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MCP Server

QueueSim

com.queuesim/public
Data & Analytics Science & Engineering Public & reachable MCP 2026-07-28

What this MCP does

Runs M/M/c queue simulations, staffing calculations, scenario comparisons, and queueing-theory analyses.

Capabilities

Tools

11 tools
compare_analytical_vs_simulated
Run the same M/M/c configuration through BOTH the closed-form Erlang-C formula AND the discrete-event simulator, returning a side-by-side comparison with deltas. Use this when the user is validating QueueSim's engine against textbook values, learning queueing theory by watching simulation converge on the formula, or auditing a result that 'feels off' — agreement within ~5%% is the canonical sanity check for an M/M/c run. Pure-Exponential M/M/c only; the closed-form Erlang-C is undefined for other service distributions. Large deltas usually mean the simulation run was too short for steady-state — raise simulationDays. ANTI-FABRICATION: both sides come from real computation — closed-form is deterministic, simulation is stochastic but engine-backed. Quote both verbatim. Do not synthesize an 'average of the two' or recompute the formula from training-data recall.
Read only
Input schema 
{'type': 'object', 'required': ['arrivalRate', 'serviceRate', 'servers'], 'properties': {'servers': {'type': 'integer', 'default': 2, 'maximum': 50, 'minimum': 1, 'description': 'Number of parallel servers (c). Integer 1-50.'}, 'arrivalRate': {'type': 'number', 'default': 20, 'maximum': 200, 'description': 'Mean arrivals per hour (λ).', 'exclusiveMinimum': 0}, 'serviceRate': {'type': 'number', 'default': 12, 'description': 'Mean customers one server can finish per hour (μ). Must be > 0.', 'exclusiveMinimum': 0}, 'simulationDays': {'type': 'integer', 'default': 7, 'maximum': 30, 'minimum': 1, 'description': 'Days to simulate on the DES side. Closed-form is instant. Range 1-30; longer runs converge closer to the formula.'}}}
Output schema 
{'type': 'object', 'required': ['inputs', 'analytical', 'simulated', 'delta'], 'properties': {'note': {'type': 'string'}, 'delta': {'type': 'object', 'properties': {'note': {'type': 'string'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}, 'avgWaitPercent': {'type': 'number'}}}, 'inputs': {'type': 'object'}, 'simulated': {'type': 'object', 'required': ['avgWaitMinutes', 'avgQueueLength', 'offeredLoad', 'totalServed'], 'properties': {'offeredLoad': {'type': 'number'}, 'totalServed': {'type': 'integer'}, 'effectiveDays': {'type': 'integer'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}}}, 'analytical': {'type': 'object', 'required': ['avgWaitMinutes', 'avgQueueLength', 'offeredLoad', 'probabilityOfWaiting'], 'properties': {'offeredLoad': {'type': 'number'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}, 'probabilityOfWaiting': {'type': 'number'}}}}}
compare_separate_vs_pooled
Run the classic operations-research teaching demo: pooled queueing (one shared queue, c servers) vs separate queues (c independent queues, one server each, λ/c traffic to each). Both runs have identical total capacity (c × μ) and identical total arrivals (λ), so the offered load ρ is the same; the only structural difference is whether arrivals share a queue or split into c isolated streams. The pooled configuration ALWAYS produces shorter waits — that's the whole teaching point. Use this when the user asks 'should we pool our resources?' / 'should we cross-train?' / 'why do banks have one line instead of c?' / 'what's the cost of siloing my call center into specialist queues?'. Returns both runs side by side with the pooled-vs-separate wait delta. ANTI-FABRICATION: numbers come from two real DES runs. Quote them VERBATIM.
