final-work.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "source": "# BL40A2030 Wireless Communication Networks\n\n## Final work\n\n### Author: [Niki Malmsten]",
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      "source": "**The goal of the final work is to compute the Transmission Capacity (i.e. the highest achievable Spatial Throughput considering a maximum acceptable outage probability constraint), also considering aspects related to queue stability. The final work will be step-by-step.**\n\nConsider a bipolar network similar to the one presented in weeks 6 and 7, but now considering a slotted-ALOHA-style of medium access. \n\n- Network density $\\lambda>0$\n- Path-loss exponent $\\alpha>2$\n- Rayleigh fading, leading to a constant $\\kappa$\n- SIR threshold $\\beta>0$\n- TX-RX distance $d\\geq1$\n- Slotted Aloha access probability (independent for individual TXs): $p$\n- Outage probability = 1 - success probability: $1 - e^{- \\kappa p \\lambda \\pi d^2 \\beta^{2/\\alpha}} = 1 - P_s$\n- Spatial throughput: $p \\times \\lambda \\times  P_s \\times \\log_2(1+\\beta)$\n- Maximum acceptable outage probability: $\\epsilon$\n\n*Note: When the slotted-ALOHA-style MAC is used, the original Poisson point process is thinned and the process of active transmitters in the networks becomes $p\\lambda$ active nodes per square-meter.*\n\n(A)  Plot: $p \\times \\lambda \\times  P_s \\times \\log_2(1+\\beta)$ in terms of the access probability $p$, considering that $P_s = 1-\\epsilon$. Assume: $\\alpha=4$, $d=1$, $\\epsilon = 0.05$ and $\\lambda=0.02$.\n\n*Hint: First proceed with the algebraic manipulation of $P_s = 1-\\epsilon$ to find the value of $\\beta$ as a function of the other variables. Then, you ended up in a new equation.*\n\n(B) Find the value of $p$ that optimizes the spatial throughput.\n\n*Hint: It might be possible that sympy cannot find the equation analytically so a numerical solution is also acceptable (see tutorial from week 7).*\n\n(C) Test the result from (B) using the numerical values from (A) by making the plot including the optimal spatial throuhgput value. Verify also for $\\lambda=0.01$ and $\\lambda=0.025$.\n\n(D) What is the transmission capacity of this network (i.e. the maximum achievable spatial throughput)? \n\n(E) Consider that the TXs have one queue where packets arrive at each time slot with a given probability $q$ (Bernoulli arrivals). What is the necessary relation between $p$ and $q$ that guarantees queue stability (i.e. the queue size is not growing to infinity with time). To answer this question you need to simulate this queue considering 3 conditions (i) $q<p$ by setting $q = p - 0.1$, (ii) $q=p$ and (iii) $q>p$ by setting $q = p + 0.1$ for the value of $p$ found in (B) for 10000 time-slots (time-steps) and evaluate the worst case delay.\n\n*Hint: The packets will arrive with probability $q$ each time slot, while packets will leave with probability $p$ (similar to thowning a coin experiment with biased probability). This queue is similar to the \"naive\" simple queue approach taken in week 1 but now with Bernoulli arrivals and departures. To generate a sequence of Bernoulli, you can use the Binomial random number generator (numpy.random.binomial).*\n\n(F) Write a short comment about what would you expect to happen with the queue of the TXs if:\n\n- Restransmissions are allowed\n- CSMA protocol is used\n- Queues have a limited size\n",
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    {
      "cell_type": "code",
      "metadata": {
        "trusted": true,
        "cell_id": "00002-edc9e304-73da-4bd2-b077-fbc65faac148",
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        "source_hash": "ce405f50",
        "execution_millis": 1,
        "execution_start": 1616443268015,
        "deepnote_cell_type": "code"
