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Queue Theory: MMc (split & FCFS/Erlang-C), independent Observed KPIs, richer sampled metrics, and updated notebooks (#32)
* Logic to handle completed rqs move to arr-gen * Added metrics L system and Lq LB * Calculations theory vs observed indipendent * improved tests and all calculation of th. vs observed indip
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‎asyncflow_queue_limit/asyncflow_mm1.ipynb‎

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‎asyncflow_queue_limit/asyncflow_mmc.ipynb‎

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"cell_type": "code",
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" LinkEdge(id=\"lb-srv1\", source=\"lb-1\", target=\"srv-1\", ),\n",
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" LinkEdge(id=\"lb-srv2\", source=\"lb-1\", target=\"srv-2\", ),\n",
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" LinkEdge(id=\"lb-srv3\", source=\"lb-1\", target=\"srv-3\", ),\n",
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" LinkEdge(id=\"srv1-client\", source=\"srv-1\", target=\"client-1\",),\n",
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" LinkEdge(id=\"srv2-client\", source=\"srv-2\", target=\"client-1\",),\n",
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" LinkEdge(id=\"srv3-client\", source=\"srv-3\", target=\"client-1\",),\n",
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" LinkEdge(id=\"srv1-client\", source=\"srv-1\", target=\"rqs-1\",),\n",
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" LinkEdge(id=\"srv2-client\", source=\"srv-2\", target=\"rqs-1\",),\n",
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" LinkEdge(id=\"srv3-client\", source=\"srv-3\", target=\"rqs-1\",),\n",
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" ]\n",
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"\n",
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" settings = SimulationSettings(\n",
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"\n",
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"---\n",
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"\n",
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"## Observed (from simulation)\n",
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"## Observed (from simulation, FCFS/Erlang-C)\n",
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"\n",
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"After processing metrics:\n",
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"Under FCFS we derive **each KPI directly** from sampled time series or per-request buckets, **not** by deriving one observed KPI from another. This keeps Little’s law checks as **independent validations**, not definitions.\n",
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"\n",
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"1. **Arrival rate**:\n",
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"1. **Arrival rate**\n",
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"\n",
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"$$\n",
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"\\lambda_{\\text{Observed}} = \\text{mean throughput (client completions)}\n",
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"\\lambda_{\\text{obs}}=\\mathrm{mean}(\\text{Throughput RPS})\n",
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"$$\n",
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"\n",
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"2. **Service rate**:\n",
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"Mean of the throughput series (client completions per fixed window, typically 1 s).\n",
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"\n",
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"2. **Service rate**\n",
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"\n",
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"$$\n",
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"\\mu_{\\text{Observed}} = 1 / \\overline{S}, \\quad \\overline{S} = \\text{mean(service\\_time)}\n",
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"\\mu_{\\text{obs}}=\\frac{1}{\\overline{S}},\\quad \n",
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"\\overline{S}=\\mathrm{mean}(\\texttt{SERVICE\\_TIME})\n",
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"$$\n",
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"\n",
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"3. **End-to-end latency**:\n",
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"Computed from per-request server buckets, aggregated across all servers.\n",
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"\n",
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"3. **End-to-end time**\n",
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"\n",
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"$$\n",
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"W_{\\text{Observed}} = \\text{mean(client latencies)}\n",
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"W_{\\text{obs}}=\\mathrm{mean}(\\text{client latencies})\n",
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"$$\n",
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"\n",
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"4. **Waiting time**:\n",
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"From the generator’s request clocks (start/finish per request).\n",
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"\n",
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"4. **Queue waiting time (LB)**\n",
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"\n",
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"$$\n",
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"W_{q,\\text{Observed}} = \\text{mean(waiting\\_time)} \n",
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"W_{q,\\text{obs}}=\\mathrm{mean}(\\text{LB waiting times})\n",
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"$$\n",
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"\n",
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"5. **Little’s law check**:\n",
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"Mean of the **load balancer FCFS** waiting time recorded per request (not inferred from queue length).\n",
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"\n",
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"5. **Mean number in system**\n",