Read only
Input schema 
{'type': 'object', 'required': ['arrivalRate', 'serviceRate', 'servers'], 'properties': {'servers': {'type': 'integer', 'default': 3, 'maximum': 50, 'minimum': 2, 'description': "Number of servers (c). Pooled: one queue feeds all c. Separate: c independent queues, each with one server and λ/c traffic. Range 2-50 (with c=1 there's nothing to compare)."}, 'arrivalRate': {'type': 'number', 'default': 30, 'maximum': 200, 'description': 'Mean total arrival rate (λ). Pooled run takes all of it; separate run divides evenly across the c queues.', 'exclusiveMinimum': 0}, 'serviceRate': {'type': 'number', 'default': 12, 'description': 'Mean service rate per server (μ, customers/hour each server finishes). Identical across both runs.', 'exclusiveMinimum': 0}, 'simulationDays': {'type': 'integer', 'default': 7, 'maximum': 30, 'minimum': 1, 'description': 'Days to simulate (same for both runs). Range 1-30.'}}}
Output schema 
{'type': 'object', 'required': ['inputs', 'summary', 'hourly'], 'properties': {'hourly': {'type': 'array', 'items': {'type': 'object', 'properties': {'hour': {'type': 'integer'}, 'label': {'type': 'string'}, 'avgArrivals': {'type': 'number'}, 'maxArrivals': {'type': 'integer'}, 'avgDepartures': {'type': 'number'}, 'maxDepartures': {'type': 'integer'}, 'serversOnDuty': {'type': 'integer'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}, 'maxQueueLength': {'type': 'number'}, 'maxWaitMinutes': {'type': 'number'}}}}, 'inputs': {'type': 'object', 'description': "Echo of the run's parameters."}, 'summary': {'type': 'object', 'required': ['totalArrivals', 'totalServed', 'effectiveDays', 'overallAvgWaitMinutes'], 'properties': {'offeredLoad': {'type': ['number', 'null']}, 'totalServed': {'type': 'integer'}, 'effectiveDays': {'type': 'integer'}, 'totalArrivals': {'type': 'integer'}, 'overallAvgWaitMinutes': {'type': 'number'}}}}}
describe_scenario
Return full details for one preset scenario: title, description, teaching note, peak parameters, and per-hour arrival + staffing arrays. Use this before simulate_scenario to understand the default shape and what overrides make sense.
Read only Idempotent
Input schema 
{'type': 'object', 'required': ['name'], 'properties': {'name': {'enum': ['single', 'coffee', 'grocery', 'callcenter'], 'type': 'string', 'default': 'coffee', 'description': 'Scenario key from list_scenarios.'}}}
Output schema 
{'type': 'object', 'required': ['key', 'title', 'defaults'], 'properties': {'key': {'type': 'string'}, 'title': {'type': 'string'}, 'defaults': {'type': 'object'}, 'description': {'type': 'string'}, 'teachingNote': {'type': 'string'}, 'supportedOverrides': {'type': 'array', 'items': {'type': 'string'}}}}
explain_advanced_patterns
Return a textbook-level description of six queueing complexity patterns beyond basic M/M/c: abandonment/reneging, priority tiers, overflow routing, skills-based routing, compound service, and server outages. Use this when the user describes real-world complexity (customers hanging up, VIP queues, specialist escalation, agent breaks, transfers) that plain M/M/c doesn't model. The tool frames each pattern conceptually and points users at ChiAha for custom modeling.
Read only Idempotent
Input schema 
{'type': 'object', 'properties': {}}
explain_queueing_theory
Return a ~500-word educational explainer of M/M/c queueing theory: Little's Law, utilization, why averages mislead, how simulation relates to Erlang-C. No inputs. Use this when the user asks a conceptual 'why' or 'how does this work' question rather than asking for a number.
Read only Idempotent
Input schema 
{'type': 'object', 'properties': {}}
interpret_result
Given an M/M/c configuration (arrivalRate, serviceRate, servers) and optionally an observed average wait, returns a queueing-theory framed interpretation: where you sit on the utilization curve, what ρ means in plain language, what one more or fewer server would qualitatively do, and which complexity factors (priority, abandonment, skills routing) might be hiding in real data the M/M/c model can't see. Use this to TEACH while answering — when the user wants context around a number, not just the number itself. Pure text computation, no simulation, no RNG — deterministic output.
