\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.495\textwidth}
  \centering
  \includegraphics[width=\linewidth,height=0.7\linewidth,keepaspectratio]{sip_architecture.pdf}
  \caption{SIP Architecture}
  \label{fig:sip-arch}
\end{subfigure}\hfill
\begin{subfigure}[b]{0.42\textwidth}
  \centering
  \includegraphics[width=\linewidth,height=0.84\linewidth]{cube_architecture.pdf}
  \caption{CUBE Architecture}
  \label{fig:cube-arch}
\end{subfigure}

\vspace{0.6em}

\begin{subfigure}[t]{\textwidth}
  \centering
  \includegraphics[width=\linewidth]{pe_architecture.png}
  \caption{PE level (zoom-in of one PE in (b)): \textsf{PE\_CPU}
  dispatches commands to \textsf{PE\_SCHED}, which routes tile-token
  streams through \textsf{PE\_DMA}, \textsf{PE\_FETCH\_STORE}, and
  \textsf{GEMM}/\textsf{MATH} engines; \textsf{PE\_IPCQ} is the
  on-device collective control plane.}
  \label{fig:pe-arch}
\end{subfigure}
\caption{Modeled hardware graph at the SIP, CUBE, and PE levels.
This figure is an \emph{illustrative topology} chosen for
readability; the \emph{experimental configuration} used throughout
this report (port counts, slice counts, and link parameters) is
specified in \S\ref{sec:hw} / Table~\ref{tab:hw}, and numbers in the
two may differ. KernBench is not tied to this particular
arrangement: each box is a modeled component node, each line a
directed link with bandwidth and propagation attributes, and any
topology that respects those attributes is supported.}
\label{fig:hw-arch}
\end{figure*}

\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{latency_model.png}
\caption{Conceptual schematic of the latency model. Two source nodes
(Requester A/B) inject flits through a chain of routers into a
destination node; on the shared edge between routers, flits from the
two transactions are interleaved flit-by-flit by the wire's FIFO
arrival order. End-to-end latency is the sum of four contributions:
\textbf{per-node overhead} (the switch's fixed processing cost,
shown in light yellow---the same colour as the destination's
processing-logic block), \textbf{per-edge transmission}
($\textit{flit\_size}/\textit{BW}$ on each wire),
\textbf{drain} (per-flit service occupancy at the destination's
channel), and \textbf{queuing delay} (waiting in a FIFO when a
shared resource is busy). The places where queuing actually
accumulates are highlighted in green: the router output queue and
the destination's input queue. This is the model, not a
measurement; specific bandwidths and overheads are listed in
\S\ref{sec:hw} (Table~\ref{tab:hw}).}
\label{fig:latency-model}
\end{figure*}

\section{The KernBench Platform}
\label{sec:platform}

All results in this report are produced on \emph{KernBench}, a
system-level discrete-event simulator for LLM kernels running on AHBM.
This section explains why the platform exists, the device and
execution model it presents to a kernel writer, how its latency model
turns that execution into a number, and the concrete hardware
configuration used for every experiment that follows.

\subsection{Why KernBench}
\label{sec:why}

In a production end-to-end (E2E) stack, kernel performance is entangled
with every layer above the hardware: the compiler's tiling and
scheduling choices, the framework's operator dispatch, the collective
library, and the runtime. Good E2E numbers require \emph{all} of those
layers to be co-optimized, which makes it hard to answer a narrower but
more fundamental question: \emph{given the hardware, how fast can a
well-written kernel be, and which hardware features actually make it
faster?}

KernBench is built to answer exactly that question. Kernels are written
and executed at the \emph{source level}---as algorithmic descriptions
in a small tile-oriented kernel API---with no dependency on a compiler
or any other software-stack layer. The simulator takes the kernel and a
hardware topology and reports the latency the modeled hardware would
deliver. This isolation is deliberate: it lets us study algorithm-level
optimizations (how to tile a GEMM, how to schedule a collective, how to
fuse an attention kernel) and the hardware features that support them,
without the confound of compiler maturity or framework overhead. The
cost is that KernBench numbers are \emph{not} E2E latencies; they are
the achievable-kernel latencies an ideal software stack would expose.

