AsyncFlow-Sim
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9428ca92",
+   "metadata": {},
+   "source": [
+    "# AsyncFlow — MMc Theory vs Simulation (Guided Notebook)\n",
+    "\n",
+    "This notebook shows how to:\n",
+    "\n",
+    "1. Make imports work inside a notebook (src-layout or package install)\n",
+    "2. Build a **multi-server** scenario compatible with **M/M/c** assumptions\n",
+    "3. Run the simulation and collect results\n",
+    "4. Compare theory vs observed KPIs (pretty-printed table)\n",
+    "5. Plot the standard dashboards (latency, throughput, server time series)\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 96,
+   "id": "3e168d4a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sys, importlib\n",
+    "\n",
+    "\n",
+    "for m in list(sys.modules):\n",
+    "    if m.startswith(\"asyncflow\"):\n",
+    "        del sys.modules[m]\n",
+    "\n",
+    "\n",
+    "from asyncflow import AsyncFlow, SimulationRunner\n",
+    "from asyncflow.analysis import MMc, ResultsAnalyzer\n",
+    "from asyncflow.components import (\n",
+    "    Client, Server, LinkEdge, Endpoint, LoadBalancer, ArrivalsGenerator\n",
+    ")\n",
+    "from asyncflow.settings import SimulationSettings\n",
+    "\n",
+    "import simpy"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 97,
+   "id": "dd39a8e3",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Imports OK.\n"
+     ]
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import simpy\n",
+    "\n",
+    "# Public AsyncFlow API\n",
+    "from asyncflow import AsyncFlow, SimulationRunner, Sweep\n",
+    "from asyncflow.components import Client, Server, LinkEdge, Endpoint, LoadBalancer, ArrivalsGenerator\n",
+    "from asyncflow.settings import SimulationSettings\n",
+    "from asyncflow.analysis import  ResultsAnalyzer, SweepAnalyzer, MMc\n",
+    "from asyncflow.enums import Distribution\n",
+    "\n",
+    "print(\"Imports OK.\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48fbf4f3",
+   "metadata": {},
+   "source": [
+    "## 1) Build an M/M/c split-friendly scenario\n",
+    "\n",
+    "* **Multiple identical servers with exponential CPU service**\n",
+    "  Topology includes **\\$c \\geq 2\\$ identical servers**, each exposing exactly **one endpoint** with exactly **one CPU-bound step**.\n",
+    "  Service times follow an **Exponential** distribution with mean \\$E\\[S]\\$ (service rate \\$\\mu = 1/E\\[S]\\$). No RAM/IO steps are included in the pipeline.\n",
+    "\n",
+    "* **Load balancer with FCFS dispatch**\n",
+    "\n",
+    "* **“Poisson arrivals” via the generator**\n",
+    "  \n",
+    "  \n",
+    "\n",
+    "---\n",
+    "\n",
+    "```mermaid\n",
+    "graph LR;\n",
+    "    rqs1[\"<b>RqsGenerator</b><br/>id: rqs-1\"]\n",
+    "    client1[\"<b>Client</b><br/>id: client-1\"]\n",
+    "    lb1[\"<b>LoadBalancer</b><br/>id: lb-1<br/>Policy: round_robin\"]\n",
+    "    app1[\"<b>Server</b><br/>id: app-1<br/>Endpoint: /api\"]\n",
+    "    app2[\"<b>Server</b><br/>id: app-2<br/>Endpoint: /api\"]\n",
+    "\n",
+    "    rqs1 -- \"Edge: gen-client<br/>Latency: 0.0001\" --> client1;\n",
+    "    client1 -- \"Request<br/>Edge: client-lb<br/>Latency: 0.0001\" --> lb1;\n",
+    "    lb1 -- \"Dispatch<br/>Edge: lb-app1<br/>Latency: 0.0001\" --> app1;\n",
+    "    lb1 -- \"Dispatch<br/>Edge: lb-app2<br/>Latency: 0.0001\" --> app2;\n",
+    "    app1 -- \"Response<br/>Edge: app1-client<br/>Latency: 0.0001\" --> client1;\n",
+    "    app2 -- \"Response<br/>Edge: app2-client<br/>Latency: 0.0001\" --> client1;\n",
+    "```\n",
+    "\n",
+    "---\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 98,
+   "id": "d2937e5e",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def build_payload():\n",
+    "    generator = ArrivalsGenerator(\n",
+    "        id=\"rqs-1\",\n",
+    "        lambda_rps=270,\n",
+    "        model=Distribution.POISSON\n",
+    "    )\n",
+    "\n",
+    "    client = Client(id=\"client-1\")\n",
+    "\n",
+    "    endpoint = Endpoint(\n",
+    "        endpoint_name=\"/api\",\n",
