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Queue Theory: MMc (split & FCFS/Erlang-C), independent Observed KPIs, richer sampled metrics, and updated notebooks #32

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Merged
GioeleB00 merged 4 commits into from
Sep 20, 2025
Merged

Queue Theory: MMc (split & FCFS/Erlang-C), independent Observed KPIs, richer sampled metrics, and updated notebooks #32

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98fc0fd
Logic to handle completed rqs move to arr-gen
GioeleB00 Sep 19, 2025
2284363
Added metrics L system and Lq LB
GioeleB00 Sep 19, 2025
667e435
Calculations theory vs observed indipendent
GioeleB00 Sep 19, 2025
f0cdd46
improved tests and all calculation of th. vs observed indip
GioeleB00 Sep 20, 2025
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401 changes: 65 additions & 336 deletions asyncflow_queue_limit/asyncflow_mm1.ipynb
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96 changes: 61 additions & 35 deletions asyncflow_queue_limit/asyncflow_mmc.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -20,7 +20,7 @@
},
{
"cell_type": "code",
"execution_count": 96,
"execution_count": 19,
"id": "3e168d4a",
"metadata": {},
"outputs": [],
Expand All @@ -45,7 +45,7 @@
},
{
"cell_type": "code",
"execution_count": 97,
"execution_count": 20,
"id": "dd39a8e3",
"metadata": {},
"outputs": [
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},
{
"cell_type": "code",
"execution_count": 98,
"execution_count": 21,
"id": "d2937e5e",
"metadata": {},
"outputs": [],
Expand Down Expand Up @@ -168,9 +168,9 @@
" 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",
" LinkEdge(id=\"srv1-client\", source=\"srv-1\", target=\"rqs-1\",),\n",
" LinkEdge(id=\"srv2-client\", source=\"srv-2\", target=\"rqs-1\",),\n",
" LinkEdge(id=\"srv3-client\", source=\"srv-3\", target=\"rqs-1\",),\n",
" ]\n",
"\n",
" settings = SimulationSettings(\n",
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},
{
"cell_type": "code",
"execution_count": 99,
"execution_count": 22,
"id": "d0634bc8",
"metadata": {},
"outputs": [
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"\n",
"---\n",
"\n",
"## Observed (from simulation)\n",
"## Observed (from simulation, FCFS/Erlang-C)\n",
"\n",
"After processing metrics:\n",
"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",
"\n",
"1. **Arrival rate**:\n",
"1. **Arrival rate**\n",
"\n",
"$$\n",
"\\lambda_{\\text{Observed}} = \\text{mean throughput (client completions)}\n",
"\\lambda_{\\text{obs}}=\\mathrm{mean}(\\text{Throughput RPS})\n",
"$$\n",
"\n",
"2. **Service rate**:\n",
"Mean of the throughput series (client completions per fixed window, typically 1 s).\n",
"\n",
"2. **Service rate**\n",
"\n",
"$$\n",
"\\mu_{\\text{Observed}} = 1 / \\overline{S}, \\quad \\overline{S} = \\text{mean(service\\_time)}\n",
"\\mu_{\\text{obs}}=\\frac{1}{\\overline{S}},\\quad \n",
"\\overline{S}=\\mathrm{mean}(\\texttt{SERVICE\\_TIME})\n",
"$$\n",
"\n",
"3. **End-to-end latency**:\n",
"Computed from per-request server buckets, aggregated across all servers.\n",
"\n",
"3. **End-to-end time**\n",
"\n",
"$$\n",
"W_{\\text{Observed}} = \\text{mean(client latencies)}\n",
"W_{\\text{obs}}=\\mathrm{mean}(\\text{client latencies})\n",
"$$\n",
"\n",
"4. **Waiting time**:\n",
"From the generator’s request clocks (start/finish per request).\n",
"\n",
"4. **Queue waiting time (LB)**\n",
"\n",
"$$\n",
"W_{q,\\text{Observed}} = \\text{mean(waiting\\_time)} \n",
"W_{q,\\text{obs}}=\\mathrm{mean}(\\text{LB waiting times})\n",
"$$\n",
"\n",
"5. **Little’s law check**:\n",
"Mean of the **load balancer FCFS** waiting time recorded per request (not inferred from queue length).\n",
"\n",
"5. **Mean number in system**\n",
"\n",
"$$\n",
"L_{\\text{Observed}} = \\lambda_{\\text{Observed}} W_{\\text{Observed}}, \\qquad\n",
"L_{q,\\text{Observed}} = \\lambda_{\\text{Observed}} W_{q,\\text{Observed}}\n",
"L_{\\text{obs}}=\\mathrm{mean}(\\texttt{L\\_SYSTEM})\n",
"$$\n",
"\n",
"6. **Utilization**:\n",
"Time average of the sampled series of **concurrent requests in the system** (sampled every $\\Delta t$).\n",
"*Fallback if the series is unavailable in tests:* $L_{\\text{obs}}=\\lambda_{\\text{obs}} W_{\\text{obs}}$.\n",
"\n",
"6. **Mean number in queue (LB)**\n",
"\n",
"$$\n",
"\\rho_{\\text{Observed}} = \\lambda_{\\text{Observed}}/(c\\,\\mu_{\\text{Observed}})\n",
"L_{q,\\text{obs}}=\\mathrm{mean}(\\texttt{LQ\\_LB})\n",
"$$\n",
"\n",
"Time average of the **LB queue length** series.\n",
"*Fallback:* $L_{q,\\text{obs}}=\\lambda_{\\text{obs}} W_{q,\\text{obs}}$.\n",
"\n",
"7. **Global utilization**\n",
"\n",
"$$\n",
"\\rho_{\\text{obs}}=\\mathrm{mean}(\\texttt{SERVER\\_UTILIZATION})\n",
"$$\n",
"\n",
"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",
"*Fallback:* $\\rho_{\\text{obs}}=\\lambda_{\\text{obs}}/(c\\,\\mu_{\\text{obs}})$.\n",
"\n",
"> 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",
"\n",
"\n",
"---\n",
"\n",
"## Comparison\n",
Expand All @@ -339,27 +365,27 @@
},
{
"cell_type": "code",
"execution_count": 100,
"execution_count": 23,
"id": "ccd7379b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"=================================================================\n",
"===================================================================\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"
"-------------------------------------------------------------------\n",
"sym metric theory observed abs rel%\n",
"-------------------------------------------------------------------\n",
"λ Arrival rate (1/s) 270.000000 269.751667 -0.248333 -0.09\n",
"μ Service rate (1/s) 100.000000 100.058789 0.058789 0.06\n",
"rho Utilization 0.900000 0.898517 -0.001483 -0.16\n",
"L Mean items in sys 10.053549 9.865831 -0.187718 -1.87\n",
"Lq Mean items in queue 7.353549 7.170280 -0.183269 -2.49\n",
"W Mean time in sys (s) 0.037235 0.036556 -0.000680 -1.83\n",
"Wq Mean waiting (s) 0.027235 0.026562 -0.000674 -2.47\n",
"===================================================================\n"
]
}
],
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