# Latency Model

## Overview

kernbench uses a discrete-event simulation (SimPy) to compute end-to-end latency.
Every request flows through a graph of **components** connected by **wires**.
The total latency reported is the **actual SimPy wall-clock** (`env.now` delta),
not a static formula—so contention and queueing are captured automatically.

```text
total_ns (actual) = wire_prop + component_overhead + drain + queueing
                    ├── deterministic ──────────────────┘       │
                    └── contention-dependent ────────────────────┘
```

## Three Deterministic Cost Components

### 1. Wire Propagation

```text
wire_ns = distance_mm × ns_per_mm       (global: 0.01 = 10 ps/mm)
```

Every edge in the topology graph has a `distance_mm`. A SimPy wire process
delays each message by `wire_ns` before delivering it to the next component.
For on-chip silicon this is ~10 ps/mm; the same constant applies everywhere
since all links are on-die or interposer. Wire propagation is typically <1 ns
and negligible compared to other costs.

### 2. Component Overhead (`overhead_ns`)

```text
component_ns = node.attrs["overhead_ns"]
```

Each component on the path adds a fixed processing delay via `yield env.timeout(overhead_ns)`.
This models arbitration, protocol processing, pipeline stages, etc.

| Component | overhead_ns | Meaning |
|-----------|-------------|---------|
| pcie_ep | 5.0 | PCIe protocol processing |
| io_cpu | 10.0 | Command decode / dispatch |
| m_cpu | 5.0 | DMA scheduling |
| fabric switch | 5.0 | Packet arbitration |
| xbar | 2.0 | Crossbar arbitration |
| xbar bridge | 1.0 | Bridge traversal between xbar halves |
| ucie | 8.0 | UCIe protocol overhead per port (TX or RX; 16ns per crossing) |
| noc (2D mesh) | 0.0 | Hop delay modeled internally via manhattan distance |
| hbm_ctrl | 0.0 | Access time via drain_ns; efficiency=0.8 reduces edge BW (256→204.8) |
| pe_cpu | 2.0 | Command dispatch |
| pe_scheduler | 1.0 | PE-internal scheduling |
| pe_gemm/math | 0.0 | Placeholder; will use flops-based model |

### 3. Drain (Serialization Delay)

```text
drain_ns = nbytes / bottleneck_bw_gbs
```

**Wormhole (cut-through) model**: data flows through intermediate nodes as a
pipeline. Serialization cost is paid **once** at the terminal node, not at
every hop. The bottleneck is the minimum `bw_gbs` across all edges in the path.

Example: 4096 bytes through a path with bottleneck 128 GB/s → `4096 / 128 = 32.0 ns`.

### Formula (Theoretical Lower Bound)

```text
formula_ns = Σ(wire_prop) + Σ(overhead_ns) + drain_ns
```

This is the latency with **zero contention**—no other request competing for
any resource. The engine provides `_formula_latency()` for verification.
With no contention: `actual == formula`. With contention: `actual > formula`.

### Diagram: PE DMA Read (pe0 → local slice0, 4096 bytes)

```mermaid
sequenceDiagram
    participant D as pe_dma
    participant X as xbar.pe0
    participant H as hbm_ctrl.slice0

    D->>X: txn (4096B)
    Note over X: overhead 2.0 ns
    X->>H: txn (wire 0.025 ns)
    Note over H: acquire Resource
    Note over H: overhead 0 ns
    Note over H: drain 4096/256 = 16.0 ns
    Note over H: release Resource
    H-->>D: done.succeed()

    Note over D,H: total_ns = 18.09 ns<br/>formula = wire(0.025) + ovhd(2.0) + drain(16.0) = 18.025 ns<br/>actual ≈ formula (no contention)
```

### Diagram: Two Requests — No Contention vs HOL Blocking

#### Case 1: Different slices (parallel, no contention)

```mermaid
sequenceDiagram
    participant A as Request A
    participant S0 as hbm_ctrl.slice0<br/>Resource(cap=1)
    participant S1 as hbm_ctrl.slice1<br/>Resource(cap=1)

    Note over A,S1: t=2 ns — both requests arrive at their own slice
    A->>S0: A (4KB)
    A->>S1: B (4KB)
    Note over S0: acquire (immediate)
    Note over S1: acquire (immediate)
    Note over S0: drain 16.0 ns
    Note over S1: drain 16.0 ns
    Note over S0: t=18 release
    Note over S1: t=18 release

    Note over A,S1: A actual = 18 ns, B actual = 18 ns<br/>No waiting — separate Resources
```

#### Case 2: Same slice (HOL blocking)

```mermaid
sequenceDiagram
    participant A as Request A (4KB)
    participant Q as hbm_ctrl.slice0<br/>Resource(cap=1)
    participant B as Request B (64B)

    Note over A,B: t=0 — A arrives first
    A->>Q: acquire (immediate)
    Note over Q: drain A = 16.0 ns

    Note over B,Q: t=5 — B arrives, yield req → BLOCKED
    B--xQ: waiting...

    Note over Q: t=16 — A drain done, release
    Q->>B: B acquires resource
    Note over Q: drain B = 0.25 ns
    Note over Q: t=16.25 — B done, release

    Note over A,B: A actual = 16.0 ns (== formula)<br/>B actual = 11.25 ns (formula 0.25 + queueing 11.0)<br/>HOL blocking: short request waits behind long drain
```