+    "        probability=1.0,\n",
+    "        steps=[\n",
+    "            {\n",
+    "                \"kind\": \"initial_parsing\",\n",
+    "                \"step_operation\": {\n",
+    "                    \"cpu_time\": {\"mean\": 0.01, \"distribution\": \"exponential\"},\n",
+    "                },\n",
+    "            },\n",
+    "        ],\n",
+    "    )\n",
+    "\n",
+    "    srv1 = Server(\n",
+    "        id=\"srv-1\",\n",
+    "        server_resources={\"cpu_cores\": 1, \"ram_mb\": 2048},\n",
+    "        endpoints=[endpoint],\n",
+    "    )\n",
+    "    srv2 = Server(\n",
+    "        id=\"srv-2\",\n",
+    "        server_resources={\"cpu_cores\": 1, \"ram_mb\": 2048},\n",
+    "        endpoints=[endpoint],\n",
+    "    )\n",
+    "    \n",
+    "    srv3 = Server(\n",
+    "        id=\"srv-3\",\n",
+    "        server_resources={\"cpu_cores\": 1, \"ram_mb\": 2048},\n",
+    "        endpoints=[endpoint],\n",
+    "    )\n",
+    "\n",
+    "    lb = LoadBalancer(\n",
+    "        id=\"lb-1\",\n",
+    "        algorithms=\"fcfs\",  \n",
+    "        server_covered={\"srv-1\", \"srv-2\", \"srv-3\"},\n",
+    "    )\n",
+    "\n",
+    "    edges = [\n",
+    "        LinkEdge(id=\"gen-client\",  source=\"rqs-1\",  target=\"client-1\",),\n",
+    "        LinkEdge(id=\"client-lb\",   source=\"client-1\", target=\"lb-1\",  ),\n",
+    "        LinkEdge(id=\"lb-srv1\",     source=\"lb-1\",   target=\"srv-1\",   ),\n",
+    "        LinkEdge(id=\"lb-srv2\",     source=\"lb-1\",   target=\"srv-2\",   ),\n",
+    "        LinkEdge(id=\"lb-srv3\",     source=\"lb-1\",   target=\"srv-3\",   ),\n",
+    "        LinkEdge(id=\"srv1-client\", source=\"srv-1\",  target=\"client-1\",),\n",
+    "        LinkEdge(id=\"srv2-client\", source=\"srv-2\",  target=\"client-1\",),\n",
+    "        LinkEdge(id=\"srv3-client\", source=\"srv-3\",  target=\"client-1\",),\n",
+    "    ]\n",
+    "\n",
+    "    settings = SimulationSettings(\n",
+    "        total_simulation_time=3600,\n",
+    "        sample_period_s=0.05,\n",
+    "    )\n",
+    "\n",
+    "    payload = (\n",
+    "        AsyncFlow()\n",
+    "        .add_arrivals_generator(generator)\n",
+    "        .add_client(client)\n",
+    "        .add_servers(srv1, srv2, srv3)\n",
+    "        .add_load_balancer(lb)\n",
+    "        .add_edges(*edges)\n",
+    "        .add_simulation_settings(settings)\n",
+    "    ).build_payload()\n",
+    "\n",
+    "    return payload\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7682861f",
+   "metadata": {},
+   "source": [
+    "## 2) Run the simulation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 99,
+   "id": "d0634bc8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Done.\n"
+     ]
+    }
+   ],
+   "source": [
+    "payload = build_payload()\n",
+    "env = simpy.Environment()\n",
+    "runner = SimulationRunner(env=env, simulation_input=payload)\n",
+    "results: ResultsAnalyzer = runner.run()\n",
+    "print(\"Done.\")\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e5fe2a4a",
+   "metadata": {},
+   "source": [
+    "# 3) M/M/c (FCFS) — theory vs observed comparison\n",
+    "\n",
+    "This section shows how we compute the **theoretical Erlang-C KPIs** (pooled queue, FCFS) and compare them against **simulation estimates**.\n",
+    "\n",
+    "---\n",
+    "\n",
+    "## Variables\n",
+    "\n",
+    "* **$c$**: number of identical servers.\n",
+    "* **$\\lambda$**: global arrival rate (req/s).\n",
+    "* **$\\mu$**: per-server service rate (req/s), $\\mu = 1/\\mathbb{E}[S]$.\n",
+    "* **$\\rho$**: global utilization, $\\rho = \\lambda/(c\\mu)$.\n",
+    "* **$W$**: mean time in system (queue + service).\n",
+    "* **$W_q$**: mean waiting time in queue.\n",
+    "* **$L$**: mean number in system.\n",
+    "* **$L_q$**: mean number in queue.\n",
+    "\n",
+    "---\n",
+    "\n",
+    "## Theory (Erlang-C formulas)\n",
+    "\n",
+    "We assume **Poisson arrivals** for $\\lambda$ (taken directly from the payload).\n",
+    "\n",
+    "1. Offered load:\n",
+    "\n",
+    "$$\n",