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"\n",
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"$$\n",
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"L_{\\text{Observed}} = \\lambda_{\\text{Observed}} W_{\\text{Observed}}, \\qquad\n",
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"L_{q,\\text{Observed}} = \\lambda_{\\text{Observed}} W_{q,\\text{Observed}}\n",
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"L_{\\text{obs}}=\\mathrm{mean}(\\texttt{L\\_SYSTEM})\n",
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"$$\n",
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"\n",
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"6. **Utilization**:\n",
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"Time average of the sampled series of **concurrent requests in the system** (sampled every $\\Delta t$).\n",
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"*Fallback if the series is unavailable in tests:* $L_{\\text{obs}}=\\lambda_{\\text{obs}} W_{\\text{obs}}$.\n",
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"\n",
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"6. **Mean number in queue (LB)**\n",
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"\n",
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"$$\n",
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"\\rho_{\\text{Observed}} = \\lambda_{\\text{Observed}}/(c\\,\\mu_{\\text{Observed}})\n",
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"L_{q,\\text{obs}}=\\mathrm{mean}(\\texttt{LQ\\_LB})\n",
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"$$\n",
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"\n",
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"Time average of the **LB queue length** series.\n",
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"*Fallback:* $L_{q,\\text{obs}}=\\lambda_{\\text{obs}} W_{q,\\text{obs}}$.\n",
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"\n",
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"7. **Global utilization**\n",
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"\n",
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"$$\n",
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"\\rho_{\\text{obs}}=\\mathrm{mean}(\\texttt{SERVER\\_UTILIZATION})\n",
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"$$\n",
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"\n",
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"Each server samples a 0/1 “busy” indicator; $\\rho_{\\text{obs}}$ is the time average (with 1 core/server it matches the busy fraction).\n",
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"*Fallback:* $\\rho_{\\text{obs}}=\\lambda_{\\text{obs}}/(c\\,\\mu_{\\text{obs}})$.\n",
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"\n",
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"> Because each quantity comes from its own primary measurement (series or buckets), cross-relations like $L=\\lambda W$ and $L_q=\\lambda W_q$ are genuine **consistency checks**, not tautologies.\n",
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"\n",
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"\n",
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"---\n",
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"\n",
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"## Comparison\n",
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"execution_count": 23,
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"=================================================================\n",
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"===================================================================\n",
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"MMc (FCFS/Erlang-C) — Theory vs Observed\n",
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"-----------------------------------------------------------------\n",
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"sym metric theory observed abs rel%\n",
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"-----------------------------------------------------------------\n",
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"λ Arrival rate (1/s) 270.000000 270.258333 0.258333 0.10\n",
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"μ Service rate (1/s) 100.000000 100.036707 0.036707 0.04\n",
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"rho Utilization 0.900000 0.900531 0.000531 0.06\n",
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"L Mean items in sys 10.053549 10.073544 0.019994 0.20\n",
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"Lq Mean items in queue 7.353549 7.371934 0.018385 0.25\n",
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"W Mean time in sys (s) 0.037235 0.037274 0.000038 0.10\n",
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"Wq Mean waiting (s) 0.027235 0.027277 0.000042 0.15\n",
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"=================================================================\n"
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"-------------------------------------------------------------------\n",
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"sym metric theory observed abs rel%\n",
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"-------------------------------------------------------------------\n",
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"λ Arrival rate (1/s) 270.000000 269.751667 -0.248333 -0.09\n",
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"μ Service rate (1/s) 100.000000 100.058789 0.058789 0.06\n",
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"rho Utilization 0.900000 0.898517 -0.001483 -0.16\n",
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"L Mean items in sys 10.053549 9.865831 -0.187718 -1.87\n",
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"Lq Mean items in queue 7.353549 7.170280 -0.183269 -2.49\n",
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"W Mean time in sys (s) 0.037235 0.036556 -0.000680 -1.83\n",
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"Wq Mean waiting (s) 0.027235 0.026562 -0.000674 -2.47\n",
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"===================================================================\n"
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]
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}
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],

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