Traffic_Flow_Modelling.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Q2SxNwfXhFvT"
      },
      "source": [
        "# **MARKOV JUMP PROCESS APPLIED TO TRAFFIC FLOW MODELLING**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "lib41cn_iKEC"
      },
      "source": [
        "#### AUTHORS: Hoodaty Soumodeep, Kumar Akshita"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "nNNOoBVUhNaq"
      },
      "source": [
        "### **Jump Process**\n",
        "A jump process is a random variable $X: (\\Omega, A, \\mathbb{P}) \\rightarrow (\\chi, \\tau)$, where:\n",
        "$A$ is a $\\sigma$ algebra on $\\Omega$ and $\\mathbb{P}$ a probability measure on $(\\Omega, A)$\n",
        "The space $\\chi$ is the set of functions from $[0,\\infty)$ to the countable state space E, which are cadlag, i.e\n",
        "$$\\chi = \\{f:[0,\\infty) \\rightarrow E, \\; \\lim_{s→t, s<t}f(s) = f(t_{-}), \\; \\lim_{s→t, s\\geq t}f(s) = f(t) \\}$$\n",
        "The $\\sigma$ field $\\tau$ is the smallest $\\sigma$ field of $\\chi$ such that $f \\in \\chi → f(t)$ is measurable for any t $\\geq$ 0.\n",
        "\n",
        "\n",
        "### **Markov Jump Process**\n",
        "A markov jump process is a jump process that satisfies the Markov property, i.e $\\forall 0 \\leq t_{0} < \\ldots < t_{n}$, $s \\geq 0$, $x_{0}, \\ldots, x_{n}$, and $y \\in E$,\n",
        "$$\\mathbb{P}(X_{t_{n}+s}|X_{t_{n}} = x_{n}, X_{t_{n-1} = x_{n-1}}, ..., X_{t_{0}}=x_{0}) = \\mathbb{P}(X_{t_{n}+s}=y|X_{t_{n}} = x_{n}) = \\mathbb{P}(X_{s}=y|X_{0} = x_{n})$$\n",
        "\n",
        "\n",
        "### **Mesanthrope Process**\n",
        "A mesanthrope process is a jump process describing particles that evolve on a set of sites, and the movement is governed by their aversion to each other, or the tendency to avoid being in close proximity with one another.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uJ1fdsFR0KuL"
      },
      "source": [
        "### **Description of the Model**\n",
        "\n",
        "The chosen model is a totally asymmetric misanthrope markov jump process that describes the evolution of the number of vehicles on different sections of a road. The model is based on a mesoscopic approach, which is an intermediate approach between macroscopic and microscopic models that describe the traffic flow dynamics in an aggregate way but represent the individual\n",
        "driver-vehicle behaviors (i.e vehicles are individually considered like in a microscopic approach but their dynamics are aggregated per section as in a macroscopic one).\n",
        "\n",
        "In this model, vehicles act as particles and road sections as sites. The maximum number of vehicles per section is denoted as $N \\in \\mathbb{N}^{*}$. We consider the case where the road has a finite number of sections M, so that the state space $E = \\{0,1,,\\ldots,N\\}^{M}$. A vehicle can jump from one section of the road to the next, with a jump rate that depends on the number of vehicles in both the departure and arrival sections.\n",
        "\n",
        "The evolution of the system takes place as:\n",
        "If at a time instant t, $\\eta \\in E$ represents the state of the system, $x \\in \\{0,1,\\ldots,M\\}$ represents the section number and $\\eta(x)$ denotes the number of vehicles in section $x$, the probability that a vehicle moves from section x to x+1 during the time interval $[t,t+\\delta t]$ id $b_{x}(\\eta(x),\\eta(x+1) + \\epsilon(\\delta t)) x \\delta t$, where $\\lim_{\\delta t →0}\\epsilon(\\delta t)=0$. The times between jumps of a section x have an exponential distribution with parameter $b_{x}(x,x+1)$, where $b_{x}(0,n) = b_{x}(n,N)$ = 0. At every step, the section with the minimum jump time is chosen, a vehicle is transferred from this section to the next, the global time of the system updated with the obtained jump time, and new jump times of modified sections are calculated based on the new jump rates.\n",
        "\n",
        "\n",
        "#### **Assumptions:**\n",
        "**1)** The maximum number of vehicles in each section is N. Further, length of each vehicle is assumed to be 1 so that length of a section is considered in terms of the maximum number of vehicles it can hold.\n",
        "\n",
        "**2)** A vehicle can only jump from one section to the next, and two vehicles cannot jump simultaneously. Vehicles in the last section jump to the previous section.\n",
        "\n",
        "**3)** Jump times of a section x containing stricly positive number of vehicles follow an exponential distribution with parameter $b_{x}$ which is a function of the number of vehicles in the section of departure (x) and the section of arrival (x+1). It is in fact an increasing function of the former while a decreasing function of the latter.\n",
        "\n",
        "**4)** If the section with minimum jump time has no vehicles or if the next section contains maximum possible vehicles (that is has no more capacity), we choose the section with the next least jump time and follow the same procedure."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "sjGwREd9nPlt"
      },
      "source": [
        "#### **Parameters:**\n",
        "\n",
        "**1)** Jump rate function $b_{x}$\n",
        "\n",
        "**2)** Section size $D_{x}$\n",
        "\n",
        "**3)** Maximum number of vehicles (capacity) of the section $N_{x}$.\n",
        "\n",
        "For simplicity, we will assume $D_{x}=N_{x}$.\n",
        "\n",
        "The jump rate, which depends on the number of vehicles in the section of departure and arrival, is defined as a function of supply and demand functions of the current and next section. Every section x applies a pressure on the next section x+1. If $\\Delta_{x}$ denotes the demand expressed by section x, and $\\Sigma_{x+1}$ denotes the supply expressed by the next section x+1, the jump rate is then defined as:\n",
        "\n",
        "$$b_{x}(\\eta(x),\\eta(x+1) = min\\left\\{\\Delta_{x}\\left(\\frac{\\eta(x)}{D_{x}}\\right),\\Sigma_{x+1}\\left(\\frac{\\eta(x+1)}{D_{x+1}}\\right)\\right\\}$$\n",
        "\n",
        "where $\\Delta_{x}$ and $\\Sigma_{x}$ (denoting the demand and supply respectively as expressed by section x) are given as:\n",
        "\n",
        "$$\\Delta_{x}(\\rho) = \\begin{cases}\n",
        "  \\begin{eqnarray*}\n",
        "  Q_{x}(\\rho) && \\text{ if } \\rho < \\rho_{x}^{cr}\\\\\n",
        "  \\text{max}_{\\rho}Q_{x}(\\rho) && \\text{ otherwise}\n",
        "  \\end{eqnarray*}\n",
        "  \\end{cases}$$\n",
        "\n",
        "$$\\Sigma_{x}(\\rho) = \\begin{cases}\n",
        "  \\begin{eqnarray*}\n",
        "  \\text{max}_{\\rho}Q_{x}(\\rho) && \\text{ if } \\rho < \\rho_{x}^{cr}\\\\\n",
        "  Q_{x}(\\rho) && \\text{ otherwise}\n",
        "  \\end{eqnarray*}\n",
        "  \\end{cases}$$\n",
        "\n",
        "Here, $Q_{x}(\\rho)$ denotes the flow volume as a function of the density $\\rho$, and we use its definition from the macroscopic model by Edie and Underwood as:\n",
        "$$Q_{x}(\\rho) = V\\rho e^{-\\lambda \\rho}$$\n",
        "\n",
        "where $\\lambda$ is a qualified parameter of sensitivity, inverse of relaxation time. $\\rho_{cr}$ denotes the argmax of the flow volumn $Q_{x}$, and is thus the only critical point of the flow volume. Upon solving $\\frac{dQ_{x}}{dt}(\\rho) = 0$, we get $\\rho^{cr} = \\frac{1}{\\lambda}$.\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Qg_h2VsrnUaw"
      },
      "source": [
        "## **SIMULATION OF THE PROCESS**"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "TnElyi_gU0WQ"
      },
      "outputs": [],
      "source": [
        "#importing the required libraries\n",
        "import numpy as np\n",
        "import scipy as sc\n",
        "import random\n",
        "import scipy.stats as scs\n",
        "import matplotlib.pyplot as plt\n",
        "from numpy import zeros, cumsum, arange, append\n",
        "from numpy.random import poisson, rand, exponential\n",
        "from matplotlib.pyplot import figure, step, title, legend\n",
        "from scipy.optimize import minimize\n",
        "import math\n",
        "import scipy.linalg as linalg\n",
        "import sympy as sp"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import warnings\n",
        "warnings.filterwarnings(\"ignore\", category=DeprecationWarning)"
      ],
      "metadata": {
        "id": "rJ4mdK2ahWt1"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "First of all, we define a function 'jump' that defines the jump markov process and takes as an input the parameters eta (denoting initial state or initial number of vehicles in each road section), D (denoting length or maximum number of vehicles in each section, since length of each vehicle is assumed to be 1), V (denoting the average speeds of vehicles in each section) and T (denoting the time limit for the process)."
