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CLAIRE vs Human Sales - Queueing Theory Simulation

M/M/c Model with Poisson Arrivals and Real Queue Dynamics

Based on Scientific Queueing Theory - Shows Realistic Customer Flow, Waiting Times & System Utilization

⚠️ System Stability Warning

Simulation Configuration

Time Settings

Simulation Duration (days): Simulation Speed: 10x

Customer Arrival Rate (λ)

Customers per Hour: Revenue per Client (£):

Note: Arrival rate (λ) represents how many customers arrive per hour. This simulation uses Poisson arrivals for realism.

Human Salesperson Configuration

Team Size (c): Working Hours per Day: Call Duration (minutes) - determines μ: Service Rate (μ): 4.0 customers/hour Conversion Rate (%): Annual Salary (£): Office Cost per Month (£), per Human:

Advanced Human Parameters

Working Days per Week:

CLAIRE AI Agent Configuration

Agent Count (c): Availability (hours per day): Interaction Duration (minutes) - determines μ: Service Rate (μ): 60.0 customers/hour Conversion Rate (%): Cost per Agent per Month (£): Infrastructure Cost per Year (£): System Uptime (%):

Simulation Progress Stopped

Ready to start - Time: 0 minutes

System Utilization (ρ = λ/(μc))

🧑 Human Team

ρ = 0.00

🤖 CLAIRE Agents

ρ = 0.00

Live Queue Visualization

🧑 Human Sales Team

Servers:

Queue (0 waiting):

🤖 CLAIRE AI Agents

Servers:

Queue (0 waiting):

Performance Metrics

🧑 Human Sales Team

Customers Served: 0

Avg Queue Length (Lq): 0.0

Avg Wait Time (Wq): 0.0 min

Conversions: 0

                        Revenue:
                        £0
                    

Total Costs: £0

                        Net Profit:
                        £0
                    

ROI: 0.0%

🤖 CLAIRE AI Agents

Customers Served: 0

Avg Queue Length (Lq): 0.0

Avg Wait Time (Wq): 0.0 min

Conversions: 0

                        Revenue:
                        £0
                    

Total Costs: £0

                        Net Profit:
                        £0
                    

ROI: 0.0%

📊 Performance Comparison

Service Speed Ratio: 0.0x

Wait Time Reduction: 0.0%

Queue Length Reduction: 0.0%

Capacity Multiplier: 0.0x

Profit Difference (CLAIRE - Human): £0

💰 Cost to Match Human Performance:

CLAIRE Agents Needed: 0

Total CLAIRE Cost: £0

💼 Cost to Match CLAIRE Performance:

Humans Needed: 0

Total Human Cost: £0

                        💡 Cost Savings with CLAIRE:
                        £0
                    

Performance Visualization

📚 About This Simulation: Queueing Theory (M/M/c Model)

This simulation is based on the M/M/c queueing model, a proven mathematical framework from operations research used to analyze waiting lines and service systems. It provides scientifically accurate predictions of customer wait times, queue lengths, and system performance.

🔬 What is M/M/c?

The M/M/c model describes a queueing system with three key characteristics:

First M (Markovian arrivals): Customers arrive randomly following a Poisson process with rate λ (lambda). This means arrivals are independent and the time between arrivals follows an exponential distribution.
Second M (Markovian service): Service times are exponentially distributed with rate μ (mu). This models situations where service duration is variable but averages to a known value.
c (servers): The number of parallel servers (salespeople or AI agents) that can serve customers simultaneously.

This model is widely used in telecommunications, call centers, hospital emergency rooms, and customer service operations.

📊 Key Metrics Explained

Symbol	Metric	Description
λ (lambda)	Arrival Rate	Number of customers arriving per hour (e.g., 20 customers/hour)
μ (mu)	Service Rate	Number of customers one server can handle per hour (e.g., 4 customers/hour for 15-min calls)
c	Server Count	Number of parallel servers (team members or AI agents)
ρ (rho)	Utilization	ρ = λ/(μc) — Fraction of total capacity being used. Must be < 1.0 for stability.
L_q	Queue Length	Average number of customers waiting in the queue (not being served)
W_q	Wait Time	Average time a customer spends waiting in the queue (in minutes)
C (Erlang-C)	Probability of Waiting	The probability that an arriving customer will have to wait (vs. being served immediately)

⚖️ Stability Condition: ρ < 1

The most critical requirement for a stable queueing system is that utilization (ρ) must be less than 1.0. This means:

λ < μ × c

"Arrival rate must be less than total service capacity"

What happens when ρ ≥ 1.0?

Customers arrive faster than they can be served
The queue grows infinitely over time
Wait times increase exponentially (minutes → hours → days)
Customer abandonment and service failure occur

⚠️ Best Practice: Target ρ < 0.85 (85% utilization) to maintain good service quality with buffer capacity for demand spikes.

🎯 How This Simulation Works

This is a discrete event simulation that processes time minute-by-minute:

Customer Arrivals: Each minute, new customers arrive according to a Poisson process with rate λ. The number of arrivals is randomly generated to match the statistical properties of real-world arrival patterns.
Queue Management: Arriving customers join the queue if all servers are busy. The queue is processed in FIFO (First-In-First-Out) order.
Service Processing: When a server becomes available, it takes the next customer from the queue. Service time is generated using an exponential distribution with mean = interaction duration.
Metric Tracking: The simulation continuously tracks queue length, wait times, customers served, conversions (based on conversion rate %), revenue, and costs.
Team Constraints: Human teams only operate during working hours (configurable), while CLAIRE agents operate 24/7 with configurable uptime %.

💡 Realism: This simulation uses Poisson arrivals and exponential service times—the same assumptions used by telecom companies, hospitals, and call centers worldwide for capacity planning.

🔗 Further Reading & References

To learn more about queueing theory and the M/M/c model:

Erlang-C Formula: Named after Danish mathematician A.K. Erlang (1917), used to calculate the probability of waiting in a multi-server queue. Widely used in call center workforce planning.
Little's Law: L = λW (average queue length = arrival rate × average wait time). A fundamental relationship in queueing theory.
Applications: Telecommunications networks, emergency services, retail checkout lines, IT help desks, restaurant seating, airport security, hospital emergency departments.

Note: This simulation is designed for business planning and capacity analysis. Results are based on well-established mathematical models and provide realistic estimates for decision-making.

CLAIRE - Cognitive Layer for Agentic Intelligence in Retail Enterprises

by Essential AI Solutions LTD.

Scientific Foundation Article (PDF)

CLAIRE vs Human Sales Performance Simulation | Based on M/M/c Queueing Theory

Built with Plotly.js for visualization | All calculations performed client-side

info@essentialai.uk

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⚠️ System Stability Warning

Simulation Configuration

Time Settings

Customer Arrival Rate (λ)

Human Salesperson Configuration

CLAIRE AI Agent Configuration

Simulation Progress Stopped

System Utilization (ρ = λ/(μc))

Live Queue Visualization

Performance Metrics

🧑 Human Sales Team

🤖 CLAIRE AI Agents

📊 Performance Comparison

Performance Visualization