\subsection{Device and execution model}
\label{sec:exec}

KernBench models AHBM as a hierarchy of SIPs, CUBEs, and processing
elements (PEs). At the system level, multiple SIPs are connected
through inter-package links, while each SIP contains a collection of
CUBEs joined by an on-package interconnect
(Fig.~\ref{fig:hw-arch}\subref{fig:sip-arch}).

Each CUBE contains eight PEs, shared SRAM, HBM controllers, an
\textsf{M\_CPU} control processor, and an intra-CUBE router mesh
(Fig.~\ref{fig:hw-arch}\subref{fig:cube-arch}). Together these
components form the execution substrate for all kernels evaluated in
this report. This organization reflects the memory-centric nature of
AHBM: each PE is paired with a dedicated slice of HBM bandwidth and
local on-PE storage (TCM), so compute lives next to the data it
consumes rather than fetching it through a far-away memory
controller. The platform's performance question is therefore not
``how many FLOPs can the chip do'' but ``how well can a kernel keep
each compute engine fed from its locally-attached memory while
moving the unavoidable traffic between PEs efficiently.''

Within a PE, commands are dispatched by \textsf{PE\_CPU} to
\textsf{PE\_SCHED}, which routes work to specialized execution
engines---DMA, FETCH/STORE, GEMM, vector-math, and IPCQ
(Fig.~\ref{fig:hw-arch}\subref{fig:pe-arch}). Commands come in two
flavours. \emph{Atomic} commands target a single engine---a plain
DMA read, a GEMM tile, a vector-math op, an IPCQ send/receive---and
are the natural unit for short or special-purpose work; the
\textsf{PE\_CPU} itself also runs control-plane work directly.
\emph{Composite} commands, by contrast, carry an ordered pipeline of
operations across multiple engines (\textsf{DMA\_READ} $\rightarrow$
\textsf{FETCH} $\rightarrow$ \textsf{GEMM}/\textsf{MATH} $\rightarrow$
\textsf{STORE} $\rightarrow$ \textsf{DMA\_WRITE}) that the
\textsf{PE\_SCHED} tiles and streams without per-tile redispatch. The
composite form is the substrate for the GEMM optimization
(\S\ref{sec:gemm}) and, combined with on-PE collectives, for fused
attention (\S\ref{sec:gqa}).

KernBench is layered along the flow of a request:

\begin{itemize}
  \item The \textbf{runtime API} is host-facing and
  topology-agnostic---it deploys tensors and launches kernels but knows
  nothing about routing or interconnect.
  \item The \textbf{simulation engine} schedules discrete events and
  routes every request through the modeled graph.
  \item The \textbf{components} are device-side nodes that model
  hardware behaviour: the per-PE blocks (scheduler, DMA, GEMM and
  vector-math engines, TCM, IPCQ), the NoC routers, the HBM
  controllers, and the inter-chiplet links.
\end{itemize}

Data and timing are handled in two passes, so that a kernel's
numeric results and its latency are computed consistently but
independently. \textbf{Pass~1 (timing)} runs the kernel under
the discrete-event engine: memory ops (\textsf{tl.load},
\textsf{tl.store}) execute against a host-side \textsf{MemoryStore}
and return real tensor data, while compute ops (GEMM, vector-math)
only emit records into an op-log carrying their operands, shapes,
dtypes, and scheduled time. \textbf{Pass~2 (data)} replays that
op-log offline in numpy, producing the actual numeric outputs and
comparing them against a reference computation under per-dtype
tolerances (e.g.\ \texttt{rtol}/\texttt{atol} $=10^{-3}$ for f16).
Pass~2 is optional---runs that need only latency skip it---but when
enabled it guarantees that every reported timing number corresponds
to a kernel whose numeric output has been verified end-to-end.