+    "a = \\frac{\\lambda}{\\mu}\n",
+    "$$\n",
+    "\n",
+    "2. Probability system is empty:\n",
+    "\n",
+    "$$\n",
+    "P_0 = \\left[\\sum_{n=0}^{c-1}\\frac{a^n}{n!} + \\frac{a^c}{c!\\,(1-\\rho)}\\right]^{-1}\n",
+    "$$\n",
+    "\n",
+    "3. Probability of waiting (Erlang-C):\n",
+    "\n",
+    "$$\n",
+    "P_w = \\frac{a^c}{c!\\,(1-\\rho)} \\, P_0\n",
+    "$$\n",
+    "\n",
+    "4. Queue length and waiting:\n",
+    "\n",
+    "$$\n",
+    "L_q = P_w \\cdot \\frac{\\rho}{1-\\rho}, \\qquad\n",
+    "W_q = \\frac{L_q}{\\lambda}\n",
+    "$$\n",
+    "\n",
+    "5. Total response time and system size:\n",
+    "\n",
+    "$$\n",
+    "W = W_q + \\frac{1}{\\mu}, \\qquad\n",
+    "L = \\lambda W\n",
+    "$$\n",
+    "\n",
+    "If $\\rho \\ge 1$, the system is unstable and all metrics diverge to $+\\infty$.\n",
+    "\n",
+    "---\n",
+    "\n",
+    "## Observed (from simulation)\n",
+    "\n",
+    "After processing metrics:\n",
+    "\n",
+    "1. **Arrival rate**:\n",
+    "\n",
+    "$$\n",
+    "\\lambda_{\\text{Observed}} = \\text{mean throughput (client completions)}\n",
+    "$$\n",
+    "\n",
+    "2. **Service rate**:\n",
+    "\n",
+    "$$\n",
+    "\\mu_{\\text{Observed}} = 1 / \\overline{S}, \\quad \\overline{S} = \\text{mean(service\\_time)}\n",
+    "$$\n",
+    "\n",
+    "3. **End-to-end latency**:\n",
+    "\n",
+    "$$\n",
+    "W_{\\text{Observed}} = \\text{mean(client latencies)}\n",
+    "$$\n",
+    "\n",
+    "4. **Waiting time**:\n",
+    "\n",
+    "$$\n",
+    "W_{q,\\text{Observed}} = \\text{mean(waiting\\_time)} \n",
+    "$$\n",
+    "\n",
+    "5. **Little’s law check**:\n",
+    "\n",
+    "$$\n",
+    "L_{\\text{Observed}} = \\lambda_{\\text{Observed}} W_{\\text{Observed}}, \\qquad\n",
+    "L_{q,\\text{Observed}} = \\lambda_{\\text{Observed}} W_{q,\\text{Observed}}\n",
+    "$$\n",
+    "\n",
+    "6. **Utilization**:\n",
+    "\n",
+    "$$\n",
+    "\\rho_{\\text{Observed}} = \\lambda_{\\text{Observed}}/(c\\,\\mu_{\\text{Observed}})\n",
+    "$$\n",
+    "\n",
+    "---\n",
+    "\n",
+    "## Comparison\n",
+    "\n",
+    "The analyzer builds a table with two columns — **Theory** (Erlang-C closed forms) and **Observed** (empirical estimates) — and reports absolute and relative deltas.\n",
+    "\n",
+    "This allows us to verify whether AsyncFlow reproduces the textbook M/M/c (FCFS) predictions under Poisson arrivals and exponential service.\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 100,
+   "id": "ccd7379b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "=================================================================\n",
+      "MMc (FCFS/Erlang-C) — Theory vs Observed\n",
+      "-----------------------------------------------------------------\n",
+      "sym  metric                    theory    observed       abs  rel%\n",
+      "-----------------------------------------------------------------\n",
+      "λ    Arrival rate (1/s)    270.000000  270.258333  0.258333  0.10\n",
+      "μ    Service rate (1/s)    100.000000  100.036707  0.036707  0.04\n",
+      "rho  Utilization             0.900000    0.900531  0.000531  0.06\n",
+      "L    Mean items in sys      10.053549   10.073544  0.019994  0.20\n",
+      "Lq   Mean items in queue     7.353549    7.371934  0.018385  0.25\n",
+      "W    Mean time in sys (s)    0.037235    0.037274  0.000038  0.10\n",
+      "Wq   Mean waiting (s)        0.027235    0.027277  0.000042  0.15\n",
+      "=================================================================\n"
+     ]
+    }
+   ],
+   "source": [
+    "mmc = MMc()\n",
+    "if mmc.is_compatible(payload):\n",
+    "   mmc.print_comparison(payload, results)  \n",
+    "else:\n",
+    "    print(\"Payload is not compatible with M/M/c:\")\n",
+    "    for reason in mmc.explain_incompatibilities(payload):\n",
+    "        print(\" -\", reason)\n",
+    "   \n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "asyncflow-sim-py3.12 (3.12.3)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.12.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}