      },
      "source": "import numpy as np \nimport matplotlib.pyplot as plt\nfrom matplotlib import style\nimport math\nfrom scipy.spatial import distance\nimport warnings\nimport sympy as sp\nwarnings.filterwarnings('ignore')",
      "execution_count": 22,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00003-a9792083-bfd0-470f-97cf-8108f16f6dfe",
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        "source_hash": "c9053fe",
        "execution_millis": 0,
        "execution_start": 1616448054593,
        "deepnote_cell_type": "code"
      },
      "source": "## The symbols\np2, lamb2, B2, a2, d2, e2, k2 = sp.symbols('p lambda beta alpha d epsilon kappa', real=True, positive=True)\na = 4\nd = 1\ne = 0.05\nlamb = 0.02\nPs = 1-e\nB = 0\nk = math.gamma(1 + 2/a) * math.gamma(1- 2/a)",
      "execution_count": 191,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": "# test shit\n\n#funktio = p2 * lamb2 * (1 - e2) * sp.log(1 - B2,2)\nf = 1*a2*lamb2*((-sp.log(1-e2)/(sp.pi*d2**2*k2*lamb2*p2))**(a2/2))*(1-e2)\n\nf2 = ((-sp.log(1-e2)/(sp.pi*d2**2*k2*lamb2*p2))**(a2/2)+1)*sp.log(2)*2\n\nf5 = (-sp.log(1-e2)/(sp.pi*d2**2*k2*lamb2*p2))**(a2/2)+1\n\nf3 = lamb2*(1-e2)*sp.log(f5)/sp.log(2)\n\nf4 = -f/f2+f3\nf4\nf6 = sp.integrate(f4, p2)\nf6",
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        "source_hash": "c915ef8b",
        "execution_millis": 134,
        "execution_start": 1616449229313,
        "deepnote_cell_type": "code"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 256,
          "data": {
            "text/plain": "                     α                          ⎛              α    ⎞\n                     ─                          ⎜              ─    ⎟\n                     2                          ⎜              2    ⎟\n       ⎛-log(1 - ε) ⎞                           ⎜⎛-log(1 - ε) ⎞     ⎟\n   α⋅λ⋅⎜────────────⎟ ⋅(1 - ε)     λ⋅(1 - ε)⋅log⎜⎜────────────⎟  + 1⎟\n       ⎜    2       ⎟                           ⎜⎜    2       ⎟     ⎟\n       ⎝ π⋅d ⋅κ⋅λ⋅p ⎠                           ⎝⎝ π⋅d ⋅κ⋅λ⋅p ⎠     ⎠\n- ────────────────────────────── + ──────────────────────────────────\n    ⎛              α    ⎞                        log(2)              \n    ⎜              ─    ⎟                                            \n    ⎜              2    ⎟                                            \n    ⎜⎛-log(1 - ε) ⎞     ⎟                                            \n  2⋅⎜⎜────────────⎟  + 1⎟⋅log(2)                                     \n    ⎜⎜    2       ⎟     ⎟                                            \n    ⎝⎝ π⋅d ⋅κ⋅λ⋅p ⎠     ⎠                                            ",
            "image/png": 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\n",
            "text/latex": "$\\displaystyle - \\frac{\\alpha \\lambda \\left(- \\frac{\\log{\\left(1 - \\epsilon \\right)}}{\\pi d^{2} \\kappa \\lambda p}\\right)^{\\frac{\\alpha}{2}} \\left(1 - \\epsilon\\right)}{2 \\left(\\left(- \\frac{\\log{\\left(1 - \\epsilon \\right)}}{\\pi d^{2} \\kappa \\lambda p}\\right)^{\\frac{\\alpha}{2}} + 1\\right) \\log{\\left(2 \\right)}} + \\frac{\\lambda \\left(1 - \\epsilon\\right) \\log{\\left(\\left(- \\frac{\\log{\\left(1 - \\epsilon \\right)}}{\\pi d^{2} \\kappa \\lambda p}\\right)^{\\frac{\\alpha}{2}} + 1 \\right)}}{\\log{\\left(2 \\right)}}$"
          },
          "metadata": {}
        }
      ],
      "execution_count": 256
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00004-149f118e-6ef2-41fb-a932-f8bd5d5decc4",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "d8c7f0ed",
        "execution_millis": 211,
        "execution_start": 1616447223246,
        "deepnote_cell_type": "code"
      },
      "source": "## The function\nsp.init_printing() \n\nfunktio = p2 * lamb2 * sp.exp(-k2*p2*lamb2*sp.pi*d2**2*B2**(2/a2))*sp.log(1+B2)/sp.log(2)\nfunktio\n\n",
