Mean and Variance of Queue Length for M-M-1 Queue.ipynb

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    "Calculate expected stationary mean and variance of M/M/1 queue length. \n",
    "====================================\n",
    "The probability of the queue having length $n$ is $p_{n} = (1-\\rho)\\rho^n$ so the expected mean is:\n",
    "$$\n",
    "\\begin{align}\n",
    "E(\\bar{n}) &= \\sum_{n=0}^\\infty np_{n} \\\\\n",
    "&= \\sum_{n=0}^\\infty n(1-\\rho)\\rho^n \\\\\n",
    "&= (1-\\rho) \\rho \\sum_{n=0}^\\infty n\\rho^{n-1} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "Use this standard result\n",
    "$$\n",
    "\\boxed{\n",
    "\\begin{align}\n",
    "\\sum_{n=0}^\\infty n\\rho^{n-1} = \\frac{1}{(1-\\rho)^2}\n",
    "\\end{align}\n",
    "}\n",
    "$$\n",
    "so\n",
    "$$\n",
    "\\begin{align}\n",
    "&= (1-\\rho) \\rho \\sum_{n=0}^\\infty n\\rho^{n-1} \\\\\n",
    "&= (1-\\rho) \\rho \\frac{1}{(1-\\rho)^2} \\\\\n",
    "&= \\frac{\\rho}{1-\\rho} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "Finally we have the result:\n",
    "$$\n",
    "\\boxed{\n",
    "\\begin{align}\n",
    "E(\\bar{n}) = \\frac{\\rho}{1-\\rho}\n",
    "\\end{align}\n",
    "}\n",
    "$$\n",
    "\n",
    "To find the variance we first need to find this relation:\n",
    "$$\n",
    "\\begin{align}\n",
    "\\sum_{n=0}^\\infty n^2\\rho^n &= \\sum_{n=0}^\\infty (n^2\\rho^n-n\\rho^n+n\\rho^n) \\\\\n",
    "&= \\sum_{n=0}^\\infty (n(n-1)\\rho^n+n\\rho^n) \\\\\n",
    "&= \\sum_{n=0}^\\infty n(n-1)\\rho^n+ \\sum_{n=0}^\\infty n\\rho^n \\\\\n",
    "&= 2\\rho^2 \\sum_{n=0}^\\infty \\frac{n(n-1)}{2} \\rho^{n-2} + \\rho \\sum_{n=0}^\\infty n\\rho^{n-1} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "Use these standard results\n",
    "$$\n",
    "\\boxed{\n",
    "\\begin{align}\n",
    "\\sum_{n=0}^\\infty \\frac{n(n-1)}{2} \\rho^{n-2} = \\frac{1}{(1-\\rho)^3} \\\\\n",
    "\\sum_{n=0}^\\infty n\\rho^{n-1} = \\frac{1}{(1-\\rho)^2}\n",
    "\\end{align}\n",
    "}\n",
    "$$\n",
    "so\n",
    "$$\n",
    "\\begin{align}\n",
    "& 2\\rho^2 \\sum_{n=0}^\\infty \\frac{n(n-1)}{2} \\rho^{n-2} + \\rho \\sum_{n=0}^\\infty n\\rho^{n-1} \\\\\n",
    "&= \\frac{2\\rho^2}{(1-\\rho)^3} + \\frac{\\rho}{(1-\\rho)^2} \\\\\n",
    "&= \\frac{\\rho}{(1-\\rho)^2} \\left( \\frac{2\\rho}{1-\\rho} + 1 \\right) \\\\\n",
    "&= \\frac{\\rho}{(1-\\rho)^2} \\left( \\frac{2\\rho + (1-\\rho)}{1-\\rho} \\right) \\\\\n",
    "&= \\frac{\\rho}{(1-\\rho)^2} \\left( \\frac{1 + \\rho}{1-\\rho} \\right) \\\\\n",
    "&= \\frac{\\rho(1 + \\rho)}{(1-\\rho)^3}\n",
    "\\end{align}\n",
    "$$\n",
    "Giving the relation:\n",
    "$$\n",
    "\\boxed{\n",
    "\\begin{align}\n",
    "\\sum_{n=0}^\\infty n^2\\rho^n &= \\frac{\\rho(1 + \\rho)}{(1-\\rho)^3} \\\\\n",
    "\\end{align}\n",
    "}\n",
    "$$\n",
    "\n",
    "The variance then is\n",
    "$$\n",
    "\\begin{align}\n",
    "Var(\\bar{n}) &= \\sum_{n=0}^\\infty n^2 p_{n} - E(\\bar{n})^2 \\\\\n",
    "&= \\sum_{n=0}^\\infty n^2(1-\\rho)\\rho^n - \\left(\\frac{\\rho}{1-\\rho}\\right)^2 \\\\\n",
    "&= (1-\\rho) \\sum_{n=0}^\\infty n^2\\rho^n - \\left(\\frac{\\rho}{1-\\rho}\\right)^2 \\\\\n",
    "&= (1-\\rho) \\frac{\\rho(1 + \\rho)}{(1-\\rho)^3} - \\frac{\\rho^2}{(1-\\rho)^2} \\\\\n",
    "&= \\frac{\\rho + \\rho^2}{(1-\\rho)^2} - \\frac{\\rho^2}{(1-\\rho)^2} \\\\\n",
    "&= \\frac{\\rho}{(1-\\rho)^2}\\\\\n",
    "\\end{align}\n",
    "$$\n",
    "Finally we have the result:\n",
    "$$\n",
    "\\boxed{\n",
    "\\begin{align}\n",
    "Var(\\bar{n}) &= \\frac{\\rho}{(1-\\rho)^2}\n",
    "\\end{align}\n",
    "}\n",
    "$$\n"
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