From 123f1926c6447e7d25def6fdf67eb5c6bbc9e76c Mon Sep 17 00:00:00 2001 From: ti_mo Date: Fri, 8 May 2026 14:07:45 +0200 Subject: [PATCH] Finished lab_2 --- lab_2/SlottedAloha.svg | 290 +++++++++++++++++++++++++++++++++ lab_2/Slotted_Aloha_Report.md | 67 ++++++++ lab_2/Slotted_Aloha_Report.pdf | Bin 0 -> 168655 bytes lab_2/lab_2.asv | 29 ++-- lab_2/lab_2.m | 19 ++- 5 files changed, 387 insertions(+), 18 deletions(-) create mode 100644 lab_2/SlottedAloha.svg create mode 100644 lab_2/Slotted_Aloha_Report.md create mode 100644 lab_2/Slotted_Aloha_Report.pdf diff --git a/lab_2/SlottedAloha.svg b/lab_2/SlottedAloha.svg new file mode 100644 index 0000000..064cf42 --- /dev/null +++ b/lab_2/SlottedAloha.svg @@ -0,0 +1,290 @@ + + +Qt SVG Document +MATLAB, The MathWorks, Inc. Version 26.1.0.3203278 R2026a + + + + + + + + + + + + + +access probability + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +0 + + + +0.1 + + + +0.2 + + + +0.3 + + + +0.4 + + + +0.5 + + + +0.6 + + + +0.7 + + + +0.8 + + + +0.9 + + + +1 + + + +throughput + + + + + + + + + + + + + + + + + + + + + + + + + + + + +0 + + + +0.05 + + + +0.1 + + + +0.15 + + + +0.2 + + + +0.25 + + + +0.3 + + + +0.35 + + + +0.4 + + + +0.45 + + + + + + + + + + + + + + + + + + + + + + + +theoretical + + + +simulation + + + + + + + diff --git a/lab_2/Slotted_Aloha_Report.md b/lab_2/Slotted_Aloha_Report.md new file mode 100644 index 0000000..22f0f4d --- /dev/null +++ b/lab_2/Slotted_Aloha_Report.md @@ -0,0 +1,67 @@ +# Simulation of Slotted ALOHA Throughput Report + +**Timo Niemann** + +## Goal + +The goal of this simulation is to analyze the throughput behavior of Slotted ALOHA for a fixed number of senders. For every access probability `p`, each sender independently decides whether to transmit in the current time slot. A slot is successful only if exactly one sender transmits. Empty slots and collisions do not contribute to the throughput. + +## Simulation Setup + +| Parameter | Value | +|---|---| +| `number_of_senders` | `4` | +| `number_of_timeslots` | `1000` | +| `access_probability` | `0:0.01:1` | +| slot result | `1 = successful transmission`, `0 = no successful transmission` | + +## Throughput Formula + +For `n` independent senders, the theoretical throughput is the probability that exactly one sender transmits in a slot: + +```text +throughput = np(1 - p)^(n - 1) +``` + +With `n = 4`, this becomes: + +```text +throughput = 4p(1 - p)^3 +``` + +The theoretical maximum is reached at: + +```text +p = 1 / n = 0.25 +``` + +Therefore, the maximum theoretical throughput is: + +```text +throughput(0.25) = 4 * 0.25 * 0.75^3 = 0.421875 +``` + +## Throughput Curve Plot + +![Simulated and theoretical throughput curve for four senders](SlottedAloha.svg) + +**Figure 1:** Simulated and theoretical throughput curve for four senders. + +## Maximum Coordinates + +| Curve | Maximum coordinate | +|---|---| +| Theoretical | `p = 0.25`, throughput `= 0.421875` | +| Simulation | approximately `p = 0.28` to `0.30`, throughput `≈ 0.43` in the shown run | + +## Comparison with the Theoretical Formula + +The theoretical curve is smooth because it is calculated directly from the probability formula. The simulated curve follows the same overall shape, but it is not perfectly smooth because only `1000` time slots are simulated for each probability value. Therefore, random variation is still visible. + +The important behavior matches the theory. At low access probabilities, the throughput is low because most slots are empty. As `p` increases, successful transmissions become more likely and the throughput rises. Around `p = 0.25`, the system reaches its best operating