      "execution_count": 130,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 130,
          "data": {
            "text/plain": "         2                    \n         ─                    \n         α  2                 \n     -π⋅β ⋅d ⋅κ⋅λ⋅p           \nλ⋅p⋅ℯ              ⋅log(β + 1)\n──────────────────────────────\n            log(2)            ",
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\frac{\\lambda p e^{- \\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda p} \\log{\\left(\\beta + 1 \\right)}}{\\log{\\left(2 \\right)}}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00005-7bd9447b-42e3-4145-9792-110a9402adc1",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "c5fe737a",
        "execution_millis": 406,
        "execution_start": 1616447692057,
        "deepnote_cell_type": "code"
      },
      "source": "## Derivation\n\nfunktio2 = sp.diff(funktio, p2)\n#funktio2 = sp.simplify(funktio2)\nfunktio2\n",
      "execution_count": 150,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 150,
          "data": {
            "text/plain": "                      2                              2                    \n     2                ─                              ─                    \n     ─                α  2                           α  2                 \n     α  2    2    -π⋅β ⋅d ⋅κ⋅λ⋅p                 -π⋅β ⋅d ⋅κ⋅λ⋅p           \n  π⋅β ⋅d ⋅κ⋅λ ⋅p⋅ℯ              ⋅log(β + 1)   λ⋅ℯ              ⋅log(β + 1)\n- ───────────────────────────────────────── + ────────────────────────────\n                    log(2)                               log(2)           ",
            "image/png": "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\n",
            "text/latex": "$\\displaystyle - \\frac{\\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda^{2} p e^{- \\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda p} \\log{\\left(\\beta + 1 \\right)}}{\\log{\\left(2 \\right)}} + \\frac{\\lambda e^{- \\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda p} \\log{\\left(\\beta + 1 \\right)}}{\\log{\\left(2 \\right)}}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": "## Numerical solution\n#funktio2 = funktio2.subs(a2, a).subs(d2, d).subs(k2, k).subs(e2, e).subs(lamb2, lamb)\nfunktio3 = sp.solve(funktio2, B2)\nfunktio3[0]\n",
      "metadata": {
        "tags": [],
        "cell_id": "00006-25ac6fc4-cfb5-4639-9a6c-1ad08317f70b",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "1a54dc07",
        "execution_millis": 562,
        "execution_start": 1616447694661,
        "deepnote_cell_type": "code"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 151,
          "data": {
            "text/plain": "               α\n⎛      1      ⎞ \n⎜─────────────⎟ \n⎝√π⋅d⋅√κ⋅√λ⋅√p⎠ ",
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\left(\\frac{1}{\\sqrt{\\pi} d \\sqrt{\\kappa} \\sqrt{\\lambda} \\sqrt{p}}\\right)^{\\alpha}$"
          },
          "metadata": {}
        }
      ],
      "execution_count": 151
    },
    {
      "cell_type": "code",
      "source": "fun = p2 * lamb2 * (1-e2) * sp.log(1-B2, 2)\nfun = fun.subs(B2, funktio3[0])\nfun = sp.diff(fun, p2)\n\n#fun = funktio.subs(B2, funktio3[0])\n#fun = sp.diff(fun, p2)\n#fun = sp.simplify(fun)\n#sp.solveset(fun,p2)\nfun = fun.subs(a2, a).subs(d2, d).subs(k2, k).subs(e2, e).subs(lamb2, lamb)\nfun = sp.nsolve(fun, p2, )\nfun",
      "metadata": {
        "tags": [],
        "cell_id": "00008-38a7cae6-0dab-49c8-a3e2-2850be5213f5",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "ce8f2b62",
        "execution_millis": 34,
        "execution_start": 1616447804655,
        "deepnote_cell_type": "code"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 185,
          "data": {
            "text/plain": "1.38494386387969e-25 + 5.1164787534149⋅ⅈ",
