refactor(simulation): Commented out prints

Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>
feat(simulation): Event based eviction instead of iterative
2024-12-05 21:58:15 +01:00 · 2024-12-05 21:46:52 +01:00 · 2024-12-05 15:53:31 +01:00 · 2024-12-05 14:49:41 +01:00 · 2024-12-05 14:40:20 +01:00 · 2024-12-05 13:48:20 +01:00
101 changed files with 1957425 additions and 3261 deletions
--- a/00_aoi_caching_simulation/.gitignore
+++ b/00_aoi_caching_simulation/.gitignore
@@ -0,0 +1 @@
+.aoi_cache/
--- a/00_aoi_caching_simulation/00-aoi_cache_simulation.ipynb
+++ b/00_aoi_caching_simulation/00-aoi_cache_simulation.ipynb
--- a/00_aoi_caching_simulation/01-aoi_cache_experiment_eval.ipynb
+++ b/00_aoi_caching_simulation/01-aoi_cache_experiment_eval.ipynb
--- a/00_aoi_caching_simulation/02-multi_aoi_cache_simulation.ipynb
+++ b/00_aoi_caching_simulation/02-multi_aoi_cache_simulation.ipynb
--- a/00_aoi_caching_simulation/aoi_cache_simulation.py
+++ b/00_aoi_caching_simulation/aoi_cache_simulation.py
--- a/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/details.csv
@@ -0,0 +1,101 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/hit_age.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_0_5s_ttl/overall_hit_age.csv
@@ -0,0 +1,9 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/details.csv
@@ -0,0 +1,101 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/hit_age.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_1_0s_ttl/overall_hit_age.csv
@@ -0,0 +1,9 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/details.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/hit_age.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_2_0s_ttl/overall_hit_age.csv
@@ -0,0 +1,9 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/details.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/hit_age.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3000000000_0s_ttl/overall_hit_age.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/details.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/hit_age.csv
@@ -0,0 +1,101 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_3_0s_ttl/overall_hit_age.csv
@@ -0,0 +1,9 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/details.csv
@@ -0,0 +1,101 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/hit_age.csv
@@ -0,0 +1,101 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_4_0s_ttl/overall_hit_age.csv
@@ -0,0 +1,9 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/details.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/details.csv
@@ -0,0 +1,101 @@
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/hit_age.csv
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--- a/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/lambda_distribution.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/lambda_distribution.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/lambda_vs_access_count.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/lambda_vs_access_count.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/objects_in_cache_over_time.pdf
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/objects_in_cache_over_time.pdf
--- a/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/overall_hit_age.csv
+++ b/00_aoi_caching_simulation/experiments/No_Refresh_5_0s_ttl/overall_hit_age.csv
@@ -0,0 +1,9 @@
+,hit_rate,expected_hit_rate,avg_cache_time,avg_age,expected_age
+count,100.0,100.0,100.0,100.0,100.0
+mean,0.9953443206249202,0.9958147334586499,0.9954451062793003,460.0120641644132,421.2475605352718
+std,0.0037838417482891156,0.003277078465023217,0.003941487312951466,438.5456330279936,459.7399440940381
+min,0.988,0.9932620530009145,0.9880895961230295,66.15867367479018,41.41515761234073
+25%,0.9923759662896949,0.9932620530009145,0.9921726947842899,108.59918894112637,65.33498084879865
+50%,0.9941648428352272,0.9932620530009145,0.9947945690963905,153.08896903666576,85.43676847106119
+75%,0.9997647330829824,0.9999546000702375,0.9998047459271276,1040.05534548247,1040.1249849775975
+max,0.9999992967215974,1.0,0.9999998745822094,1061.668255053678,1082.3749855637955
--- a/00_aoi_caching_simulation/experiments/avg_ages.md
+++ b/00_aoi_caching_simulation/experiments/avg_ages.md
@@ -0,0 +1,9 @@
+|               |   avg_ages |
+|:--------------|-----------:|
+| 0.5s          |    32.1964 |
+| 1.0s          |    77.6952 |
+| 2.0s          |   131.396  |
+| 3.0s          |   274.829  |
+| 4.0s          |   442.732  |
+| 5.0s          |   460.012  |
+| 3000000000.0s |  1028.55   |
--- a/00_aoi_caching_simulation/experiments/hit_rates.md
+++ b/00_aoi_caching_simulation/experiments/hit_rates.md
@@ -0,0 +1,9 @@
+|               |   hit_rates |
+|:--------------|------------:|
+| 0.5s          |    0.564884 |
+| 1.0s          |    0.769828 |
+| 2.0s          |    0.916617 |
+| 3.0s          |    0.967932 |
+| 4.0s          |    0.989788 |
+| 5.0s          |    0.995344 |
+| 3000000000.0s |    0.999625 |
--- a/00_aoi_caching_simulation/experiments/hr_and_age_vs_ttl.png
+++ b/00_aoi_caching_simulation/experiments/hr_and_age_vs_ttl.png
--- a/00_aoi_caching_simulation/note.md
+++ b/00_aoi_caching_simulation/note.md
@@ -83,4 +83,15 @@ CPU times: user 3min 46s, sys: 43 s, total: 4min 29s
 Wall time: 4min 29s
 for ACCESS_COUNT_LIMIT = 10_000 # Total time to run the simulation

-## Notes 11/27/2024
+## Notes 11/29/2024
+
+C_m = cost for cache miss
+C_delta = cost for refresh
+C = Cm + C_delta over all objects summarized
+
+We wanna minimize cost function
+
+N = number of objects
+B is cache size
+
+C_f roughly equals C_m
--- a/01_nb_cncf_optimization/00-hitrate_optimization.ipynb
+++ b/01_nb_cncf_optimization/00-hitrate_optimization.ipynb
@@ -0,0 +1,566 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "ab5cd7d1-1a57-46fc-8282-dae0a6cc2944",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import random\n",
+    "import pandas as pd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "3d1ad0b9-f6a8-4e98-84aa-6e02e4279954",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "DATABASE_OBJECT_COUNT = 100\n",
+    "CACHE_SIZE = DATABASE_OBJECT_COUNT/2\n",
+    "ZIPF_CONSTANT = 2\n",
+    "\n",
+    "CACHE_MISS_COST = 2\n",
+    "CACHE_REFRESH_COST = 1\n",
+    "\n",
+    "SEED = 42\n",
+    "np.random.seed(SEED)\n",
+    "random.seed(SEED)\n",
+    "\n",
+    "LAMBDA_VALUES = np.array([np.random.zipf(ZIPF_CONSTANT) for i in np.arange(1, DATABASE_OBJECT_COUNT + 1,1)])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "9cc83cf6-5c78-4f0d-b7cb-08cdb80c362e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# LAMBDA_VALUES = np.array([0.03, 0.04,0.05,0.06,0.07,1,1.1,1.2,1.3,1.4,1.5])\n",
+    "# DATABASE_OBJECT_COUNT = len(LAMBDA_VALUES)\n",
+    "# CACHE_SIZE = 4.4\n",
+    "# CACHE_MISS_COST = 7\n",
+    "# CACHE_REFRESH_COST = 1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "3dc07233-0b56-4fee-a93b-212836c18b42",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "db_object_count = DATABASE_OBJECT_COUNT\n",
+    "cache_sz = CACHE_SIZE\n",
+    "\n",
+    "lambda_vals = LAMBDA_VALUES\n",
+    "c_f = CACHE_MISS_COST\n",
+    "c_delta = CACHE_REFRESH_COST"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "5a27d416-8f98-4814-af9e-6c6bef95f4ef",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def eta_star(db_object_count, c_f, cache_sz, c_delta, lambda_vals):\n",
+    "    num = (db_object_count * c_f - cache_sz * c_delta)\n",
+    "    denom = np.sum(1.0/lambda_vals)\n",
+    "    return max(0, num/denom)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "6276a9ce-f839-4fe6-90f2-2195cf065fc8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def h_i_star(c_f, eta, lambda_vals, c_delta):\n",
+    "    optimized_hitrate = (c_f - (eta/lambda_vals)) / c_delta\n",
+    "    return optimized_hitrate"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "dcd31a8c-6864-4b9a-8bb3-998f0c32baf6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
+    "    lower_bound_violation =  hitrates[(hitrates < 0)]\n",
+    "    upper_bound_violation = hitrates[(hitrates > 1)]\n",
+    "    smallest_delta = np.abs(np.min(lower_bound_violation))\n",
+    "    biggest_delta = np.max(upper_bound_violation) - 1\n",
+    "    if smallest_delta > biggest_delta:\n",
+    "        print(smallest_delta)\n",
+    "        index = np.where(hitrates == np.min(local_hitrates))[0][0]\n",
+    "        return index\n",
+    "    else:\n",
+    "        \n",
+    "        index = np.where(hitrates == np.max(local_hitrates))[0][0]\n",
+    "        return index"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "9d774304-ae68-43b3-a76a-e970c06c5236",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
+    "    outside_bounds = (hitrates < 0) | (hitrates > 1)\n",
+    "    distances = np.where(outside_bounds, np.maximum(np.abs(hitrates - 0), np.abs(hitrates - 1)), -np.inf)\n",
+    "    index = np.argmax(distances)\n",
+    "    return index"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "19678083-15e1-439b-be8c-42033d501644",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([ 1,  3,  1,  1,  2,  1,  5,  1,  1,  1,  2,  1,  1,  1,  2,  2,  1,\n",
+       "        1,  3,  1,  1,  1,  1,  2,  1,  1,  1,  5,  1,  1,  1,  4,  1,  4,\n",
+       "        1,  1,  1,  3,  8,  1,  4,  4,  2,  1,  1,  1, 10,  1,  1,  1,  5,\n",
+       "        9,  1,  1,  1,  1,  1, 17,  2,  1, 26,  1,  1,  2,  1, 10,  1, 69,\n",
+       "        1,  1,  2,  1,  1,  1,  3,  2,  2,  3, 15,  1,  1,  5,  2,  1,  1,\n",
+       "        2,  1,  2,  1,  1,  2,  2,  3,  1,  2,  1,  1, 37,  4,  2])"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "lambda_vals"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "ccd4b95d-1cdd-4c99-a22e-4b31338993cf",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2.1159070575516945\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "array([-0.11590706,  1.29469765, -0.11590706, -0.11590706,  0.94204647,\n",
+       "       -0.11590706, -0.11590706, -0.11590706, -0.11590706,  0.94204647,\n",
+       "       -0.11590706, -0.11590706, -0.11590706,  0.94204647,  0.94204647,\n",
+       "       -0.11590706, -0.11590706,  1.29469765, -0.11590706, -0.11590706,\n",
+       "       -0.11590706, -0.11590706,  0.94204647, -0.11590706, -0.11590706,\n",
+       "       -0.11590706, -0.11590706, -0.11590706, -0.11590706,  1.47102324,\n",
+       "       -0.11590706,  1.47102324, -0.11590706, -0.11590706, -0.11590706,\n",
+       "        1.29469765,  1.73551162, -0.11590706,  1.47102324,  1.47102324,\n",
+       "        0.94204647, -0.11590706, -0.11590706, -0.11590706,  1.78840929,\n",
+       "       -0.11590706, -0.11590706, -0.11590706,  1.76489922, -0.11590706,\n",
+       "       -0.11590706, -0.11590706, -0.11590706, -0.11590706,  1.87553488,\n",
+       "        0.94204647, -0.11590706,  1.91861896, -0.11590706, -0.11590706,\n",
+       "        0.94204647, -0.11590706,  1.78840929, -0.11590706,  1.96933468,\n",
+       "       -0.11590706, -0.11590706,  0.94204647, -0.11590706, -0.11590706,\n",
+       "       -0.11590706,  1.29469765,  0.94204647,  0.94204647,  1.29469765,\n",
+       "        1.85893953, -0.11590706, -0.11590706,  0.94204647, -0.11590706,\n",
+       "       -0.11590706,  0.94204647, -0.11590706,  0.94204647, -0.11590706,\n",
+       "       -0.11590706,  0.94204647,  0.94204647,  1.29469765, -0.11590706,\n",
+       "        0.94204647, -0.11590706, -0.11590706,  1.94281332,  1.47102324,\n",
+       "        0.94204647])"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "eta = eta_star(db_object_count, c_f, cache_sz, c_delta, lambda_vals[lambda_vals != lambda_vals[6]])\n",
+    "print(eta)\n",
+    "optimized_hitrates = (c_f - eta / lambda_vals[lambda_vals != lambda_vals[6]]) / c_delta\n",
+    "optimized_hitrates"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "05b17074-719f-4bca-8434-2aaee26094d0",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div>\n",
+       "<style scoped>\n",
+       "    .dataframe tbody tr th:only-of-type {\n",
+       "        vertical-align: middle;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe tbody tr th {\n",
+       "        vertical-align: top;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe thead th {\n",
+       "        text-align: right;\n",
+       "    }\n",
+       "</style>\n",
+       "<table border=\"1\" class=\"dataframe\">\n",
+       "  <thead>\n",
+       "    <tr style=\"text-align: right;\">\n",
+       "      <th></th>\n",