Read only Idempotent
Input schema 
{'type': 'object', 'required': ['arrivalRate', 'serviceRate', 'servers'], 'properties': {'servers': {'type': 'integer', 'default': 2, 'maximum': 50, 'minimum': 1, 'description': 'Server count (c).'}, 'arrivalRate': {'type': 'number', 'default': 20, 'maximum': 200, 'description': 'Mean arrivals per hour (λ).', 'exclusiveMinimum': 0}, 'serviceRate': {'type': 'number', 'default': 12, 'description': 'Mean customers one server finishes per hour (μ).', 'exclusiveMinimum': 0}, 'observedAvgWaitMinutes': {'type': 'number', 'minimum': 0, 'description': 'Optional. The avg wait the user observed (from simulate_mmc, an Erlang-C calculator, or real measurements). If omitted, the tool computes ρ from the inputs and gives a parameter-only interpretation.'}}}
list_scenarios
List the four pre-built QueueSim scenarios. Returns key, title, and one-line description for each (Single Server, Coffee Shop, Grocery Checkout, Call Center). Call this when the user's problem matches one of the preset shapes — use describe_scenario for more detail and simulate_scenario to run one.
Read only Idempotent
Input schema 
{'type': 'object', 'properties': {}}
Output schema 
{'type': 'object', 'required': ['scenarios'], 'properties': {'scenarios': {'type': 'array', 'items': {'type': 'object', 'required': ['key', 'title', 'description'], 'properties': {'key': {'type': 'string'}, 'title': {'type': 'string'}, 'description': {'type': 'string'}}}}}}
recommend_staffing
INVERSE of simulate_mmc — given an arrival rate, service rate, and a target average wait time, returns the SMALLEST number of servers needed to meet the target. Use this when the user asks 'how many servers do I need?' / 'what staffing keeps wait under N minutes?'. The tool runs a binary search over candidate server counts (up to maxServers, default 50), invoking the simulator for each candidate. Saves Claude from iterating simulate_mmc 3-5 times by hand. If even maxServers servers can't meet the target, the recommendation is null and the response includes the achieved wait so Claude can explain that the target is infeasible at the given load. ANTI-FABRICATION: `recommendedServers` and `achievedAvgWaitMinutes` come from real DES runs. Quote them VERBATIM. Do not propose a different number you think 'feels right'; this tool already binary-searches for the minimum that meets the target. If the user asks 'what if c=N?' for a specific N, call simulate_mmc with that c.
Read only
Input schema 
{'type': 'object', 'required': ['arrivalRate', 'serviceRate', 'targetAvgWaitMinutes'], 'properties': {'maxServers': {'type': 'integer', 'default': 50, 'maximum': 50, 'minimum': 1, 'description': "Search ceiling (default 50, max 50). If even this many servers can't meet the target, the tool returns null with the achieved wait."}, 'serviceCoV': {'type': 'number', 'maximum': 5, 'minimum': 0}, 'arrivalRate': {'type': 'number', 'default': 20, 'maximum': 200, 'description': 'Mean arrivals per hour (λ).', 'exclusiveMinimum': 0}, 'serviceRate': {'type': 'number', 'default': 12, 'description': 'Mean customers one server can finish per hour (μ). Must be > 0.', 'exclusiveMinimum': 0}, 'simulationDays': {'type': 'integer', 'default': 7, 'maximum': 30, 'minimum': 1, 'description': 'Days to simulate per candidate (default 7). Range 1-30. Lower = faster search; higher = less seed-to-seed variance.'}, 'arrivalDistribution': {'enum': ['Exponential', 'Constant'], 'type': 'string', 'default': 'Exponential'}, 'serviceDistribution': {'enum': ['Exponential', 'Constant', 'Normal', 'LogNormal'], 'type': 'string', 'default': 'Exponential'}, 'targetAvgWaitMinutes': {'type': 'number', 'default': 3, 'minimum': 0, 'description': "Maximum average wait time you're willing to accept, in minutes. The tool returns the smallest server count that meets this target."}}}
Output schema 
{'type': 'object', 'required': ['achievedAvgWaitMinutes', 'targetAvgWaitMinutes', 'trials', 'note'], 'properties': {'note': {'type': 'string'}, 'trials': {'type': 'array', 'items': {'type': 'object', 'required': ['servers', 'avgWaitMinutes'], 'properties': {'servers': {'type': 'integer'}, 'avgWaitMinutes': {'type': 'number'}}}}, 'recommendedServers': {'type': ['integer', 'null'], 'description': "Smallest c meeting the target. null when even maxServers can't meet it."}, 'targetAvgWaitMinutes': {'type': 'number'}, 'achievedAvgWaitMinutes': {'type': 'number'}, 'offeredLoadAtRecommendation': {'type': 'number', 'description': 'Present only when a feasible recommendation was found.'}}}
simulate_mmc