            "image/png": "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\n",
            "text/latex": "$\\displaystyle 1.38494386387969 \\cdot 10^{-25} + 5.1164787534149 i$"
          },
          "metadata": {}
        }
      ],
      "execution_count": 185
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00005-9dedacbf-a698-4507-8066-5562942d8ab2",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "f804c555",
        "execution_millis": 75,
        "execution_start": 1616424290085,
        "deepnote_cell_type": "code"
      },
      "source": "#e = 0.05\n#λ = 0.02\n#d = 1\n#a = 4\n#k = math.gamma(1 + 2/a) * math.gamma(1- 2/a)\np, l, B, λ, a, d, e, k = sp.symbols('p l B λ a d e k')\nb = sp.exp(-k*p*λ*sp.pi*d**2*B**(2/a))\na = sp.solve(p*λ*b*sp.log(1+B,2),B)\na\n#c = sp.diff(a, B)\n\n",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 37,
          "data": {
            "text/plain": "[0]",
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\left[ 0\\right]$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00004-69068284-d872-48e6-9203-9118226fbcd0",
        "deepnote_cell_type": "code"
      },
      "source": "",
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00006-fae0fb61-b0ca-4cd5-99df-995fed9fa329",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "ec3940a5",
        "execution_millis": 48,
        "execution_start": 1616423558322,
        "deepnote_cell_type": "code"
      },
      "source": "sp.init_printing() \np, l, e, B = sp.symbols('p l e B')\na = p*l*(1-e)*sp.log(1+B,2)\na\nb = sp.diff(a,B)\nb",
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 17,
          "data": {
            "text/plain": " l⋅p⋅(1 - e)  \n──────────────\n(B + 1)⋅log(2)",
            "image/png": "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\n",
            "text/latex": "$\\displaystyle \\frac{l p \\left(1 - e\\right)}{\\left(B + 1\\right) \\log{\\left(2 \\right)}}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "tags": [],
        "cell_id": "00007-400341b4-826b-40b3-9f34-9ae44606b0bc",
        "deepnote_to_be_reexecuted": false,
        "source_hash": "c964a004",
        "execution_millis": 2027,
        "output_cleared": false,
        "execution_start": 1616442158651,
        "deepnote_cell_type": "code"
      },
      "source": "%reset",
      "outputs": [],
      "execution_count": null
    },
    {
      "cell_type": "markdown",
      "source": "<a style='text-decoration:none;line-height:16px;display:flex;color:#5B5B62;padding:10px;justify-content:end;' href='https://deepnote.com?utm_source=created-in-deepnote-cell&projectId=cc035cc0-996d-4fa3-af0c-6c39ba437636' target=\"_blank\">\n<img style='display:inline;max-height:16px;margin:0px;margin-right:7.5px;' src='data:image/svg+xml;base64,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' > </img>\nCreated in <span style='font-weight:600;margin-left:4px;'>Deepnote</span></a>",
      "metadata": {
        "tags": [],
        "created_in_deepnote_cell": true,
        "deepnote_cell_type": "markdown"
      }
    }
  ],
  "nbformat": 4,
  "nbformat_minor": 2,
  "metadata": {
    "kernelspec": {
      "name": "python36",
      "display_name": "Python 3.6",
      "language": "python"
    },
    "language_info": {
      "mimetype": "text/x-python",
      "nbconvert_exporter": "python",
      "name": "python",
      "pygments_lexer": "ipython3",
      "version": "3.6.6",
      "file_extension": ".py",
      "codemirror_mode": {
        "version": 3,
        "name": "ipython"
      }
    },
    "deepnote_notebook_id": "0b0144a5-339a-4556-a340-fbd597530eef",
    "deepnote": {},
    "deepnote_execution_queue": [
      {
        "cellId": "00004-5d224c9d-70ee-466b-8a7f-240708a758b9",
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