+       "      <th>0</th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>count</th>\n",
+       "      <td>96.000000</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>mean</th>\n",
+       "      <td>0.437500</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>std</th>\n",
+       "      <td>0.726101</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>min</th>\n",
+       "      <td>-0.115907</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>25%</th>\n",
+       "      <td>-0.115907</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>50%</th>\n",
+       "      <td>-0.115907</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>75%</th>\n",
+       "      <td>0.942046</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>max</th>\n",
+       "      <td>1.969335</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "               0\n",
+       "count  96.000000\n",
+       "mean    0.437500\n",
+       "std     0.726101\n",
+       "min    -0.115907\n",
+       "25%    -0.115907\n",
+       "50%    -0.115907\n",
+       "75%     0.942046\n",
+       "max     1.969335"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pd.DataFrame(optimized_hitrates).describe()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "0e21c26f-058a-4e56-a5ad-1c47bf28656c",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Optimized: 67 1.97 // [ 1.79077042 -0.09229584  1.         -0.09229584 -0.09229584]\n",
+      "Optimized: 97 1.94 // [-0.07876743 -0.07876743  1.          1.48030814  0.96061628]\n",
+      "Optimized: 60 1.92 // [ 0.96720258 -0.06559484  1.         -0.06559484 -0.06559484]\n",
+      "Optimized: 57 1.88 // [-0.05274002 -0.05274002  1.          0.97362999 -0.05274002]\n",
+      "Optimized: 78 1.86 // [ 0.97977406  1.31984937  1.         -0.04045188 -0.04045188]\n",
+      "Optimized: 46 1.80 // [-0.02836604 -0.02836604  1.         -0.02836604 -0.02836604]\n",
+      "Optimized: 65 1.80 // [ 0.99140044 -0.01719911  1.         -0.01719911  1.        ]\n",
+      "Optimized: 51 1.78 // [-0.00600086  1.59879983  1.         -0.00600086 -0.00600086]\n",
+      "Optimized: 38 1.75 // [0.00491746 1.33497249 1.         0.00491746 1.50122936]\n",
+      "Optimized: 6 1.60 // [1.00774103 0.01548205 1.         0.01548205 0.01548205]\n",
+      "Optimized: 27 1.60 // [0.02399435 0.02399435 1.         0.02399435 0.02399435]\n",
+      "Optimized: 50 1.61 // [0.03255485 0.03255485 1.         1.         0.03255485]\n",
+      "Optimized: 81 1.61 // [0.04116395 0.04116395 1.         1.02058197 0.04116395]\n",
+      "Optimized: 31 1.51 // [0.04982206 0.04982206 1.         0.04982206 1.51245552]\n",
+      "Optimized: 33 1.51 // [1.         0.05714286 1.         0.05714286 0.05714286]\n",
+      "Optimized: 40 1.52 // [1.         0.06451613 1.         1.51612903 1.03225806]\n",
+      "Optimized: 41 1.52 // [0.07194245 1.         1.         1.03597122 0.07194245]\n",
+      "Optimized: 98 1.52 // [0.07942238 1.         1.         1.03971119]\n",
+      "Optimized: 1 1.36 // []\n",
+      "Optimized: 18 1.36 // [0.09223301 0.09223301 1.         0.09223301 0.09223301]\n",
+      "Optimized: 37 1.37 // [0.09756098 0.09756098 1.         1.         0.09756098]\n",
+      "Optimized: 74 1.37 // [0.10294118 0.10294118 1.         1.05147059 1.05147059]\n",
+      "Optimized: 77 1.37 // [1.05418719 1.05418719 1.         1.         0.10837438]\n",
+      "Optimized: 92 1.37 // [1.05693069 1.05693069 1.         0.11386139 1.05693069]\n",
+      "Optimized: 4 1.06 // [0.11940299 0.11940299 1.         0.11940299 1.        ]\n",
+      "Optimized: 10 1.06 // [0.12030075 0.12030075 1.         0.12030075 0.12030075]\n",
+      "Optimized: 14 1.06 // [0.12121212 0.12121212 1.         1.06060606 0.12121212]\n",
+      "Optimized: 15 1.06 // [0.1221374 1.        1.        0.1221374 0.1221374]\n",
+      "Optimized: 23 1.06 // [0.12307692 0.12307692 1.         0.12307692 0.12307692]\n",
+      "Optimized: 42 1.06 // [1.         1.         1.         0.12403101 0.12403101]\n",
+      "Optimized: 58 1.06 // [0.125 1.    1.    0.125 1.   ]\n",
+      "Optimized: 63 1.06 // [0.12598425 0.12598425 1.         0.12598425 1.        ]\n",
+      "Optimized: 70 1.06 // [0.12698413 0.12698413 1.         0.12698413 0.12698413]\n",
+      "Optimized: 75 1.06 // [0.128 1.    1.    1.064 1.   ]\n",
+      "Optimized: 76 1.06 // [1. 1. 1. 1. 1.]\n",
+      "Optimized: 82 1.07 // [0.1300813 1.        1.        0.1300813 0.1300813]\n",
+      "Optimized: 85 1.07 // [0.13114754 0.13114754 1.         0.13114754 1.06557377]\n",
+      "Optimized: 87 1.07 // [1.        0.1322314 1.        0.1322314 0.1322314]\n",
+      "Optimized: 90 1.07 // [0.13333333 0.13333333 1.         1.06666667 1.        ]\n",
+      "Optimized: 91 1.07 // [0.13445378 1.         1.         1.         0.13445378]\n",
+      "Optimized: 94 1.07 // [1.         0.13559322 1.         0.13559322 0.13559322]\n",
+      "Optimized: 99 1.07 // [1. 1. 1.]\n",
+      "All values optimized.\n"
+     ]
+    }
+   ],
+   "source": [
+    "\"\"\"\n",
+    "Perform theoretical optimization to compute optimal hit probabilities.\n",
+    "\n",
+    "Parameters:\n",
+    "- lambda_vals (numpy array): Request rates for each item.\n",
+    "- B (float): Total cache size.\n",
+    "- c_f (float): Fetching linear cost (cache miss cost).\n",
+    "- c_delta (float): Age linear cost.\n",
+    "\n",
+    "Returns:\n",
+    "- h_opt (numpy array): Optimal hit probabilities for each item.\n",
+    "\"\"\"\n",
+    "optimized_hitrates = np.zeros(DATABASE_OBJECT_COUNT)\n",
+    "current_db_object_count = DATABASE_OBJECT_COUNT\n",
+    "current_cache_size = CACHE_SIZE\n",
+    "\n",
+    "differenc_set = np.arange(DATABASE_OBJECT_COUNT)\n",
+    "fix_i = []\n",
+    "\n",
+    "while True:\n",
+    "    if current_db_object_count == 0:\n",
+    "        print(\"No objects left to optimize.\")\n",
+    "        if current_cache_size > 0:\n",
+    "            print(\"Add obj with optimized hitrate 0 and add them to optimization pool for re-optimization.\")\n",
+    "            # Redistribute unused cache size among items with zero hit probability\n",
+    "            differenc_set = np.where(optimized_hitrates == 0)[0]\n",
+    "            fix_i = np.setdiff1d(np.arange(DATABASE_OBJECT_COUNT), differenc_set).tolist()\n",
+    "            current_db_object_count = len(differenc_set)\n",
+    "            continue\n",
+    "        else:\n",
+    "            \"Reset\"\n",
+    "            optimized_hitrates[differenc_set] = 0\n",
+    "            break\n",
+    "    # Compute Lagrangian multiplier and optimal hit probabilities\n",
+    "    eta = eta_star(current_db_object_count, c_f, current_cache_size, c_delta, lambda_vals[differenc_set])\n",
+    "    optimized_hitrates[differenc_set] = (c_f - eta / lambda_vals[differenc_set]) / c_delta\n",
+    "    if eta < 0:\n",
+    "        print(\"eta was negative.\")\n",
+    "        current_cache_size = current_db_object_count * c_f / c_delta  # Adjust cache size for next iteration\n",
+    "        continue\n",
+    "    \n",
+    "    if len((optimized_hitrates[differenc_set])[((optimized_hitrates[differenc_set]) < 0) | ((optimized_hitrates[differenc_set])> 1)]) == 0:\n",
+    "        print(\"All values optimized.\")\n",
+    "        break\n",
+    "    \n",
+    "    max_outbound_index = get_index_of_furthest_hitrate_from_boundary(optimized_hitrates)\n",
+    "    fix_i.append(max_outbound_index)\n",
+    "    differenc_set = np.setdiff1d(np.arange(DATABASE_OBJECT_COUNT), fix_i)\n",
+    "\n",
+    "    old_hitrate = optimized_hitrates[max_outbound_index]\n",
+    "    optimized_hitrates[max_outbound_index] = (1 if optimized_hitrates[max_outbound_index] > 1 else 0)\n",
+    "    \n",
+    "    print(f\"Optimized: {max_outbound_index} {old_hitrate:.2f} // {optimized_hitrates[max_outbound_index-2:max_outbound_index+3]}\")\n",
+    "    \n",
+    "    current_db_object_count -= 1\n",
+    "    current_cache_size -= optimized_hitrates[max_outbound_index]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "f559ee7a-be2f-4076-b01c-f08950ad5a88",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.13793103, 1.        , 0.13793103, 0.13793103, 1.        ,\n",
+       "       0.13793103, 1.        , 0.13793103, 0.13793103, 0.13793103,\n",
+       "       1.        , 0.13793103, 0.13793103, 0.13793103, 1.        ,\n",
+       "       1.        , 0.13793103, 0.13793103, 1.        , 0.13793103,\n",
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+       "       0.13793103, 1.        , 0.13793103, 0.13793103, 0.13793103,\n",
+       "       1.        , 1.        , 0.13793103, 0.13793103, 0.13793103,\n",
+       "       0.13793103, 0.13793103, 1.        , 1.        , 0.13793103,\n",
+       "       1.        , 0.13793103, 0.13793103, 1.        , 0.13793103,\n",
+       "       1.        , 0.13793103, 1.        , 0.13793103, 0.13793103,\n",
+       "       1.        , 0.13793103, 0.13793103, 0.13793103, 1.        ,\n",
+       "       1.        , 1.        , 1.        , 1.        , 0.13793103,\n",
+       "       0.13793103, 1.        , 1.        , 0.13793103, 0.13793103,\n",
+       "       1.        , 0.13793103, 1.        , 0.13793103, 0.13793103,\n",
+       "       1.        , 1.        , 1.        , 0.13793103, 1.        ,\n",
+       "       0.13793103, 0.13793103, 1.        , 1.        , 1.        ])"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "optimized_hitrates"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "8b2d3cea-1cc0-476e-92bf-2ac4344a9b1b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<div>\n",
+       "<style scoped>\n",
+       "    .dataframe tbody tr th:only-of-type {\n",
+       "        vertical-align: middle;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe tbody tr th {\n",
+       "        vertical-align: top;\n",
+       "    }\n",
+       "\n",
+       "    .dataframe thead th {\n",
+       "        text-align: right;\n",
+       "    }\n",
+       "</style>\n",
+       "<table border=\"1\" class=\"dataframe\">\n",
+       "  <thead>\n",
+       "    <tr style=\"text-align: right;\">\n",
+       "      <th></th>\n",
+       "      <th>0</th>\n",
+       "    </tr>\n",
+       "  </thead>\n",
+       "  <tbody>\n",
+       "    <tr>\n",
+       "      <th>count</th>\n",
+       "      <td>100.000000</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>mean</th>\n",
+       "      <td>0.500000</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>std</th>\n",
+       "      <td>0.427625</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>min</th>\n",
+       "      <td>0.137931</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>25%</th>\n",
+       "      <td>0.137931</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>50%</th>\n",
+       "      <td>0.137931</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>75%</th>\n",
+       "      <td>1.000000</td>\n",
+       "    </tr>\n",
+       "    <tr>\n",
+       "      <th>max</th>\n",
+       "      <td>1.000000</td>\n",
+       "    </tr>\n",
+       "  </tbody>\n",
+       "</table>\n",
+       "</div>"
+      ],
+      "text/plain": [
+       "                0\n",
+       "count  100.000000\n",
+       "mean     0.500000\n",
+       "std      0.427625\n",
+       "min      0.137931\n",
+       "25%      0.137931\n",
+       "50%      0.137931\n",
+       "75%      1.000000\n",
+       "max      1.000000"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pd.DataFrame(optimized_hitrates).describe()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7a998837-72b8-4039-95a5-ca8d9c8e65ab",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "graphs",
+   "language": "python",
+   "name": "graphs"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--- a/01_nb_cncf_optimization/01-objective_gridsearch.ipynb
+++ b/01_nb_cncf_optimization/01-objective_gridsearch.ipynb
@@ -0,0 +1,287 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "ab5cd7d1-1a57-46fc-8282-dae0a6cc2944",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import random\n",
+    "import pandas as pd\n",
+    "import itertools\n",
+    "from tqdm import tqdm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "3d1ad0b9-f6a8-4e98-84aa-6e02e4279954",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "SEED = 42\n",
+    "np.random.seed(SEED)\n",
+    "random.seed(SEED)\n",
+    "\n",
