A.3.3.CC:5.2 - Buffer counts, external arrivals and job order

Two buffers each hold at most two jobs. Exactly three jobs are distributed between them. The product of count ranges is {0,1,2} x {0,1,2}; the stock condition is q1 + q2 = 3.

Testing the nine pairs leaves Q = {(1,2), (2,1)}. Equivalently, choose q1 from {1,2} and reconstruct q2 = 3 - q1. The description now supplies both possible occupancy arrangements.

An external arrival can raise the total to four. Keeping q2 = 3 - q1 would omit the possible arrangement (2,2). Retain both counts and use the total applicable to the new question, or include total N as a variable with q1 + q2 = N. Which arrival can occur, and when, comes from an arrival and transition account.

Suppose the next question asks which job a first-in-first-out buffer serves. The sequences [A,B] and [B,A] have the same count, yet they give different next jobs under that rule. Represent the ordered job identities in the relevant buffer. Counts remain recoverable as sequence lengths; the receiving operation gains the distinction it lacked.