القائمة
Queuing theory analysis skill for analytical evaluation of waiting line systems.
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Reference for querying the Atlas knowledge graph through its MCP tools — the SECONDARY enrichment/comparison layer that adds best-practice context to systems you have ALREADY scanned from your real sources (`az`, repos, dirs). Use when you need to look up nodes, edges, kinds, clusters, stats, or wiki pages in Atlas to compare against your real inventory. (atlas graph, query atlas, atlas mcp, search the graph, graph neighbors, atlas record, atlas kinds, enrichment layer)
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| name | queuing-analyzer |
| description | Queuing theory analysis skill for analytical evaluation of waiting line systems. |
| allowed-tools | Bash(*) Read Write Edit Glob Grep WebFetch |
| metadata | {"author":"babysitter-sdk","version":"1.0.0","category":"simulation","backlog-id":"SK-IE-007"} |
| graph | {"domains":["domain:industrial-engineering"],"skillAreas":["skill-area:statistical-analysis","skill-area:organizational-design","skill-area:data-analysis"],"roles":["role:operations-analyst","role:research-engineer"]} |
You are queuing-analyzer - a specialized skill for analytical evaluation of waiting line systems using queuing theory.
This skill enables AI-powered queuing analysis including:
def mm1_queue(arrival_rate, service_rate):
"""
M/M/1 queue performance measures
- Poisson arrivals, exponential service, single server
"""
lambda_ = arrival_rate
mu = service_rate
# Utilization
rho = lambda_ / mu
if rho >= 1:
return {"error": "System unstable (rho >= 1)"}
# Performance measures
L = rho / (1 - rho) # Expected number in system
Lq = rho**2 / (1 - rho) # Expected number in queue
W = 1 / (mu - lambda_) # Expected time in system
Wq = rho / (mu - lambda_) # Expected time in queue
# Probabilities
P0 = 1 - rho # Probability system empty
Pn = lambda n: (1 - rho) * rho**n # Probability of n in system
return {
"model": "M/M/1",
"arrival_rate": lambda_,
"service_rate": mu,
"utilization": rho,
"L": L,
"Lq": Lq,
"W": W,
"Wq": Wq,
"P0": P0,
"P_wait": rho,
"stable": rho < 1
}
from scipy.special import factorial
import numpy as np
def mmc_queue(arrival_rate, service_rate, num_servers):
"""
M/M/c queue performance measures
- Multiple parallel servers
"""
lambda_ = arrival_rate
mu = service_rate
c = num_servers
rho = lambda_ / (c * mu)
if rho >= 1:
return {"error": "System unstable (rho >= 1)"}
# Calculate P0
sum_term = sum((c * rho)**n / factorial(n) for n in range(c))
last_term = (c * rho)**c / (factorial(c) * (1 - rho))
P0 = 1 / (sum_term + last_term)
# Erlang C formula (probability of waiting)
C = ((c * rho)**c / factorial(c)) * (1 / (1 - rho)) * P0
# Performance measures
Lq = C * rho / (1 - rho)
L = Lq + lambda_ / mu
Wq = Lq / lambda_
W = Wq + 1 / mu
return {
"model": "M/M/c",
"arrival_rate": lambda_,
"service_rate": mu,
"servers": c,
"utilization": rho,
"L": L,
"Lq": Lq,
"W": W,
"Wq": Wq,
"P0": P0,
"P_wait": C,
"stable": rho < 1
}
def mg1_queue(arrival_rate, service_mean, service_variance):
"""
M/G/1 queue using Pollaczek-Khinchin formula
- General service time distribution
"""
lambda_ = arrival_rate
Es = service_mean
Var_s = service_variance
# Second moment of service time
Es2 = Var_s + Es**2
rho = lambda_ * Es
if rho >= 1:
return {"error": "System unstable (rho >= 1)"}
# Pollaczek-Khinchin formula
Lq = (lambda_**2 * Es2) / (2 * (1 - rho))
L = Lq + rho
Wq = Lq / lambda_
W = Wq + Es
return {
"model": "M/G/1",
"arrival_rate": lambda_,
"service_mean": Es,
"service_variance": Var_s,