Run a generic M/M/c queue simulation. Provide an arrival rate (λ, arrivals/hour), a service rate per server (μ, customers/hour each server can finish), and a server count (c). Optional: distribution shapes, service coefficient of variation, run length. Returns per-hour metrics and an overall summary (avg wait, queue length, offered load, throughput). This is the primary tool for 'how many servers do I need?' / 'what's my average wait?' style questions. ALSO preferred over simulate_scenario for what-if questions about scheduled scenarios (Coffee Shop) when the user wants flat uniform numbers — pull the peak params from describe_scenario and run them here. That usually matches user intent better than collapsing a schedule. ANTI-FABRICATION: the returned numbers come from a real discrete-event simulation run. Quote them VERBATIM in your reply. Do not round, estimate, or compute derived figures from training-data recall. If the user asks a follow-up about the same configuration, re-call this tool rather than recalling numbers from earlier in the conversation.
Read only
Input schema 
{'type': 'object', 'required': ['arrivalRate', 'serviceRate', 'servers'], 'properties': {'servers': {'type': 'integer', 'default': 2, 'maximum': 50, 'minimum': 1, 'description': 'Number of parallel servers (c). Integer 1-50.'}, 'serviceCoV': {'type': 'number', 'maximum': 5, 'minimum': 0, 'description': "Coefficient of variation for service time — used when serviceDistribution is 'Normal' or 'LogNormal'. Ignored for Exponential/Constant. Range 0-5."}, 'arrivalRate': {'type': 'number', 'default': 20, 'maximum': 200, 'description': 'Mean arrivals per hour (λ). Any positive value up to 200.', 'exclusiveMinimum': 0}, 'serviceRate': {'type': 'number', 'default': 12, 'description': 'Mean customers one server can finish per hour (μ). Must be > 0.', 'exclusiveMinimum': 0}, 'simulationDays': {'type': 'integer', 'default': 7, 'maximum': 30, 'minimum': 1, 'description': 'Days to simulate (default 7). Range 1-30 — longer runs are not supported on the public surface; for production-scale studies contact queuesim@chiaha.com.'}, 'arrivalDistribution': {'enum': ['Exponential', 'Constant'], 'type': 'string', 'default': 'Exponential', 'description': "Shape of inter-arrival times. 'Exponential' = Poisson process (default). 'Constant' = evenly-spaced."}, 'serviceDistribution': {'enum': ['Exponential', 'Constant', 'Normal', 'LogNormal'], 'type': 'string', 'default': 'Exponential', 'description': "Shape of service-time distribution. 'Exponential' = classical M/M/c (default)."}}}
Output schema 
{'type': 'object', 'required': ['inputs', 'summary', 'hourly'], 'properties': {'hourly': {'type': 'array', 'items': {'type': 'object', 'properties': {'hour': {'type': 'integer'}, 'label': {'type': 'string'}, 'avgArrivals': {'type': 'number'}, 'maxArrivals': {'type': 'integer'}, 'avgDepartures': {'type': 'number'}, 'maxDepartures': {'type': 'integer'}, 'serversOnDuty': {'type': 'integer'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}, 'maxQueueLength': {'type': 'number'}, 'maxWaitMinutes': {'type': 'number'}}}}, 'inputs': {'type': 'object', 'description': "Echo of the run's parameters."}, 'summary': {'type': 'object', 'required': ['totalArrivals', 'totalServed', 'effectiveDays', 'overallAvgWaitMinutes'], 'properties': {'offeredLoad': {'type': ['number', 'null']}, 'totalServed': {'type': 'integer'}, 'effectiveDays': {'type': 'integer'}, 'totalArrivals': {'type': 'integer'}, 'overallAvgWaitMinutes': {'type': 'number'}}}}}
simulate_scenario
Run one of the four preset scenarios (single, coffee, grocery, callcenter) with optional overrides. Overrides apply UNIFORMLY across open hours — e.g. setting servers=5 on 'coffee' replaces the 4/6/4 staffing pattern with a flat 5 during open hours (closed hours stay at zero). Use this for (a) faithful reproduction of a scenario's defaults, or (b) uniform scaling (everywhere it was open, use these new numbers). Do NOT use this when the user wants to keep a scheduled scenario's shape but tweak just one part — there's no per-hour override here, and collapsing a 4/6/4 pattern to 5 often isn't what the user meant. For flat what-if analysis on scheduled scenarios, prefer simulate_mmc using peak params from describe_scenario. ANTI-FABRICATION: returned numbers come from a real discrete-event simulation run. Quote them VERBATIM in your reply. Do not round, estimate, or compute derived figures from training-data recall. If the user asks a follow-up about the same scenario+overrides, re-call this tool rather than recalling numbers from earlier in the conversation.