      ],
      "metadata": {
        "id": "eraARfssZfnh"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "Bq2-KmvIU8Uw"
      },
      "outputs": [],
      "source": [
        "def jump(eta1,D,V,T):\n",
        "    for elem1, elem2 in zip(eta1,D):\n",
        "      assert (elem1<=elem2),\"Initial number of vehicles in a section cannot be greater than section capacity\"\n",
        "    eta=eta1\n",
        "    cst=2 #parameter of sensitivity for definition of Q\n",
        "    rho_cr=1/cst #density threshold for which flow volume Q(rho) is maximal\n",
        "\n",
        "    def Q(x,rho): #defining the flow volume function\n",
        "        return V[x]*rho*math.exp(-cst*rho)\n",
        "\n",
        "    def delta(x,rho): #defining the demand function\n",
        "        initial_g=0\n",
        "        if rho<rho_cr:\n",
        "            return Q(x,rho)\n",
        "        else:\n",
        "            result = minimize(lambda y: -Q(x,y), initial_g)\n",
        "            return -result.fun\n",
        "\n",
        "    def sigma(x,rho): #defining the supply function\n",
        "        initial_g=0\n",
        "        if rho<rho_cr:\n",
        "            result = minimize(lambda y: -Q(x,y), initial_g)\n",
        "            return -result.fun\n",
        "        else:\n",
        "            return Q(x,rho)\n",
        "\n",
        "    def b(x): #defining the parameter of the exponential distribution\n",
        "        if x<len(eta)-1:\n",
        "            if delta(x+1,eta[x+1]/D[x+1])<=sigma(x,eta[x]/D[x]):\n",
        "                return delta(x+1,eta[x+1]/D[x+1])\n",
        "            else:\n",
        "                return sigma(x,eta[x]/D[x])\n",
        "        else:\n",
        "            if delta(x-1,eta[x-1]/D[x-1])<=sigma(x,eta[x]/D[x]):\n",
        "                return delta(x-1,eta[x-1]/D[x-1])\n",
        "            else:\n",
        "                return sigma(x,eta[x]/D[x])\n",
        "\n",
        "    lst=[] #list to store states after each iteration\n",
        "    cumtime = []#list to store the time increment arrays\n",
        "    gtime = 0 #variable to store global time of the system\n",
        "    T_n=np.zeros(len(eta)) #empty array to store jump times\n",
        "    ar=[] #empty array to store jump time increments\n",
        "    iterations = 0\n",
        "    max_it=1000\n",
        "    while(gtime<=T and iterations<=max_it):\n",
        "        times=[]\n",
        "        #generating jump time increments for all sections using random exponential distribution\n",
        "        for i in range(len(eta)):\n",
        "            if b(i)>0:\n",
        "                times.append(np.random.exponential(1/b(i)))\n",
        "            if b(i)<=0:\n",
        "                times.append(0)\n",
        "        ar.append(times)\n",
        "        np.array(ar)\n",
        "        T_n=np.sum(np.array(ar),axis=0)\n",
        "        lT_n = list(T_n)\n",
        "        lst.append(eta.copy())\n",
        "        idx=np.argmin(T_n) #selecting the state with the minimum jump time\n",
        "        gtime += lT_n[idx] #updating global time of system with the jump time of selected section\n",
        "        #updating the state eta\n",
        "        if idx<len(eta)-1: #if the section with minimum jump time is not the last section\n",
        "            if eta[idx]>0 and eta[idx+1]<D[idx+1]: #checking if selected section has >0 vehicles and if next section has capacity\n",
        "              eta[idx]=eta[idx]-1\n",
        "              eta[idx+1]=eta[idx+1]+1\n",
        "            elif eta[idx]==0: #if selected section has no vehicles choose another section with the next least jump time\n",
        "              tm=sorted(T_n)\n",
        "              for i in range (1,len(lT_n)):\n",
        "                smin = tm[i]\n",
        "                if eta[lT_n.index(smin)] == 0:\n",
        "                  continue\n",
        "                else:\n",
        "                  if lT_n.index(smin)<len(eta)-1 and  eta[lT_n.index(smin)+1]<D[lT_n.index(smin)+1]:\n",
        "                    eta[lT_n.index(smin)]=eta[lT_n.index(smin)]-1\n",
        "                    eta[lT_n.index(smin)+1]=eta[lT_n.index(smin)+1]+1\n",
        "                    break\n",
        "                  elif lT_n.index(smin)==len(eta)-1 and  eta[lT_n.index(smin)-1]<D[lT_n.index(smin)-1]:\n",
        "                    eta[lT_n.index(smin)]=eta[lT_n.index(smin)]-1\n",
        "                    eta[lT_n.index(smin)-1]=eta[lT_n.index(smin)-1]+1\n",
        "                    break\n",
        "\n",
        "        else: #if the section with minimum jump time is the last section\n",
        "            if eta[idx]>0 and eta[idx-1]<D[idx-1]:\n",
        "                eta[idx]=eta[idx]-1\n",
        "                eta[idx-1]=eta[idx-1]+1\n",
        "            elif eta[idx]==0:\n",
        "              tm=sorted(T_n)\n",
        "              for i in range (1,len(eta)):\n",
        "                smin = tm[i]\n",
        "                if eta[lT_n.index(smin)] == 0:\n",
        "                  continue\n",
        "                elif eta[lT_n.index(smin)]>0 and eta[lT_n.index(smin)+1]<D[lT_n.index(smin)+1]:\n",
        "                   eta[lT_n.index(smin)]=eta[lT_n.index(smin)]-1\n",
        "                   eta[lT_n.index(smin)+1]=eta[lT_n.index(smin)+1]+1\n",
        "                   break\n",
        "\n",
        "        cumtime.append(np.cumsum(ar,axis=0))\n",
        "        iterations+=1\n",
        "    lst.append(eta.copy())\n",
        "\n",
        "    return lst, cumtime"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Now that we have defined our jump process, we try to simulate it using different values of initial state eta, the list D containing lengths (or vehicle capacities) of sections, the average speeds of vehicles in each section contained in the list V."
      ],
      "metadata": {
        "id": "Oj-pS6AMtkFC"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "jwBuJBCDVFhy"
      },
      "outputs": [],
      "source": [
        "eta = np.array([10,6,9,6,8,4,5,8,1,7]) #initial state of the process for a road with 10 sections, eta represents the\n",
        "#array denoting initial number of vehicles on each of the 10 sections\n",
        "D = np.array([10,10,10,10,10,10,10,10,10,10]) #array representing length of each section, assumed to be equal to the\n",
        "#maximum number of vehicles on each section\n",
        "V = [20,30,30,69,34,56,23,48,89,17] #array representing average speed of vehicles on each of the 10 sections\n",
        "T = 50\n",
        "states,times = jump(eta,D,V,T)"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We observe the evolution of the number of vehicles on each section by means of subplots for each section."
      ],
      "metadata": {
        "id": "edHGaek35MVT"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "fb0b9AwPj9HD",
        "outputId": "a26e2800-3974-4fce-babd-b6d1a6964bd7"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1000x2500 with 10 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "\n",
        "# Create x-axis values (time steps)\n",
        "x = range(len(states))\n",
        "\n",
        "# Concatenate values for each section\n",
        "concatenated_states = np.array([np.array([state[i] for state in states]) for i in range(len(states[0]))])\n",
        "# Plot each section in a separate subplot\n",
        "num_sections = len(states[0])\n",
        "fig, axs = plt.subplots(num_sections, 1, figsize=(10, 25), sharex=True)\n",
        "\n",
        "for i, section in enumerate(concatenated_states):\n",
        "    axs[i].step(x, section, where='post')\n",
        "    axs[i].set_ylabel('Number of Cars')\n",
        "    axs[i].set_title(f'Section {i+1}')\n",
        "    axs[i].set_xlabel('Time Step')  # Set x-label for each subplot\n",
        "    axs[i].set_ylim(0, 20)\n",
        "\n",
        "plt.tight_layout()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We observe from the simulation that while some sections observe a lot of changes in the number of vehicles over the chosen time duration, some others do not change at all from the initial states. The observed behavior changes as the values of eta, D and V are changed as these influence the values of the flow volume function and consequently the demand and supply functions and hence the function for parameter of the exponential distribution."