\subsection{Latency model: graph traversal and contention}
\label{sec:latency}

The modeled hardware hierarchy described above is represented
internally as a directed graph. Nodes correspond to hardware
components---PEs, routers, memory controllers, SRAM blocks, IO
chiplets---while edges represent communication links with associated
bandwidth (\si{\giga\byte\per\second}) and propagation
(\si{\nano\second}) attributes. Every operation in KernBench---DMA
transfers, remote memory accesses, collective communication, command
dispatch---is modelled as a traversal through this graph. End-to-end
latency is decomposed into four contributions accumulated along the
traversal path (Fig.~\ref{fig:latency-model}): \textbf{per-node
overhead} at each component, \textbf{per-edge transmission} on each
wire, \textbf{drain} (per-flit service occupancy) at the destination,
and \textbf{queuing delay} at the shared FIFOs that the wire and the
destination share between concurrent transactions.

\paragraph{The hardware as a graph.} The topology is compiled once at
configuration time into this graph and is never mutated during a run.
There are no hidden shortcuts, implicit bypasses, or magic paths: if a
request reaches its destination, the path it took is explicit in the
graph, and the latency it incurred is the sum of the per-node and
per-edge costs paid along that path. The same graph representation
applies recursively at every hierarchy level---system, SIP, CUBE, and
PE (Fig.~\ref{fig:hw-arch}).

\paragraph{From graph to discrete-event simulation.} The graph is
driven by a discrete-event engine. Two kinds of events advance
simulation time: \emph{node events} (component switching overhead,
service completions such as an HBM channel commit or a GEMM tile
finish) and \emph{edge events} (the flit-by-flit serialization of a
payload across a bandwidth-limited link). The engine maintains a
priority queue of pending events ordered by their scheduled time and
fires them one at a time, with ties broken under a deterministic
policy so that the same kernel on the same topology always yields the
same trace. Per-request correlation IDs are stamped at injection and
carried through every hop, so the path from injection to completion
is fully traceable. Every modeled latency contribution
corresponds to exactly one of these events on exactly one node or
edge---there is no slack in the budget.

\paragraph{Latency contributions.} Three kinds of latency accumulate
along a traversal: (i) \emph{per-node fixed overhead}---each component
carries a small switching cost (router decode, controller pickup,
scheduler handoff); (ii) \emph{per-edge transfer time}---each link's
payload is decomposed into fixed-size flits (default
\SI{256}{\byte}), and each flit arrives at
$\text{prop}+\text{flit\_bytes}/\text{bw}$ after the previous one,
giving wormhole semantics across multi-hop paths; and (iii)
\emph{per-service occupancy}---memory controllers, GEMM stages, and
collective engines hold the request for their service time before
releasing it downstream. Each of these is attached to a specific node
or edge in the graph; together they make up the entire latency budget.

Beyond data movement and execution latency, KernBench
also models the control-plane cost required to issue work
to accelerator engines. Since every DMA, GEMM, vector-
math, and IPCQ operation is initiated through the
\textsf{PE\_CPU} $\rightarrow$ \textsf{PE\_SCHED} path,
command dispatch latency contributes to the end-to-end
execution time of all kernels.

