 Circular buffer operation costs

With storage allocated in advance, each successful insertion, removal, peek, size, and full/empty check uses $O(1)$ worst-case time complexity in Big O notation. This assumes constant-cost indexed array access, bounded machine word index arithmetic and fixed-cost scalar item reads and writes. Each operation touches a bounded number of slots and fields regardless of $n$ and $C$; no shifting, scanning or resizing occurs. Rejected operations also take constant work. Producing a copy of all $n$ live items takes $O(n)$ work under the same scalar-cost assumptions.

The preallocated array occupies $O(C)$ scalar slots plus constant bookkeeping. Allocation and any initial slot construction are separate setup costs. Copying a large item, running an expensive destructor or callback, allocating an item, or growing a dynamic array changes the operation cost and must be charged separately. The bound is worst-case structural work, with no need for amortized analysis, and says nothing about a measured wall-clock deadline.

These formulas describe sequential calls. Wrapping array indices implies neither atomic operations nor safe access from multiple software threads. Publication of an item's contents and reuse of its slot require a protocol justified under the relevant memory model, including any necessary acquire-release synchronization. The kernel documentation discusses barriers under explicit producer/consumer restrictions; Boost's documentation separately requires mutual exclusion for shared access involving writes. Neither property follows merely from choosing a circular buffer.^[ref][ref]