+    "ZIPF_CONSTANT = 2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "5a27d416-8f98-4814-af9e-6c6bef95f4ef",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def eta_star(db_object_count, c_f, cache_sz, c_delta, lambda_vals):\n",
+    "    num = (db_object_count * c_f - cache_sz * c_delta)\n",
+    "    denom = np.sum(1.0/lambda_vals)\n",
+    "    if denom == 0:\n",
+    "        print(\"sum(1.0/lambda_vals) == 0\")\n",
+    "        print(db_object_count, c_f, cache_sz, c_delta, lambda_vals)\n",
+    "    return max(0, num/denom)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "6276a9ce-f839-4fe6-90f2-2195cf065fc8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def h_i_star(c_f, eta, lambda_vals, c_delta):\n",
+    "    optimized_hitrate = (c_f - (eta/lambda_vals)) / c_delta\n",
+    "    return optimized_hitrate"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "dcd31a8c-6864-4b9a-8bb3-998f0c32baf6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
+    "    lower_bound_violation =  hitrates[(hitrates < 0)]\n",
+    "    upper_bound_violation = hitrates[(hitrates > 1)]\n",
+    "    smallest_delta = np.abs(np.min(lower_bound_violation))\n",
+    "    biggest_delta = np.max(upper_bound_violation) - 1\n",
+    "    if smallest_delta > biggest_delta:\n",
+    "        print(smallest_delta)\n",
+    "        index = np.where(hitrates == np.min(local_hitrates))[0][0]\n",
+    "        return index\n",
+    "    else:\n",
+    "        \n",
+    "        index = np.where(hitrates == np.max(local_hitrates))[0][0]\n",
+    "        return index"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "9d774304-ae68-43b3-a76a-e970c06c5236",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
+    "    outside_bounds = (hitrates < 0) | (hitrates > 1)\n",
+    "    distances = np.where(outside_bounds, np.maximum(np.abs(hitrates - 0), np.abs(hitrates - 1)), -np.inf)\n",
+    "    index = np.argmax(distances)\n",
+    "    return index"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "0e21c26f-058a-4e56-a5ad-1c47bf28656c",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "def optimize_hitrates(db_object_count, cache_size, c_f, c_delta, lambda_vals):\n",
+    "    optimized_hitrates = np.zeros(db_object_count)\n",
+    "    current_db_object_count = db_object_count\n",
+    "    current_cache_size = cache_size\n",
+    "    \n",
+    "    differenc_set = np.arange(db_object_count)\n",
+    "    fix_i = []\n",
+    "    while True:\n",
+    "        if current_db_object_count == 0:\n",
+    "            if current_cache_size > 0:\n",
+    "                # print(\"Re-optimize objects with optimal hitrate of 0.\")\n",
+    "                differenc_set = np.where(optimized_hitrates == 0)[0]\n",
+    "                fix_i = np.setdiff1d(np.arange(db_object_count), differenc_set).tolist()\n",
+    "                current_db_object_count = len(differenc_set)\n",
+    "                continue\n",
+    "            else:\n",
+    "                # print(\"Stop optimization.\")\n",
+    "                optimized_hitrates[differenc_set] = 0\n",
+    "                break\n",
+    "        \n",
+    "        eta = eta_star(current_db_object_count, c_f, current_cache_size, c_delta, lambda_vals[differenc_set])\n",
+    "        optimized_hitrates[differenc_set] = h_i_star(c_f, eta, lambda_vals[differenc_set], c_delta)\n",
+    "\n",
+    "        if eta < 0:\n",
+    "            # print(\"eta was negative.\")\n",
+    "            current_cache_size = current_db_object_count * c_f / c_delta  # Adjust cache size for next iteration\n",
+    "            continue\n",
+    "        \n",
+    "        if len((optimized_hitrates[differenc_set])[((optimized_hitrates[differenc_set]) < 0) | ((optimized_hitrates[differenc_set])> 1)]) == 0:\n",
+    "            # print(\"All values optimized.\")\n",
+    "            break\n",
+    "        \n",
+    "        max_outbound_index = get_index_of_furthest_hitrate_from_boundary(optimized_hitrates)\n",
+    "        fix_i.append(max_outbound_index)\n",
+    "        differenc_set = np.setdiff1d(np.arange(db_object_count), fix_i)\n",
+    "    \n",
+    "        old_hitrate = optimized_hitrates[max_outbound_index]\n",
+    "        optimized_hitrates[max_outbound_index] = (1 if optimized_hitrates[max_outbound_index] > 1 else 0)\n",
+    "        \n",
+    "        current_db_object_count -= 1\n",
+    "        current_cache_size -= optimized_hitrates[max_outbound_index]\n",
+    "    return optimized_hitrates"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "b6bf3329-3a63-4807-ab8b-8a54f824f47e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def objective_function(optimized_hitrates, c_f, c_delta, lambda_vals):\n",
+    "    return np.sum(lambda_vals*(1-optimized_hitrates)*c_f+0.5*np.power(optimized_hitrates,2)*c_delta)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "7a998837-72b8-4039-95a5-ca8d9c8e65ab",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Perform grid search\n",
+    "def grid_search(db_object_counts, cache_sizes, c_f_values, c_delta_values):\n",
+    "    best_objective = float('inf')\n",
+    "    best_params = None\n",
+    "\n",
+    "    # Iterate through all combinations of parameters\n",
+    "    for db_object_count, cache_size, c_f, c_delta in tqdm(itertools.product(db_object_counts, cache_sizes, c_f_values, c_delta_values), total=len(db_object_counts) * len(cache_sizes) * len(c_f_values) * len(c_delta_values), desc=\"Grid Search Progress\"):\n",
+    "        if db_object_count < cache_size:\n",
+    "            continue\n",
+    "        lambda_vals = np.array([np.random.zipf(ZIPF_CONSTANT) for i in np.arange(1, db_object_count + 1,1)])\n",
+    "        # print(db_object_count, cache_size, c_f, c_delta)\n",
+    "        # Call the optimization function\n",
+    "        optimized_hitrates = optimize_hitrates(db_object_count, cache_size, c_f, c_delta, lambda_vals)\n",
+    "\n",
+    "        # Compute the objective function\n",
+    "        objective = objective_function(optimized_hitrates, c_f, c_delta, lambda_vals)\n",
+    "        \n",
+    "        # Track the best (minimum) objective and corresponding parameters\n",
+    "        if objective < best_objective:\n",
+    "            best_objective = objective\n",
+    "            best_params = (db_object_count, cache_size, c_f, c_delta)\n",
+    "\n",
+    "    return best_objective, best_params"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "a271b52d-1f24-4670-ae3f-af5dd9096a2f",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "Grid Search Progress: 100%|█████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| 64152/64152 [12:27<00:00, 85.87it/s]"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 12min 16s, sys: 11.5 s, total: 12min 28s\n",
+      "Wall time: 12min 27s\n"
+     ]
+    },
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "%%time\n",
+    "\n",
+    "# Define the grid search space\n",
+    "test_ratios = np.array([0.1, 0.2, 0.5, 0.7, 1, 1.5, 2, 5, 10])\n",
+    "db_object_count_values = np.round(np.array([10, 15, 30, 100, 200, 500]))\n",
+    "cache_size_values = np.unique(np.round(np.array([db_object_count_values * i for i in test_ratios]).flatten()))\n",
+    "c_f_values = np.array([0.1, 0.2, 0.5, 0.7, 1, 1.5, 2, 5, 10])\n",
+    "c_delta_values = np.unique(np.array([c_f_values * i for i in test_ratios]).flatten())\n",
+    "\n",
+    "best_objective, best_params = grid_search(db_object_count_values, cache_size_values, c_f_values, c_delta_values)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "b2f625d0-ebe0-4a5d-92ff-7de03942ef51",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0.05000000000000002, (10, 10.0, 1.5, 0.010000000000000002))"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "best_objective, best_params "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "86a23d02-6f14-4d4d-ad8a-39084ea69151",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "graphs",
+   "language": "python",
+   "name": "graphs"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--- a/01_nb_cncf_optimization/02-objective_multi-core_gridsearch.ipynb
+++ b/01_nb_cncf_optimization/02-objective_multi-core_gridsearch.ipynb
@@ -0,0 +1,360 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "ab5cd7d1-1a57-46fc-8282-dae0a6cc2944",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import random\n",
+    "import pandas as pd\n",
+    "import itertools\n",
+    "from joblib import Parallel, delayed\n",
+    "import os.path"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "3d1ad0b9-f6a8-4e98-84aa-6e02e4279954",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "SEED = 42\n",
+    "np.random.seed(SEED)\n",
+    "random.seed(SEED)\n",
+    "\n",
+    "ZIPF_CONSTANT = 2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "5a27d416-8f98-4814-af9e-6c6bef95f4ef",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def eta_star(db_object_count, c_f, cache_sz, c_delta, lambda_vals):\n",
+    "    num = (db_object_count * c_f - cache_sz * c_delta)\n",
+    "    denom = np.sum(1.0/lambda_vals)\n",
+    "    return max(0, num/denom)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "6276a9ce-f839-4fe6-90f2-2195cf065fc8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def h_i_star(c_f, eta, lambda_vals, c_delta):\n",
+    "    optimized_hitrate = (c_f - (eta/lambda_vals)) / c_delta\n",
+    "    return optimized_hitrate"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "dcd31a8c-6864-4b9a-8bb3-998f0c32baf6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
+    "    lower_bound_violation =  hitrates[(hitrates < 0)]\n",
+    "    upper_bound_violation = hitrates[(hitrates > 1)]\n",
+    "    smallest_delta = np.abs(np.min(lower_bound_violation))\n",
+    "    biggest_delta = np.max(upper_bound_violation) - 1\n",
+    "    if smallest_delta > biggest_delta:\n",
+    "        index = np.where(hitrates == np.min(local_hitrates))[0][0]\n",
+    "        return index\n",
+    "    else:\n",
+    "        \n",
+    "        index = np.where(hitrates == np.max(local_hitrates))[0][0]\n",
+    "        return index"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "9d774304-ae68-43b3-a76a-e970c06c5236",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
+    "    outside_bounds = (hitrates < 0) | (hitrates > 1)\n",
+    "    distances = np.where(outside_bounds, np.maximum(np.abs(hitrates - 0), np.abs(hitrates - 1)), -np.inf)\n",
+    "    index = np.argmax(distances)\n",
+    "    return index"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "0e21c26f-058a-4e56-a5ad-1c47bf28656c",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "def optimize_hitrates(db_object_count, cache_size, c_f, c_delta, lambda_vals):\n",
+    "    optimized_hitrates = np.zeros(db_object_count)\n",
+    "    current_db_object_count = db_object_count\n",
+    "    current_cache_size = cache_size\n",
+    "    \n",
+    "    differenc_set = np.arange(db_object_count)\n",
+    "    fix_i = []\n",
+    "    while True:\n",
+    "        if current_db_object_count == 0:\n",
+    "            if current_cache_size > 0:\n",
+    "                # print(\"Re-optimize objects with optimal hitrate of 0.\")\n",
+    "                differenc_set = np.where(optimized_hitrates == 0)[0]\n",
+    "                fix_i = np.setdiff1d(np.arange(db_object_count), differenc_set).tolist()\n",
+    "                current_db_object_count = len(differenc_set)\n",
+    "                continue\n",
+    "            else:\n",
+    "                # print(\"Stop optimization.\")\n",
+    "                optimized_hitrates[differenc_set] = 0\n",
+    "                break\n",
+    "        \n",
+    "        eta = eta_star(current_db_object_count, c_f, current_cache_size, c_delta, lambda_vals[differenc_set])\n",
+    "        optimized_hitrates[differenc_set] = h_i_star(c_f, eta, lambda_vals[differenc_set], c_delta)\n",
+    "\n",
+    "        if eta < 0:\n",
+    "            # print(\"eta was negative.\")\n",
+    "            current_cache_size = current_db_object_count * c_f / c_delta  # Adjust cache size for next iteration\n",
+    "            continue\n",
+    "        \n",
+    "        if len((optimized_hitrates[differenc_set])[((optimized_hitrates[differenc_set]) < 0) | ((optimized_hitrates[differenc_set])> 1)]) == 0:\n",
+    "            # print(\"All values optimized.\")\n",
+    "            break\n",
+    "        \n",
+    "        max_outbound_index = get_index_of_furthest_hitrate_from_boundary(optimized_hitrates)\n",