\paragraph{Command dispatch overhead model.} The cost incurred by
the \textsf{PE\_CPU} when it dispatches a command to one of the
accelerator engines is modelled structurally rather than with a
per-operation calibration table. The \textsf{PE\_CPU} charges, per
command,
\[
  d_{\text{cmd}} = \textsf{FIXED} + b_{\text{logical}} \cdot R,
\]
This linear-in-size form reflects the serialization cost of writing a
command descriptor into the scheduler queue: a fixed per-command
bookkeeping cost plus a byte-wise descriptor-transfer cost.
Concretely, $b_{\text{logical}}$ is the command's hardware-logical byte
size, \textsf{FIXED} captures the fixed per-command cost (queue-tail
update, completion registration) and $R$ captures the per-byte cost of
serializing the command descriptor into the scheduler queue. This
\textsf{FIXED} is distinct from Table~\ref{tab:hw}'s
\SI{2}{\nano\second} / \SI{1}{\nano\second} \textsf{PE\_CPU} /
\textsf{PE\_SCHED} fixed costs: the latter are per-component
traversal overheads each command pays when it transits those nodes,
whereas the \textsf{FIXED} term here is the command-descriptor
issue cost. \textsf{FIXED} depends on the command class: a single-op
command --- one engine operation, i.e.\ a single DMA descriptor, GEMM,
or elementwise issue --- carries a lighter 8-cycle \textsf{FIXED},
while the composite command, which the scheduler expands into a
multi-stage tile-feeder plan, carries 40 cycles
(\S\ref{sec:gemm-vs-async}). With the composite \textsf{FIXED} $= 40$
cycles and $R = 0.0625$ cycles/byte (i.e.\ \SI{16}{\byte\per\cycle},
at \SI{1}{\giga\hertz}) a typical composite lands at roughly
\SI{43}{\nano\second}, and a hard cap on a composite's descriptor size
prevents the model from rewarding arbitrarily large fused commands
beyond what real descriptor queues accept. In the configurations measured here, command issue is
not the bottleneck---data movement is---so this term stays small
relative to DMA and collective time.

\begin{table*}[t]
\centering
\caption{PE\,$\to$\,HBM DMA latency probe at varying hop distances
(\SI{32}{\kibi\byte} transfer; output captured from \texttt{kernbench
probe}). \emph{Util\%} is the achieved bandwidth as a fraction of the
path's bottleneck-edge bandwidth.}
\label{tab:probe-pe-dma}
\small
\begin{tabular}{@{}lrrr@{}}
\toprule
Case & Latency~(\si{\nano\second}) & Util\% (\,32\,KiB) & Util\% (\,1\,MiB) \\
\midrule
PE\,$\to$\,local HBM             & 141.0 & 90.8 & 100.0 \\
PE\,$\to$\,same-half HBM         & 147.9 & 86.6 & ~99.9 \\
PE\,$\to$\,cross-half HBM        & 161.2 & 79.4 & ~99.8 \\
PE\,$\to$\,cross-CUBE (best)     & 330.5 & 77.5 & ~99.6 \\
PE\,$\to$\,cross-CUBE (worst)    & 677.1 & 37.8 & ~97.8 \\
\bottomrule
\end{tabular}
\end{table*}

\subsubsection{Congestion and contention modeling}
\label{sec:congestion}

The simulator's
sharpness comes from how it models contention for those nodes and
edges. \emph{Every directed edge has a FIFO}: an arriving flit takes
its bandwidth-limited transfer time on top of whatever earlier flits
are still being served, so a busy link queues later traffic behind
earlier traffic rather than transferring everything at peak BW.
\emph{HBM is modelled with per-pseudo-channel parallelism}: a
stateless array of channel-availability timestamps with address-based
channel selection captures the bank-level concurrency that real HBM
exposes, so 64 channels per CUBE deliver real parallelism on uniform
addresses but a hot channel surfaces as the bottleneck on skewed ones.
\emph{Every component has a serial worker}: a router carrying two
heavy streams interleaves them at flit granularity in arrival order
rather than fanning out for free, so two concurrent collectives sharing
a link share its bandwidth, not double it. Without these mechanisms
the simulator would simply re-confirm the peak-BW roofline; with them,
it reveals where the real bottlenecks form and which hardware levers
actually relieve them---which is exactly the question the codesign
work in this report turns on.