region. After that point, too many senders transmit in the same slot, so collisions become more frequent and the throughput decreases. + +The simulated maximum is slightly shifted and slightly higher than the theoretical maximum in the shown run. This is expected for a finite simulation. With more time slots per probability value, the simulated curve should become smoother and the maximum should move closer to the theoretical coordinate `p = 0.25`, `S = 0.421875`. + +## Conclusion + +The simulation correctly reproduces the characteristic Slotted ALOHA throughput curve. The best throughput is obtained when the access probability is close to `1 / n`. For four senders, this is `p = 0.25`. Below this value, the channel is underused. Above this value, collisions dominate increasingly. 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zpsmmx9w<&Mj)x;90<}G#kC{+9Xo$tJIUt>b;vzZ%Lgkr;P!fmn@$n#!j@6Ttza2zD6 NfUlyWYos^kzW~WCs%HQI literal 0 HcmV?d00001 diff --git a/lab_2/lab_2.asv b/lab_2/lab_2.asv index 764d029..e95330b 100644 --- a/lab_2/lab_2.asv +++ b/lab_2/lab_2.asv @@ -4,25 +4,34 @@ access_probability = 0:0.01:1; access_probability_length = length(access_probability); -sender_rands = zeros(number_of_senders); -% | slot1 | slot -% prob1 | +sender_access = zeros(number_of_senders); + +% | slot1 | slot2 | ... +% prob1 | | | ... +% prob2 | | | ... +% ... | ... | ...  | ... result = zeros(access_probability_length, number_of_timeslots); % simulate all time slots on each probability for probability_idx = 1 : access_probability_length - current_result = zeros(access_probability_length); + current_result = zeros(1, number_of_timeslots); probability = access_probability(probability_idx); - for i = 1 : number_of_timeslots - % generate random sender accesses - for j = 1 : number_of_senders - sender_rands(j) = rand(); + for time_slot = 1 : number_of_timeslots + % generate random sender accesses by probability + sender_access = zeros(1, number_of_senders); + for sender = 1 : number_of_senders + if rand() < access_probability(probability_idx) + sender_access(1, sender) = 1; + end end - + current_result(1, time_slot) = sum(sender_access(1, :)) == 1; end result(probability_idx, :) = current_result; -end \ No newline at end of file +end + +disp(result(floor(access_probability_length / 2), 1:100)) +plot(result); \ No newline at end of file diff --git a/lab_2/lab_2.m b/lab_2/lab_2.m index 52828ef..322b871 100644 --- a/lab_2/lab_2.m +++ b/lab_2/lab_2.m @@ -6,11 +6,8 @@ access_probability_length = length(access_probability); sender_access = zeros(number_of_senders); -% | slot1 | slot2 | ... -% prob1 | | | ... -% prob2 | | | ... -% ... | ... | ...  | ... -result = zeros(access_probability_length, number_of_timeslots); +simulated_throughput = zeros(1, access_probability_length); +estimated_throughput = zeros(1, access_probability_length); % simulate all time slots on each probability for probability_idx = 1 : access_probability_length @@ -29,8 +26,14 @@ for probability_idx = 1 : access_probability_length current_result(1, time_slot) = sum(sender_access(1, :)) == 1; end - - result(probability_idx, :) = current_result; + + simulated_throughput(probability_idx) = mean(current_result); + estimated_throughput(probability_idx) = number_of_senders * probability * (1 - probability)^(number_of_senders - 1); end -disp(result(floor(access_probability_length / 2), 1:100)) \ No newline at end of file +plot(access_probability, estimated_throughput) +hold on +plot(access_probability, simulated_throughput) +legend("theoretical", "simulation") +xlabel("access probability") +ylabel("throughput") \ No newline at end of file