+    "        fix_i.append(max_outbound_index)\n",
+    "        differenc_set = np.setdiff1d(np.arange(db_object_count), fix_i)\n",
+    "    \n",
+    "        old_hitrate = optimized_hitrates[max_outbound_index]\n",
+    "        optimized_hitrates[max_outbound_index] = (1 if optimized_hitrates[max_outbound_index] > 1 else 0)\n",
+    "        \n",
+    "        current_db_object_count -= 1\n",
+    "        current_cache_size -= optimized_hitrates[max_outbound_index]\n",
+    "    return optimized_hitrates"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "b6bf3329-3a63-4807-ab8b-8a54f824f47e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def objective_function(optimized_hitrates, c_f, c_delta, lambda_vals):\n",
+    "    return np.sum(lambda_vals*(1-optimized_hitrates)*c_f+0.5*np.power(optimized_hitrates,2)*c_delta)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "a289bb1a-0385-4835-bc92-88304c1834df",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def optimize_ttl(optimized_hitrates, lambda_vals):\n",
+    "    result = []\n",
+    "    for i in range(len(lambda_vals)):\n",
+    "        if optimized_hitrates[i] < 1:\n",
+    "            result.append(-1 / lambda_vals[i] * np.log(1 - optimized_hitrates[i]))\n",
+    "        else:\n",
+    "            result.append(np.inf)\n",
+    "    # ti_values = np.where(\n",
+    "    #     optimized_hitrates < 1,\n",
+    "    #     -1 / lambda_vals * np.log(1 - optimized_hitrates),\n",
+    "    #     np.inf\n",
+    "    # )\n",
+    "    \n",
+    "    return np.array(result)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "bd4536e9-273b-4f49-b06c-2f00605e0f7d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Define the task to be parallelized\n",
+    "def grid_search_task(db_object_count, cache_size, c_f, c_delta):\n",
+    "    if db_object_count < cache_size:\n",
+    "        return None  # Skip this combination if db_object_count < cache_size\n",
+    "    \n",
+    "    # Generate lambda_vals\n",
+    "    lambda_vals = np.array([np.random.zipf(ZIPF_CONSTANT) for _ in np.arange(1, db_object_count + 1, 1)])\n",
+    "    \n",
+    "    # Call the optimization function\n",
+    "    optimized_hitrates = optimize_hitrates(db_object_count, cache_size, c_f, c_delta, lambda_vals)\n",
+    "\n",
+    "    optimized_ttl = optimize_ttl(optimized_hitrates, lambda_vals)\n",
+    "    \n",
+    "    # Compute the objective function\n",
+    "    objective = objective_function(optimized_hitrates, c_f, c_delta, lambda_vals)\n",
+    "\n",
+    "    return (objective, optimized_ttl, db_object_count, cache_size, c_f, c_delta, optimized_hitrates)\n",
+    "\n",
+    "# Perform grid search with parallelization and tqdm progress bar\n",
+    "def grid_search(db_object_counts, cache_sizes, c_f_values, c_delta_values):\n",
+    "    results = []  # List to collect the results (objective, parameters)\n",
+    "    total_combinations = len(db_object_counts) * len(cache_sizes) * len(c_f_values) * len(c_delta_values)\n",
+    "    \n",
+    "    # Use Parallel from joblib to parallelize the grid search\n",
+    "    task_results = Parallel(n_jobs=-1, verbose=1)(\n",
+    "        delayed(grid_search_task)(db_object_count, cache_size, c_f, c_delta)\n",
+    "        for db_object_count, cache_size, c_f, c_delta in itertools.product(db_object_counts, cache_sizes, c_f_values, c_delta_values)\n",
+    "    )\n",
+    "\n",
+    "    # Collect valid results\n",
+    "    for result in task_results:\n",
+    "        if result is not None:\n",
+    "            results.append(result)\n",
+    "    \n",
+    "    # Convert the results into a pandas DataFrame\n",
+    "    df = pd.DataFrame(results, columns=[\"Objective\", \"Optimal TTL\", \"db_object_count\", \"cache_size\", \"c_f (Miss Cost)\", \"c_delta (Refresh Cost)\", \"optimized_hitrates\"])\n",
+    "    \n",
+    "    return df\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "a92c6772-6609-41a8-a3d1-4d640b69a864",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "[Parallel(n_jobs=-1)]: Using backend LokyBackend with 12 concurrent workers.\n",
+      "[Parallel(n_jobs=-1)]: Done  26 tasks      | elapsed:    0.4s\n",
+      "[Parallel(n_jobs=-1)]: Done 1420 tasks      | elapsed:    0.7s\n",
+      "[Parallel(n_jobs=-1)]: Done 64152 out of 64152 | elapsed:  1.4min finished\n"
+     ]
+    },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "CPU times: user 42 s, sys: 731 ms, total: 42.7 s\n",
+      "Wall time: 2min 5s\n"
+     ]
+    }
+   ],
+   "source": [
+    "%%time\n",
+    "# Define the grid search space\n",
+    "test_ratios = np.array([0.1, 0.2, 0.5, 0.7, 1, 1.5, 2, 5, 10])\n",
+    "db_object_count_values = np.round(np.array([10, 15, 30, 100, 200, 500]))\n",
+    "cache_size_values = np.unique(np.round(np.array([db_object_count_values * i for i in test_ratios]).flatten()))\n",
+    "c_f_values = np.array([0.1, 0.2, 0.5, 0.7, 1, 1.5, 2, 5, 10])\n",
+    "c_delta_values = np.unique(np.array([c_f_values * i for i in test_ratios]).flatten())\n",
+    "\n",
+    "objective_result_file = \"./objective_grid-search_multi-core.csv\"\n",
+    "\n",
+    "results_df = None\n",
+    "if not os.path.isfile(objective_result_file):\n",
+    "    # Call the grid search function\n",
+    "    results_df = grid_search(db_object_count_values, cache_size_values, c_f_values, c_delta_values)\n",
+    "    results_df.to_csv(objective_result_file,index=False)\n",
+    "else:\n",
+    "    results_df = pd.read_csv(objective_result_file)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "e79e6ed1-d6a5-4b04-a2b2-b3f0984e0fbe",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Best Result:\n",
+      "Objective                                                              0.05\n",
+      "Optimal TTL               [inf, inf, inf, inf, inf, inf, inf, inf, inf, ...\n",
+      "db_object_count                                                          10\n",
+      "cache_size                                                             10.0\n",
+      "c_f (Miss Cost)                                                         0.7\n",
+      "c_delta (Refresh Cost)                                                 0.01\n",
+      "optimized_hitrates        [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, ...\n",
+      "Name: 2376, dtype: object\n"
+     ]
+    }
+   ],
+   "source": [
+    "# After performing the grid search and obtaining the DataFrame 'results_df'\n",
+    "best_row = results_df.loc[results_df['Objective'].idxmin()]\n",
+    "\n",
+    "# Display the best row\n",
+    "print(\"Best Result:\")\n",
+    "print(best_row)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "95af94b4-05c0-488c-9561-50fc4e7cc3d4",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0.049999999999999996,\n",
+       " array([inf, inf, inf, inf, inf, inf, inf, inf, inf, inf]),\n",
+       " 10,\n",
+       " 10,\n",
+       " 1.5,\n",
+       " 0.01,\n",
+       " array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1.]))"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "grid_search_task(10, 10, 1.5, 0.01)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "20c943b4-b32b-4294-949b-0f3abe2fb97a",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "graphs",
+   "language": "python",
+   "name": "graphs"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--- a/01_nb_cncf_optimization/03-plot_objective_optimization.ipynb
+++ b/01_nb_cncf_optimization/03-plot_objective_optimization.ipynb
--- a/01_nb_cncf_optimization/NB_Tuan.pdf
+++ b/01_nb_cncf_optimization/NB_Tuan.pdf
--- a/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Main-checkpoint.ipynb
+++ b/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Main-checkpoint.ipynb
@@ -0,0 +1,258 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d0996120-bb17-4476-b912-ce155100b2cb",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Hit probabilities (numerical and theoretical):\n",
+      " [[0.4  0.  ]\n",
+      " [0.4  0.08]\n",
+      " [0.4  0.16]\n",
+      " [0.4  0.24]\n",
+      " [0.4  0.32]\n",
+      " [0.4  0.4 ]\n",
+      " [0.4  0.48]\n",
+      " [0.4  0.56]\n",
+      " [0.4  0.64]\n",
+      " [0.4  0.72]\n",
+      " [0.4  0.8 ]]\n",
+      "Objective function values (numerical, theoretical): [33.17, 22.944080000000007]\n",
+      "Constraint violations (numerical, theoretical): [0.0, 0.0]\n"
+     ]
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "from scipy.optimize import minimize\n",
+    "import numpy as np\n",
+    "\n",
+    "# Define Parameters\n",
+    "lambda_vals = np.array([0.03, 0.04, 0.05, 0.06, 0.07, 1, 1.1, 1.2, 1.3, 1.4, 1.5])  # Request rates ascendingly\n",
+    "N = len(lambda_vals)\n",
+    "B = 4.4  # Cache size\n",
+    "c_delta = 1  # Age linear cost\n",
+    "c_f = 7  # Fetching linear cost (cache miss cost)\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fcf0c13c-5b2c-457e-9aa6-8d349fcf13fa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "def theoretical_opt(lambda_vals, B, c_f, c_delta):\n",
+    "    \"\"\"\n",
+    "    Perform theoretical optimization to compute optimal hit probabilities.\n",
+    "    \n",
+    "    Parameters:\n",
+    "    - lambda_vals: Array of request rates.\n",
+    "    - B: Cache size (constraint on total hit probabilities).\n",
+    "    - c_f: Cost of fetching (cache miss cost).\n",
+    "    - c_delta: Cost of caching (hit cost).\n",
+    "\n",
+    "    Returns:\n",
+    "    - h_optt: Optimal hit probabilities.\n",
+    "    \"\"\"\n",
+    "    N = len(lambda_vals)\n",
+    "    h_optt = np.zeros(N)  # Initialize optimal hit probabilities\n",
+    "    differenc_set = np.arange(N)  # Set of variables to optimize\n",
+    "    fix_i = []  # Set of fixed variables (those already optimized)\n",
+    "    n = N\n",
+    "    b = B\n",
+    "    flag = True\n",
+    "\n",
+    "    while flag:\n",
+    "        if n == 0:  # If no variables left to optimize\n",
+    "            if b > 0:  # If there is leftover cache size, redistribute it\n",
+    "                differenc_set = np.where(h_optt == 0)[0]  # Find zero hit probability variables\n",
+    "                fix_i = np.setdiff1d(np.arange(N), differenc_set)\n",
+    "                n = len(differenc_set)\n",
+    "                continue\n",
+    "            else:  # No variables to optimize, finalize\n",
+    "                h_optt[differenc_set] = 0\n",
+    "                break\n",
+    "        \n",
+    "        # Calculate the optimal Lagrange multiplier (mu) and hit probabilities for the set of variables\n",
+    "        mu = max(0, (n * c_f - b * c_delta) / np.sum(1 / lambda_vals[differenc_set]))\n",
+    "        h_optt[differenc_set] = (c_f - mu / lambda_vals[differenc_set]) / c_delta\n",
+    "        \n",
+    "        # If mu < 0, adjust the cache size to set mu to zero in the next iteration\n",
+    "        if mu < 0:\n",
+    "            b = (n * c_f / c_delta)\n",
+    "            continue\n",
+    "        \n",
+    "        # Identify violations of the hit probability constraints (h > 1 or h < 0)\n",
+    "        larger_i = np.where(h_optt > 1)[0]\n",
+    "        smaller_i = np.where(h_optt < 0)[0]\n",
+    "\n",
+    "        # If no violations, the optimal solution is reached\n",
+    "        if len(smaller_i) == 0 and len(larger_i) == 0:\n",
+    "            break\n",
+    "        \n",
+    "        # Find the furthest object from the boundary (either 0 or 1)\n",
+    "        min_viol = 0\n",
+    "        min_viol_i = -1\n",
+    "        if len(smaller_i) > 0:\n",
+    "            min_viol, min_viol_i = np.min(h_optt[smaller_i]), np.argmin(h_optt[smaller_i])\n",
+    "\n",
+    "        max_viol = 0\n",
+    "        max_viol_i = -1\n",
+    "        if len(larger_i) > 0:\n",
+    "            max_viol, max_viol_i = np.max(h_optt[larger_i] - 1), np.argmax(h_optt[larger_i] - 1)\n",
+    "        \n",
+    "        # Choose the variable with the largest violation to adjust\n",
+    "        if max_viol > abs(min_viol):\n",
+    "            viol_i = max_viol_i\n",
+    "            min_viol_flag = 0\n",
+    "        else:\n",
+    "            viol_i = min_viol_i\n",
+    "            min_viol_flag = 1\n",
+    "        \n",
+    "        # Set the furthest object to the nearest boundary (0 or 1)\n",
+    "        if min_viol_flag:\n",
+    "            h_optt[viol_i] = 0\n",
+    "        else:\n",