      ],
      "metadata": {
        "id": "jbW28QViaIA-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Particular Case: Road Containing Three Sections"
      ],
      "metadata": {
        "id": "YFe5PCWtDoD0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We now limit ourselves to the analysis of the case where the road contains 3 sections.\n",
        "\n",
        "We try to find the expected number of cars in each section using Monte-Carlo approximation.\n",
        "\n",
        "###**Monte Carlo Approximation**\n",
        "The Monte-Carlo approach to approximate the expectation of a random variable involves the usage of the law of large numbers. The law of large numbers states that if we have a sequence of n independent and identically distributed random variables $(X_{n})_{n \\geq 0}$, then the empirical average of these iid random variables converges almost surely as $n → \\infty$ to $\\mathbb{E}(X_{1})$, i.e.,\n",
        "\n",
        "$$\\frac{\\sum_{i=1}^{n}X_{i}}{n} \\xrightarrow{n \\to \\infty} \\mathbb{E}(X_{1}) \\quad \\text{a.s.}$$.\n",
        "\n",
        "So, we simulate n independent paths of our process $(\\eta_{t})_{t\\geq 0}$ for large n (representing number of simulations) and compute the empirical average of the obtained values of the process at some time instant t. For large n, this approximates the expectation of the number of vehicles in each section, i.e gives us an approximation of $\\mathbb{E}(\\eta_{t})$."
      ],
      "metadata": {
        "id": "gdCKjlPs7PeB"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "eta = np.array([10,6,3]) #list representing initial number of vehicles on each of the three sections\n",
        "D = np.array([10,10,10]) # list representing length (or max number of vehicles) of each section\n",
        "V = [20,30,69] # list representing average speeds of vehicles on each section\n",
        "T = 50 #global time value\n",
        "states,times = jump(eta,D,V,T)"
      ],
      "metadata": {
        "id": "eAIXLhc170pm"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "hKsnpDSpVK9i"
      },
      "outputs": [],
      "source": [
        "def monte_carlo_expectation(eta, D, V, T, num_simulations):\n",
        "    final_states = []\n",
        "    for _ in range(num_simulations):\n",
        "        states,times = jump(eta, D, V, T)\n",
        "        final_state=states[-1]  # Get the final state of the jump process\n",
        "        final_states.append(final_state)\n",
        "    # Calculating the average final state\n",
        "    avg_final_state = np.mean(final_states, axis=0)\n",
        "    return avg_final_state"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eR5ymc_5VMVm",
        "outputId": "a9eeeaf1-ffbc-475f-d355-33d731c0f16b"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([0.266, 9.732, 9.002])"
            ]
          },
          "metadata": {},
          "execution_count": 8
        }
      ],
      "source": [
        "monte_carlo_expectation(eta,D,V,T,500)"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "##Infinitesimal Generator $A$ and Matrix of Transition Probabilities $P_{t}$\n",
        "\n",
        "The **infinitesimal generator** of the process is defined as the matrix of transition rates, and can be expressed as\n",
        "$$Af(\\eta) = \\sum_{x=1}^{M}b_{x}(\\eta(x),\\eta(x+1))[f(\\eta^{x})-f(\\eta)]$$ for all bounded functions $f$, where $\\eta$ denotes the current state and $\\eta^{x}$ denotes the next state, i.e\n",
        "\n",
        "$$\\begin{eqnarray*}\n",
        "\\eta^{x}(z) && = && \\eta(z), && \\text{if} && z\\neq x,x+1\\\\\n",
        "\\eta^{x}(x) && = && \\eta(x)-1 \\\\\n",
        "\\eta^{x}(x+1) && = && \\eta(x)+1\n",
        "\\end{eqnarray*}$$\n",
        "\n",
        "\n",
        "The elements of the infinitesimal generator matrix $A$ represent the rates of transition from one section to the other if that is possible, taking into considerations all of the necessary assumptions. We now describe the construction of $A$ for 3 sections.\n",
        "\n",
        "Let $\\eta = (i,j,k)$ denote the current state, i.e section 1 contains i vehicles initially, section 2 contains j and section 3 contains k vehicles. There are 3 possibilities that we need to consider all of them greater than or equal to 0 and less than or equal to the maximum number of vehicles possible in each section:\n",
        "\n",
        "**1)** The minimum jump time is for section 1, and the next state $\\eta^{x}$ has $[i-1,j+1,k]$ vehicles section-wise, given that $j+1$ is less than or equal to the maximum capacity in the second section.\n",
        "\n",
        "**2)** The minimum jump time is for section 2, and the next state $\\eta^{x}$ has $[i,j-1,k+1]$ vehicles section-wise, given that $k+1$ is less than or equal to the maximum capacity in the third section.\n",
        "\n",
        "**3)** The minimum jump time is for section 3, and the next state $\\eta^{x}$ has $[i,j+1,k-1]$ vehicles section-wise, given that $j+1$ is less than or equal to the maximum capacity in the second section.\n",
        "\n",
        "For simplicity and faster computations, we take D=2, i.e the maximum number of vehicles in each of the sections is 2. We have generated the state space which contains 27 elements, since each of the three sections can contain 0,1 or 2 elements.\n",
        "\n",
        "In this case, for any $x, \\eta = [i\\hspace{0.3cm}j\\hspace{0.3cm}k]$.\n",
        "\n",
        "\n",
        "$$A_{\\eta \\eta^{x}} = \\begin{cases}\n",
        "b_{0}(\\eta(0),\\eta(1)); \\; \\eta^{x} = [i-1\\hspace{0.3cm}j+1\\hspace{0.3cm}k]\\\\\n",
        "\\\\\n",
        "b_{1}(\\eta(1),\\eta(2)); \\; \\eta^{x} = [i\\hspace{0.3cm}j-1\\hspace{0.3cm}k+1]\\\\\n",
        "\\\\\n",
        "b_{2}(\\eta(2),\\eta(1)); \\; \\eta^{x} = [i\\hspace{0.3cm}j+1\\hspace{0.3cm}k-1] \\\\\n",
        "\\\\\n",
        "- \\sum_{j=1, j \\neq i}^{27}A_{ij}; \\; \\eta^{x} = [i\\hspace{0.3cm}j\\hspace{0.3cm}k]\\\\\n",
        "\\\\\n",
        "0; \\; \\text{otherwise}\n",
        "\\end{cases}\n",
        "$$\n",
        "\n",
        "The diagonal elements of $A$ are generated such that $A$ follows the property of the infinitesimal generator that the elements in each row sum up to 0, i.e the diagonal elements are negative of the sum of the elements over the non-diagonal elements."