"utilization": rho,
"L": L,
"Lq": Lq,
"W": W,
"Wq": Wq,
"stable": rho < 1
}
def erlang_c_staffing(arrival_rate, service_rate, target_service_level,
target_wait_time):
"""
Determine minimum servers for service level target
"""
lambda_ = arrival_rate
mu = service_rate
# Minimum servers for stability
min_servers = int(np.ceil(lambda_ / mu))
for c in range(min_servers, min_servers + 100):
result = mmc_queue(lambda_, mu, c)
if result.get('error'):
continue
# Service level: P(wait <= target)
# SL = 1 - C * exp(-(c*mu - lambda) * target_wait)
C = result['P_wait']
exp_term = np.exp(-(c * mu - lambda_) * target_wait_time)
service_level = 1 - C * exp_term
if service_level >= target_service_level:
return {
"recommended_servers": c,
"achieved_service_level": service_level,
"target_service_level": target_service_level,
"P_wait": C,
"utilization": result['utilization'],
"avg_wait": result['Wq']
}
return {"error": "Could not achieve target service level"}
def finite_population_queue(arrival_rate, service_rate, num_servers,
population_size):
"""
Finite population queue (machine repair model)
"""
lambda_ = arrival_rate # Per-customer arrival rate
mu = service_rate
c = num_servers
K = population_size
# State probabilities using recursion
P = np.zeros(K + 1)
P[0] = 1 # Temporary
for n in range(1, K + 1):
if n <= c:
P[n] = P[n-1] * (K - n + 1) * lambda_ / (n * mu)
else:
P[n] = P[n-1] * (K - n + 1) * lambda_ / (c * mu)
# Normalize
P = P / P.sum()
# Performance measures
L = sum(n * P[n] for n in range(K + 1))
Lq = sum((n - c) * P[n] for n in range(c + 1, K + 1))
# Effective arrival rate
lambda_eff = sum((K - n) * lambda_ * P[n] for n in range(K))
W = L / lambda_eff if lambda_eff > 0 else 0
Wq = Lq / lambda_eff if lambda_eff > 0 else 0
return {
"model": "M/M/c/K/K",
"servers": c,
"population": K,
"L": L,
"Lq": Lq,
"W": W,
"Wq": Wq,
"effective_arrival_rate": lambda_eff,
"state_probabilities": P.tolist()
}
def jackson_network(arrival_rates, service_rates, routing_matrix):
"""
Open Jackson network analysis
arrival_rates: external arrivals to each node
service_rates: service rate at each node
routing_matrix: probability of routing from i to j
"""
n_nodes = len(service_rates)
# Solve for effective arrival rates
# lambda_i = gamma_i + sum_j(lambda_j * r_ji)
R = np.array(routing_matrix)
gamma = np.array(arrival_rates)
# lambda = gamma + lambda * R => lambda = gamma * (I - R)^-1
I = np.eye(n_nodes)
lambdas = np.linalg.solve((I - R.T), gamma)
# Analyze each queue as M/M/1
results = []
for i in range(n_nodes):
result = mm1_queue(lambdas[i], service_rates[i])
result['node'] = i
result['effective_arrival_rate'] = lambdas[i]
results.append(result)
# Network totals
L_total = sum(r['L'] for r in results if 'L' in r)
return {
"model": "Jackson_Network",
"effective_arrival_rates": lambdas.tolist(),
"node_results": results,
"total_L": L_total
}
This skill integrates with the following processes:
queuing-system-analysis.jscapacity-planning-analysis.jsdiscrete-event-simulation-modeling.js{
"model": "M/M/c",
"parameters": {
"arrival_rate": 10,
"service_rate": 4,
"servers": 3
},
"performance_measures": {
"utilization": 0.833,
"L": 6.01,
"Lq": 3.51,
"W": 0.601,
"Wq": 0.351
},
"probabilities": {
"P0": 0.045,
"P_wait": 0.702
},
"service_level": {
"P_wait_less_5min": 0.82
},
"recommendations": [
"High utilization - consider adding server"
]
}
| Library | Description | Use Case |
|---|---|---|
| scipy | Scientific computing | Core calculations |
| queueing | Python package | Queue analysis |
| queuecomputer (R) | R package | Advanced models |
| Custom | Hand-coded | Specific needs |