Journal of Intelligent Communication

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Information Data Length Theory for the Transient M/M/∞ Queueing System

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https://doi.org/10.54963/jic.v4i1.257
A Mageed, I. (2024). Information Data Length Theory for the Transient M/M/∞ Queueing System. Journal of Intelligent Communication, 4(1), 63–78. https://doi.org/10.54963/jic.v4i1.257
Volume 4 Issue 1 (2024): In Progress
Copyright (c) 2024 Ismail A Mageed
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Authors

The current paper provides a cutting-edge information data length-theoretic approach to the transient M/M/∞ queueing system. This is the first investigation that unifies information data length and queueing theories. Notably, an exposition of a significant real-life application of the M/M/∞ queueing system was addressed, namely the computation of the common average time for unsaturated site visitor flows beneath double-parking situations. This adds another dimension of the significance of the current work. On another note, the significant impact of both time and the number of states for the transient M/M/∞ queue on both the upper and lower bounds of the obtained information data length is observed and noted, which is a completely unprecedented innovative research methodology. The undertaken analytical technique in this paper is based on calculating the information data length for the M/M/∞ transient queue rather than going into higher complexities to go through a non-standard integral to be accomplished. It was a necessity to find both the upper and lower bounds of such a desired-to-be-calculated integration. The data collection process to carry out the numerical validation of the key analytic findings was conducted by choosing values for the parameters of the M/M/∞ transient queue to reveal both obtained upper and lower bounds numerically. The paper concludes with some challenging open problems, combined with concluding remarks and future research pathways.

Keywords:

information data length transient queues M/M/∞ queueing system

Author Biography

Dr. Ismail A Mageed, University of Bradford, United Kingdom.

Dr. Ismail A Mageed obtained his doctorate in Applied Probability at The University of Bradford, United Kingdom. Dr Mageed has been nominated by numerous high-profile academic institutions to the world prestigious ABEL PRIZE(NOBLE PRIZE OF MATHEMATICS) FOR THE ACADEMIC YEAR 2025, based on his great services to humanity through revolutionary mathematical applications to advance several scientific disciplines, including Engineering, Computer Science and much more. His current research interests include the unification of queueing theory with information theory and information geometry. His leading research on the relativisation of queuing theory and discovering the geodesic  equation of motion for transient queues was greatly received by the world research community, based on spotlighting novel avenues for a UNIFIED THEOREM ON EVERYTHING. Mageed’s research on finding the analytic solutions of the longstanding simulative approach of The Pointwise Stationary Fluid Flow Approximation theory (PSFFA) was an exceptional discovery to advance PSFFA theory. Dr Mageed has published numerous papers in many highly reputable journals and IEEE conferences. He is also a reviewer  and a member of the editorial board to many international prominent journals. Mageed’s research has been internationally recognized as being revolutionary by providing several breakthroughs and solving many longstanding open problems. He is currently an active member at the NetPen Research Group, which is the strongest research group in queueing networks in the world. Dr Mageed has published a chapter in a book of the best eight queueing theorists in the world, entitled:

Queueing Theory 2: Advanced Trends

By the world-renowned Publishing Company, ISTE WILLEY, which was translated into French by the same Publishing Company. He has also published another chapter in a high-profile book, entitled:

Fractal Analysis - Applications and Updates

By the world leading open access publishing house, IntechOpen. He is currently coaching numerous volunteering  several research teams worldwide to deliver more insights on employing research to serve humanity. He is also a fellow of the Royal Statistical Society (RSS),the OR Society of the United Kingdom, a member of INTISCC (Austria), IEANG (world council of engineering) and a life member of the Islamic Society of Statistical Sciences. 

Highlights

  • Introducing a revolutionary breakthrough  by unifying information data length and transient queuing systems theories  by revealing the significant impact of both time and the number of states for the transient queue on both the upper and lower bounds of the obtained information data length is observed and noted.
  • An exposition of a significant real-life application of queueing system was addressed, namely the computation of the common average time for unsaturated site visitor flows beneath double-parking situations.
  • The practical implication for the field includes the provision of some challenging open problems, which would open a plethora of several gates to the establishment of contemporary information data length theory of transient queueing systems combined with concluding remarks and future research pathways.