Read only
Input schema 
{'type': 'object', 'required': ['name'], 'properties': {'name': {'enum': ['single', 'coffee', 'grocery', 'callcenter'], 'type': 'string', 'default': 'coffee', 'description': 'Scenario key from list_scenarios.'}, 'overrides': {'type': 'object', 'properties': {'servers': {'type': 'integer', 'maximum': 50, 'minimum': 1, 'description': 'Override server count applied to every hour the scenario had staff.'}, 'serviceCoV': {'type': 'number', 'maximum': 5, 'minimum': 0}, 'arrivalRate': {'type': 'number', 'maximum': 200, 'description': 'Override arrival rate (customers/hr) applied to every hour the scenario was open.', 'exclusiveMinimum': 0}, 'simulationDays': {'type': 'integer', 'default': 7, 'maximum': 30, 'minimum': 1, 'description': 'Days to simulate (default 7). Range 1-30 — longer runs are not supported on the public surface.'}, 'serviceTimeMinutes': {'type': 'number', 'description': 'Override mean service time (minutes per customer).', 'exclusiveMinimum': 0}, 'arrivalDistribution': {'enum': ['Exponential', 'Constant'], 'type': 'string'}, 'serviceDistribution': {'enum': ['Exponential', 'Constant', 'Normal', 'LogNormal'], 'type': 'string'}}, 'description': "Optional overrides applied uniformly across open hours (closed hours preserved at zero for scheduled scenarios). All fields optional — leave empty to run the scenario's published defaults."}}}
Output schema 
{'type': 'object', 'required': ['inputs', 'summary', 'hourly'], 'properties': {'hourly': {'type': 'array', 'items': {'type': 'object', 'properties': {'hour': {'type': 'integer'}, 'label': {'type': 'string'}, 'avgArrivals': {'type': 'number'}, 'maxArrivals': {'type': 'integer'}, 'avgDepartures': {'type': 'number'}, 'maxDepartures': {'type': 'integer'}, 'serversOnDuty': {'type': 'integer'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}, 'maxQueueLength': {'type': 'number'}, 'maxWaitMinutes': {'type': 'number'}}}}, 'inputs': {'type': 'object', 'description': "Echo of the run's parameters."}, 'summary': {'type': 'object', 'required': ['totalArrivals', 'totalServed', 'effectiveDays', 'overallAvgWaitMinutes'], 'properties': {'offeredLoad': {'type': ['number', 'null']}, 'totalServed': {'type': 'integer'}, 'effectiveDays': {'type': 'integer'}, 'totalArrivals': {'type': 'integer'}, 'overallAvgWaitMinutes': {'type': 'number'}}}}}
simulate_schedule
Run a queueing simulation against an arbitrary 24-hour staffing schedule. Take this when the user describes a custom day shape that doesn't match a preset (e.g., 'my coffee shop is open 6am–10pm with 4 baristas off-peak, 7 at the 8am rush, 5 at the 4pm rush'). Inputs: `arrivalRates` (24-element array of customers/hr per hour-of-day) and `staffing` (24-element array of servers/hr); optional uniform `serviceTimeMinutes`. Use 0 in both arrays for closed hours (terminating system). Returns the same per-hour metrics + summary shape as simulate_mmc / simulate_scenario. Stronger fit than simulate_scenario when the user's shape doesn't match the four presets; stronger fit than simulate_mmc when they need per-hour variation. ANTI-FABRICATION: numbers come from a real DES run. Quote them VERBATIM. Do not round, estimate, or derive from training-data recall.
