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A tensor has no fixed serving cost. Its impact depends on the operations applied to it, the bytes moved, how those operations run on particular GPUs, and how the serving system schedules requests. Here is an illustrative trace of one activation through a decoder-only Transformer, from matrix multiplication to memory capacity and cost accounting.
What tensor are we tracing?
Use a hypothetical decoder-only Transformer with hidden width 4,096, 32 attention heads, and 32 layers. Assume BF16 activations and weights, a batch of one prompt containing 512 tokens, and no grouped-query attention. These are illustrative dimensions—not a specification for a particular model.
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At one layer, let X be the residual-stream activation entering the query projection. Its shape is [batch, sequence, hidden], or [1, 512, 4096]. At two bytes per BF16 value, it occupies 4 MiB. The mathematical query projection is Q = XW, where W has shape [4096, 4096]; the result Q has shape [1, 512, 4096], also 4 MiB.
For each output value, the projection combines 4,096 input values with 4,096 weights. Across all 512 positions and 4,096 output channels, that is 8,589,934,592 multiply-accumulates. If one multiply-add is counted as two FLOPs, the operation is about 17.18 GFLOPs. That counting convention is described in NVIDIA’s GPU Performance Background User’s Guide; it is a way to count mathematical work, not a runtime prediction.
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How much data moves, and what does arithmetic intensity tell us?
A simple lower-bound traffic estimate for this prompt projection counts one read of the 32 MiB weight matrix, one read of the 4 MiB input, and one write of the 4 MiB output: 40 MiB total. Under those assumptions, 17.18 GFLOPs divided by 40 MiB is roughly 410 FLOPs per byte. This estimate assumes weights are fetched once and does not include extra traffic from intermediate tensors, padding, cache behavior, or other layers and operators.
For one decode position instead of 512 prompt positions, the same projection performs about 33.55 MFLOPs. If the weight matrix must be read and the 8 KiB input and 8 KiB output are read or written, traffic is about 32 MiB; the ratio is roughly 1 FLOP per byte. The weight matrix dominates this small operation, while the 512-position case reuses those weights across many more rows. Actual traffic depends on whether data is cached or reused and on the chosen implementation.
Arithmetic intensity—operations per byte moved—helps explain whether an operation is more likely limited by math throughput or memory bandwidth. NVIDIA’s guide frames performance limits as math bandwidth, memory bandwidth, or latency. Its V100-era FP16 examples classify a linear layer with 4,096 outputs and 1,024 inputs as arithmetic-limited at batch 512 (315 FLOPs/byte) and memory-limited at batch 1 (1 FLOP/byte). Those examples illustrate the batch-size effect under their stated assumptions; they are not predictions for every GPU or model.
How does the framework turn this operation into GPU work?
The equation Q = XW describes the model math. It does not dictate a single GPU kernel. A framework may lower the operation to a matrix-multiplication kernel, combine it with neighboring operations through fusion, or compile a larger region. The resulting performance depends on the GPU, data types, matrix sizes, kernel implementation, and how much work is available to run in parallel.
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- Launch and latency: For small workloads, the time to launch work and other fixed overheads can matter as much as the arithmetic.
- Parallelism and occupancy: A kernel needs enough independent work to keep GPU execution units busy. Small batches or awkward dimensions can leave capacity unused or create tail effects.
- Fusion and compiler boundaries: Combining operations may avoid extra launches or intermediate reads and writes. But graph breaks can limit what a compiler can optimize together. PyTorch’s Llama 2 inference report describes graph breaks associated with unsupported operations and distributed collectives: PyTorch’s report.
- Communication: When work spans devices, moving intermediate results between them adds time that a FLOP count does not capture.
Why are prompt prefill and token decode different workloads?
Prompt prefill
During prefill, the model processes the prompt’s existing tokens. In this example, the query projection handles 512 positions together, allowing each weight to be reused across those rows. The arithmetic-intensity estimate above is for that projection alone; attention and other layer operations also contribute to prefill time and memory traffic.
Autoregressive decode
During decode, the model generates tokens sequentially: each new token depends on the preceding context. Rather than recomputing keys and values for all earlier tokens at every step, inference commonly keeps them in a key/value (KV) cache. In this example, the cache for one sequence at 512 positions is approximately 8 MiB per layer: 512 positions × 4,096 values for each of K and V × 2 bytes. Across 32 layers that is about 256 MiB per sequence, before allocator overhead or implementation-specific storage. At 4,096 positions, the same assumptions imply about 2 GiB per sequence.
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Prompt length, generated length, batch size, and concurrency therefore change both the amount of work and the active cache. Variable sequence dimensions also complicate execution. PyTorch/XLA describes bucketing or padding prompts and using fixed-shape KV-cache updates to manage dynamic shapes: PyTorch/XLA’s inference report. These are implementation techniques, not requirements for every serving stack.
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A published result belongs to its full setup. PyTorch and IBM Research contributors reported 29 ms/token in 2023 for a single-user Llama 2 70B configuration on eight NVIDIA A100 GPUs; the reported experiment used a 512-token input and generated 50 tokens. That is a configuration-specific report, not a speed guarantee for another model, batch, hardware setup, or serving target (report details).
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What changes when the model is deployed across GPUs?
First account for model weights, runtime allocations, and the KV cache for the active workload. A model may fit by weights alone yet lack room for the cache and concurrent requests. For the illustrative model, the 32-layer KV-cache estimate is 256 MiB per 512-position sequence; real models can differ substantially because of width, layer count, cache dtype, and attention layout.
If a model or its working set does not fit on one GPU, deployment may distribute work. Tensor parallelism splits operations across GPUs, commonly within a node; pipeline parallelism assigns different layers to different stages and can span nodes. Both introduce communication and topology considerations. vLLM’s deployment guidance discusses parallelism choices, and its logs can report KV-cache token capacity and an estimated maximum concurrency. Those estimates help assess fit; they are not a monetary bill or a guarantee of achieved throughput: vLLM parallelism and scaling.
| Deployment choice | What it distributes | What to account for |
|---|---|---|
| One GPU | Weights and execution stay on one device if they fit. | Usable memory for weights, active KV cache, runtime needs, and the desired request concurrency. |
| Tensor parallelism | Parts of operations are spread across GPUs. | GPU topology and communication between devices, alongside memory and workload fit. |
| Pipeline parallelism | Layers are assigned to different stages or devices. | Communication between stages and how the request workload uses those stages. |
Compare deployment options at the workload you expect to serve—not from peak FLOPs alone. Relevant measures include time to first token (TTFT), inter-token latency, throughput at target concurrency, usable memory and KV-cache headroom, GPU count and interconnect, and quality constraints imposed by numeric format. A system that looks faster at one batch size or prompt length may not be the better fit at another.
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Only after defining the workload and the cost basis. A FLOP estimate does not supply a price, and there is no general dollar cost per token established here. For a provider, use the applicable dated machine rate; for owned hardware, use a stated amortization method and period. Then relate that cost to useful completed work under the service-level objective.
- Workload: model, input and output length distributions, batch size, concurrency, and request mix.
- Capacity and performance: GPU configuration, memory headroom, utilization, throughput, TTFT, and inter-token latency.
- Cost basis: machine or internal cost, billing unit, time period, and how idle capacity is treated.
- Unit of comparison: cost per request or per generated token, with the workload and latency target stated.
Use measured serving data for that configuration to calculate cost per useful request or token. If utilization falls, fixed machine cost is spread over fewer completed requests; if longer contexts or concurrency force more GPUs, the cost basis changes. The model equation alone cannot settle either outcome.
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