Retail systems operate under conditions of stochastic demand uncertainty, where customer arrivals are inherently random, service capacity is finite, and congestion emerges as an endogenous property of the system. Traditional deterministic models fail to capture this variability.
This project develops a stochastic framework to model demand as a probabilistic arrival process and evaluate operational performance through queueing theory.
Customer arrivals are modeled using a Poisson process, defined as:
P(N = k) = (λ^k e^{-λ}) / k!
where λ represents the arrival rate (demand intensity), and k denotes the number of arrivals within a given time interval.
To evaluate system performance, the model adopts an M/M/1 queueing structure, assuming:
- Poisson arrivals (λ)
- Exponentially distributed service times (μ)
- A single service channel
Key performance metrics include:
- Utilization: ρ = λ / μ
- Expected number of customers in the system: L = ρ / (1 - ρ)
- Expected time spent in the system: W = 1 / (μ - λ)
The repository is structured as a modular stochastic modeling pipeline:
data/→ synthetic store data generationmodels/→ Poisson demand and queueing modelssimulation/→ system behavior simulationutils/→ performance metricsmain.py→ integrated execution pipeline
The system implements:
- Demand estimation using Poisson processes
- Weekday-based approximation of non-homogeneous demand
- Queue performance evaluation via M/M/1 theory
- Simulation of congestion dynamics
- Modular architecture enabling extensibility
Typical execution produces:
Demand forecast: [18, 21, 19, 25, 30, 28, 17...]
Queue metrics: ρ = 0.78 L = 3.54 W = 0.18
Simulation results: avg_queue = 3.2 max_queue = 9
Install dependencies and run the model:
pip install -r requirements.txt python main.py
The system reveals a fundamental operational contradiction:
- Increasing demand (λ ↑) improves revenue potential
- But simultaneously increases congestion (ρ → 1)
System stability requires:
λ < μ
If λ ≥ μ, the system becomes unstable, leading to theoretically unbounded queue growth.
The current implementation assumes:
- Exponential service time distribution
- Single-server system
- No behavioral modeling of customers
- No integration with inventory dynamics
These assumptions simplify analysis but limit realism in complex environments.
Potential extensions include:
- Non-homogeneous Poisson processes λ(t)
- Multi-server queueing systems (M/M/c)
- Bayesian inference for demand estimation
- Reinforcement learning for dynamic staffing
- High-fidelity discrete-event simulation (digital twin)
This project integrates concepts from:
- Operations Research
- Stochastic Processes
- Service Systems Engineering
- Supply Chain Analytics
Its applications extend to:
- Retail operations
- Call centers
- Logistics and distribution systems
- E-commerce fulfillment
Developed as a stochastic systems modeling project focused on integrating probabilistic demand modeling, queueing theory, and simulation-based analysis into a unified analytical framework.