Read only
Input schema 
{'type': 'object', 'required': ['arrivalRates', 'staffing'], 'properties': {'staffing': {'type': 'array', 'items': {'type': 'integer', 'maximum': 50, 'minimum': 0}, 'maxItems': 24, 'minItems': 24, 'description': '24 hourly staffing counts (servers/hr, one per hour-of-day 0-23). Use 0 for closed hours. Range 0-50 per hour.'}, 'serviceCoV': {'type': 'number', 'maximum': 5, 'minimum': 0, 'description': 'CoV for service time when distribution is Normal or LogNormal. Range 0-5.'}, 'arrivalRates': {'type': 'array', 'items': {'type': 'number', 'maximum': 200, 'minimum': 0}, 'maxItems': 24, 'minItems': 24, 'description': '24 hourly arrival rates (customers/hr, one per hour-of-day 0-23). Use 0 for closed hours (terminating system). Range 0-200 per hour.'}, 'simulationDays': {'type': 'integer', 'default': 7, 'maximum': 30, 'minimum': 1, 'description': 'Days to simulate. Range 1-30.'}, 'serviceTimeMinutes': {'type': 'number', 'default': 5, 'maximum': 180, 'description': 'Mean service time in minutes per customer. Applies uniformly across hours. Range 0.1-180.', 'exclusiveMinimum': 0}, 'arrivalDistribution': {'enum': ['Exponential', 'Constant'], 'type': 'string', 'default': 'Exponential'}, 'serviceDistribution': {'enum': ['Exponential', 'Constant', 'Normal', 'LogNormal'], 'type': 'string', 'default': 'Exponential'}}}
Output schema 
{'type': 'object', 'required': ['inputs', 'summary', 'hourly'], 'properties': {'hourly': {'type': 'array', 'items': {'type': 'object', 'properties': {'hour': {'type': 'integer'}, 'label': {'type': 'string'}, 'avgArrivals': {'type': 'number'}, 'maxArrivals': {'type': 'integer'}, 'avgDepartures': {'type': 'number'}, 'maxDepartures': {'type': 'integer'}, 'serversOnDuty': {'type': 'integer'}, 'avgQueueLength': {'type': 'number'}, 'avgWaitMinutes': {'type': 'number'}, 'maxQueueLength': {'type': 'number'}, 'maxWaitMinutes': {'type': 'number'}}}}, 'inputs': {'type': 'object', 'description': "Echo of the run's parameters."}, 'summary': {'type': 'object', 'required': ['totalArrivals', 'totalServed', 'effectiveDays', 'overallAvgWaitMinutes'], 'properties': {'offeredLoad': {'type': ['number', 'null']}, 'totalServed': {'type': 'integer'}, 'effectiveDays': {'type': 'integer'}, 'totalArrivals': {'type': 'integer'}, 'overallAvgWaitMinutes': {'type': 'number'}}}}}
History

Recent tool changes

Added
compare_analytical_vs_simulated
Sept. 17, 2026, 12:37 p.m.
Added
interpret_result
Sept. 17, 2026, 12:37 p.m.
Added
recommend_staffing
Sept. 17, 2026, 12:37 p.m.
Added
explain_advanced_patterns
Sept. 17, 2026, 12:37 p.m.
Added
explain_queueing_theory
Sept. 17, 2026, 12:37 p.m.
Added
compare_separate_vs_pooled
Sept. 17, 2026, 12:37 p.m.
Added
simulate_schedule
Sept. 17, 2026, 12:37 p.m.
Added
simulate_scenario
Sept. 17, 2026, 12:37 p.m.
Added
describe_scenario
Sept. 17, 2026, 12:37 p.m.
Added
list_scenarios
Sept. 17, 2026, 12:37 p.m.
Added
simulate_mmc
Sept. 17, 2026, 12:37 p.m.
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