+    "            h_optt[viol_i] = min(1, b)\n",
+    "        \n",
+    "        # Update cache size and fix the selected variable\n",
+    "        b -= h_optt[viol_i]\n",
+    "        fix_i.append(viol_i)\n",
+    "        differenc_set = np.setdiff1d(np.arange(N), fix_i)\n",
+    "        n = N - len(fix_i)\n",
+    "    \n",
+    "    return h_optt\n",
+    "\n",
+    "\n",
+    "# Example usage\n",
+    "lambda_vals = np.array(\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5d223324-d193-416a-b2c3-1e143e981a37",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "def numerical_opt(lambda_vals, B, c_f, c_delta):\n",
+    "    \"\"\"\n",
+    "    Perform numerical optimization to compute optimal hit probabilities.\n",
+    "\n",
+    "    Parameters:\n",
+    "    - lambda_vals: Array of request rates.\n",
+    "    - B: Cache size (constraint on total hit probabilities).\n",
+    "    - c_f: Cost of fetching (cache miss cost).\n",
+    "    - c_delta: Cost of caching (hit cost).\n",
+    "\n",
+    "    Returns:\n",
+    "    - x_opt: Optimal hit probabilities.\n",
+    "    \"\"\"\n",
+    "    N = len(lambda_vals)  # Number of items\n",
+    "\n",
+    "    # Initial guess: Even distribution of cache capacity\n",
+    "    x_init = np.full(N, B / N)\n",
+    "\n",
+    "    # Objective function\n",
+    "    def objective(x):\n",
+    "        return np.sum(lambda_vals * ((1 - x) * c_f + x**2 * c_delta / 2))\n",
+    "\n",
+    "    # Constraint: Sum of hit probabilities <= cache size (B)\n",
+    "    def constraint_total_hit(x):\n",
+    "        return B - np.sum(x)  # Non-negative means constraint satisfied\n",
+    "\n",
+    "    # Bounds for hit probabilities: 0 <= h_i <= 1\n",
+    "    bounds = [(0, 1) for _ in range(N)]\n",
+    "\n",
+    "    # Optimization\n",
+    "    constraints = [{'type': 'ineq', 'fun': constraint_total_hit}]  # Inequality constraint\n",
+    "    result = minimize(\n",
+    "        objective, \n",
+    "        x_init, \n",
+    "        method='SLSQP',  # Sequential Least Squares Quadratic Programming\n",
+    "        bounds=bounds, \n",
+    "        constraints=constraints, \n",
+    "        options={'disp': True}  # Set to True for optimization output\n",
+    "    )\n",
+    "\n",
+    "    # Optimal solution\n",
+    "    if result.success:\n",
+    "        return result.x  # Optimal hit probabilities\n",
+    "    else:\n",
+    "        raise ValueError(\"Optimization failed: \" + result.message)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "073be740-dc97-454b-87e7-2f8f93c8f137",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "# Example usage\n",
+    "lambda_vals = np.array([0.03, 0.04, 0.05, 0.06, 0.07, 1, 1.1, 1.2, 1.3, 1.4, 1.5])\n",
+    "B = 4.4\n",
+    "c_f = 7\n",
+    "c_delta = 1\n",
+    "\n",
+    "optimal_hit_probs = numerical_opt(lambda_vals, B, c_f, c_delta)\n",
+    "print(\"Optimal Hit Probabilities:\", optimal_hit_probs)\n",
+    "\n",
+    "\n",
+    "# Optimization\n",
+    "h_numerical = numerical_opt(lambda_vals, B, c_f, c_delta)\n",
+    "h_theoretical = theoretical_opt(lambda_vals, B, c_f, c_delta)\n",
+    "\n",
+    "# Comparison\n",
+    "hit_opt = np.vstack((h_numerical, h_theoretical)).T  # Combine for comparison\n",
+    "\n",
+    "# Objective Function Calculation\n",
+    "obj_1 = np.sum(lambda_vals * ((1 - h_numerical) * c_f + h_numerical**2 * c_delta / 2))\n",
+    "obj_2 = np.sum(lambda_vals * ((1 - h_theoretical) * c_f + h_theoretical**2 * c_delta / 2))\n",
+    "obj = [obj_1, obj_2]\n",
+    "\n",
+    "# Constraints\n",
+    "const_1 = np.sum(h_numerical) - B\n",
+    "const_2 = np.sum(h_theoretical) - B\n",
+    "constraint = [const_1, const_2]\n",
+    "\n",
+    "# Outputs\n",
+    "print(\"Hit probabilities (numerical and theoretical):\\n\", hit_opt)\n",
+    "print(\"Objective function values (numerical, theoretical):\", obj)\n",
+    "print(\"Constraint violations (numerical, theoretical):\", constraint)\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "graphs",
+   "language": "python",
+   "name": "graphs"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--- a/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Main-checkpoint.m
+++ b/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Main-checkpoint.m
@@ -0,0 +1,27 @@
+clc
+close all
+clear all
+
+
+%% Define parameters
+lambda = [0.03, 0.04,0.05,0.06,0.07,1,1.1,1.2,1.3,1.4,1.5]';  % Request rates ascendingly
+N=length(lambda);
+B = 4.4;  % Cache size
+c_delta=1; % age linear cost
+c_f=7; % fetching linear cost (caching miss cost)
+
+
+%% Optimization 
+[h_numerical ]=Numerical_opt(lambda,B,c_f,c_delta)
+[h_theo] = Theoritical_opt(lambda,B,c_f,c_delta)
+
+
+%% Comparison
+hit_opt=[h_numerical h_theo]
+obj_1=sum(lambda .* ((1-h_numerical)*c_f+h_numerical.^2*c_delta/2));
+obj_2=sum(lambda .* ((1-h_theo)*c_f+h_theo.^2*c_delta/2));
+obj=[obj_1 obj_2]
+const_1=sum(h_numerical)-B;
+const_2=sum(h_theo)-B;
+constraint=[const_1 const_2]
+
--- a/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Numerical_opt-checkpoint.m
+++ b/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Numerical_opt-checkpoint.m
@@ -0,0 +1,17 @@
+function [x_opt] = Numerical_opt(lambda,B,c_f,c_delta)
+% Numerical optimization
+% x_opt are the optimal hit probabilities
+ 
+N = length(lambda);  % Number of variables
+h_init = ones(N, 1) * B / N;  % Initial guess for h_i, evenly distributed
+x_init=h_init;
+objective1 = @(x) sum(lambda .*( (1-x)*c_f+x.^2*c_delta/2));  % Objective function
+constraint1 = @(x) sum(x) - B; % cache size constraint
+% constraint on hit prob. 0<=h_i<=1
+nonlcon1 = @(x) deal(constraint1(x), []);  % 
+lb = zeros(N, 1);  % Lower bounds
+ub = ones(N, 1);     % Upper bounds
+options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');
+[x_opt, fval_h] = fmincon(objective1, x_init, [], [], [], [], lb, ub, nonlcon1, options);
+end
+
--- a/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Theoritical_opt-Copy1-checkpoint.m
+++ b/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Theoritical_opt-Copy1-checkpoint.m
@@ -0,0 +1,171 @@
+function [h_optt] = Theoritical_opt(lambda,B,c_f,c_delta)
+%% Theoritical optimization 
+
+
+%% Iterative identification of active constraints
+N=length(lambda)
+flag=1;
+h_optt=zeros(N,1); %optimal hit prob
+differenc_set=1:N; % the set of variables to optimize 
+fix_i=[]; % set of variables that reached optimality and are excluded from the optimization
+n=N; 
+b=B;
+
+%%
+while flag
+    if(n==0) 
+        if(b>0) % if there is left over cache size and mu is not zero (the loop would break), redistribute it among the zero hit probability
+            differenc_set=find(h_optt==0)';
+            fix_i=setdiff(1:N,differenc_set)';
+            n=length(differenc_set);
+            continue;     
+        else           
+             h_optt(differenc_set)=0;
+             break;
+        end
+    end
+    % Optimal Lagrangian mult. and hit prob. calculated theoritically for the set of variables in  differenc_set
+    mu=max(0,(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set)));  %optimal lagrangian mult.
+    h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta %optimal hit prob
+    
+    % mu has to be >=0
+    if(mu<0)
+        b=(n*c_f/c_delta); % this sets mu to zero in the next iteration
+        continue;
+    end
+    
+    % check the violation of the hit_prob const  
+    larger_i=find(h_optt>1); % h>1
+    smaller_i=find(h_optt<0); % h<0
+
+%     smaller=h(find(differenc_set<0))-0;
+    % no violation means optimal solution is reached for all objects
+    if(length(smaller_i)==0 && length(larger_i)==0)
+%         flag=0;
+  
+        break;
+    end
+    
+    % find the furthest object from the 0 boundary
+    min_viol=0;
+    min_viol_i=-1;
+    if(length(smaller_i)>0)
+        [min_viol, min_viol_i]=min(h_optt);
+    end
+     % find the furthest object from the 1 boundary
+    max_viol=0; 
+    max_viol_i=-1;
+    if(length(larger_i)>0)
+       larger=h_optt-1;
+       [max_viol ,max_viol_i]=max(h_optt-1);
+
+    end
+    
+     % compare both furthest objects from both boundaries
+    viol_i=min_viol_i;
+    min_viol_flag=1; % True if the furthest one is from the left
+    if(max_viol>abs(min_viol))
+        viol_i= max_viol_i;
+        min_viol_flag=0;
+
+    end
+     % set the furthest object to the nearest boundary
+     if(min_viol_flag)
+        h_optt(viol_i)=0;
+    else
+        h_optt(viol_i)=min(1,b);
+     end
+    
+    %calculate the new parameters after removing the furthest object from
+    %the decision variables
+    B_new=b-(h_optt(viol_i));
+    b=B_new;
+    fix_i=[fix_i' viol_i']';
+    differenc_set=setdiff(1:N,fix_i)   ; 
+    n=N-length(fix_i);
+    
+    
+    %     % Identify the most violating object from the right side  h>1
+%     if(length(larger_i)>0)
+%        larger=h_optt-1;
+%        [max_viol ,max_viol_i]=max(h_optt-1); % maximum violating object
+%        h_optt(max_viol_i)=min(1,b); %project to the feasible range
+%        b=max(b-1,0); % update the cache size 
+%        fix_i=[fix_i' max_viol_i']'; %exclude i from the set of decision variables
+%        differenc_set=setdiff(1:N,fix_i); % obtain the set of decision variables 
+%        n=N-length(fix_i); % update the number of decision variables 
+%        continue;
+%     end
+%     
+%     if(length(smaller_i)>0)
+%         [min_viol, min_viol_i]=min(h_optt);
+%         h_optt(min_viol_i)=0; 
+%         fix_i=[fix_i' min_viol_i']';
+%         differenc_set=setdiff(1:N,fix_i)   ; 
+%         n=N-length(fix_i);
+%     end
+%     
+% end
+
+end
+
+%% Identfying the active constraints collectively
+% flag=1;
+% h_optt=zeros(N,1);
+% differenc_set=1:N;
+% fix_i=[];
+% n=N;
+% b=B;
+% while flag
+%     mu=(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set));
+%     h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta
+% 
+%     larger_i=find(h_optt>1);    
+% %     larger=h_optt(larger_i)-1;
+%     smaller_i=find(h_optt<0);
+% %     smaller=h(find(differenc_set<0))-0;
+%     mult=solve_multipliers(lambda,B,c_f,c_delta,larger_i)
+% 
+%     if(length(larger_i)+length(smaller_i)==0)
+%         flag=0;
+%         break;
+%     end
+%     if(length(smaller_i)>0)
+%         h_optt(smaller_i)=0;
+%         fix_i=[fix_i' smaller_i' ]';
+%         differenc_set=setdiff(1:N,fix_i)    
+%         n=N-length(fix_i);
+%         continue
+%     end
+% %     h_optt(smaller_i)=0;
+%     if(length(larger_i)>b)
+%         [~,index]=maxk(h_optt,b)
+%         h_optt(index)=1;
+%         B_new=b-sum(h_optt(index));
+%         fix_i=[fix_i' smaller_i' index']';
+%     else
+%          h_optt(larger_i)=1;
+%          B_new=b-sum(h_optt(larger_i));
+%          fix_i=[fix_i' smaller_i' larger_i']';
+%     end
+% %     mult=solve_multipliers(lambda,B,c_f,c_delta,larger_i)
+% 
+%     b=B_new;
+%    
+%     differenc_set=setdiff(1:N,fix_i)    
+%     n=N-length(fix_i);
+% end
+% % h_optt=zeros(N,1);
+% % h_optt(end-B+1:end)=1;
+% optimal=[h_opt h_optt]
+% obj_1=sum(lambda .* ((1-h_opt)*c_f+h_opt.^2*c_delta/2));
+% obj_2=sum(lambda .* ((1-h_optt)*c_f+h_optt.^2*c_delta/2));
+% objective=[obj_1 obj_2]
+% const_1=sum(h_opt)-B;
+% const_2=sum(h_optt)-B;
+% constraint=[const_1 const_2]
+% 
+
+
+end
+
--- a/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Theoritical_opt-checkpoint.m
+++ b/01_nb_cncf_optimization/matlab/.ipynb_checkpoints/Theoritical_opt-checkpoint.m
@@ -0,0 +1,171 @@
+function [h_optt] = Theoritical_opt(lambda,B,c_f,c_delta)
+%% Theoritical optimization 
+
+
+%% Iterative identification of active constraints
+N=length(lambda)
+flag=1;
+h_optt=zeros(N,1); %optimal hit prob
+differenc_set=1:N; % the set of variables to optimize 
+fix_i=[]; % set of variables that reached optimality and are excluded from the optimization
+n=N; 
+b=B;
+
+%%
+while flag
+    if(n==0) 
+        if(b>0) % if there is left over cache size and mu is not zero (the loop would break), redistribute it among the zero hit probability
+            differenc_set=find(h_optt==0)';
+            fix_i=setdiff(1:N,differenc_set)';
+            n=length(differenc_set);
+            continue;     
+        else           
+             h_optt(differenc_set)=0;
+             break;
+        end
+    end
+    % Optimal Lagrangian mult. and hit prob. calculated theoritically for the set of variables in  differenc_set
+    mu=max(0,(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set)));  %optimal lagrangian mult.