\subsection{Accuracy}
\label{sec:accuracy}

The model is precise about the effects that
dominate kernel latency on this class of hardware: per-edge bandwidth
occupancy and flit-level serialization, HBM pseudo-channel parallelism,
and per-component switching overhead. Two independent cross-checks
drawn from the experiments in this report confirm that this precision
translates into physically reasonable kernel latencies.

First, in the GEMM study (\S\ref{sec:gemm}), simulator-measured MAC
efficiency tracks an analytic ideal-pipeline model within
\SI{1.4}{ppt} across every swept shape---from a single-tile
$M{=}K{=}N{=}32$ at $\sim\SI{7.7}{\percent}$ up to the deep-$K$
$K{=}3072$ case at $\sim\SI{90}{\percent}$. The residual gap is
fill/tail overhead the closed-form pipeline model omits.

Second, in the all-reduce study (\S\ref{sec:allreduce},
Fig.~\ref{fig:allreduce-cmp}), simulator latency for a 2D-torus over
six devices follows the expected startup-plus-per-packet shape across
the entire payload sweep---tight at small payloads where startup
dominates, and within a single-digit multiplicative factor at the
largest payloads, where the residual gap is explained by per-router
switching the analytic shape elides.

Third, the simulator's per-traversal behaviour can be probed
directly. Running \texttt{kernbench probe} on the modelled topology
issues a sequence of PE-to-HBM DMA reads at progressively greater
hop distances and reports the per-component overhead, per-edge
serialization, and per-PC drain that the model charges
(Table~\ref{tab:probe-pe-dma}). Three properties stand out. First,
the reported latency increases \emph{monotonically} with hop
count---from \SI{141}{\nano\second} at the local HBM slice to
\SI{677}{\nano\second} at the worst-case remote-CUBE slice---with the
increment per added hop matching the per-router overhead and the
UCIe-link cost in Table~\ref{tab:hw}. Second, the bandwidth-saturation
curves track the wire's serialization model: at \SI{1}{\mebi\byte}
transfers the local PE DMA reaches \SI{100}{\percent} of the per-edge
bandwidth limit, while the longest cross-CUBE path saturates at
\SI{97.8}{\percent}---exactly the gap that the per-edge propagation
and per-router overhead model would predict. Third, the simulator
passes the internal-consistency invariants the probe builds in
(monotonic hop progression; D2H reads at least as long as the
equivalent H2D writes because of the reverse-path acknowledgement;
cross-CUBE best less than cross-CUBE worst). These are properties
that hold \emph{only} when the underlying graph traversal, FIFO
contention, and propagation models are internally
self-consistent---a model error in any of them would surface as a
non-monotonic or under-utilising curve here long before it polluted a
kernel-level measurement.

The known simplifications---FIFO router arbitration (instead of
round-robin), HBM scheduler without write-buffer reordering, no bank
conflict, no refresh or thermal effects, and no upstream
backpressure---are the price of a deterministic, inspectable model.
These simplifications bound the absolute accuracy, but the agreement
with both analytic models and the probe's internal-consistency
checks above indicates that KernBench is sufficiently accurate for
evaluating the \emph{relative} hardware--software design trade-offs
(tiling A vs.\ B, topology X vs.\ Y, with vs.\ without composite
command, mesh vs.\ torus) that are the primary objective of this
work.



\subsection{Modeled hardware configuration}
\label{sec:hw}

Table~\ref{tab:hw} summarizes the hardware configuration
used throughout this report. The intent of this
configuration is not to model a specific product, but to
represent a realistic memory-centric accelerator and to
provide a consistent baseline for evaluating hardware--
software co-design mechanisms.

