1. Read λ , T and n simulations;

2. Create an empty list N values

3. For i ranging from 1 to n simulations, do

4. Calculate λ T = λ ∗ T

5. Generate a value N t according to a Poisson distribution ( λ T )

6. Add N t to the list N v a l u e s

7. end for

8. Calculate the empirical mean: e m p i r i c a l m e a n = M e a n ( N v a l u e s )

9. Calculate the empirical variance: e m p i r i c a l v a r = V a r i a n c e ( N v a l u e s )

10. Calculate the theoretical mean: t h e o r e t i c a l m e a n = λ ∗ T

11. Calculate the theoretical variance: t h e o r e t i c a l v a r = λ ∗ T

12. DISPLAY: −   e m p i r i c a l m e a n − t h e o r e t i c a l m e a n − e m p i r i c a l v a r − t h e o r e t i c a l v a r

13. Construct a histogram of the values of N v a l u e s

14. End

N ( t ) : number of requests/transactions arriving before t . λ : average arrival rate (transactions per unit of time). k : number of events.

λ ( t ) : instantaneous arrival intensity. μ : base rate (normal traffic). t i : times of previous arrivals. α : impact of an event on the future rate. β : rate of decay of the impact.

λ : average arrival rate of requests. μ : average service rate of the server. ρ : system occupancy rate. W : average time spent by a request in the system. L : average number of requests in the system.

c : number of parallel servers. λ : average arrival rate. μ : average service rate per server. ρ : overall occupancy rate.

X t : state of the system at time t . p i j : probability of transition from state i to state j .

π : vector of stationary probabilities. P : transition matrix ( p i j ) .

Q = ( q i j ) : generator matrix. P ( t ) : transition probability matrix at time t . I : identity matrix.

π : stationary distribution. Q : generator.

Results for λ = 2.0, T = 5.0.Empirical mean: 9.9535. Theoretical mean: 10.0. Empirical variance: 9.974324912456227. Theoretical variance: 10.0.

The empirical average calculated is 9.9535, while the theoretical average is 10.0. This result indicates a very small difference between the two values, with a deviation of only 0.0465. As shown in Figure 1, the histogram of the simulated Poisson process closely follows the expected theoretical distribution, confirming that the arrival process in the mobile payment system is well modeled by a Poisson process. This strong agreement suggests that the system behaves in line with theoretical predictions and that the proposed traffic modeling approach is effective. This suggests that the mobile payment system behaves in line with theoretical predictions and that the methods used to model the average payment are effective. This proximity reinforces the validity of our model, indicating that current payment management practices are relatively well aligned with theoretical expectations. The empirical variance is 9.974324912456227, which is also close to the theoretical variance of 10.0. Here again, the difference is minimal (0.025675087543773), showing that the observed dispersion of payments closely follows the predictions. A variance lower than the theoretical value may imply that the system exhibits greater homogeneity in transaction amounts than expected. This finding could mean that there are adjustment mechanisms in place that minimize extreme differences in payments, which would be desirable for the proper functioning of a fair payment system.

Figure 1. Simulation of the Poisson process.

The results suggest that the theoretical model used to analyze Max-Mean dispersion is well calibrated to represent payment behavior in the system studied. The small discrepancies observed between empirical and theoretical values may be due to random factors inherent in mobile transactions, such as variations in user behavior and economic fluctuations. These results are important for developers and managers of mobile payment systems, as they indicate that the mechanisms in place to stabilize payments are working effectively. However, it is essential to continue monitoring these measures dynamically to ensure that they remain consistent with theoretical expectations as new users and payment methods enter the system. The results obtained demonstrate a good fit between the empirical and theoretical aspects of the Max-Mean model, thereby reinforcing confidence in the methods used to analyze mobile payment systems and transaction dispersion. However, further investigation may be necessary to explore different environments and time periods in order to ensure greater robustness of the model. by virtually zero total interference.

The transition matrix-based analysis of the Max-Mean dispersion problem in mobile payment systems demonstrates that an effective balance between spatial dispersion, computational efficiency, and system robustness can be achieved when optimization is guided by state-transition dynamics rather than purely heuristic exploration. The results show that incorporating transition probabilities allows the system to converge toward stable configurations that preserve performance even under dense device deployment and fluctuating transaction loads, which are characteristic of IoT-enabled mobile payment environments. From a practical perspective, a mobile payment provider could implement these findings through dynamic server allocation or adaptive transaction routing mechanisms. For instance, transaction requests could be probabilistically redirected toward servers or gateways that maximize dispersion while minimizing cryptographic and communication overhead, based on real-time state transitions inferred from network load and device distribution. Such an approach would enable the system to dynamically adapt to peak usage periods, reduce congestion, and maintain post-quantum security guarantees without excessive resource consumption. Moreover, the integration of post-quantum cost factors—such as increased computational time, message size, and cryptographic load—reveals that security constraints are now a central component of dispersion optimization.

The study shows that a well-calibrated compromise between dispersion, resource usage, and quantum-resistant cryptography can significantly enhance the resilience and scalability of mobile transactions in financial IoT ecosystems. Hybrid optimization strategies, combining matrix-based modeling with heuristic methods, emerge as particularly suitable for real-world deployments. Finally, this study has certain limitations that must be acknowledged. The underlying Markov modeling framework assumes a memoryless property, implying that future system states depend only on the current state and not on historical behavior. While this assumption simplifies analysis and enables tractable modeling, it may not fully capture long-term dependencies or correlated transaction patterns observed in real payment networks. Future work could extend this framework by incorporating higher-order Markov models or learning-based approaches to better reflect temporal dependencies and evolving attack or usage patterns.

This matrix transition study opens up numerous research prospects for the Max-Mean dispersion problem in mobile payment systems.

1) It would be relevant to explore dynamic models where transition matrices evolve according to users’ temporal behavior.

2) The integration of security and confidentiality constraints could enhance the robustness of the model.

3) A natural extension is to combine our approach with machine learning techniques to refine interference prediction.

In addition, a major avenue for future work will be the development of the project entitled: Minimizing interference between mobile payment objects via hybrid metaheuristics, which aims to design an optimization framework capable of simultaneously reducing proximity conflicts and operational latencies. This future work will enable the testing of more advanced scalable hybrids, including adaptive and multi-agent strategies. Finally, large-scale validation on real payment platforms will be a key step in confirming the practical relevance of the proposed approaches.