP.14 when communications or interleaved updates need a computational interaction model. It supplies allowed steps, interference, coordination and progress arguments. For ordinary schedules or algebraic combinations, the simpler construction can be sufficient.