Construct a deterministic GPipe schedule with equal-duration stage operations and no overlap between the forward and backward waves. Let $p$ be num_stages and $m$ be num_microbatches. Each wave lasts $T$ time slots:

In forward time slot $t$, stage $s$ processes microbatch $i_{\mathrm{forward}}$:

When include_backward is true, the backward wave follows the completed forward wave. At backward offset $u$, stage $s$ processes microbatch $i_{\mathrm{backward}}$:

All indices start at zero: $s$ ranges from 0 through $p-1$, while $t$ and $u$ range from 0 through $T-1$. A microbatch index outside 0 through $m-1$ means that stage is idle. Label work F0, F1, and so on for forward, or B0, B1, and so on for backward; use None for idle cells. With $q=2$ when backward is included and $q=1$ otherwise, useful and total stage-slot counts are:

Return a Python dictionary with schedule, a time-ordered list of stage-ordered lists containing strings or None; bubble_slots and total_slots, Python integer cell counts; and utilization, a Python float equal to useful cells divided by total cells.