--- title: "GoS and optimal N for 15 dB S/I (n = 3)" subject: "Wireless Communication" chapter: 2 key: "semester_7/wireless_communication:gos-optimal-cluster-size-n3" source: https://www.bctengineer.com/q/semester_7/wireless_communication/gos-optimal-cluster-size-n3 site: BCT Engineering --- # GoS and optimal N for 15 dB S/I (n = 3) Wireless Communication · Chapter 2: Cellular Mobile Communication Concepts - Asked in: 2079 Bhadra, 3+5 marks - Video: [Cellular System Numerical Example-2 Find S/I Ratio, Cluster Size](https://www.youtube.com/watch?v=z5pb8MaoB7g) — ECIT Engineering - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/wireless_communication/gos-optimal-cluster-size-n3 - Whole chapter: https://www.bctengineer.com/md/semester_7/wireless_communication/chapter-2 ## Question Define Grade of Service (GoS) and explain how can it be measured in a 'blocked call cleared' type of trunking system. A cellular service provider decides to use a digital TDMA scheme that can tolerate a signal-to-interference ratio of 15 dB in the worst case. The mobile radio channel provided a propagation path loss exponent of n = 3. Find the optimal value of N for (a) Omni-directional antennas, (b) 120° sectoring, and (c) 60° sectoring. Comment on your results. ## Short answer GoS is the blocking probability in a trunked system, measured by Erlang B. With n=3 and required S/I=15 dB: omnidirectional antennas need N=12 (15.56 dB), 120° sectoring N=7 (16.82 dB), and 60° sectoring N=4 (16.19 dB) because sectoring cuts co‑channel interferers, enabling smaller clusters. ## Answer #### Grade of Service (GoS) **GoS** is a measure of the ability of a user to access a trunked system during the busiest hour. It is usually expressed as the **probability that a call is blocked** (or delayed beyond a set time). Example: GoS = 2 % means 2 calls in 100 are blocked during the busy hour. **Measurement in a blocked-calls-cleared system:** a blocked call is not queued; the user must try again later. Assuming Poisson call arrivals, exponential holding times and infinite users, the blocking probability is given by the **Erlang B formula**: ```text Aᶜ / C! Pr[block] = ───────────── Σ (k=0..C) Aᵏ / k! ``` where A = offered traffic (Erlangs) = U·λ·H and C = number of channels. In practice GoS is measured as the ratio of blocked call attempts to total call attempts in the busy hour, and the Erlang B chart/table gives A for a given C and GoS. #### Optimal N for S/I ≥ 15 dB, n = 3 Using S/I = (√(3N))ⁿ / i₀, with i₀ = 6 (omni), 2 (120°), 1 (60°) first-tier interferers. Valid cluster sizes are N = i² + ij + j² = 3, 4, 7, 9, 12, … **(a) Omnidirectional (i₀ = 6)** ```text N = 7 : (√21)³ / 6 = 96.23 / 6 = 16.04 → 12.05 dB < 15 ✗ N = 9 : (√27)³ / 6 = 140.30 / 6 = 23.38 → 13.69 dB < 15 ✗ N = 12: (√36)³ / 6 = 216 / 6 = 36.00 → 15.56 dB ≥ 15 ✓ ``` Optimal **N = 12**. **(b) 120° sectoring (i₀ = 2)** ```text N = 4 : (√12)³ / 2 = 41.57 / 2 = 20.78 → 13.18 dB < 15 ✗ N = 7 : 96.23 / 2 = 48.12 → 16.82 dB ≥ 15 ✓ ``` Optimal **N = 7**. **(c) 60° sectoring (i₀ = 1)** ```text N = 3 : (√9)³ = 27.00 → 14.31 dB < 15 ✗ N = 4 : (√12)³ = 41.57 → 16.19 dB ≥ 15 ✓ ``` Optimal **N = 4**. | Antenna | i₀ | Optimal N | S/I | |---|---|---|---| | Omni | 6 | 12 | 15.56 dB | | 120° | 2 | 7 | 16.82 dB | | 60° | 1 | 4 | 16.19 dB | #### Comment Sectoring reduces the number of first-tier co-channel interferers, so the required S/I is met with a smaller cluster. A smaller N gives more channels per cell (S/N) and hence more capacity: going from N = 12 to N = 4 gives three times as many channels per cell. The cost is more antennas, more handoffs between sectors and lower trunking efficiency. With n = 3 (less path loss than n = 4), interference falls off more slowly, so larger clusters are needed than in the usual n = 4 case. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Grade of Service is: A. The ratio of signal to interference B. The probability that a call is blocked during the busy hour C. The number of channels per cell D. The average call duration Answer: B. GoS measures the user's ability to access the system in the busy hour, typically as blocking probability. (Key point: Definition of GoS) 2. The Erlang B formula for blocked calls cleared assumes: A. Blocked calls wait in a queue B. Constant arrival times C. Only a small number of users D. Poisson arrivals, exponential holding times and infinite users, with blocked calls not queued Answer: D. Blocked calls are cleared, i.e. the user must retry. (Key point: Erlang B assumptions) 3. For S/I >= 15 dB, n = 3 and omni antennas (i0 = 6), the optimal N is: A. 12 B. 7 C. 9 D. 4 Answer: A. N = 7 gives 12.05 dB and N = 9 gives 13.69 dB, both below 15; N = 12 gives 15.56 dB. (Key point: Omni optimal N) 4. Going from omni to 60 degree sectoring, the optimal N changes from 12 to 4 because: A. Sectoring raises transmit power B. Sectoring increases path loss exponent C. Fewer first-tier interferers (i0 = 1 instead of 6) means a smaller cluster meets the S/I D. Sectoring increases D/R Answer: C. Fewer interferers allow a smaller N, giving three times more channels per cell. (Key point: Effect of sectoring) ## Related in Wireless Communication - [Prove Q = √3N; optimal N for 15 dB S/I (n = 4)](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Areuse-ratio-optimal-n-n4) — Wireless Communication, chapter 2 - [Numerical: trunking capacity loss with sectoring](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Atrunking-loss-sectoring) — Wireless Communication, chapter 2 - [Numerical: 12-cell system capacity at 2% GOS](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Acellular-capacity-12-cell-system) — Wireless Communication, chapter 2 ## Related in other subjects - [Traffic intensity, GOS and call duration probability](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Atraffic-intensity-call-duration-probability) — Telecommunications, chapter 6