i-manager Publications

Transient Analysis of M/M/1 Fractional Queue

Nanduri Sujatha*, G. V. S. R. Deekshitulu**

* Department of Mathematics, Aditya Engineering College, Surampalem, Kakinada, India.

** Department of Mathematics, University College of Engineering, Jawaharlal Nehru Technological University, Kakinada, Andhra Pradesh, India.

Periodicity:July - December'2023
DOI : https://doi.org/10.26634/jmat.12.2.20067

Abstract

Queuing models consisting of servers which may not work with full efficiency are modelled with the help of fractional differential equations. Such problems with partial activity of the server are designed with the help of differentialdifference equations involving fractional derivatives whose solutions are expressed in terms of Mittag-Leffler function. In this paper, we propose an alternative approach which gives a transient solution of fractional M/M/1 queue in matrix form. The results obtained by this new approach are justified by comparing them with solutions of classical queue which are available in the literature. Efficacy of the model can be assessed by computing its state probabilities and also measures such as expected number of customers in the system etc. Also, the variations in these measures with respect to partial activity of the server have been presented graphically and numerically. Further, a mathematical procedure to find optimal efficiency of the server has been discussed.

Keywords

Transient Analysis, Fractional Derivatives, Mittag- Leffler Function, Optimal Efficiency.

How to Cite this Article?

Sujatha, N., and Deekshitulu, G. V. S. R. (2023). Transient Analysis of M/M/1 Fractional Queue. i-manager’s Journal on Mathematics, 12(2), 36-45. https://doi.org/10.26634/jmat.12.2.20067

References

[1]. Abate, J., & Whitt, W. (1988). Transient behavior of the M/M/1 queue via Laplace transforms. Advances in Applied Probability, 20(1), 145-178.

[2]. Ammar, S.I. (2015). Transient analysis of an M/M/1 queue with impatient behavior and multiple vacations. Applied Mathematics and Computation, 260, 97-105.

[3]. Ascione, G., Leonenko, N., & Pirozzi, E. (2018). Fractional queues with catastrophes and their transient behaviour. Mathematics, 6(9), 159.

[4]. Ascione, G., Leonenko, N., & Pirozzi, E. (2020). Fractional Erlang queues. Stochastic Processes and their Applications, 130(6), 3249-3276.

[5]. Beghin, L., & Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electronic Journal of Probability, 14, 1790-1826.

[6]. Cahoy, D. O., Polito, F., & Phoha, V. (2015). Transient behavior of fractional queues and related processes. Methodology and Computing in Applied Probability, 17, 739-759.

[7]. Cahoy, D. O., Uchaikin, V. V., & Woyczynski, W. A. (2010). Parameter estimation for fractional Poisson processes. Journal of Statistical Planning and Inference, 140(11), 3106-3120.

[8]. Demirci, E., & Ozalp, N. (2012). A method for solving differential equations of fractional order. Journal of Computational and Applied Mathematics, 236(11), 2754-2762.

[9]. Griffiths, J. D., Leonenko, G. M., & Williams, J. E. (2006). The transient solution to M/Ek/1 queue. Operations Research Letters, 34(3), 349-354.

[10]. Jain, J. L., & Meitei, A. J. (2007). Computation of the transient solution of M/M/1 queue. Opsearch, 44, 100-113.

[11]. Jain, J.L., Mohanty, S.G., Jiran Meitei, A. (2004). Tran- sient solution to M/M/1 Queue - A Probabilistic Approach, APORS, vol.I, 74-81.

[12]. Jelicic, Z. D., & Petrovacki, N. (2009). Optimality conditions and a solution scheme for fractional optimal control problems. Structural and Multidisciplinary Optimization, 38(6), 571-581.

[13]. Kalidass, K., Gnanaraj, J., Gopinath, S., & Kasturi, R. (2014). Transient analysis of an M/M/1 queue with a repairable server and multiple vacations. International Journal of Mathematics in Operational Research, 6(2), 193-216.

[14]. Khandakar, M., & Kuldeep Kumar, K. (2023). Some compound fractional poisson processes. Fractal and Fractional, 7(1), 15.

[15]. Kilicman, A., & Ahmood, W.A. (2017). On matrix fractional differential equations. Advances in Mechanical Engineering, 9(1), 1-7.

[16]. Kumar, B. K., & Arivudainambi, D. (2000). Transient solution of an M/M/1 queue with catastrophes. Computers & Mathematics with Applications, 40(10-11), 1233-1240.

[17]. Kumar, B. K., Parthasarathy, P. R., & Sharafali, M. (1993). Transient solution of an M/M/1 queue with balking. Queueing Systems, 13, 441-448.

[18]. Laskin, N. (2003). Fractional poisson process. Communications in Nonlinear Science and Numerical Simulation, 8(3-4), 201-213.

[19]. Mohan, J. J., & Deekshitulu, G. V. S. R. (2012). Fractional order difference equations. International Journal of Differential Equations, 2012.

[20]. Mohan, J. J., & Deekshitulu, G. V. S. R. (2014). Solutions of nabla fractional difference equations using N-transforms. Communications in Mathematics and Statistics, 1(2), 1-16.

[21]. Podlubny, I. (1998). Fractional Differential Equations. Academic Press, New York.

[22]. Rida, S. Z., & Arafa, A. A. M. (2011). New method for solving linear fractional differential equations. International Journal of Differential Equations, 2011.

[23]. Sharma, O. P., Tarabia, A. M. K. (2000). A simple transient analysis of an M/M/1/N queue. Sankhya: The Indian Journal of Statistics, Series A, 62(2), 273-281.

[24]. Srinivasan, A. V. (2008). Managing a Modern Hospital. SAGE Publications, India.

[25]. Srivastava, H. M., & Saxena, R. K. (2001). Operators of fractional integration and their applications. Applied Mathematics and Computation, 118(1), 1-52.

[26]. Sundari, S. M., & Srinivasan, S. (2012). Analysis of transient behaviour of M/G/1 queue with single vacation. International Journal of Pure and Applied Mathematics, 76(1), 149-156.

[27]. Tarabia, A. M. K. (2001). Transient analysis of a non-empty M/M/1/N queue an alternative approach. Opsearch, 38, 431-440.

[28]. Veillette, M., & Taqqu, M. S. (2010). Numerical computation of first-passage times of increasing Lévy processes. Methodology and Computing in Applied Probability, 12(4), 695-729.

Transient Analysis of M/M/1 Fractional Queue

Abstract

Keywords

How to Cite this Article?

References

If you have access to this article please login to view the article or kindly login to purchase the article

Purchase Instant Access

Options for accessing this content:

	North Americas,UK, Middle East,Europe		India	Rest of world
	USD	EUR	INR	USD-ROW
Pdf	35	35	200	20
Online	35	35	200	15
Pdf & Online	35	35	400	25