Formulation of the simple Markovian model using fractional calculus approach and its application to analysis of queue behavior of severe patients

Soma Dhar; Lipi B. Mahanta; Kishore Kumar Das

doi:10.21307/stattrans-2019-007

Formulation of the simple Markovian model using fractional calculus approach and its application to analysis of queue behavior of severe patients

Soma Dhar Gauhati University , Lipi B. Mahanta Institute of Advanced Study in Science and Technology. E-mail:. ORCID:https://orcid.org/0000-0002-7733-5461 , Kishore Kumar Das Department of Statistics, Gauhati University, Guwahati, India Statistics in Transition new series, vol. 20, 2019, 1, pages: 117-129 Published online: 25 April 2019 DOI 10.21307/stattrans-2019-007

997 Views 27 Downloads

ARTICLE

(English) PDF

ABSTRACT

In this paper, we introduce a fractional order of a simple Markovian model where the arrival rate of the patient is Poisson, i:e: independent of the patient size. Fraction is obtained by replacing the first order time derivative in the difference differential equations which govern the probability law of the process with the Mittag-Leffler function. We derive the probability distribution of the number N(t) of patients suffering from severe disease at an arbitrary time t. We also obtain the mean size (number) of the patients suffering from severe disease waiting for service at any given time t, in the form of E^V_0.5,0.5(t), for different fractional values of server activity status, n = 1,0.95,0.90 and for arrival rates α = β = 0.5. A numerical example is also evaluated and analysed by using the simple Markovian model with the help of simulation techniques.

KEYWORDS

fractional order, arrival rate, patients, fractional calculus

REFERENCES

ABEL, N. H., (1823). Solution de quelques problemesa laide dintegrales definies. Mag. Naturvidenskaberne, 2, pp. 63–68.

BAILEY, N. T., (1954). Queueing for medical care. Applied Statistics, pp. 137–145.

BAILEY, N. T., (1990). The elements of stochastic processes with applications to the natural sciences, volume 25. John Wiley & Sons.

BEGHIN, L., ORSINGHER, E., (2009). Fractional poisson processes and related planar random motions. Electronic Journal of Probability, 14 (61), pp. 1790– 1826.

CAHOY, D. O., POLITO, F., PHOHA, V., (2013). Transient behavior of fractional queues and related processes. Methodology and Computing in Applied Probability, pp. 1–21.

CAHOY, D. O., UCHAIKIN, V. V., WOYCZYNSKI, W. A., (2010). Parameter estimation for fractional poisson processes. Journal of Statistical Planning and Inference, 140 (11), pp. 3106–3120.

DHAR, S, DAS, K. K., MAHANTA, L. B., (2014). Comparative study of waiting and service costs of single and multiple server system: A case study on an outpatient department. International Journal of Scientific Footprints, 3 (2), pp. 18–30.

DHAR, S., DAS, K. K., MAHANTA, L. B., (2017). An infinite server queueing model with varying arrival and departures rates for health care system.International Journal of Pure and Applied Mathematics, 113 (5), pp. 583–593.

GORENO, R., MAINARDI, F., (1997). Fractional calculus. Springer.

HAUBOLD, H. J., MATHAI, A. M., SAXENA, R. K., (2011). Mittag-Leffler functions and their applications. Journal of Applied Mathematics, 2011.

HERRMANN, R., (2014). Fractional calculus: an introduction for physicists. World Scientific.

HILFER, R., (2000). Applications of fractional calculus in physics. World Scientific.

HILFER, R., et al., (2008). Threefold introduction to fractional derivatives. Anomalous transport: Foundations and applications, pp. 17–73.

KIRYAKOVA, V., (1994). Generalized fractional calculus and applications longman (pitman res. notes in math. ser. 301).

LASKIN, N., (2003). Fractional poisson process. Communications in Nonlinear Science and Numerical Simulation, 8 (3), pp. 201–213.

MAHANTA, L. B., DAS, K. K., DHAR, S., (2016). A queuing model for dealing with patients with severe disease. Electronic Journal of Applied Statistical Analysis, 9 (2), pp. 362–370.

MAINARDI, F., (2010). Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific.

MATHAI, A. M., HAUBOLD, H. J., (2008). Special functions for applied scientists, Vol. 4. Springer.

MILLER, K. S., ROSS, B., (1993). An introduction to the fractional calculus and fractional differential equations.

OLDHAM, K., SPANIER, J., (1974). The fractional calculus. 1974.

ORSINGHER, E., POLITO, F., et al., (2011). On a fractional linear birth-death process. Bernoulli, 17 (1), pp. 114–137.

ORSINGHER, E., POLITO, F., SAKHNO, L., (2010). Fractional non-linear, linear and sublinear death processes. Journal of Statistical Physics, 141 (1), pp. 68– 93.

PODLUBNY, I., (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198. Academic press.

RUBIN, B., (1996). Fractional integrals and potentials, pitman monogr. Surv. Pure Appl. Math, 82.

SAMKO, S. G., KILBAS, A. A., MARICHEV, O. I., et al., (1993). Fractional integrals and derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

SAXENA, R., SAIGO, M., (2005). Certain properties of fractional calculus operators associated with generalized Mittag-Leffer function. Fractional calculus and applied analysis, 8 (2), pp. 141–154.

SRINIVASAN, A. V., (2008). Managing a modern hospital. SAGE Publications, India.

SRIVASTAVA, H. M., SAXENA, R. K., (2001). Operators of fractional integration and their applications. Applied Mathematics and Computation, 118 (1), pp. 1– 52.

UCHAIKIN, V. V., CAHOY, D. O., SIBATOV, R. T., (2008). Fractional processes: from poisson to branching one. International Journal of Bifurcation and Chaos, 18 (09), pp. 2717–2725.

VEILLETTE, M., TAQQU, M. S., (2010). Numerical computation of first-passage times of increasing Levy processes. Methodology and Computing in Applied Probability, 12 (4), pp. 695–729.