Shannonian Maximum Entropy Balking Threshold Mechanism (BTM) for a Stable M/G/1 Queue with Significant Applications of M/G/1 Queue Theory to Augmented Reality (AR)

Yıl 2024, Cilt: 12 Sayı: 2, 137 - 143, 30.08.2024

Ismail A Mageed

https://doi.org/10.17694/bajece.1302056

Öz

An exposition is undertaken to analytically derive validate the Shannonian Maximum Entropy BTM for the underlying stable queue. Most importantly, the analytic derivation of the upper and lower bounds 0f the absolute difference between Shannonian Cumulative service time distribution functions (CDFs) with and without balking. Typical numerical experiments are provided. Additionally, some applications of M/G/1 queue theory to AR are given. Some challenging open problems are addressed combined with closing remarks and future research directions.

Anahtar Kelimeler

BTM, M/G/1 queue theory, ME, AR, CDF

Kaynakça

• [1] I.A.Mageed, et al, “M/G/1 queue with Balking Shannonian Maximum Entropy Closed Form Expression with Some Potential Queueing Applications to Energy”, 2022 Global Energy Conference (GEC). IEEE, 2022.
• [2] E.T.Jaynes, “Information Theory and Statistical Mechanics”, Physical Review, 106, 1957, 620 - 630.
• [3] E.T.Jaynes, E.T., “Where do we Stand on Maximum Entropy?”, in Proc. The Maximum Entropy Formalism Conference, M.I.T., USA, 1978.
• [4]S.-C.Fang,et al, “Entropy optimization and mathematical programming,1997, Kluwer Academic Publishers, Boston.
• [5] J.Shore, and R. Johnson, “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy”, IEEE Transactions onInformation Theory,1980, 26, 26-37.
• [6] J.Shore, “Information theoretic approximations for M/G/1 and G/G/1 queuing systems”,1982, Acta Informatica, 17, 43.
• [7] J.Cantor,et al, “Information theoretic analysis for a general queueing system at equilibrium with application to queues in tandem”, Acta Informatica,1986, 23 ,657-678.
• [8] F.A.Haight, “Queueing with Balking”, Biometrika, 1957,44 (1957) 360-369.
• [9] L.Liu, “ Service Systems with Balking Based on Queueing Time ,PhD Thesis”, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, USA, 2007.
• [10] O.J.Boxma, and B.J. Prabhu, “Analysis of an M/G/1 Queue with Customer Impatience and an Adaptive Arrival Process”, 2009,Technical Report Eurandom; Vol. 2009028, Eindhoven University of Technology, Eindhoven, The Netherlands.
• [11] F.Baccelli,et al, “Single-Server Queues with Impatient Customers”, 1984, Advances in Applied Probability, 16, 887-905.
• [12] P.M.Morse, “ Queues, Inventories and Maintenance: The Analysis of Operational Systems with Variable Demand and Supply”, 1958,First ed., John Wiley & Sons, Inc.
• [13] A.W.Kemp, “Steady-state Markov chain models for certain q-confluent hypergeometric distributions”, 2005,Journal of Statistical Planning and Inference, 135, 107.
• [14] A.W.Kemp, “The Discrete Half-Normal Distribution”, in: Arnold, B.C., Balakrishnan,N., J.M. Sarabia,J.M. & Minguez, R. (Eds.) Advances in Mathematical and Statistical Modeling, Birkhäuser Boston,2008, pp. 353-360.
• [15] M.A.El-Affendi,and D.D.Kouvatsos,“ A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium”, 1983,Acta Informatica, 19,339.
• [16] D.D.Kouvatsos, “A Maximum Entropy Analysis of the G/G/1 Queue at Equilibrium”, 1988, The Journal of the Operational Research Society, 39, 183-200.
• [17] N.Shah, “Entropy Maximisation and Queues With or Without Balking”, 2014,PhD Thesis, Department of Computing, University of Bradford.
• [18] P.V.Gapeev , et al, “ Optimal double stopping problems for maxima and minima of geometric Brownian motions”, Methodology and Computing in Applied Probability. 2022 Jun;24(2):789-813.
• [19] C.Chaccour and W.Saad, “On the ruin of age of information in augmented reality over wireless terahertz (THz) networks”, InGLOBECOM 2020-2020 IEEE Global Communications Conference 2020 Dec 7 (pp. 1-6). IEEE.
• [20] M.Liubogoshchev, et al, “Adaptive cloud-based extended reality: Modeling and optimization”, IEEE Access. 2021 Feb 26;9:35287-99.
• [21] J.Xu , et al, “ Joint service caching and task offloading for mobile edge computing in dense networks”,In IEEE INFOCOM 2018-IEEE Conference on Computer Communications 2018 Apr 16 (pp. 207-215). IEEE