---

## How SimPy Tracks Latency

### Measurement

```python
start_ns = env.now
yield txn_done          # wait for the transaction to complete
total_ns = env.now - start_ns     # ← this is what probe reports
```

`env.now` is SimPy's simulation clock. It only advances when a process `yield`s
a timeout or waits on a resource/store. The delta between start and done captures
**everything**: wire delays, component overheads, drain, and any queueing.

### Component Pipeline

Each component is a SimPy process:

```text
_fan_in (per in_port)  →  _inbox (Store)  →  _worker  →  out_ports
```

1. **`_fan_in`**: relays messages from each `in_port` into a shared `_inbox` Store.
2. **`_worker`**: pulls from `_inbox`, spawns `_forward_txn` per message.
3. **`_forward_txn`**: calls `run()` (overhead), then puts to `out_ports[next_hop]`.

The worker uses `env.process()` (pipeline model), so multiple messages can be
in-flight through the same component concurrently. Contention happens when
they compete for shared resources (e.g., `simpy.Resource` in hbm_ctrl).

### Wire Process

```python
while True:
    msg = yield out_port.get()      # wait for sender
    yield env.timeout(prop_ns)      # propagation delay
    yield in_port.put(msg)          # deliver to receiver
```

Each directed edge has its own wire process. Messages are delayed by exactly
`distance_mm × ns_per_mm`.

---

## Contention and Queueing

Queueing delay is **not a separate formula term**—it emerges from SimPy's
event scheduling when multiple requests compete for the same resource.

### Where Contention Occurs

| Resource | SimPy Type | Capacity | Effect |
|----------|-----------|----------|--------|
| hbm_ctrl | `simpy.Resource` | 1 | Serializes HBM access |
| m_cpu DMA read engine | `simpy.Resource` | 1 | Serializes DMA reads |
| m_cpu DMA write engine | `simpy.Resource` | 1 | Serializes DMA writes |
| pe_dma channels | `simpy.Resource` | configurable | Serializes PE DMA ops |
| component inbox | `simpy.Store` | unbounded | No backpressure (FIFO) |

### How Queueing Works

```python
# hbm_ctrl._worker
with self._resource.request() as req:
    yield req                     # ← BLOCKS if resource is occupied
    yield from self.run(env, txn.nbytes)
    yield env.timeout(drain_ns)
```

If request A holds the resource and request B arrives:
- B's `yield req` blocks until A releases the resource
- SimPy advances B's `env.now` by A's remaining service time
- This "extra" time shows up in B's `total_ns` automatically

```text
No contention:  actual_ns == formula_ns
Contention:     actual_ns  > formula_ns
                queueing_delay = actual_ns - formula_ns
```

### Head-of-Line (HOL) Blocking at hbm_ctrl

The `simpy.Resource` is held for the **entire** `with` block—both overhead and
drain. The resource is NOT released between overhead and drain:

```python
with self._resource.request() as req:
    yield req                              # acquire (or wait)
    yield from self.run(env, txn.nbytes)   # overhead_ns  ─┐
    yield env.timeout(drain_ns)            # drain_ns      │ resource held
# ← resource released here ───────────────────────────────┘
```

This means a short request arriving during a long request's drain must wait
for the full remaining drain time—classic head-of-line blocking:

```text
Request A: 4 KB,  drain = 16.0 ns   (arrives at t=0)
Request B: 64 B,  drain = 0.25 ns   (arrives at t=5)

Timeline:
  t=0.00   A acquires resource
  t=0.00   A: overhead (0 ns)
  t=0.00   A: drain starts (16.0 ns)
  t=5.00   B arrives → yield req → BLOCKED (A holds resource)
  t=16.00  A: drain done → resource released
  t=16.00  B acquires resource
  t=16.00  B: overhead (0 ns)
  t=16.25  B: drain done → resource released

  B actual  = 11.25 ns (waited 11.0 + own 0.25)
  B formula = 0.25 ns
  B queueing = 11.0 ns  ← HOL blocking penalty
```

**Why this is physically realistic**: An HBM channel processes one burst at a
time. While data is being serialized onto the channel (drain), no other request
can use that channel. The FIFO ordering (`simpy.Resource` default) reflects
the simplest controller scheduling policy.

**Alternative: priority scheduling**: If needed, `simpy.PriorityResource` can
prioritize shorter requests (Shortest Job First), but this is not currently
used since FIFO matches typical HBM controller behavior.

---

## Worked Example: Two Concurrent PE DMA Reads

Setup: PE0 and PE1 in cube0 both read 4096 bytes from their local HBM slices
(slice0 and slice1), submitted to the **same engine** at the same time.