The compute capability within each CUBE is provisioned
such that inference workloads can effectively saturate
the available HBM bandwidth. The aggregate compute
throughput is therefore balanced against memory-system
bandwidth rather than being intentionally over- or
under-provisioned. Concretely, 8 PEs $\times$
\SI{8}{\tera\flop\per\second} give roughly
\SI{64}{\tera\flop\per\second} of f16 compute per CUBE
against \SI{2048}{\giga\byte\per\second} of HBM bandwidth,
which places the balanced point at about
$\sim$31~FLOP/byte; inference decode kernels typically
operate well below that, so HBM bandwidth---not compute---is
the natural ceiling on this configuration. For reference, the
GQA decode kernels evaluated in \S\ref{sec:gqa} operate at
arithmetic intensity well below this balance point, so their
ceiling is set by HBM and inter-PE traffic rather than by GEMM
throughput. HBM bandwidth
is distributed evenly across the PEs within a CUBE, with
each PE responsible for servicing approximately one-eighth
of the CUBE's memory bandwidth through its dedicated HBM
channels.

The on-chip interconnect is configured using bandwidth
and latency parameters representative of commercially
available mesh-network IPs. The goal is not to study a
particular NoC implementation, but rather to evaluate
kernel behavior under a realistic baseline communication
fabric.

For die-to-die communication, UCIe-A bandwidth
characteristics are used as the reference point for
inter-CUBE links. Inter-SIP communication is modeled
using PCIe-class links. Together, these assumptions
provide a representative communication hierarchy for
evaluating collective communication and distributed
kernel execution.

Unless otherwise stated, all experiments use this
configuration. Workload-specific parameters such as
matrix dimensions, sequence lengths, and collective
payload sizes are introduced in their respective sections.

\begin{table}[t]
\centering
\caption{Modeled hardware configuration}
\label{tab:hw}
\small
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Parameter} & \textbf{Value} \\
\midrule
\multicolumn{2}{@{}l}{\emph{Hierarchy}} \\
SIPs & 2 (1D ring) \\
CUBEs per SIP & 16 ($4\times4$ mesh) \\
PEs per CUBE & 8 (4 corners $\times$ 2) \\
PEs total & 256 \\
\midrule
\multicolumn{2}{@{}l}{\emph{Processing element (PE)}} \\
GEMM engine peak & \SI{8}{\tera\flop\per\second} (f16) \\
TCM (on-PE) & \SI{16}{\mega\byte}, \SI{512}{\giga\byte\per\second} R/W \\
\quad kernel scratch & \SI{1}{\mega\byte} \\
DMA engines & 1 read + 1 write \\
\textsf{PE\_CPU} fixed cost   & \SI{2}{\nano\second} \\
\textsf{PE\_SCHED} fixed cost & \SI{1}{\nano\second} \\
\midrule
\multicolumn{2}{@{}l}{\emph{Memory (per CUBE)}} \\
HBM capacity & \SI{48}{\giga\byte} (8 slices) \\
HBM aggregate BW & \SI{2048}{\giga\byte\per\second} \\
HBM pseudo-channels & 64 (8 per PE), \SI{32}{\giga\byte\per\second} each \\
SRAM (shared) & \SI{32}{\mega\byte}, \SI{128}{\giga\byte\per\second} link \\
HBM burst & \SI{256}{\byte} \\
\midrule
\multicolumn{2}{@{}l}{\emph{Interconnect}} \\
Intra-CUBE NoC link & \SI{256}{\giga\byte\per\second}, \SI{2}{\nano\second}/router \\
Inter-CUBE (UCIe PHY) & \SI{512}{\giga\byte\per\second}, \SI{8}{\nano\second}, XY routing \\
Inter-SIP (PCIe) & \SI{768}{\giga\byte\per\second} per endpoint \\
\midrule
\multicolumn{2}{@{}l}{\emph{Command-issue cost model (defaults)}} \\
FIXED per single-op command & 8 cycles \\
FIXED per composite command & 40 cycles \\
per-byte rate $R$ & 0.0625 cycles/byte (\SI{16}{\byte\per\cycle}) \\
composite size cap & \SI{1024}{\byte} \\
\bottomrule
\end{tabular}
\end{table}