References

  1. Jones, R.P. Addressing the Knowledge Deficit in Hospital Bed Planning and Defining an Optimum Region for the Number of Different Types of Hospital Beds in an Effective Health Care System. Int. J. Environ. Res. Public Health 2023, 20, 7171.
  2. Wang, J. Patient Flow Modelling and Optimal Staffing for Emergency Departments: A Petri Net Approach. IEEE Trans. Comput. Soc. Syst. 2022, 10, 2022–2032.
  3. Izady, N.; Arabzadeh, B.; Sands, N.; Adams, J. Reconfiguration of Inpatient Services to Reduce Bed Pressure in Hospitals. Eur. J. Oper. Res. 2024, 316, 680–693.
  4. Kim, W.; Wang, X. Double Parking in New York City: A Comparison Between Commercial Vehicles and Passenger Vehicles. Transportation 2022, 49, 1315–1337.
  5. Dharmaprani, D.; Jenkins, E.; Aguilar, M.; Quah, J.X.; Lahiri, A.; Tiver, K.; Mitchell, L.;Kuklik, P.; Meyer, C.; Willems, S.; et al. M/M/Infinity Birth-Death Process- A Quantitative Representational Framework to Summarize and Explain Phase Singularity and Wavelet Dynamics in Atrial Fibrillation. Front. Physiol. 2021, 11, 616866.
  6. Mageed, I.A. Extended Entropy Maximisation and Queueing Systems with Heavy-Tailed Distributions. Doctoral dissertation, University of Bradford, Bradford, UK, 6 October 2022.
  7. Quah, J.; Dharmaprani, D.; Lahiri, A.; Schopp, M.; Mitchell, L.; Selvanayagam, J.B.; Perry, R.;Chahadi, F.; Tung, M.; Ahmad, W.; et al. Prospective Cross‐Sectional Study Using Poisson Renewal Theory to Study Phase Singularity Formation and Destruction Rates in Atrial Fibrillation (RENEWAL‐AF): Study Design. J. Arrhythmia 2020, 36, 660–667.
  8. Chamorro, H.R.; Guel-Cortez, A.J.; Kim, E.J.; Gonzalez-Longatt, F.; Ortega, A.; Martinez, W. Information Length Quantification and Forecasting of Power Systems Kinetic Energy. IEEE Trans. Power Syst. 2022, 37, 4473–4484.
  9. Mageed, I.A.; Yin, X.; Liu, Y.; Zhang, Q. of the Stable Five-Dimensional Queue Manifold Formalism's Info-Geometric Structure with Potential Info-Geometric Applications to Human Computer Collaborations and Digital Twins. In Proceedings of the 28th International Conference on Automation and Computing, Birmingham, UK, 30 August–1 September 2023.
  10. Mageed, I.A.; Zhang, Q.; Akinci, T.C.; Yilmaz, M.; Sidhu, M.S. Towards Abel Prize: The Generalized Brownian Motion Manifold's Fisher Information Matrix with Info-Geometric Applications to Energy Works. In Proceedings of the 2022 Global Energy Conference, Batman, Turkey, 26–29 October 2022.
  11. Mageed, I.A.; Zhou, Y.; Liu, Y.; Zhang, Q. Towards a Revolutionary Info-Geometric Control Theory with Potential Applications of Fokker Planck Kolmogorov (FPK) Equation to System Control, Modelling and Simulation. In Proceedings of 28th International Conference on Automation and Computing, Birmingham, UK, 30 August–1 September 2023.
  12. Mageed, I.A.; Kouvatsos, D.D. The Impact of Information Geometry on the Analysis of the Stable M/G/1 Queue Manifold. In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems, Online, 4–6 February 2021.
  13. Bouhlel, N.; Rousseau, D. Exact R´ enyi and Kullback-Leibler Divergences Between Multivariate t-Distributions. IEEE Signal Process. Lett. 2023, 30, 1672–1676.
  14. Mageed, I.A. A Unified Information Data Length (IDL) Theoretic Approach to Information-Theoretic Pathway Model Queueing Theory (QT) with Rényi Entropic Applications to Fuzzy Logic. In Proceedings of the 2023 International Conference on Computer and Applications, Cairo, Egypt, 28–30 November 2023.
  15. Wang, M.; Perez-Morelo, D.J.; Lopez, D.; Aksyuk, V.A. Persistent Nonlinear Phase-Locking and Nonmonotonic Energy Dissipation in Micromechanical Resonators. Phys. Rev. X 2022, 12, 041025.
  16. Nicholson, S.B.; García-Pintos, L.P.; del Campo, A.; Green, J.R. Time-Information Uncertainty Relations in Thermodynamics. Nat. Phys. 2020, 16, 1211–1215.
  17. Kim, E. Information Geometry and Non-Equilibrium Thermodynamic Relations in the Over-Damped Stochastic Processes. J. Stat. Mech.: Theory Exp. 2021, 9, 093406.
  18. Deb, S.; Dutta, P.S. Critical Transitions in Spatial Systems Induced by Ornstein–Uhlenbeck Noise: Spatial Mutual Information as a Precursor. Proc. R. Soc. A 2024, 480, 20230594.