+    h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta %optimal hit prob
+    
+    % mu has to be >=0
+    if(mu<0)
+        b=(n*c_f/c_delta); % this sets mu to zero in the next iteration
+        continue;
+    end
+    
+    % check the violation of the hit_prob const  
+    larger_i=find(h_optt>1); % h>1
+    smaller_i=find(h_optt<0); % h<0
+
+%     smaller=h(find(differenc_set<0))-0;
+    % no violation means optimal solution is reached for all objects
+    if(length(smaller_i)==0 && length(larger_i)==0)
+%         flag=0;
+  
+        break;
+    end
+    
+    % find the furthest object from the 0 boundary
+    min_viol=0;
+    min_viol_i=-1;
+    if(length(smaller_i)>0)
+        [min_viol, min_viol_i]=min(h_optt);
+    end
+     % find the furthest object from the 1 boundary
+    max_viol=0; 
+    max_viol_i=-1;
+    if(length(larger_i)>0)
+       larger=h_optt-1;
+       [max_viol ,max_viol_i]=max(h_optt-1);
+
+    end
+    
+     % compare both furthest objects from both boundaries
+    viol_i=min_viol_i;
+    min_viol_flag=1; % True if the furthest one is from the left
+    if(max_viol>abs(min_viol))
+        viol_i= max_viol_i;
+        min_viol_flag=0;
+
+    end
+     % set the furthest object to the nearest boundary
+     if(min_viol_flag)
+        h_optt(viol_i)=0;
+    else
+        h_optt(viol_i)=min(1,b);
+     end
+    
+    %calculate the new parameters after removing the furthest object from
+    %the decision variables
+    B_new=b-(h_optt(viol_i));
+    b=B_new;
+    fix_i=[fix_i' viol_i']';
+    differenc_set=setdiff(1:N,fix_i)   ; 
+    n=N-length(fix_i);
+    
+    
+    %     % Identify the most violating object from the right side  h>1
+%     if(length(larger_i)>0)
+%        larger=h_optt-1;
+%        [max_viol ,max_viol_i]=max(h_optt-1); % maximum violating object
+%        h_optt(max_viol_i)=min(1,b); %project to the feasible range
+%        b=max(b-1,0); % update the cache size 
+%        fix_i=[fix_i' max_viol_i']'; %exclude i from the set of decision variables
+%        differenc_set=setdiff(1:N,fix_i); % obtain the set of decision variables 
+%        n=N-length(fix_i); % update the number of decision variables 
+%        continue;
+%     end
+%     
+%     if(length(smaller_i)>0)
+%         [min_viol, min_viol_i]=min(h_optt);
+%         h_optt(min_viol_i)=0; 
+%         fix_i=[fix_i' min_viol_i']';
+%         differenc_set=setdiff(1:N,fix_i)   ; 
+%         n=N-length(fix_i);
+%     end
+%     
+% end
+
+end
+
+%% Identfying the active constraints collectively
+% flag=1;
+% h_optt=zeros(N,1);
+% differenc_set=1:N;
+% fix_i=[];
+% n=N;
+% b=B;
+% while flag
+%     mu=(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set));
+%     h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta
+% 
+%     larger_i=find(h_optt>1);    
+% %     larger=h_optt(larger_i)-1;
+%     smaller_i=find(h_optt<0);
+% %     smaller=h(find(differenc_set<0))-0;
+%     mult=solve_multipliers(lambda,B,c_f,c_delta,larger_i)
+% 
+%     if(length(larger_i)+length(smaller_i)==0)
+%         flag=0;
+%         break;
+%     end
+%     if(length(smaller_i)>0)
+%         h_optt(smaller_i)=0;
+%         fix_i=[fix_i' smaller_i' ]';
+%         differenc_set=setdiff(1:N,fix_i)    
+%         n=N-length(fix_i);
+%         continue
+%     end
+% %     h_optt(smaller_i)=0;
+%     if(length(larger_i)>b)
+%         [~,index]=maxk(h_optt,b)
+%         h_optt(index)=1;
+%         B_new=b-sum(h_optt(index));
+%         fix_i=[fix_i' smaller_i' index']';
+%     else
+%          h_optt(larger_i)=1;
+%          B_new=b-sum(h_optt(larger_i));
+%          fix_i=[fix_i' smaller_i' larger_i']';
+%     end
+% %     mult=solve_multipliers(lambda,B,c_f,c_delta,larger_i)
+% 
+%     b=B_new;
+%    
+%     differenc_set=setdiff(1:N,fix_i)    
+%     n=N-length(fix_i);
+% end
+% % h_optt=zeros(N,1);
+% % h_optt(end-B+1:end)=1;
+% optimal=[h_opt h_optt]
+% obj_1=sum(lambda .* ((1-h_opt)*c_f+h_opt.^2*c_delta/2));
+% obj_2=sum(lambda .* ((1-h_optt)*c_f+h_optt.^2*c_delta/2));
+% objective=[obj_1 obj_2]
+% const_1=sum(h_opt)-B;
+% const_2=sum(h_optt)-B;
+% constraint=[const_1 const_2]
+% 
+
+
+end
+
--- a/01_nb_cncf_optimization/matlab/Main.ipynb
+++ b/01_nb_cncf_optimization/matlab/Main.ipynb
@@ -0,0 +1,248 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d0996120-bb17-4476-b912-ce155100b2cb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "from scipy.optimize import minimize\n",
+    "import numpy as np\n",
+    "\n",
+    "# Define Parameters\n",
+    "lambda_vals = np.array([0.03, 0.04, 0.05, 0.06, 0.07, 1, 1.1, 1.2, 1.3, 1.4, 1.5])  # Request rates ascendingly\n",
+    "N = len(lambda_vals)\n",
+    "B = 4.4  # Cache size\n",
+    "c_delta = 1  # Age linear cost\n",
+    "c_f = 7  # Fetching linear cost (caching miss cost)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "fcf0c13c-5b2c-457e-9aa6-8d349fcf13fa",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "def theoretical_opt(lambda_vals, B, c_f, c_delta):\n",
+    "    \"\"\"\n",
+    "    Perform theoretical optimization to compute optimal hit probabilities.\n",
+    "    \n",
+    "    Parameters:\n",
+    "    - lambda_vals: Array of request rates.\n",
+    "    - B: Cache size (constraint on total hit probabilities).\n",
+    "    - c_f: Cost of fetching (cache miss cost).\n",
+    "    - c_delta: Cost of caching (hit cost).\n",
+    "\n",
+    "    Returns:\n",
+    "    - h_optt: Optimal hit probabilities.\n",
+    "    \"\"\"\n",
+    "    N = len(lambda_vals)\n",
+    "    h_optt = np.zeros(N)  # Initialize optimal hit probabilities\n",
+    "    differenc_set = np.arange(N)  # Set of variables to optimize\n",
+    "    fix_i = []  # Set of fixed variables (those already optimized)\n",
+    "    n = N\n",
+    "    b = B\n",
+    "    flag = True\n",
+    "\n",
+    "    while flag:\n",
+    "        if n == 0:  # If no variables left to optimize\n",
+    "            if b > 0:  # If there is leftover cache size, redistribute it\n",
+    "                differenc_set = np.where(h_optt == 0)[0]  # Find zero hit probability variables\n",
+    "                fix_i = np.setdiff1d(np.arange(N), differenc_set)\n",
+    "                n = len(differenc_set)\n",
+    "                continue\n",
+    "            else:  # No variables to optimize, finalize\n",
+    "                h_optt[differenc_set] = 0\n",
+    "                break\n",
+    "        \n",
+    "        # Calculate the optimal Lagrange multiplier (mu) and hit probabilities for the set of variables\n",
+    "        mu = max(0, (n * c_f - b * c_delta) / np.sum(1 / lambda_vals[differenc_set]))\n",
+    "        h_optt[differenc_set] = (c_f - mu / lambda_vals[differenc_set]) / c_delta\n",
+    "        \n",
+    "        # If mu < 0, adjust the cache size to set mu to zero in the next iteration\n",
+    "        if mu < 0:\n",
+    "            b = (n * c_f / c_delta)\n",
+    "            continue\n",
+    "        \n",
+    "        # Identify violations of the hit probability constraints (h > 1 or h < 0)\n",
+    "        larger_i = np.where(h_optt > 1)[0]\n",
+    "        smaller_i = np.where(h_optt < 0)[0]\n",
+    "\n",
+    "        # If no violations, the optimal solution is reached\n",
+    "        if len(smaller_i) == 0 and len(larger_i) == 0:\n",
+    "            break\n",
+    "        \n",
+    "        # Find the furthest object from the boundary (either 0 or 1)\n",
+    "        min_viol = 0\n",
+    "        min_viol_i = -1\n",
+    "        if len(smaller_i) > 0:\n",
+    "            min_viol, min_viol_i = np.min(h_optt[smaller_i]), np.argmin(h_optt[smaller_i])\n",
+    "\n",
+    "        max_viol = 0\n",
+    "        max_viol_i = -1\n",
+    "        if len(larger_i) > 0:\n",
+    "            max_viol, max_viol_i = np.max(h_optt[larger_i] - 1), np.argmax(h_optt[larger_i] - 1)\n",
+    "        \n",
+    "        # Choose the variable with the largest violation to adjust\n",
+    "        if max_viol > abs(min_viol):\n",
+    "            viol_i = max_viol_i\n",
+    "            min_viol_flag = 0\n",
+    "        else:\n",
+    "            viol_i = min_viol_i\n",
+    "            min_viol_flag = 1\n",
+    "        \n",
+    "        # Set the furthest object to the nearest boundary (0 or 1)\n",
+    "        if min_viol_flag:\n",
+    "            h_optt[viol_i] = 0\n",
+    "        else:\n",
+    "            h_optt[viol_i] = min(1, b)\n",
+    "        \n",
+    "        # Update cache size and fix the selected variable\n",
+    "        b -= h_optt[viol_i]\n",
+    "        fix_i.append(viol_i)\n",
+    "        differenc_set = np.setdiff1d(np.arange(N), fix_i)\n",
+    "        n = N - len(fix_i)\n",
+    "    \n",
+    "    return h_optt"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "5d223324-d193-416a-b2c3-1e143e981a37",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "\n",
+    "def numerical_opt(lambda_vals, B, c_f, c_delta):\n",
+    "    \"\"\"\n",
+    "    Perform numerical optimization to compute optimal hit probabilities.\n",
+    "\n",
+    "    Parameters:\n",
+    "    - lambda_vals: Array of request rates.\n",
+    "    - B: Cache size (constraint on total hit probabilities).\n",
+    "    - c_f: Cost of fetching (cache miss cost).\n",
+    "    - c_delta: Cost of caching (hit cost).\n",
+    "\n",
+    "    Returns:\n",
+    "    - x_opt: Optimal hit probabilities.\n",
+    "    \"\"\"\n",
+    "    N = len(lambda_vals)  # Number of items\n",
+    "\n",
+    "    # Initial guess: Even distribution of cache capacity\n",
+    "    x_init = np.full(N, B / N)\n",
+    "\n",
+    "    # Objective function\n",
+    "    def objective(x):\n",
+    "        return np.sum(lambda_vals * ((1 - x) * c_f + x**2 * c_delta / 2))\n",
+    "\n",
+    "    # Constraint: Sum of hit probabilities <= cache size (B)\n",
+    "    def constraint_total_hit(x):\n",
+    "        return B - np.sum(x)  # Non-negative means constraint satisfied\n",
+    "\n",
+    "    # Bounds for hit probabilities: 0 <= h_i <= 1\n",
+    "    bounds = [(0, 1) for _ in range(N)]\n",
+    "\n",
+    "    # Optimization\n",
+    "    constraints = [{'type': 'ineq', 'fun': constraint_total_hit}]  # Inequality constraint\n",
+    "    result = minimize(\n",
+    "        objective, \n",
+    "        x_init, \n",
+    "        method='SLSQP',  # Sequential Least Squares Quadratic Programming\n",
+    "        bounds=bounds, \n",
+    "        constraints=constraints, \n",
+    "        options={'disp': True}  # Set to True for optimization output\n",
+    "    )\n",
+    "\n",
+    "    # Optimal solution\n",
+    "    if result.success:\n",
+    "        return result.x  # Optimal hit probabilities\n",
+    "    else:\n",
+    "        raise ValueError(\"Optimization failed: \" + result.message)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "073be740-dc97-454b-87e7-2f8f93c8f137",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Optimization terminated successfully    (Exit mode 0)\n",
+      "            Current function value: 16.15721739129548\n",
+      "            Iterations: 4\n",
+      "            Function evaluations: 48\n",
+      "            Gradient evaluations: 4\n",
+      "Hit probabilities (numerical and theoretical):\n",
+      " [[0.00000000e+00 0.00000000e+00]\n",
+      " [0.00000000e+00 0.00000000e+00]\n",
+      " [0.00000000e+00 0.00000000e+00]\n",
+      " [0.00000000e+00 0.00000000e+00]\n",
+      " [0.00000000e+00 0.00000000e+00]\n",
+      " [4.58923826e-13 1.00000000e+00]\n",
+      " [4.26087031e-01 0.00000000e+00]\n",
+      " [9.73912969e-01 4.00000000e-01]\n",
+      " [1.00000000e+00 0.00000000e+00]\n",
+      " [1.00000000e+00 0.00000000e+00]\n",
+      " [1.00000000e+00 0.00000000e+00]]\n",
+      "Objective function values (numerical, theoretical): [16.15721739129548, 44.486]\n",
+      "Constraint violations (numerical, theoretical): [1.241673430740775e-12, -3.0]\n"
+     ]
+    }
+   ],
+   "source": [
+    "\n",
+    "# Optimization\n",
+    "h_numerical = numerical_opt(lambda_vals, B, c_f, c_delta)\n",
+    "h_theoretical = theoretical_opt(lambda_vals, B, c_f, c_delta)\n",
+    "\n",
+    "# Comparison of Hit Probabilities\n",
+    "hit_opt = np.vstack((h_numerical, h_theoretical)).T  # Combine numerical and theoretical hit probabilities\n",
+    "\n",
+    "# Objective Function Calculation\n",
+    "obj_1 = np.sum(lambda_vals * ((1 - h_numerical) * c_f + h_numerical**2 * c_delta / 2))\n",
+    "obj_2 = np.sum(lambda_vals * ((1 - h_theoretical) * c_f + h_theoretical**2 * c_delta / 2))\n",
+    "obj = [obj_1, obj_2]  # Store objective function values for both methods\n",
+    "\n",
+    "# Constraints\n",
+    "const_1 = np.sum(h_numerical) - B\n",
+    "const_2 = np.sum(h_theoretical) - B\n",
+    "constraint = [const_1, const_2]  # Check if the cache size constraint is satisfied\n",
+    "\n",
+    "# Outputs\n",
+    "print(\"Hit probabilities (numerical and theoretical):\\n\", hit_opt)\n",
+    "print(\"Objective function values (numerical, theoretical):\", obj)\n",
+    "print(\"Constraint violations (numerical, theoretical):\", constraint)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "graphs",
+   "language": "python",
+   "name": "graphs"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
--- a/01_nb_cncf_optimization/matlab/Main.m
+++ b/01_nb_cncf_optimization/matlab/Main.m
@@ -0,0 +1,27 @@
+clc
+close all
+clear all
+
+
+%% Define parameters
+lambda = [0.03, 0.04,0.05,0.06,0.07,1,1.1,1.2,1.3,1.4,1.5]';  % Request rates ascendingly
+N=length(lambda);
+B = 4.4;  % Cache size
+c_delta=1; % age linear cost
+c_f=7; % fetching linear cost (caching miss cost)
+
+
+%% Optimization 
+[h_numerical ]=Numerical_opt(lambda,B,c_f,c_delta)
+[h_theo] = Theoritical_opt(lambda,B,c_f,c_delta)
+
+
+%% Comparison
+hit_opt=[h_numerical h_theo]
+obj_1=sum(lambda .* ((1-h_numerical)*c_f+h_numerical.^2*c_delta/2));
+obj_2=sum(lambda .* ((1-h_theo)*c_f+h_theo.^2*c_delta/2));
+obj=[obj_1 obj_2]
+const_1=sum(h_numerical)-B;
+const_2=sum(h_theo)-B;
+constraint=[const_1 const_2]
+
--- a/01_nb_cncf_optimization/matlab/Numerical_opt.m
+++ b/01_nb_cncf_optimization/matlab/Numerical_opt.m
@@ -0,0 +1,17 @@
+function [x_opt] = Numerical_opt(lambda,B,c_f,c_delta)
+% Numerical optimization
+% x_opt are the optimal hit probabilities
+ 
+N = length(lambda);  % Number of variables
+h_init = ones(N, 1) * B / N;  % Initial guess for h_i, evenly distributed
+x_init=h_init;
+objective1 = @(x) sum(lambda .*( (1-x)*c_f+x.^2*c_delta/2));  % Objective function
+constraint1 = @(x) sum(x) - B; % cache size constraint
+% constraint on hit prob. 0<=h_i<=1
+nonlcon1 = @(x) deal(constraint1(x), []);  % 
+lb = zeros(N, 1);  % Lower bounds
+ub = ones(N, 1);     % Upper bounds
+options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');
+[x_opt, fval_h] = fmincon(objective1, x_init, [], [], [], [], lb, ub, nonlcon1, options);
+end
+
--- a/01_nb_cncf_optimization/matlab/Theoritical_opt-Copy1.m
+++ b/01_nb_cncf_optimization/matlab/Theoritical_opt-Copy1.m
@@ -0,0 +1,87 @@
+function [h_optt] = Theoritical_opt(lambda,B,c_f,c_delta)
+%% Theoritical optimization
+
+
+%% Iterative identification of active constraints
+N=length(lambda)
+flag=1;
+h_optt=zeros(N,1); %optimal hit prob
+differenc_set=1:N; % the set of variables to optimize
+fix_i=[]; % set of variables that reached optimality and are excluded from the optimization
+n=N;
+b=B;
+
+%%
+while flag
+    if(n==0)
+        if(b>0) % if there is left over cache size and mu is not zero (the loop would break), redistribute it among the zero hit probability
+            differenc_set=find(h_optt==0)';
+            fix_i=setdiff(1:N,differenc_set)';
+            n=length(differenc_set);
+            continue;
+        else
+             h_optt(differenc_set)=0;
+             break;
+        end
+    end
+    % Optimal Lagrangian mult. and hit prob. calculated theoritically for the set of variables in  differenc_set
+    mu=max(0,(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set)));  %optimal lagrangian mult.
