{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "b4c02b95-328c-44b7-8c78-0b4b9c64ab8d",
   "metadata": {},
   "source": [
    "Math 420/620 Final Question 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c966a5e5-6512-4c16-95f0-c7486e12fafa",
   "metadata": {},
   "source": [
    "A Markov chain model of floods uses the state variable\n",
    "$X_n=0,1,2,3,4$ where state 0 means the average daily flow\n",
    "is below 1000 cubic feet per second, state 1 is 1000-2000 cfs,\n",
    "state 2 is 2000-5000 cfs, 3 is 5000-10000 cfs and 4 is\n",
    "over 10000 cfs.\n",
    "The state transition probability matrix for this model is\n",
    "$$\n",
    "    P=\\left[\\matrix{\n",
    "        0.9& 0.05&0.025&0.015&0.01\\cr\n",
    "        0.3&0.7&0&0&0\\cr\n",
    "        0&0.4&0.6&0&0\\cr\n",
    "        0&0&0.6&0.4&0\\cr\n",
    "        0&0&0&0.8&0.2\n",
    "    }\\right].\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0c6be044-c802-4d04-8977-3b8c9c3eec88",
   "metadata": {},
   "source": [
    "(i) Draw the state transition probability diagram for this model."
   ]
  },
  {
   "attachments": {
    "5794e50c-7ce4-4f14-baa6-773ef820f7d5.png": {
     "image/png": 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16k5Y06ZNt2zZMnPmTD6ff/78eV1dXbKdLGUuTRwOp8LeJJRZVfy7J2ilTE7x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l4qy5cvV1VVbfnkTZs2IYfyyZMnly5dIt0b5gRfjvxZZ+qGhoZnz56tW7fuZ1OYQkJCKioqKioq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\nAElFTkSuQmCC\n"
    }
   },
   "cell_type": "markdown",
   "id": "a317a866-c5f2-4869-9fa6-78831124c176",
   "metadata": {},
   "source": [
    "![q3states.png](attachment:5794e50c-7ce4-4f14-baa6-773ef820f7d5.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6abfaace-9326-4c0a-931e-dcad81b40b77",
   "metadata": {},
   "source": [
    "(ii) Find the steady-state probability distribution for this model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "52ca14a8-0e84-468e-bf0b-bb3568034839",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Matrix{Float64}:\n",
       " 0.9  0.05  0.025  0.015  0.01\n",
       " 0.3  0.7   0.0    0.0    0.0\n",
       " 0.0  0.4   0.6    0.0    0.0\n",
       " 0.0  0.0   0.6    0.4    0.0\n",
       " 0.0  0.0   0.0    0.8    0.2"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P=[0.9 0.05 0.025 0.015 0.01;\n",
    "    0.3 0.7 0 0 0;\n",
    "    0 0.4 0.6 0 0;\n",
    "    0 0 0.6 0.4 0;\n",
    "    0 0 0 0.8 0.2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8c218d0b-c3c2-42fe-98b2-9e205cf2bcbd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{ComplexF64}:\n",
       "    -0.941292284529958 + 0.0im\n",
       "   -0.3137640948433191 + 0.0im\n",
       "   -0.1176615355662446 + 0.0im\n",
       " -0.039220511855414884 + 0.0im\n",
       "  -0.01176615355662446 + 0.0im"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "K5=eigvecs(P')[:,5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d9c2e33d-4267-4722-b978-0471f49658a4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The steady state is as follows:\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5-element Vector{Float64}:\n",
       " 0.6611570247933886\n",
       " 0.22038567493112934\n",
       " 0.08264462809917347\n",
       " 0.027548209366391168\n",
       " 0.008264462809917347"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K5=real.(K5)\n",
    "println(\"The steady state is as follows:\")\n",
    "PI=K5/sum(K5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc9e789e-eaf9-442c-b8eb-56deed6a8594",
   "metadata": {},
   "source": [
    "(iii) How often are severe floods (over 10000 cfs) expected to occur?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b36c16e4-351f-439a-939d-e2253f041905",
   "metadata": {},
   "source": [
    "Since\n",
    "$$\n",
    "    \\lim_{N\\to\\infty} {1\\over N}\\sum_{n=1}^N {\\bf P}\\{ X_n=4 \\}=\\pi_4\\approx 0.008264462809917349.\n",
    "$$\n",
    "I would expect 0.8\\% chance each day for there to be a sever flood.\n",
    "\n",
    "Since there are 365 days in a year that means about $3.0165$ times a year."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8acad15e-d4d2-444c-a5de-1f75654d10ba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0165289256198315"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "365*PI[5]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c661861d-a51e-4df3-942b-369a15fba141",
   "metadata": {},
   "source": [
    "(iv) A reservoir used for drought storage depends on the flow of\n",
    "this river.  When the flow is over 5000 cfs, the reservoir is allowed\n",
    "to store 1000 acre-feet per day.  When the flow is under 1000 cfs\n",
    "the reservoir is required to release 100 acre-feet per day back\n",
    "into the river.  Find the expected annual number of acre-feet\n",
    "of water stored in the reservoir.\n",
    "Is this number positive or negative?  What does this mean?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a75c27e4-8e94-4d1b-bdfa-5768a4f66325",
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   "source": [
    "Let $R_n$ represent water flowing in or\n",
    "out of the reservoir on day $n$.  Thus,\n",
    "$$\n",
    "    R_n=\\cases{ 1000 & if $X_n\\ge 3$\\cr\n",
    "        -100 & if $X_n=0$\\cr\n",
    "        0 & otherwise.}\n",
    "$$\n",
    "The expected value of $R_n$ per day is\n",
    "$$\n",
    "    {\\bf E}[R_n]=\n",
    "        1000{\\bf P}\\{X_n\\ge 3\\}-100{\\bf P}\\{X_n=0\\}.\n",
    "$$\n",
    "Therefore, provided $n$ is large\n",
    "$$\n",
    "    {\\bf E}[R_n]\\approx 1000(\\pi_3+\\pi_4)-100\\pi_0.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "e7cc349d-e3ef-40e0-8a2f-6892b0170ab2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-30.30303030303034"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# All the indices are off by one\n",
    "ER=1000*(PI[4]+PI[5])-100*PI[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "185cff77-a4f4-4687-b167-c888117ea0f3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-11060.606060606075"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cf=ER*365"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e67f5259-5fa3-45c9-bb5f-39da91351915",
   "metadata": {},
   "source": [
    "The annual number of acre-feet of water stored in the reservoir\n",
    "is negative.  This means we expect to release 11060.61 acre-feet of water\n",
    "from the reservoir per year and the reservoir will eventually run out of water."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db45efec-ce21-4462-b3dc-4a7727f529ab",
   "metadata": {},
   "source": []
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