### Paths

```text
DMA A: pe0.pe_dma → xbar.pe0 → hbm_ctrl.slice0
DMA B: pe1.pe_dma → xbar.pe1 → hbm_ctrl.slice1
```

### No Contention (different HBM slices)

Since slice0 and slice1 are **separate** hbm_ctrl instances, each with its own
`simpy.Resource(capacity=1)`, there is no resource competition.

```text
DMA A timeline:
  t=0.00   pe_dma dequeues txn
  t=0.00   xbar.pe0: overhead_ns=2.0 → t=2.00
  t=2.025  wire prop (2.5mm × 0.01) → t=2.025
  t=2.025  hbm_ctrl.slice0: yield req → immediate (no contention)
  t=2.025  hbm_ctrl.slice0: overhead_ns=0 → t=2.025
  t=18.025 drain_ns = 4096/256 = 16.0 → t=18.025
  t=18.025 done

DMA B timeline: (identical, on its own slice)
  t=0.00   → ... → t=18.09  done
```

Both complete at ~18.09 ns. `actual == formula` for both.

### With Contention (same HBM slice)

Now suppose both PE0 and PE1 read from **slice0**:

```text
DMA A: pe0.pe_dma → xbar.pe0 → hbm_ctrl.slice0
DMA B: pe1.pe_dma → xbar.pe1 → xbar.pe0 → hbm_ctrl.slice0
                                (chain traversal to reach slice0)
```

```text
DMA A timeline:
  t=0.00   xbar.pe0(2.0) → wire → hbm_ctrl.slice0
  t=2.025  yield req → immediate (first to arrive)
  t=18.025 drain 16.0 → release resource → done
  actual_A = 18.025 ns (== formula)

DMA B timeline:
  t=0.00   xbar.pe1(2.0) → xbar.pe0(2.0) → wire → hbm_ctrl.slice0
  t=4.035  yield req → BLOCKED (A holds resource until t=18.025)
  t=18.025 acquire resource
  t=34.025 drain 16.0 → release → done
  actual_B = 34.035 ns

  formula_B = wire(0.035) + overhead(4.0) + drain(32.0) = 36.035 ns
  But actual_B is different because drain uses bottleneck BW of B's path (128 GB/s)
  while A's path has BW 256 GB/s. Let's recalculate:

  B's bottleneck: xbar_x_bw = 128 GB/s → drain = 4096/128 = 32.0 ns
  formula_B = 0.035 + 4.0 + 32.0 = 36.035 ns
  actual_B  = 36.035 + queueing ≈ 50+ ns
  queueing  = time waiting for A to release hbm_ctrl
```

The key insight: **queueing delay is not in the formula**. It only appears in
the actual SimPy simulation when resources are contested. The probe reports
`actual_ns`, which includes all queueing. To see pure queueing overhead,
compare `actual_ns` vs `formula_ns` (available in PE DMA traces).

---

## Probe Output Explained

```text
=== PE DMA Latency ===
Case                Target              Actual  Ovhd  Drain  Wire  Ovhd% Drain%  Eff.BW   BN.BW   Util%
pe-local-hbm        c0.pe0->c0.slice0    18.09   2.0  16.0  0.08  11.1% 88.5%   226.49   256.0   88.5%
pe-cross-half-hbm   c0.pe0->c0.slice4    37.14   5.0  32.0  0.14  13.5% 86.1%   110.27   128.0   86.1%
```

| Column | Meaning |
|--------|---------|
| **Actual** | SimPy measured `env.now` delta (includes contention if any) |
| **Ovhd** | Sum of `overhead_ns` for all components on the forward path |
| **Drain** | `nbytes / bottleneck_bw` — serialization at terminal |
| **Wire** | Sum of `distance_mm × ns_per_mm` for all edges |
| **Ovhd%** | `Ovhd / Actual × 100` — fraction of time spent in component processing |
| **Drain%** | `Drain / Actual × 100` — fraction of time spent in data transfer |
| **Eff.BW** | `nbytes / Actual` — achieved bandwidth |
| **BN.BW** | Bottleneck bandwidth (min `bw_gbs` on path) |
| **Util%** | `Eff.BW / BN.BW × 100` — how close to theoretical max BW |

### Why Util% < 100%

`Util% = Drain% = drain_ns / actual_ns`. The gap from 100% is the overhead
fraction. For small transfers (4KB), overhead is significant relative to drain.
For large transfers, drain dominates and utilization approaches 100%.

```text
  4 KB:  Ovhd=2.0, Drain=16.0  → Util=88.5%   (overhead is 11% of time)
 64 KB:  Ovhd=2.0, Drain=256.0 → Util=99.2%   (overhead is <1% of time)
```

### H2D Path: Why Ovhd% is ~40%

H2D traverses many components (pcie_ep → io_cpu → ucie → noc → m_cpu → noc →
xbar → hbm_ctrl + response path). Total forward overhead is ~23 ns vs drain
of 32 ns for 4KB, so overhead is comparable to data transfer time—resulting
in ~55% utilization. This is expected for small command-path transfers.