      ],
      "metadata": {
        "id": "gxBYS3R-0dVj"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# redefining the flow volume, demand, supply and transition rate functions to be accessed outside the jump process function\n",
        "\n",
        "cst=2\n",
        "rho_cr=1/cst #critical threshold for vehicle density\n",
        "\n",
        "def Q(x,rho): #defining the flow volume function\n",
        "    return V[x]*rho*math.exp(-cst*rho)\n",
        "\n",
        "def delta(x,rho): #defining the demand function\n",
        "    initial_g=0\n",
        "    if rho<rho_cr:\n",
        "        return Q(x,rho)\n",
        "    else:\n",
        "        result = minimize(lambda y: -Q(x,y), initial_g)\n",
        "        return -result.fun\n",
        "def sigma(x,rho): #defining the supply function\n",
        "    initial_g=0\n",
        "    if rho<rho_cr:\n",
        "        result = minimize(lambda y: -Q(x,y), initial_g)\n",
        "        return -result.fun\n",
        "    else:\n",
        "        return Q(x,rho)\n",
        "\n",
        "def b(x,eta): #defining the parameter of the exponential distribution\n",
        "        if x<len(eta)-1:\n",
        "            if delta(x+1,eta[x+1]/D[x+1])<=sigma(x,eta[x]/D[x]):\n",
        "                return delta(x+1,eta[x+1]/D[x+1])\n",
        "            else:\n",
        "                return sigma(x,eta[x]/D[x])\n",
        "        else:\n",
        "            if delta(x-1,eta[x-1]/D[x-1])<=sigma(x,eta[x]/D[x]):\n",
        "                return delta(x-1,eta[x-1]/D[x-1])\n",
        "            else:\n",
        "                return sigma(x,eta[x]/D[x])"
      ],
      "metadata": {
        "id": "RXrWoF6rKfm6"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "We now define the infinitesimal generator matrix according to the definition above, based on the transition rates from one state to another over the whole state space of 27 states. The resulting matrix has a dimension of 27x27.\n"
      ],
      "metadata": {
        "id": "-MG-1fzCz7cC"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#Defining the transition matrix\n",
        "D=[2,2,2]\n",
        "state_space = [(i,j,k) for i in range(3) for j in range(3) for k in range(3)]#generating the state space\n",
        "generator = np.zeros((len(state_space),len(state_space)))#initialising the generator matrix\n",
        "for p in range(1,len(state_space)):\n",
        "  s = 0\n",
        "  for q in range (0,len(state_space)):\n",
        "    if (state_space[q][0] == state_space[p][0]-1) and (state_space[q][1] == state_space[p][1]+1) and (state_space[q][2] == state_space[p][2]): #(i-1,j+1,k)\n",
        "      generator[p,q] = b(0,state_space[p])\n",
        "      s += generator[p,q]\n",
        "    if (state_space[q][0] == state_space[p][0]) and (state_space[q][1] == state_space[p][1]-1) and (state_space[q][2] == state_space[p][2]+1): #(i,j-1,k+1)\n",
        "      generator[p,q] = b(1,state_space[p])\n",
        "      s += generator[p,q]\n",
        "    if (state_space[q][0] == state_space[p][0]) and (state_space[q][1] == state_space[p][1]+1) and (state_space[q][2] == state_space[p][2]-1): #(i,j+1,k-1)\n",
        "      generator[p,q] = b(2,state_space[p])\n",
        "      s += generator[p,q]\n",
        "  generator[p,p] = -s"
      ],
      "metadata": {
        "id": "Eym66bBH1tH0"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"The infinitesimal generator A is\\n\",generator)"
      ],
      "metadata": {
        "id": "goo3Db3h5TrP",
        "outputId": "7bd2a744-2bc7-42aa-9311-4b8f30e7ada7",
        "colab": {
          "base_uri": "https://localhost:8080/"
        }
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The infinitesimal generator A is\n",
            " [[  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.          -0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.          -0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.          -0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           5.51819162   0.         -11.03638324\n",
            "    0.           5.51819162   0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "   -5.51819162   0.           5.51819162   0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.          -0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    4.0600585    0.          -4.0600585    0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.          -0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "   -0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.          -0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           3.67879441   0.           0.           0.\n",
            "    0.           0.          -3.67879441   0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           3.67879441   0.           0.\n",
            "    0.           5.51819162   0.         -14.71517765   0.\n",
            "    5.51819162   0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           3.67879441   0.\n",
            "    0.           0.           0.           0.          -9.19698603\n",
            "    0.           5.51819162   0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "   -0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           4.0600585\n",
            "    0.          -4.0600585    0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.          -0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.          -0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "   -0.           0.           0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    2.70670566   0.           0.           0.           0.\n",
            "    0.          -2.70670566   0.           0.           0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           2.70670566   0.           0.           0.\n",
            "    5.51819162   0.         -13.7430889    0.           5.51819162\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           2.70670566   0.           0.\n",
            "    0.           0.           0.          -8.22489728   0.\n",
            "    5.51819162   0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.          -0.\n",
            "    0.           0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           4.0600585    0.\n",
            "   -4.0600585    0.        ]\n",
            " [  0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.           0.           0.           0.\n",
            "    0.           0.        ]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "The zero rows in this matrix are due to the transition rate $b_{x}(\\eta(x),\\eta(x+1))$ taking value 0 for some values of $x$ and $\\eta$. This is often due to the supply or demand functions $\\Sigma_{x}$ and $\\Delta_{x}$ taking the values 0 here."
      ],
      "metadata": {
        "id": "N1YAHn-lkw7B"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Next, we wish to define the matrix of transition probabilities $P_{t}$.\n",
        "\n",
        "---\n",
        "\n",
        "**Theorem**\n",
        "\n",
        "Let $u(t) = (\\mathbb{P}(\\eta_{t}=p))_{p \\in E}$. Then $u(t)$ satisfies the differential equation $u'(t) = u(t)A$, where $A$ is the infinitesimal generator such that $A_{xy} = \\lambda(x)P(x,y)$ if $y \\neq x$ and $A_{xx} = -\\lambda(x)$.\n",
        "\n",
        "---\n",
        "**Corollary**\n",
        "\n",
        "$\\forall t \\geq 0$,\n",
        "$$u(t) = u(0)e^{At}$$\n",
        "Hence $\\forall t \\geq 0, \\forall j \\geq 0$,\n",
        "$$\\mathbb{P}(X_{t}=j) = \\sum_{i \\in E}^{} \\mu_{0}(i) (e^{At})_{ij}$$\n",
        "which means\n",
        "$$\\mathbb{P}(X_{t}=j|X_{0}=i) = P_{t}(i,j) = \\mu_{0}(i) (e^{At})_{ij}$$\n",
        "where $\\mu_{0}$ denotes the initial distribution of the process\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "aJFRNaOB7hl2"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We define a function to compute the matrix exponential $e^A$, acting on a vector $x$, using a series expansion and recursion for convergence. Then we utilise that function to generate the value of $e^{tA}$ for any given $t$."
      ],
      "metadata": {
        "id": "z5qPakecgXmF"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def exp(A, x):\n",
        "\n",
        "    Ax = A * x\n",
        "    if np.max(np.absolute(Ax)) > 1e-4:\n",
        "        sqrt = exp(A, x / 2) # Recursively call exp with A and x divided by 2\n",
        "        return sqrt @ sqrt\n",
        "    sq = Ax @ Ax\n",
        "    return np.eye(A.shape[0], dtype=np.float128) + Ax + sq / 2 + sq @ Ax / 6 # Return the exponential approximation using the Taylor series expansion\n",
        "\n",
        "exptA = exp(generator,2)"
      ],
      "metadata": {
        "id": "XLhf3DWJ1-dy"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(exptA)"
      ],
      "metadata": {
        "id": "oao_rbB52A0x",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "af48705a-d0d4-41db-e22d-f2897a4903a6"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 1.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 1.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 1.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 5.00000000e-01 0.00000000e+00\n",
            "  2.59369749e-10 0.00000000e+00 5.00000000e-01 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 4.23883118e-01 0.00000000e+00 5.76116882e-01\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 1.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 4.23883113e-01 0.00000000e+00 5.76116887e-01\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 1.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 1.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 1.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 9.99362266e-01 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  6.37734297e-04 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 1.05970777e-01 0.00000000e+00 1.44029223e-01\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 3.75000000e-01\n",
            "  0.00000000e+00 1.65408985e-13 0.00000000e+00 3.75000000e-01\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  9.35613173e-01 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 2.17595052e-02 0.00000000e+00\n",
            "  4.26273218e-02 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 1.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  9.07194958e-01 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 3.13634307e-02 0.00000000e+00\n",
            "  6.14416110e-02 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 1.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 1.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 1.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  1.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 9.95543588e-01\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 4.45641196e-03 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  1.76854384e-01 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 6.79141206e-03 0.00000000e+00\n",
            "  1.33045171e-02 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  4.01524843e-01 0.00000000e+00 1.15585845e-12 0.00000000e+00\n",
            "  4.01524843e-01 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 8.80729051e-01 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 4.28021515e-02\n",
            "  0.00000000e+00 7.64687976e-02 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  1.00000000e+00 0.00000000e+00 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 8.43220653e-01 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 5.62625970e-02\n",
            "  0.00000000e+00 1.00516750e-01 0.00000000e+00]\n",
            " [0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 0.00000000e+00 0.00000000e+00\n",
            "  0.00000000e+00 0.00000000e+00 1.00000000e+00]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We can multiply the entries of the obtained matrix exponential above with $\\mu_{0}(i)$ (where i denotes the initial state) to obtain the transition matrix $P_{t}$."