  19. Kim, E.J.; Guel-Cortez, A.J. Causal Information Rate. Entropy 2021, 23, 1087.
  20. Guel-Cortez, A.J.; Kim, E.J. Information Length Analysis of Linear Autonomous Stochastic Processes. Entropy 2020, 22, 1265.
  21. Guel-Cortez, A.J.; Kim, E.J. Information Geometric Theory in the Prediction of Abrupt Changes in System Dynamics. Entropy 2021, 23, 694.
  22. Caselle, M.; Cellini, E.; Nada, A.; Panero, M. Stochastic Normalizing Flows as Non-Equilibrium Transformations. J. High Energy Phys. 2022, 2022, 1–31.
  23. Rolandi, A.; Perarnau-Llobet, M. Finite-Time Landauer Principle Beyond Weak Coupling. Quantum 2023, 7, 1161.
  24. Guel-Cortez, A.J.; Kim, E.J. Relations Between Entropy Rate, Entropy Production and Information Geometry in Linear Stochastic Systems. J. Stat. Mech.: Theory Exp. 2023, 2023, 033204.
  25. Grimmett, G.; Straker, D. Probability and Random Processes, 4th ed.; Oxford University Press: Oxford, UK, 2020; 668p.
  26. Qi, H.; Sparks, E.R.; Talwalkar, A. Paleo: A Performance Model for Deep Neural Networks. In Proceedings of the International Conference on Learning Representations, San Juan, Puerto Rico, 2–4 May 2016.
  27. Thilaka, B.; Poorani, B.; Udayabaskaran, S. Performance Analysis of an M/M/2 Queue with Partially Active Server Subject to Catastrophe. In Applications of Operational Research in Business and Industries: Proceedings of 54th Annual Conference of ORSI, 1st ed.; Gunasekaran, A., Sharma, J.K., Kar, S., Eds.; Springer Nature: Singapore, Singapore, 2023; pp. 233–251.
  28. Adeen, M.; Milano, F. Modelling of Correlated Stochastic Processes for the Transient Stability Analysis of Power Systems. IEEE Trans. Power Syst. 2021, 36, 4445–4456.
  29. Greiner-Petter, A.; Cohl, H.S.; Youssef, A.; Schubotz, M.; Trost, A.; Dey, R.; Aizawa, A.; Gipp, B. Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems. In Proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, Munich, Germany, 2–7 April 2022.
  30. Rodrigo, M. An Alternative Way of Defining Integration in Multivariable Calculus. Int. J. Math. Educ. Sci. Tech. 2024, 55, 1277–1290.
  31. Peng, J. Shape of Numbers and Calculation Formula of Stirling Numbers. Open Access Libr. J. 2020, 7, 1–11.
  32. Parthasarathy, P.R. A Transient Solution to an M /M /1 Queue: A Simple Approach. Adv. Appl. Prob. 1987, 19, 997–998.
  33. Angstmann, C.N.; Henry, B.I. Time Fractional Fisher–KPP and Fitzhugh–Nagumo Equations. Entropy 2020, 22, 1035.
  34. Hilfer, R.; Anton, L. Fractional Master Equations and Fractal Time Random Walks. Phys. Rev. E 1995, 51, R848.
  35. Compte, A. Stochastic Foundations of Fractional Dynamics. Phys. Rev. E 1996, 53, 4191–4193.
  36. Montroll, E.W.; Weiss, G.H. Random Walks on Lattices. II. J. Math. Phys. 1965, 6, 167–181.
  37. Li, C.; Qian, D.; Chen, Y.Q. On Riemann-Liouville and Caputo Derivatives. Discret. Dyn. Nat. Soc. 2011, 2011, 1–15.
  38. Ebadi, M.J.; Fahs, A.; Fahs, H.; Dehghani, R. Competitive Secant (BFGS) Methods Based on Modified Secant Relations for Unconstrained Optimization. Optimization 2023, 72, 1691–1706.
  39. Larijani, A.; Dehghani, F. An Efficient Optimization Approach for Designing Machine Models Based on Combined Algorithm. FinTech 2023, 3, 40–54.
  40. Shin, M.J.; Jung, Y. Using a Global Sensitivity Analysis to Estimate the Appropriate Length of Calibration Period in the Presence of High Hydrological Model Uncertainty. J. Hydrol. 2022, 607, 127546.
  41. Basijokaite, R.; Kelleher, C. Time‐Varying Sensitivity Analysis Reveals Relationships Between Watershed Climate and Variations in Annual Parameter Importance in Regions with Strong Interannual Variability. Water Resour, Res. 2021, 57, e2020WR028544.
  42. Guse, B.; Pfannerstill, M.; Fohrer, N.; Gupta, H. Improving Information Extraction from Simulated Discharge Using Sensitivity‐Weighted Performance Criteria. Water Resour. Res. 2020, 56, e2019WR025605.
  43. Li, W.; Cao, D.; Jöst, D.; Ringbeck, F.; Kuipers, M.; Frie, F.; Sauer, D.U. Parameter Sensitivity Analysis of Electrochemical Model-Based Battery Management Systems for Lithium-Ion Batteries. Appl. Energy 2020, 269, 115104.