+    h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta %optimal hit prob
+
+    % mu has to be >=0
+    if(mu<0)
+        b=(n*c_f/c_delta); % this sets mu to zero in the next iteration
+        continue;
+    end
+
+    % check the violation of the hit_prob const
+    larger_i=find(h_optt>1); % h>1
+    smaller_i=find(h_optt<0); % h<0
+
+    if(length(smaller_i)==0 && length(larger_i)==0)
+
+
+        break;
+    end
+
+    % find the furthest object from the 0 boundary
+    min_viol=0;
+    min_viol_i=-1;
+    if(length(smaller_i)>0)
+        [min_viol, min_viol_i]=min(h_optt);
+    end
+     % find the furthest object from the 1 boundary
+    max_viol=0;
+    max_viol_i=-1;
+    if(length(larger_i)>0)
+       larger=h_optt-1;
+       [max_viol ,max_viol_i]=max(h_optt-1);
+
+    end
+
+     % compare both furthest objects from both boundaries
+    viol_i=min_viol_i;
+    min_viol_flag=1; % True if the furthest one is from the left
+    if(max_viol>abs(min_viol))
+        viol_i= max_viol_i;
+        min_viol_flag=0;
+
+    end
+     % set the furthest object to the nearest boundary
+     if(min_viol_flag)
+        h_optt(viol_i)=0;
+    else
+        h_optt(viol_i)=min(1,b);
+     end
+
+    %calculate the new parameters after removing the furthest object from
+    %the decision variables
+    B_new=b-(h_optt(viol_i));
+    b=B_new;
+    fix_i=[fix_i' viol_i']';
+    differenc_set=setdiff(1:N,fix_i)   ;
+    n=N-length(fix_i);
+
+end
+end
+
--- a/01_nb_cncf_optimization/matlab/Theoritical_opt.m
+++ b/01_nb_cncf_optimization/matlab/Theoritical_opt.m
@@ -0,0 +1,171 @@
+function [h_optt] = Theoritical_opt(lambda,B,c_f,c_delta)
+%% Theoritical optimization 
+
+
+%% Iterative identification of active constraints
+N=length(lambda)
+flag=1;
+h_optt=zeros(N,1); %optimal hit prob
+differenc_set=1:N; % the set of variables to optimize 
+fix_i=[]; % set of variables that reached optimality and are excluded from the optimization
+n=N; 
+b=B;
+
+%%
+while flag
+    if(n==0) 
+        if(b>0) % if there is left over cache size and mu is not zero (the loop would break), redistribute it among the zero hit probability
+            differenc_set=find(h_optt==0)';
+            fix_i=setdiff(1:N,differenc_set)';
+            n=length(differenc_set);
+            continue;     
+        else           
+             h_optt(differenc_set)=0;
+             break;
+        end
+    end
+    % Optimal Lagrangian mult. and hit prob. calculated theoritically for the set of variables in  differenc_set
+    mu=max(0,(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set)));  %optimal lagrangian mult.
+    h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta %optimal hit prob
+    
+    % mu has to be >=0
+    if(mu<0)
+        b=(n*c_f/c_delta); % this sets mu to zero in the next iteration
+        continue;
+    end
+    
+    % check the violation of the hit_prob const  
+    larger_i=find(h_optt>1); % h>1
+    smaller_i=find(h_optt<0); % h<0
+
+%     smaller=h(find(differenc_set<0))-0;
+    % no violation means optimal solution is reached for all objects
+    if(length(smaller_i)==0 && length(larger_i)==0)
+%         flag=0;
+  
+        break;
+    end
+    
+    % find the furthest object from the 0 boundary
+    min_viol=0;
+    min_viol_i=-1;
+    if(length(smaller_i)>0)
+        [min_viol, min_viol_i]=min(h_optt);
+    end
+     % find the furthest object from the 1 boundary
+    max_viol=0; 
+    max_viol_i=-1;
+    if(length(larger_i)>0)
+       larger=h_optt-1;
+       [max_viol ,max_viol_i]=max(h_optt-1);
+
+    end
+    
+     % compare both furthest objects from both boundaries
+    viol_i=min_viol_i;
+    min_viol_flag=1; % True if the furthest one is from the left
+    if(max_viol>abs(min_viol))
+        viol_i= max_viol_i;
+        min_viol_flag=0;
+
+    end
+     % set the furthest object to the nearest boundary
+     if(min_viol_flag)
+        h_optt(viol_i)=0;
+    else
+        h_optt(viol_i)=min(1,b);
+     end
+    
+    %calculate the new parameters after removing the furthest object from
+    %the decision variables
+    B_new=b-(h_optt(viol_i));
+    b=B_new;
+    fix_i=[fix_i' viol_i']';
+    differenc_set=setdiff(1:N,fix_i)   ; 
+    n=N-length(fix_i);
+    
+    
+    %     % Identify the most violating object from the right side  h>1
+%     if(length(larger_i)>0)
+%        larger=h_optt-1;
+%        [max_viol ,max_viol_i]=max(h_optt-1); % maximum violating object
+%        h_optt(max_viol_i)=min(1,b); %project to the feasible range
+%        b=max(b-1,0); % update the cache size 
+%        fix_i=[fix_i' max_viol_i']'; %exclude i from the set of decision variables
+%        differenc_set=setdiff(1:N,fix_i); % obtain the set of decision variables 
+%        n=N-length(fix_i); % update the number of decision variables 
+%        continue;
+%     end
+%     
+%     if(length(smaller_i)>0)
+%         [min_viol, min_viol_i]=min(h_optt);
+%         h_optt(min_viol_i)=0; 
+%         fix_i=[fix_i' min_viol_i']';
+%         differenc_set=setdiff(1:N,fix_i)   ; 
+%         n=N-length(fix_i);
+%     end
+%     
+% end
+
+end
+
+%% Identfying the active constraints collectively
+% flag=1;
+% h_optt=zeros(N,1);
+% differenc_set=1:N;
+% fix_i=[];
+% n=N;
+% b=B;
+% while flag
+%     mu=(n*c_f-b*c_delta)/ sum(1./lambda(differenc_set));
+%     h_optt(differenc_set)=(c_f-mu./lambda(differenc_set))/c_delta
+% 
+%     larger_i=find(h_optt>1);    
+% %     larger=h_optt(larger_i)-1;
+%     smaller_i=find(h_optt<0);
+% %     smaller=h(find(differenc_set<0))-0;
+%     mult=solve_multipliers(lambda,B,c_f,c_delta,larger_i)
+% 
+%     if(length(larger_i)+length(smaller_i)==0)
+%         flag=0;
+%         break;
+%     end
+%     if(length(smaller_i)>0)
+%         h_optt(smaller_i)=0;
+%         fix_i=[fix_i' smaller_i' ]';
+%         differenc_set=setdiff(1:N,fix_i)    
+%         n=N-length(fix_i);
+%         continue
+%     end
+% %     h_optt(smaller_i)=0;
+%     if(length(larger_i)>b)
+%         [~,index]=maxk(h_optt,b)
+%         h_optt(index)=1;
+%         B_new=b-sum(h_optt(index));
+%         fix_i=[fix_i' smaller_i' index']';
+%     else
+%          h_optt(larger_i)=1;
+%          B_new=b-sum(h_optt(larger_i));
+%          fix_i=[fix_i' smaller_i' larger_i']';
+%     end
+% %     mult=solve_multipliers(lambda,B,c_f,c_delta,larger_i)
+% 
+%     b=B_new;
+%    
+%     differenc_set=setdiff(1:N,fix_i)    
+%     n=N-length(fix_i);
+% end
+% % h_optt=zeros(N,1);
+% % h_optt(end-B+1:end)=1;
+% optimal=[h_opt h_optt]
+% obj_1=sum(lambda .* ((1-h_opt)*c_f+h_opt.^2*c_delta/2));
+% obj_2=sum(lambda .* ((1-h_optt)*c_f+h_optt.^2*c_delta/2));
+% objective=[obj_1 obj_2]
+% const_1=sum(h_opt)-B;
+% const_2=sum(h_optt)-B;
+% constraint=[const_1 const_2]
+% 
+
+
+end
+
--- a/01_nb_cncf_optimization/notes_code.excalidraw
+++ b/01_nb_cncf_optimization/notes_code.excalidraw
--- a/01_nb_cncf_optimization/notes_paper.excalidraw
+++ b/01_nb_cncf_optimization/notes_paper.excalidraw
--- a/01_nb_cncf_optimization/objective_grid-search_multi-core.csv
+++ b/01_nb_cncf_optimization/objective_grid-search_multi-core.csv
--- a/aoi_cache_experiment_eval.ipynb
+++ b/aoi_cache_experiment_eval.ipynb
--- a/aoi_cache_simulation.ipynb
+++ b/aoi_cache_simulation.ipynb
--- a/experiments/No_Refresh_0_5s_ttl/details.csv
+++ b/experiments/No_Refresh_0_5s_ttl/details.csv
@@ -1,101 +0,0 @@
-obj_id,access_count,hits,misses,mu,lambda,hit_rate,avg_age
-1,122,41,81,0,1,33.61,0.02459016393442623
-2,382,235,147,0,3,61.52,0.15706806282722513
-3,127,38,89,0,1,29.92,0.10236220472440945
-4,113,33,80,0,1,29.2,0.08849557522123894
-5,244,121,123,0,2,49.59,0.11065573770491803
-6,116,40,76,0,1,34.48,0.09482758620689655
-7,656,474,182,0,5,72.26,0.16310975609756098
-8,128,39,89,0,1,30.47,0.046875
-9,114,37,77,0,1,32.46,0.08771929824561403
-10,115,33,82,0,1,28.7,0.06956521739130435
-11,246,129,117,0,2,52.44,0.09349593495934959
-12,132,50,82,0,1,37.88,0.08333333333333333
-13,139,50,89,0,1,35.97,0.08633093525179857
-14,120,35,85,0,1,29.17,0.058333333333333334
-15,261,134,127,0,2,51.34,0.13793103448275862
-16,225,109,116,0,2,48.44,0.10666666666666667
-17,120,39,81,0,1,32.5,0.06666666666666667
-18,117,39,78,0,1,33.33,0.07692307692307693
-19,360,213,147,0,3,59.17,0.10277777777777777
-20,117,40,77,0,1,34.19,0.09401709401709402
-21,134,48,86,0,1,35.82,0.07462686567164178
-22,147,55,92,0,1,37.41,0.12244897959183673
-23,142,50,92,0,1,35.21,0.07746478873239436
-24,264,140,124,0,2,53.03,0.10984848484848485
-25,123,42,81,0,1,34.15,0.11382113821138211
-26,141,50,91,0,1,35.46,0.0851063829787234
-27,128,45,83,0,1,35.16,0.0703125
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--- a/experiments/No_Refresh_0_5s_ttl/hit_age.csv
+++ b/experiments/No_Refresh_0_5s_ttl/hit_age.csv
@@ -1,101 +0,0 @@
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--- a/experiments/No_Refresh_0_5s_ttl/objects_in_cache_over_time.pdf
+++ b/experiments/No_Refresh_0_5s_ttl/objects_in_cache_over_time.pdf
--- a/experiments/No_Refresh_0_5s_ttl/overall_hit_age.csv
+++ b/experiments/No_Refresh_0_5s_ttl/overall_hit_age.csv
@@ -1,9 +0,0 @@
-,hit_rate,avg_age
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--- a/experiments/No_Refresh_1_0s_ttl/details.csv
+++ b/experiments/No_Refresh_1_0s_ttl/details.csv
@@ -1,101 +0,0 @@
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--- a/experiments/No_Refresh_1_0s_ttl/hit_age.csv
+++ b/experiments/No_Refresh_1_0s_ttl/hit_age.csv
@@ -1,101 +0,0 @@
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--- a/experiments/No_Refresh_1_0s_ttl/objects_in_cache_over_time.pdf
+++ b/experiments/No_Refresh_1_0s_ttl/objects_in_cache_over_time.pdf
--- a/experiments/No_Refresh_1_0s_ttl/overall_hit_age.csv
+++ b/experiments/No_Refresh_1_0s_ttl/overall_hit_age.csv
@@ -1,9 +0,0 @@
-,hit_rate,avg_age
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--- a/experiments/No_Refresh_2_0s_ttl/details.csv
+++ b/experiments/No_Refresh_2_0s_ttl/details.csv