      ],
      "metadata": {
        "id": "v3KwyXK-LCOn"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## **Special Case: Single Section With Open Boundaries**\n",
        "\n",
        "We now limit ourselves to the analysis of a single section with open boundaries, containing n vehicles and having an entry rate of $\\lambda(n)$ and an exit rate of $\\mu(n)$. We denote the markov jump process defining the number of vehicles in this section at any instant $t \\in \\mathbb{R}^{+}$ by $\\eta_{t}$, and the maximum number of vehicles (capacity) of this section by N. We define the entry and exit rates for the section respectively as:\n",
        "$$\\lambda(n)=\\text{min}\\left\\{\\alpha,\\Sigma\\left(\\frac{n}{D}\\right)\\right\\}$$\n",
        "$$\\mu(n) = \\text{min}\\left\\{\\Delta\\left(\\frac{n}{D}\\right),\\beta\\right\\}$$\n"
      ],
      "metadata": {
        "id": "rgUf9uVauVoE"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "![n.png](data:image/png;base64,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)\n"
      ],
      "metadata": {
        "id": "TeUWhBDc6zZT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "###**Assumptions:**\n",
        "**1)** We take $\\alpha$ to be the maximal fow volume of the section.(upstream demand)\n",
        "\n",
        "**2)** We take $\\beta$ to be the mean flow volume of the section.(downstream supply)\n",
        "\n",
        "**3)** $\\Delta$ and $\\Sigma$ represent the demand and supply of the section as defined earlier.\n",
        "\n",
        "**CLAIM**: The markov process is irreducible since for all states x and y, $\\mathbb{P}(\\eta_{t}=y|\\eta_{0}=x) = P_{t}(x,y) > 0 $ for some $t>0$.\n",
        "\n",
        "\n",
        "**PROOF:** Let $\\eta_{0} = x \\in E$.\n",
        "\n",
        "We want to prove that $\\forall y \\in E$ such that $y \\neq x$ and $0 \\leq y \\leq D$, where $D$ denotes the maximum capacity of the section (in terms of number of vehicles),\n",
        "$$\\mathbb{P}(\\eta_{t}=y|\\eta_{0}=x)>0 \\text{ for some } t>0$$\n",
        "We take $y \\neq x$ since this is a markov jump process and jumps can only happen to some state different from the current one.\n",
        "\n",
        "*Case 1:* $y>x$\n",
        "\n",
        "To show:\n",
        "$$\\exists t>0 \\text{ such that } \\mathbb{P}(\\eta_{t}=y|\\eta_{0}=x)>0$$\n",
        "\n",
        "We claim that this holds for $t = T_{y-x}$.\n",
        "\n",
        "$$\\mathbb{P}(\\eta_{T_{y-x}}=y|\\eta_{0}=x) > \\mathbb{P}(\\eta_{T_{y-x}}=y|\\eta_{T_{y-x-1}}=y-1, \\ldots, \\eta_{0}=x) \\mathbb{P}(\\eta_{T_{y-x-1}}=y-1|\\eta_{T_{y-x-2}}=y-2,\\ldots,\\eta_{0}=x) \\ldots \\mathbb{P}(\\eta_{T_{1}}=x+1|\\eta_{0}=x) > 0$$\n",
        "\n",
        "Using the Markov property,\n",
        "\n",
        "$$ = \\mathbb{P}(\\eta_{T_{y-x}}=y|\\eta_{T_{y-x-1}}=y-1) \\mathbb{P}(\\eta_{T_{y-x-1}}=y-1|\\eta_{T_{y-x-2}}=y-2) \\ldots \\mathbb{P}(\\eta_{T_{1}}=x+1|\\eta_{0}=x) ⟶ (*)$$\n",
        "\n",
        "The last inequality follows since $\\forall x \\in E$,\n",
        "$$\\mathbb{P}(\\eta_{T_{1}}=x+1|\\eta_{0}=x) = \\mathbb{P}(T_{1}-T_{0}<Y_{1})=\\mathbb{P}(X_{1}<Y_{1}) ⟶ (**)$$\n",
        "\n",
        "where $X_{1}=T_{1}-T_{0}$ denotes the first arrival time (first jump time increment) following $Exp(\\lambda)$ and $Y_{1}$ denotes the first departure time following $Exp(\\mu)$.\n",
        "\n",
        "The last equality in $(**)$ is true as when we are going from $x$ to $x+1$ in only one jump, then the interarrival time would be surely less than the first time to depart (since then the cars in the system would reduce by 1), i.e., the interarrival time is less than the departure time of the first car. But the interarrival time between the first two jumps would be $T_1-T_0$ and the first departure time would be $Y_1$.\n",
        "\n",
        "Also, we know that\n",
        "\n",
        "$$\\mathbb{P}(X_{1}< Y_{1}) = \\int_{0}^{\\infty}\\int_{0}^{y}\\lambda e^{-\\lambda x}\\mu e^{-\\mu y}dx dy = \\frac{\\lambda}{\\lambda+\\mu} > 0$$\n",
        "\n",
        "Again, using the Markov property, each of the terms in $(*)$ is greater than 0 and so, $\\mathbb{P}(\\eta_{T_{y-x}}=y|\\eta_{0}=x) > 0$.\n",
        "\n",
        "\n",
        "\n",
        "*Case 2:* $y<x$\n",
        "\n",
        "This can be handled similarly, using the fact that,\n",
        "\n",
        "$$\\mathbb{P}(Y_{1}< X_{1}) = \\int_{0}^{\\infty}\\int_{0}^{x}\\lambda e^{-\\lambda x}\\mu e^{-\\mu y}dx dy = \\frac{\\mu}{\\lambda+\\mu} > 0$$\n",
        "\n",
        "We have hence proved that the Markov process is irreducible.\n",
        "\n",
        "Moreover, since the state space $E = \\{0,1,\\ldots,N\\}$ is finite, the markov process is positive recurrent and hence admits a unique stationary probability $m$."
