Digit Estimation.ipynb

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    "%%html\n",
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   "source": [
    "# Purpose\n",
    "\n",
    "The purpose of this notebook is to determine the tradeoffs of compressing the encoding the Better CAT's transaction event priority field, `pr`."
   ]
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   "source": [
    "## Advantages\n",
    "\n",
    "The main advantage of compressed priority encoding would be a per-packet byte savings. Currently, we use 20 bytes to encode prioirty as a 16 decimal-digit value, e.g. \"0.1242135235235235\" - 2 for the quotation marks, 1 for the digit before the decimal, 1 for the decimal, and 16 after the decimal. If, for example, we were to change that to a 1-digit integer (i.e. a priority could only have a value between 0 to 9, so \"0.1242135235235235\" turns into just 1), we would only need 1 byte per packet.\n"
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   "source": [
    "## Disadvantages\n",
    "\n",
    "I can think of three possible disadvantages here: version incompatibility, feature obstacles, and potential dropped trace spans that would not have been dropped had the priority value been encoded with more digits.\n"
   ]
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    "* __Version Incompatibility__.\n",
    "\n",
    "  Encoding the `pr` field would cause problems if interfacing with a non-NR app that doesn't encode its priorities.\n",
    "\n",
    "  If, for example, we were to encode priority as an integer between, say, 0 and 999, and we receive requests from a system whose priorities are encoded as 16 decimal-digit floats, then those packets would be completely buried.  \n"
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    "\n",
    "* __Feature Obstacles__.\n",
    "\n",
    "  Reducing the total possible `pr` values gives less design space to implement future features.\n",
    "\n",
    "  If we were, for example, to encode priority as a 1 digit integer, then we only have 10 possible values for the priority. In order to implement the `sampled=True` feature (where some samples are assigned an increased priority to ensure all spans of the trace are recorded), we would actually be forced to only encode priority as a digit from 0 to 4, as we would need to use the other half of the available values to give space to the super-sampled values. For example, a priority whose original value was 0.5242135235235235 with `sampled=False` would need be encoded to a single-digit integer as 2 so that there would be \"room\" to allow its `sampled=True` value to be 7.  \n"
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   "source": [
    "While the first two of these problems are more abstract in nature, we can calculate an answer for the third one as a probability that a span will be improperly dropped due to compressed encoding."
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    "## Calculating the Probability of Dropping a Trace Due to Compressed Encoding\n",
    "\n",
    "First, some definitions:\n",
    "    \n",
    "$$P_{DCE}(x) = Probability\\:that\\:a\\:\"$$"
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    "#digits = [1, 5, 9]\n",
    "\n",
    "#raw_sizes = [2.5, 3, 3.5, 4]\n",
    "#sizes = [int(10**x) for x in raw_sizes]\n",
    "\n",
    "logstep_throughput = 0.25\n",
    "log_throughputs = [2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75, 4]\n",
    "throughputs = [math.floor(10**x) for x in log_throughputs]\n",
    "\n",
    "jitters = [10.0, 45.0, 70.0, 140.0]\n",
    "cluster_ratios = [1.0, 10.0, 100.0]"
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    "def EMG_cdf(x, mu, stddev, rate):\n",
    "    u = rate*(x-mu)\n",
    "    v = rate*stddev\n",
    "    \n",
    "    t1 = st.norm.cdf(u, 0, v)\n",
    "    t2 = st.norm.cdf(u, v*v, v)\n",
    "    t3 = t2*numpy.exp(-u + v*v/2)\n",
    "    \n",
    "    return t1 - t3\n",
    "\n",
    "def prob_overlap(x, jitter, throughput, cluster_ratio):\n",
    "    reqs_per_ms = throughput/60.0/1000.0\n",
    "    mu = reqs_per_ms/cluster_ratio\n",
    "    return EMG_cdf(x, mu, jitter, reqs_per_ms)"
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    "chosen_cro = cluster_ratios[2]\n",
    "xdata = numpy.arange(-4.0, 4.0, 0.01)\n",
    "ydata = prob_overlap(xdata, jitters[0], throughputs[0], chosen_cro)\n",
    "  \n",
    "x_sc = LinearScale(min=min(xdata), max=max(xdata))\n",
    "y_sc = LinearScale(min=-0.1, max=1.1)\n",
    "ax_x = Axis(label='x', scale=x_sc, grid_lines='solid')\n",
    "ax_y = Axis(label='y', scale=y_sc, orientation='vertical', side='left', grid_lines='solid')\n",
    "prob_line = Lines(x=xdata, y=ydata, colors=['Gray'], scales={'x': x_sc, 'y': y_sc}, visible=True)\n",
    "\n",
    "throughputs_slider = FloatLogSlider(value=throughputs[0], min=log_throughputs[0], max=log_throughputs[-1], step=logstep_throughput, description=\"Throughput\")\n",
    "jitters_slider = FloatSlider(value=jitters[0], min=jitters[0], max=jitters[-1], step=jitters[1]-jitters[0], description=\"Network Jitter stddev (ms)\")\n",
    "\n",
    "def on_change(change):\n",
    "    ydata = prob_overlap(xdata, jitters_slider.value, throughputs_slider.value, chosen_cro)\n",
    "    ax_y.scale = y_sc\n",
    "    prob_line.y = ydata\n",
    "    \n",
    "throughputs_slider.observe(on_change, names='value')\n",
    "jitters_slider.observe(on_change, names='value')\n",
    "\n",
    "fig = Figure(marks=[prob_line], axes=[ax_x, ax_y], title='CDF of Exponentially Modified Gaussian')\n",
    "VBox([throughputs_slider, jitters_slider, fig])"
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    "interval_steps = 100\n",
    "\n",
    "def Ej(x, num_digits, jitter, data_rate, cluster_ratio):\n",
    "    stddev = numpy.sqrt(data_rate)\n",
    "    integral_delta_bounds = numpy.linspace(-3*stddev, 3*stddev, interval_steps)\n",
    "    \n",
    "    total = []\n",
    "    for dri in integral_delta_bounds:\n",
    "        dri_rate = data_rate + dri\n",
    "        p = prob_overlap(0, jitter, dri_rate, cluster_ratio)\n",
    "        px = 10**(-2*num_digits)\n",
    "        poli = 1-st.binom.cdf(0, dri_rate, p*px)\n",
    "        total.append(poli)\n",
    "        \n",
    "    # data_rate*10**(-2*num_digits)\n",
    "    return total\n",
    "\n",
    "Ej(1, 6, 150, 10000, 100)"
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