# G, S and throughput: which number are you finding?

Subject: Computer Networks

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/cn/#tee/read/aloha/formulas

Canonical: https://deep-learning-study-room.vercel.app/read/cn/tee/read/aloha--formulas/

TEE · shared foundations and Units 3–4 · current syllabus 702AI0C023

**Definition.** Offered load G is the average number of transmission attempts per frame time. Normalized throughput S is the average number of successful frames per frame time.

**Simple meaning.** G = kitne attempts; S = unmein successful output per frame-time. Dono same nahi.

Swipe across the diagram to read its labels ↔

![ALOHA: too many attempts eventually reduce successful output](https://deep-learning-study-room.vercel.app/cn/fig/aloha-curve.58889b65f2.svg)

Diagram: ALOHA: too many attempts eventually reduce successful output. Labels: ALOHA: too many attempts eventually reduce successful output | ALOHA: too many attempts eventually reduce successful output | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 0.1 | 0.2 | 0.3 | 0.4 | Slotted max ≈ 0.368 at G=1 | Pure max ≈ 0.184 at G=0.5 | G: aggregate attempts per frame time | S

G, S and throughput: which number are you finding?

## Understand it step by step

1. Let L be frame length in bits, R the channel rate in bits/s, and λ the total attempt rate of all stations in frames/s. Then T=L/R seconds and G=λT.
2. For the standard model, attempts are modelled as Poisson, frames have equal duration, there is no capture, and overlapping transmissions fail. G includes retransmissions when they are part of the offered traffic.
3. Pure ALOHA uses S=Ge⁻²ᴳ; Slotted ALOHA uses S=Ge⁻ᴳ. Do not use arrival rate in frames/s directly in the exponent.
4. Successful frames/s is S/T. Successful frame bits/s is (S/T)L = SR. If L includes overhead, this is successful frame throughput, not application payload goodput.
5. For G>0, the success probability per attempt is S/G. S itself is not generally that probability. Under this fixed-frame ideal model, 100S is the normalized throughput as a percentage of raw bit rate.



$$
T=L/R,\qquad G=\lambda T
$$





$$
S_{\mathrm{pure}}=G e^{-2G}
$$





$$
S_{\mathrm{slotted}}=G e^{-G}
$$





$$
\text{successful frames/s}=S/T
$$





$$
\text{successful frame bits/s}=SR
$$



#### Practical example

If G=0.5 and S≈0.18394, about 0.18394 successful frames occur per frame time. The success fraction per attempt is 0.18394/0.5≈0.36788, not 0.18394.

### Practice question. Write the throughput expressions of Pure and Slotted ALOHA, defining G and S. [3 practice marks]

#### Answer

G denotes aggregate attempts per frame transmission time and S denotes successful frames per frame transmission time. Pure ALOHA: S=Ge⁻²ᴳ, with maximum 1/(2e) at G=0.5. Slotted ALOHA: S=Ge⁻ᴳ, with maximum 1/e at G=1. These results use the equal-length Poisson collision model.

#### Practical example

If G=0.5 and S≈0.18394, about 0.18394 successful frames occur per frame time. The success fraction per attempt is 0.18394/0.5≈0.36788, not 0.18394.

**Yaad rakho:** P mein pair: −2G; S mein single: −G.

## Important conversion in the faculty examples

For 200-bit frames on a 200000-bit/s channel, T = 200/200000 = 0.001 s. If 500 frames/s attempt transmission, G = 0.5 and pure-ALOHA S = 0.5e⁻¹ ≈ 0.1839397. Convert to frames/s by dividing by T: 0.1839397/0.001 ≈ **183.94 frames/s**.

Do not multiply S by 500: S is successful frames per frame-time, not success probability per attempt. Some slide totals use that incorrect conversion. At 250 attempts/s the correct pure-ALOHA output is **151.63 frames/s**. Use the definitions and units above.