      ],
      "metadata": {
        "id": "A4fXe8x_LIbK"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "From the definition of ergodic behaviour as done in class, we know that:\n",
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "***Theorem:***\n",
        "There exists $C, \\lambda >0$ such that $\\forall t \\geq 0 $ and any initial condition $\\eta_{0}$, the following holds:\n",
        "$$\\sum_{y \\in E}\\vert \\mathbb{P}(\\eta_{t}=y \\vert \\eta_{0}=x)- m(y)   \\vert \\leq Ce^{-\\lambda t}$$\n",
        "\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "As t $→ \\infty$ the error term $Ce^{-\\lambda t}$ goes to $0$ and hence we have,\n",
        "\n",
        "$$m(n) = \\lim_{{t \\to \\infty}}\\mathbb{P}_{x}({\\eta_{t} = n})\\hspace{1cm}∀ 0 \\leq n \\leq D$$\n",
        "\n",
        "The distribution is reversible for the process. One has:\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "***Definition:***\n",
        "\n",
        "A distribution $\\pi_{i}$ is said to be reversible if $\\forall i,j \\in E$,\n",
        "$$\\pi_i q_{ij} = \\pi_j q_{ji}$$\n",
        "\n",
        "Where $q_{ij}$ represents the transition rate from state *i* to state *j* in one unit of time.\n",
        "\n",
        "\n",
        "\n",
        "---\n",
        " For our process, the distribution m is reversible. Hence, we have:\n",
        "\n",
        "$$m(n-1)λ(n-1) = m(n)μ(n) \\hspace{1cm} ∀ n = 1, . . . , D$$\n",
        "\n",
        "\n",
        "One obtains : $$m(n) = m(0)\\prod_{i=1}^{n} \\frac{\\lambda(i-1)}{\\mu(i)}\\hspace{1cm} ∀1\\leq n\\leq D$$\n",
        "\n",
        " with $$m(0) = \\frac{1}{1+\\sum_{n=1}^{N}\\prod_{i=1}^{n}\\frac{\\lambda(i-1)}{\\mu(i)}}$$\n",
        "\n",
        "A is the generator matrix given by:\n",
        "\n",
        "$$A = \\begin{bmatrix}\n",
        "-\\lambda(0) & \\lambda(0) & 0\\\\\n",
        "\\mu(1) & -\\lambda(1)-\\mu(1) & \\lambda(1)\\\\\n",
        "0 & \\mu(2) & -\\mu(2)\n",
        "\\end{bmatrix}$$\n",
        "\n",
        "The columns and rows in this matrix represent the state space(number of cars possible in the section). We take one row at a time and calculate the elements corresponding to that row for every column by calculating the entry or exit rate for the transition from the state in the row to that of the column. The diagonal elements are taken as the negative sum of the rest of the elements in the row, by the usual definition of a generator matrix.\n"
      ],
      "metadata": {
        "id": "HKy_aeJC-A6r"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let us consider, $$m = [m(1)\\hspace{0.3cm}m(2)\\hspace{0.3cm}m(3)]$$\n",
        "\n",
        "---\n",
        "\n",
        "\n",
        "***Theorem:***\n",
        "Let $m$ be the stationary distribution and $A$ be the generator matrix of a continuous-time Markov process. Then,\n",
        "\n",
        "$$mA=0$$\n",
        "\n",
        "$0$ represents the zero matrix.\n",
        "\n",
        "---\n",
        "\n",
        "We calculate the value of $m$ using the equation $mA=0$, as follows:\n",
        "\n",
        "$$\\begin{bmatrix}\n",
        "m(1) & m(2) & m(3)\\\\\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}\n",
        "-\\lambda(0) & \\lambda(0) & 0\\\\\n",
        "\\mu(1) & -\\lambda(1)-\\mu(1) & \\lambda(1)\\\\\n",
        "0 & \\mu(2) & -\\mu(2)\n",
        "\\end{bmatrix}=0$$\n",
        "\n",
        "We observe that $A$ does not contain the values $\\lambda(2)$ and $\\mu(0)$.\n",
        "\n",
        "From this we get three equations:\n",
        "\n",
        "$$-\\lambda(0)m(1) + \\mu(1)m(2) = 0 ⟶(I)$$\n",
        "\n",
        "$$\\lambda(0)m(1) + (-\\lambda(1)-\\mu(1))m(2) + \\mu(2)m(3) = 0⟶(II)$$\n",
        "\n",
        "$$\\lambda(1)m(2) - \\mu(2)m(3) = 0⟶(III)$$\n",
        "\n",
        "Solving these, we get the following values:\n",
        "$$m(1) = \\frac{\\mu(1)}{\\lambda(0)\\left( 1 + \\frac{\\lambda(1)}{\\mu(2)} + \\frac{\\mu(1)}{\\lambda(0)}  \\right)}$$\n",
        "\n",
        "$$m(2) = \\frac{1}{1 + \\frac{\\lambda(1)}{\\mu(2)} + \\frac{\\mu(1)}{\\lambda(0)}}$$\n",
        "\n",
        "$$m(3) = \\frac{\\lambda(1)}{\\mu(2)\\left( 1 + \\frac{\\lambda(1)}{\\mu(2)} + \\frac{\\mu(1)}{\\lambda(0)}\\right )}$$"
      ],
      "metadata": {
        "id": "sMrDOOHbRL6j"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Using the following initial conditions we can estimate the value of $m$.\n",
        "\n",
        "We consider:\n",
        "\n",
        "\n",
        "*   $D$(maximum number of cars in the section) = $2$\n",
        "\n",
        "*   $\\eta_0$(initial number of cars in the section) $=1$\n",
        "\n",
        "*   $T $(maximum global time) $= 50$\n",
        "\n",
        "*   $\\alpha = Q(\\rho_{cr})$ (maximal flow volume)\n",
        "\n",
        "*   $\\beta = Q(\\rho_{cr})/2$ (mean flow volume)\n",
        "\n",
        "*   $n = 0,1,2$\n",
        "\n",
        "From these values we get:\n",
        "\n",
        "* $\\lambda(1) = 1.83939$\n",
        "\n",
        "* $\\lambda(0) = 1.83939$\n",
        "\n",
        "* $\\mu(1) = 0.91969$\n",
        "\n",
        "* $\\mu(2) = 0.91969$\n",
        "\n",
        "\n",
        "We have seen that the infinitesimal generator $A$ of this process contains only the values $\\lambda(0)$, $\\lambda(1)$, $\\mu(0)$ and $\\mu(1)$. From above we observe that in our case, $\\lambda(1) = \\lambda(0)$ and $\\mu(1) = \\mu(2)$. Hence, we can take the entry and departure rates to be constant equal to $\\alpha$ and $\\beta$ respectively.\n",
        "\n",
        "\n",
        "Substituting the values of $\\lambda(0), \\lambda(1), \\mu(0), \\mu(1)$ in the obtained values of m above, we get\n",
        "\n",
        "$$m(0) = 0.14285$$\n",
        "$$m(1) = 0.2857$$\n",
        "$$m(2) = 0.5714$$\n",
        "\n",
        "$\\therefore$ $$m = [0.14285 \\hspace{0.3cm} 0.2857 \\hspace{0.3cm} 0.5714]$$\n",
        "\n",
        "---"
      ],
      "metadata": {
        "id": "qwSyeQZvKOOD"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "We now try to define a function m for the stationary probability that takes as parameters n and D, that is the maximum number of vehicles that can be accommodated in the section. To do so, we use the formula as defined previously:\n",
        "$$m(n) = m(0)\\prod_{i=1}^{n} \\frac{\\lambda(i-1)}{\\mu(i)}\\hspace{1cm} ∀1\\leq n\\leq 2$$"
      ],
      "metadata": {
        "id": "mdjo69TFONWm"
      }
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "tdiLlbDOhJd5"
      },
      "outputs": [],
      "source": [
        "#Defining the flow volume, demand, supply and transition rate functions for this case of a single section of the road.\n",
        "\n",
        "cst=2 #parameter of sensitivity for definition of Q\n",
        "rho_cr=1/cst #density threshold for which flow volume Q(rho) is maximal\n",
        "V=10\n",
        "\n",
        "def Q(rho): #defining the flow volume function\n",
        "        return V*rho*math.exp(-cst*rho)\n",
        "\n",
        "def delta(rho): #defining the demand function\n",
        "        initial_g=0\n",
        "        if rho<rho_cr:\n",
        "            return Q(rho)\n",
        "        else:\n",
        "            result = minimize(lambda y: -Q(y), initial_g)\n",
        "            return -result.fun\n",
        "def sigma(rho): #defining the supply function\n",
        "        initial_g=0\n",
        "        if rho<rho_cr:\n",
        "            result = minimize(lambda y: -Q(y), initial_g)\n",
        "            return -result.fun\n",
        "        else:\n",
        "            return Q(rho)"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def m(n,D):#definition of the stationary distribution\n",
        "  alpha = Q(rho_cr)\n",
        "  beta = Q(rho_cr)/2\n",
        "  m_0 = 0\n",
        "  for i in range(1,D+1):\n",
        "    product = 1\n",
        "    for j in range (1,i+1):\n",
        "      l = np.min([alpha,sigma((j-1)/D)])#lambda\n",
        "      m = np.min([beta,delta(j/D)])#mu\n",
        "      product = product*(l/m)\n",
        "    m_0+=product\n",
        "  m_n = 1/(1+m_0)#definition fo m(0), initialising m(n)\n",
        "  for i in range(1,n+1):\n",
        "    l = np.min([alpha,sigma((i-1)/D)])\n",
        "    m = np.min([beta,delta(i/D)])\n",
        "    m_n = m_n*(l/m)\n",
        "  return m_n"
      ],
      "metadata": {
        "id": "YhWVM6IKCyBt"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "We compute the values of $m$, for $n = 0,1,2$ using the function defined here and observe that they coincide with our values of m(0), m(1), and m(2) computed by hand using the equation $mA = 0$."