@@ -1,101 +0,0 @@
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--- a/experiments/No_Refresh_2_0s_ttl/hit_age.csv
+++ b/experiments/No_Refresh_2_0s_ttl/hit_age.csv
@@ -1,101 +0,0 @@
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--- a/experiments/No_Refresh_2_0s_ttl/objects_in_cache_over_time.pdf
+++ b/experiments/No_Refresh_2_0s_ttl/objects_in_cache_over_time.pdf
--- a/experiments/No_Refresh_2_0s_ttl/overall_hit_age.csv
+++ b/experiments/No_Refresh_2_0s_ttl/overall_hit_age.csv
@@ -1,9 +0,0 @@
-,hit_rate,avg_age
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--- a/experiments/No_Refresh_3_0s_ttl/details.csv
+++ b/experiments/No_Refresh_3_0s_ttl/details.csv
@@ -1,101 +0,0 @@
-obj_id,access_count,hits,misses,mu,lambda,hit_rate,avg_age
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--- a/experiments/No_Refresh_3_0s_ttl/hit_age.csv
+++ b/experiments/No_Refresh_3_0s_ttl/hit_age.csv
@@ -1,101 +0,0 @@
-obj_id,hit_rate,avg_age
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--- a/experiments/No_Refresh_3_0s_ttl/overall_hit_age.csv
+++ b/experiments/No_Refresh_3_0s_ttl/overall_hit_age.csv
@@ -1,9 +0,0 @@
-,hit_rate,avg_age
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--- a/experiments/No_Refresh_4_0s_ttl/details.csv
+++ b/experiments/No_Refresh_4_0s_ttl/details.csv
@@ -1,101 +0,0 @@
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--- a/experiments/No_Refresh_4_0s_ttl/hit_age.csv
+++ b/experiments/No_Refresh_4_0s_ttl/hit_age.csv
@@ -1,101 +0,0 @@
-obj_id,hit_rate,avg_age
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-99,0.986648865153538,1.981957600360848
-100,0.7899159663865546,1.8829787234042554
--- a/experiments/No_Refresh_4_0s_ttl/lambda_vs_access_count.pdf
+++ b/experiments/No_Refresh_4_0s_ttl/lambda_vs_access_count.pdf
--- a/experiments/No_Refresh_4_0s_ttl/objects_in_cache_over_time.pdf
+++ b/experiments/No_Refresh_4_0s_ttl/objects_in_cache_over_time.pdf
--- a/experiments/No_Refresh_4_0s_ttl/overall_hit_age.csv
+++ b/experiments/No_Refresh_4_0s_ttl/overall_hit_age.csv
@@ -1,9 +0,0 @@
-,hit_rate,avg_age
-count,100.0,100.0
-mean,0.8461611168860607,1.966244726179581
-std,0.06904890299740231,0.12918574131498722
-min,0.75,1.588648184618492
-25%,0.7946428571428571,1.8798211706342576
-50%,0.8120158760642724,1.9792294446835987
-75%,0.8918173495078966,2.054958791208791
-max,0.9984873695356224,2.2560975609756095
--- a/experiments/No_Refresh_5_0s_ttl/details.csv
+++ b/experiments/No_Refresh_5_0s_ttl/details.csv
@@ -1,101 +0,0 @@
-obj_id,access_count,hits,misses,mu,lambda,hit_rate,avg_age
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-100,272,247,25,0,2,90.81,2.1654411764705883
--- a/experiments/No_Refresh_5_0s_ttl/hit_age.csv
+++ b/experiments/No_Refresh_5_0s_ttl/hit_age.csv
@@ -1,101 +0,0 @@
-obj_id,hit_rate,avg_age
-1,0.8308823529411765,2.504424778761062
-2,0.841726618705036,2.5982905982905984
-3,0.8410596026490066,2.1811023622047245
-4,0.935064935064935,2.5805555555555557
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-14,0.8571428571428571,2.5476190476190474
-15,0.8248175182481752,2.433628318584071
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-25,0.9114391143911439,2.445344129554656
-26,0.8253968253968254,2.25
-27,0.85625,2.72992700729927
-28,0.8114754098360656,2.5555555555555554
-29,0.9661458333333334,2.6132075471698113
-30,0.9101123595505618,2.707818930041152
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-37,0.8099173553719008,2.5
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-49,0.8264462809917356,2.5
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-93,0.9615952732644018,2.490015360983103
-94,0.8308823529411765,2.3805309734513274
-95,0.8421052631578947,2.625
-96,0.9701327433628318,2.4994298745724057
-97,0.9336734693877551,2.4918032786885247
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-99,0.9012345679012346,2.4292237442922375
-100,0.9080882352941176,2.3846153846153846
--- a/experiments/No_Refresh_5_0s_ttl/lambda_vs_access_count.pdf
+++ b/experiments/No_Refresh_5_0s_ttl/lambda_vs_access_count.pdf
--- a/experiments/No_Refresh_5_0s_ttl/objects_in_cache_over_time.pdf
+++ b/experiments/No_Refresh_5_0s_ttl/objects_in_cache_over_time.pdf
--- a/experiments/No_Refresh_5_0s_ttl/overall_hit_age.csv
+++ b/experiments/No_Refresh_5_0s_ttl/overall_hit_age.csv
@@ -1,9 +0,0 @@
-,hit_rate,avg_age
-count,100.0,100.0
-mean,0.8689161003980432,2.469801316710304
-std,0.05362014513420393,0.13549611807744597
-min,0.78,2.0483199198321485
-25%,0.8270676691729323,2.3846153846153846
-50%,0.841726618705036,2.4849237260629664
-75%,0.9100820467221729,2.5632199275958536
-max,0.9991551939924906,2.7454545454545456
--- a/experiments/README.md
+++ b/experiments/README.md
@@ -1,17 +0,0 @@
-# Experiments: No Refresh with variable TTL
-
-Explanation for files in each experiment:
-
- `details.csv`: Access Count, Hit, Miss, Mu, Lambda and Hit Rate for each object
- `hit_age.csv`: Shows hit rate/average age at time of request for each object.
- `lambda_distribution.pdf`: Lambda Distribution across all objects/discrete
-  values of the Zipf distribution
- `lambda_vs_access_count.pdf`: Displays the access count against lambda,
-  expecting a higher lambda to result in a higher access count.
- `objects_in_cache_over_time.pdf`: Amount of cache entries at given time.
- `overall_hit_age.csv`: Cumulative description of `hit_age.csv`
-
-Length of simulation doesn't change much since we're not touching the request
-rate across the simulations.
-Break condition for the simulation is when the each database object has been
-requested at least `ACCESS_COUNT_LIMIT` (i.e. 10) times.
--- a/experiments/avg_ages.md
+++ b/experiments/avg_ages.md
@@ -1,8 +0,0 @@
-|      |   avg_ages |
-|:-----|-----------:|
-| 0.5s |   0.240582 |
-| 1.0s |   0.497634 |
-| 2.0s |   1.00349  |
-| 3.0s |   1.47952  |
-| 4.0s |   1.96624  |
-| 5.0s |   2.4698   |
--- a/experiments/hit_rates.md
+++ b/experiments/hit_rates.md
@@ -1,8 +0,0 @@
-|      |   hit_rates |
-|:-----|------------:|
-| 0.5s |    0.45867  |
-| 1.0s |    0.619674 |
-| 2.0s |    0.769815 |
-| 3.0s |    0.790693 |
-| 4.0s |    0.846161 |
-| 5.0s |    0.868916 |
--- a/experiments/hr_and_age_vs_ttl.png
+++ b/experiments/hr_and_age_vs_ttl.png
--- a/Show More
+++ b/Show More
Author	SHA1	Message	Date
Tuan-Dat Tran	b166a9e64a	refactor(simulation): Commented out prints Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-05 21:58:15 +01:00
Tuan-Dat Tran	ed08b8fef3	feat(simulation): Event based eviction instead of iterative Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-05 21:46:52 +01:00
Tuan-Dat Tran	6da629f90e	fix(simulation): Fixed expected age calculation Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-05 15:53:31 +01:00
Tuan-Dat Tran	b7aaa31860	fix(simulation): Added missing imports for `00_aoi_caching_simulation/multi_aoi_cache_simulation.ipynb` and ran new experiments Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-05 14:49:41 +01:00
Tuan-Dat Tran	78345c9788	refactor(simulation): Copied changes from `aoi_cache_simulation` to `multi_aoi_cache_simulation` Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-05 14:40:20 +01:00
Tuan-Dat Tran	032251dd78	refactor/fix(simulation): Added expected hitrate for each objct and refactored Cache to handle TTL/non-TTL Cache-Types Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-05 13:48:20 +01:00
Tuan-Dat Tran	78e700a2cf	fix(simulation): event based age calculation instead of iterative Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-04 17:11:50 +01:00
Tuan-Dat Tran	7d194176f0	feat(simulation): Added time spent in cache log for each object Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-04 16:38:39 +01:00
Tuan-Dat Tran	036789cc7c	cleanup(plot): removed heatmap Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-03 15:31:03 +01:00
Tuan-Dat Tran	3787d004c1	fix(optimize ttl): manual differentiation of hitrate<1 to fix log(0) warnings Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-03 15:28:52 +01:00
Tuan-Dat Tran	0ea1fb5d07	feat(optimal ttl): Added calculation for optimal ttl of each object Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-03 15:15:56 +01:00
Tuan-Dat Tran	272f722f23	feat(objective optimization): Perform gridsearch as single-core and multi-core to find optimal parameters to minimize objective Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-03 14:19:45 +01:00
Tuan-Dat Tran	799f7b78d4	fix(h_i_opt calculation): Fix for calculation of optimized hitrate: - Assignment of current_cache_size instead of decrease Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-03 10:12:07 +01:00
Tuan-Dat Tran	4ea5505130	feat(eta_calculation): Implementing "Joint Caching and Freshness Optimization" (in progress) Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-12-02 18:07:08 +01:00
Tuan-Dat Tran	b2cc80bb09	refactor: Restructure Repository to add eta optimization Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-11-29 21:49:59 +01:00
Tuan-Dat Tran	f32588340d	feat(simulation): add expected age computation and enhance statistics reporting - Increased `ACCESS_COUNT_LIMIT` to extend simulation runtime. - Introduced `expected_age` metric based on hit rates for additional insights. - Calculated and exported `age_delta` for comparing average and expected ages. - Improved data exports to include detailed metrics for analysis. Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-11-29 14:42:52 +01:00
Tuan-Dat Tran	6672608721	feat(simulation): Single source of truth regarding avg_age and hit_rate Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-11-28 16:50:10 +01:00
Tuan-Dat Tran	ad4654dd0f	fix(simulation): Updated avg_age calculation not only for `details.csv`, but all avg_age calculation. Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-11-28 16:38:55 +01:00
Tuan-Dat Tran	3a9e3105f2	chore(simulation): Reran simulation Signed-off-by: Tuan-Dat Tran <tuan-dat.tran@tudattr.dev>	2024-11-28 16:23:45 +01:00