      ],
      "metadata": {
        "id": "68bXcwDKo0YS"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"First element of the stationary distribution m(0):\",m(0,2))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xClrs2MNo_1q",
        "outputId": "78b31e63-2531-4ae8-f391-0a6cef1fbf7c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "First element of the stationary distribution m(0): 0.14285714285714413\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"Second element of the stationary distribution m(1):\",m(1,2))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "f3xkJR5MFHLl",
        "outputId": "be6f0207-b9de-4bb3-c5ab-6aaaff0e1bb0"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Second element of the stationary distribution m(1): 0.28571428571428525\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"Third element of the stationary distribution m(2):\",m(2,2))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "kc70JeptpF7z",
        "outputId": "e05f2cba-aeb7-4a26-a6ac-23dbaa282645"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Third element of the stationary distribution m(2): 0.5714285714285705\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Monte Carlo Approximation of the Stationary Probability"
      ],
      "metadata": {
        "id": "gn40feQYP3Uf"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We now use the monte carlo approach to obtain an approximation of the stationary probability m. As we know from the theorems stated above that $\\mathbb{P}(\\eta_{t}=n|\\eta_{0}=i) \\xrightarrow{t \\to \\infty} m(n)$. Since\n",
        "\n",
        "$$\\mathbb{P}(\\eta_{t}=n|\\eta_{0}=i) = \\mathbb{E}_{i}(\\mathbb{1}_{\\eta_{t}=n})$$\n",
        "and from the law of large numbers,\n",
        "$$\\frac{1}{N}\\sum_{k=1}^{N}\\mathbb{1}_{\\eta_{t}^{k}=n} \\xrightarrow{N \\to \\infty} \\mathbb{E}_{i}(\\mathbb{1}_{\\eta_{t}^{1}=n}) \\; \\text{a.s.}$$\n",
        "\n",
        "where $\\eta_{t}^{i}$ denote independently simulated paths of the single section process. So, the obtained Monte-Carlo estimator is $\\hat{m_{n}} = \\frac{1}{N}\\sum_{k=1}^{N}\\mathbb{1}_{\\eta_{t}^{k}=n}$.\n",
        "\n",
        "---\n",
        "\n",
        "Moreover, by the Central Limit Theorem,\n",
        "$$\\frac{\\sqrt{N}}{\\sigma}(\\hat{m_{n}}-m) \\xrightarrow[N \\to \\infty]{(D)} \\mathcal{N}(0,1)$$\n",
        "which gives us an asymptotic confidence interval with level $\\alpha$ for m:\n",
        "$$m \\in \\left[\\frac{\\sqrt{N}}{\\hat{\\sigma_{n}}} \\hat{m_{n}}-q_{\\alpha/2},\\frac{\\sqrt{N}}{\\hat{\\sigma_{n}}} \\hat{m_{n}}+q_{\\alpha/2}\\right]$$\n",
        "Here $q_{\\alpha/2}$ denotes the $\\frac{\\alpha}{2}^{th}$ quantile of the standard gaussian distribution."
      ],
      "metadata": {
        "id": "hSRjCz4_P8h0"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "***Parameter List:***\n",
        "\n",
        "*  alambda($\\lambda$) = value of the parameter for arrival time\n",
        "\n",
        "*  dmu($\\mu$) = value of the parameter for departure time\n",
        "\n",
        "* eta_0($\\eta_{0}$) = intital number of cars in the section\n",
        "\n",
        "* *D* = maximum number of cars in the section (length of the section)\n",
        "\n",
        "* *T* = global time\n",
        "\n",
        "* *seed* = to incorporate reproducibility of results"
      ],
      "metadata": {
        "id": "VgWFdA9L4kAe"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#We define the jump function for the speacial case of a single section(one way) road\n",
        "def jump_1section(alambda, dmu, eta_0, D, T):\n",
        "    globalT = 0\n",
        "    jump_times = []\n",
        "    states = []\n",
        "    num_cars = eta_0\n",
        "    assert (eta_0<=D),\"Initial number of cars cannot be greater than section capacity!\"\n",
        "    while globalT < T:\n",
        "      atime = np.random.exponential(1/alambda)#arrival time\n",
        "      dtime = np.random.exponential(1/dmu)#departure time\n",
        "      globalT+=min(atime,dtime)\n",
        "      if atime < dtime:\n",
        "        if num_cars<D: #to check if the number of cars in the section has not exceeded the maximum permitted cars which is given by D\n",
        "          num_cars += 1\n",
        "          jump_times.append(atime)\n",
        "          states.append(num_cars)\n",
        "      else:\n",
        "        if num_cars>0:#to check if the number of cars is not =0, because there would beno cars to depart\n",
        "            num_cars -= 1\n",
        "            jump_times.append(dtime)\n",
        "            states.append(num_cars)\n",
        "    return jump_times, states"
      ],
      "metadata": {
        "id": "z62IIIP9f1HT"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "#We define the method which estimates the value of the probability of having n cars in the section at time t-->inf using the Monte-Carlo method.\n",
        "def monte_carlo_prob(alambda, dmu, eta_0, D, n, T, num_sim,seed):\n",
        "    np.random.seed(seed)\n",
        "    E = []\n",
        "    for i in range(1, num_sim + 1):\n",
        "        jump_times_1sec, states_1sec = jump_1section(alambda, dmu, eta_0, D, T)\n",
        "        if states_1sec and states_1sec[-1] == n: #we check if states_1sec is empty and if not compare its last element with n\n",
        "           E.append(1)\n",
        "        else:\n",
        "           E.append(0)\n",
        "    return np.mean(E), np.std(E)"
      ],
      "metadata": {
        "id": "SR8J3DJWpLQS"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "We take the arrival rate lambda to be equal to the maximal flow volume Q(rho_cr), according to the assumptions and the departure rate equal to beta which is the average flow volume equal to Q(rho_cr)/2."
      ],
      "metadata": {
        "id": "8NhOx7_jHW5L"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = Q(rho_cr)\n",
        "beta = Q(rho_cr)/2\n",
        "np.set_printoptions(precision=10)\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,0,2,0,50,10000,42)[0])) # eta_0 =0, n=0\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,0,2,1,50,10000,42)[0])) # eta_0 =0, n=1\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,0,2,2,50,10000,42)[0])) # eta_0 =0, n=2"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "x0nfEtOo6VSx",
        "outputId": "283c58d9-09f9-4f7a-c8e0-3a7a8f555a0a"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Estimated probability of 0 cars in the section at time t-->inf: 0.1431000000\n",
            "Estimated probability of 0 cars in the section at time t-->inf: 0.2878000000\n",
            "Estimated probability of 0 cars in the section at time t-->inf: 0.5691000000\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,1,2,0,50,10000,42)[0])) # eta_0 =1, n=0\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,1,2,1,50,10000,42)[0])) # eta_0 =1, n=1\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,1,2,2,50,10000,42)[0])) # eta_0 =1, n=2"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "FOaXs7n15jrt",
        "outputId": "c4dbe44c-6802-4b1c-d71d-7eeef065f1e0"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Estimated probability of 0 cars in the section at time t-->inf: 0.1431000000\n",
            "Estimated probability of 0 cars in the section at time t-->inf: 0.2878000000\n",
            "Estimated probability of 0 cars in the section at time t-->inf: 0.5691000000\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,2,2,0,50,10000,42)[0])) # eta_0 =2, n=0\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,2,2,1,50,10000,42)[0])) # eta_0 =2, n=1\n",
        "print(\"Estimated probability of 0 cars in the section at time t-->inf: {:.10f}\".format(monte_carlo_prob(alpha,beta,2,2,2,50,10000,42)[0])) # eta_0 =2, n=2"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cBgRpnW2Ib44",
        "outputId": "6aa5a9c5-f54e-4aa2-883f-0d4cea6cf023"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Estimated probability of 0 cars in the section at time t-->inf: 0.1431000000\n",
            "Estimated probability of 0 cars in the section at time t-->inf: 0.2878000000\n",
            "Estimated probability of 0 cars in the section at time t-->inf: 0.5691000000\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We observe that the value of the estimated probability does not change with change in the intial number of vehicles in the section."
      ],
      "metadata": {
        "id": "D0IUHi4M_c7C"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We can also build confidence intervals for the stationary probabilities m(0), m(1), and m(2) to get an idea about the error in the monte carlo approximation."
      ],
      "metadata": {
        "id": "jPbYDUNj-PJ-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "alpha = Q(rho_cr)\n",
        "beta = Q(rho_cr)/2\n",
        "prob,std=monte_carlo_prob(alpha,beta,0,2,0,50,10000,42)\n",
        "lenght_IC=2*1.96*std/np.sqrt(10000)\n",
        "\n",
        "print(\"Exact stationary probability m(0):\",m(0,2))\n",
        "print(\"\\n\")\n",
        "print(\"Monte Carlo Probability, 95% Asymptotic Confidence interval for m(0)\")\n",
        "print(\"Confidence Interval = {:1.10f} +/- {:1.6f} \\n\".format(prob,lenght_IC/2))\n",
        "print(\"Confidence Interval = [{:1.10f}, {:1.6f}] \\n\".format(prob-1.96*std/np.sqrt(10000),prob+1.96*std/np.sqrt(10000)))\n",
        "print(\"Relative error = lenght_CI/MonteCarloProba={:1.10f} % \\n\".format(100*lenght_IC/prob))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "CPG8qLUJ-Ajr",
        "outputId": "98f41f7e-d405-4394-d061-6390d6fb48bb"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Exact stationary probability m(0): 0.14285714285714413\n",
            "\n",
            "\n",
            "Monte Carlo Probability, 95% Asymptotic Confidence interval for m(0)\n",
            "Confidence Interval = 0.1431000000 +/- 0.006863 \n",
            "\n",
            "Confidence Interval = [0.1362365739, 0.149963] \n",
            "\n",
            "Relative error = lenght_CI/MonteCarloProba=9.5924892585 % \n",
            "\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "prob,std=monte_carlo_prob(alpha,beta,0,2,1,50,10000,42)\n",
        "lenght_IC=2*1.96*std/np.sqrt(10000)\n",
        "\n",
        "print(\"Exact stationary probability m(1):\",m(1,2))\n",
        "print(\"\\n\")\n",
        "print(\"Monte Carlo Probability, 95% Asymptotic Confidence interval for m(1)\")\n",
        "print(\"Confidence Interval = {:1.10f} +/- {:1.6f} \\n\".format(prob,lenght_IC/2))\n",
        "print(\"Confidence Interval = [{:1.10f}, {:1.6f}] \\n\".format(prob-1.96*std/np.sqrt(10000),prob+1.96*std/np.sqrt(10000)))\n",
        "print(\"Relative error = lenght_CI/MonteCarloProba = {:1.10f} % \\n\".format(100*lenght_IC/prob))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "1Wf2vLG9-kMB",
        "outputId": "14d7d683-aeb0-4d91-a836-44c81c66e5ee"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Exact stationary probability m(1): 0.28571428571428525\n",
            "\n",
            "\n",
            "Monte Carlo Probability, 95% Asymptotic Confidence interval for m(1)\n",
            "Confidence Interval = 0.2878000000 +/- 0.008874 \n",
            "\n",
            "Confidence Interval = [0.2789263468, 0.296674] \n",
            "\n",
            "Relative error = lenght_CI/MonteCarloProba = 6.1665414759 % \n",
            "\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "prob,std=monte_carlo_prob(alpha,beta,0,2,2,50,10000,42)\n",
        "lenght_IC=2*1.96*std/np.sqrt(10000)\n",
        "\n",
        "print(\"Exact stationary probability m(2):\",m(2,2))\n",
        "print(\"\\n\")\n",
        "print(\"Monte Carlo Probability, 95% Asymptotic Confidence interval for m(2)\")\n",
        "print(\"Confidence Interval = {:1.10f} +/- {:1.6f} \\n\".format(prob,lenght_IC/2))\n",
        "print(\"Confidence Interval = [{:1.10f}, {:1.6f}] \\n\".format(prob-1.96*std/np.sqrt(10000),prob+1.96*std/np.sqrt(10000)))\n",
        "print(\"Relative error = lenght_CI/MonteCarloProba={:1.10f} % \\n\".format(100*lenght_IC/prob))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AWD6520l-0tC",
        "outputId": "7a135343-79d5-4bec-e76f-67bdd34d2b62"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Exact stationary probability m(2): 0.5714285714285705\n",
            "\n",
            "\n",
            "Monte Carlo Probability, 95% Asymptotic Confidence interval for m(2)\n",
            "Confidence Interval = 0.5691000000 +/- 0.009706 \n",
            "\n",
            "Confidence Interval = [0.5593940375, 0.578806] \n",
            "\n",
            "Relative error = lenght_CI/MonteCarloProba=3.4109866626 % \n",
            "\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Finally, in order to graphically illustrate that\n",
        "$$\\sum_{y \\in E}\\vert \\mathbb{P}(X_{t}=y \\vert X_{0}=x)- m(y)   \\vert \\leq Ce^{-\\lambda t}$$\n",
        "we can plot the convergence of the logarithm of this error term, i.e the error of the transition probability of the jump process and the stationary distribution $m$, vs the time t."
      ],
      "metadata": {
        "id": "ngOhP3_E3y1X"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "## Illustration of the exponential convergence\n",
        "\n",
        "def vector_ln_err(vect_T):\n",
        "    Err=[]\n",
        "\n",
        "    for T in vect_T:\n",
        "        #sum of absolute errors of transition probabilities P_t(x,y) and the stationary probability m(y)\n",
        "        err_curr= np.abs(monte_carlo_prob(Q(rho_cr),Q(rho_cr)/2,2,2,2,T,10000,42)[0]-m(2,2)) + np.abs(monte_carlo_prob(Q(rho_cr),Q(rho_cr)/2,2,2,1,T,10000,42)[0]-m(1,2)) + np.abs(monte_carlo_prob(Q(rho_cr),Q(rho_cr)/2,2,2,0,T,10000,42)[0]-m(0,2))\n",
        "        #we sum the error terms for all values of n=0,1,2\n",
        "        Err.append(np.log(err_curr))\n",
        "    return Err\n",
        "\n",
        "vect_T=[0.1,2.6,4,6.2,10.6,13,30,50]#Create an array to find the errors for the global times in the array\n",
        "error=vector_ln_err(vect_T)\n",
        "plt.xlabel(\"time\")\n",
        "plt.ylabel(\"Logarithm of error\")\n",
        "plt.plot(vect_T, error)\n",
        "title(\"Markov Jump process L1 convergence to m\")\n",
        "legend(loc=\"best\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 527
        },
        "id": "wylEYSvy2Z6w",
        "outputId": "79a57a76-432d-4a64-de79-a5831d7b51ff"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "WARNING:matplotlib.legend:No artists with labels found to put in legend.  Note that artists whose label start with an underscore are ignored when legend() is called with no argument.\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x79942d7ea980>"
            ]
          },
          "metadata": {},
          "execution_count": 27
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We observe that the error converges to 0 as the time t goes to infinity."
      ],
      "metadata": {
        "id": "FEwe8sL5_pj4"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## REFERENCES"
      ],
      "metadata": {
        "id": "-tLUlDsXGISM"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Bodineau, T. (n.d.). Modélisation de phénomènes aléatoires: introduction aux chaînes de Markov et aux martingales. http://www.cmap.polytechnique.fr/~bodineau/MAP432.pdf\n",
        "\n",
        "Markov Chain Initial Distribution. (n.d.). Mathematics Stack Exchange. https://math.stackexchange.com/questions/566957/markov-chain-initial-distribution\n",
        "\n",
        "Steady state distribution for Markov pure jump process. (n.d.). Mathematics Stack Exchange. https://math.stackexchange.com/questions/557848/steady-state-distribution-for-markov-pure-jump-process\n",
        "\n",
        "Wang, Y., & He, Z. (2019). Mesoscopic modelling and analysis of traffic flow based on stationary observations. Procedia Computer Science, 151, 800–807. https://doi.org/10.1016/j.procs.2019.04.109\n",
        "\n",
        "JANSEN, S. (2020). MARKOV JUMP PROCESSES. https://www.mathematik.uni-muenchen.de/~jansen/jump-processes.pdf\n",
        "\n",
        "Karlin, S. (2018). Continuous time processes. In Continuous Time Processes. https://web.stanford.edu/class/stat217/Chapter3.pdf\n",
        "\n",
        "Markov Chain Initial Distribution. (n.d.-b). Mathematics Stack Exchange. https://math.stackexchange.com/questions/566957/markov-chain-initial-distribution\n",
        "\n",
        "Pardoux, É. (2006). Processus de Markov et applications (Vol. 2, pp. 11–333). https://www.association-tremplin.org/Documents/Eleves/Apres/CoursChainedeMarkovM1Pardoux.pdf\n",
        "\n",
        "Tordeux, A., Michel ROUSSIGNOL, Sylvain LASSARRE, & UNIVERSITÄT ZU KÖLN, INSTITUT FÜR THEORETISCHE PHYSIK. (2010). Traffic flow models by Markovian jump process. https://www.vzu.uni-wuppertal.de/fileadmin/site/vzu/Presentation_-_Institute_for_Theoretical_Physics.pdf\n",
        "\n",
        "Rey, C., Bansaye V., Lecture Notes. Processus Markov & Applications [MAP566B]"
      ],
      "metadata": {
        "id": "qZPKKr6T3esL"
      }
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "Z6zUuuOa5nP8"
      },
      "execution_count": null,
      "outputs": []
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}