# On a functional equation arising from two processors with coupled inputs

## DOI:

https://doi.org/10.14419/ijbas.v4i3.4760## Published:

2015-06-19## Keywords:

Functional Equation, Complex Analysis, Queueing Theory, Boundary Value Problem, Singularity, Generating Function.## Abstract

During the last few decades, a certain interesting class of functional equations arises when obtaining the generating functions of many system distributions. Such a class of equations has numerous applications in many modern disciplines like wireless networks and communications. This paper has been motivated by an issue considered by Paul E. Wright in [Advances in applied probability, (1992), 986 ô€€€ 1007]. The functional equation obtained there has been solved using elliptic functions and analytic continuation, which in turn lead to the determination of the main unknown. Unfortunately that solution seems to be a bit too general with many technical assumptions. In this paper on one hand, we introduce a solution in the symmetric case using boundary value problem approach. On the other hand, we investigate the potential singularities of the unknowns of the functional equation giving one possible application, and we compute some expectation of interest using the corresponding generating function.

## References

[1] P. K. Sahoo and P. Kannappan, Introduction to functional equations, CRC Press, 2011.

[2] M. Kuczma, "A survey of the theory of functional equations", Univ. Beograd Publ. Elektrotechn. Fak. Ser.Mat.Fiz, No.130, (1964), pp. 1-64.

[3] M. Kuczma, An introduction to the theory of functional equations and inequalities: Cauchy's equation and Jensen's inequality, Springer, 2008.

[4] C. G. Small, Functional equations and how to solve them, Springer, 2007.

[5] T. M. Rassias and J. Brzdek, Functional Equations in Mathematical Analysis, Springer, 2012.

[6] J. Acz_el, Lectures on functional equations and their applications, Courier Dover Publications, 2006.

[7] Z. Dar_oczy and Z. P_ales, Functional Equations-Results and Advances, Springer, 2002.

[8] J. Acz_el and J. G. Dhombres, Functional equations in several variables, Birkhauser, Cambridge University Press, USA, 31, 1989.

[9] J. Acz_el, Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioural Sciences, Springer Science & Business Media, Vol. 3, (1987).

[10] J. Aczel, J. C. Falmagne and R. D. Luce, "Functional equations in the behavioral sciences", Mathematica japonicae, Vol. 52, No. 3, pp. 469-512, (2000).

[11] J. Acz_el, On applications and theory of functional equations, Academic Press, 2014.

[12] L. Kindermann, A. Lewandowski, and P. Protzel, "A framework for solving functional equations with neural networks" Proceedings of Neural Information Processing (ICONIP2001), 2, (2001), pp. 1075-1078.

[13] W. Gehrig, Functional equation methods applied to economic problems: some examples, Functional Equations: History, Applications and Theory, Springer, pp. 33-52, (1984).

[14] H. Fripertinger and J. Schwaiger, Some applications of functional equations in astronomy, na, (2001)

[15] C. Blackorby, W. Bossert and D. Donaldson, "Functional equations and population ethics", aequationes mathematicae, Vol. 58, No. 3, (1999), pp. 272-284.

[16] J. Cohen, "On the asymmetric clocked bu_ered switch", Queueing systems, Springer, Vol. 30, No. 3-4, (1998), pp. 385-404.

[17] F. Guillemin and J. S. V. Leeuwaarden, "Rare event asymptotics for a random walk in the quarter plane", Queueing Systems, Springer, Vol. 67, No. 1, (2011), pp. 65-98.

[18] J. Walraevens, Discrete-time queueing models with priorities, Ghent University, (2004).

[19] P. E. Wright, "Two parallel processors with coupled inputs", Advances in applied probability, JSTOR, (1992), pp. 986-1007.

[20] V. A. Malyshev, "An analytical method in the theory of two-dimensional positive random walks", Siberian Mathematical Journal, Springer, Vol. 13, No. 6, (1972), pp. 917-929.

[21] G. Fayolle and R. Iasnogorodski, "Two coupled processors: the reduction to a Riemann-Hilbert problem", Zeitschrift ur Wahrscheinlichkeitstheorie und verwandte Gebiete, Springer, Vol. 47, No. 3, (1979), pp. 325-351.

[22] J. W. Cohen and O. J. Boxma, Boundary value problems in queueing system analysis, Elsevier, 2000.

[23] G. Fayolle, R. Iasnogorodski, and V. A. Malyshev, Random walks in the quarter-plane: algebraic methods, boundary value problems and applications, Vol. 40, Springer, (1999).

[24] N. I. Muskhelishvili, Singular integral equations: boundary problems of function theory and their application to mathematical physics, Courier Dover Publications, 2008.

[25] F. D. Gakhov, Boundary value problems, Courier Corporation, 1990.

[26] F. Guillemin, C. Knessl, and J. S. V. Leeuwaarden, "Wireless Multihop Networks with Stealing: Large Bu_er Asymptotics via the Ray Method", SIAM Journal on Applied Mathematics, SIAM, Vol. 71, No. 4, (2011), pp. 1220-1240.

[27] J. Resing and L. ORmeci, "A tandem queueing model with coupled processors", Operations Research Letters, Vol. 31, No. 5, Elsevier, (2003), pp. 383-389.

[28] I. J. Adan, O. J. Boxma, and J. Resing, "Queueing models with multiple waiting lines", Queueing Systems, Springer, Vol. 37, No. 1-3, (2001), pp. 65-98.

[29] H. Nassar, H. Al Mahdi, "Queueing analysis of an ATM multimedia multiplexer with non-pre-emptive priority", IEE Proceedings-Communications, IET, Vol. 150, No. 3, (2003), pp. 189-196.

[30] H. Nassar, "Two-dimensional queueing model for a LAN gateway", WSEAS Transactions on Communications, WSEAS PRESS, Vol. 5, No. 9, (2006), pp. 1585.

[31] L. Flatto and S. Hahn, "Two parallel queues created by arrivals with two demands I", SIAM Journal on Applied Mathematics, Vol. 44, No. 5, (1984), pp. 1041-1053.

[32] L. Flatto, "Two parallel queues created by arrivals with two demands II", SIAM Journal on Applied Mathematics, Vol. 45, No. 5, pp.861-878, (1985).

[33] Z. Nehari, Conformal mapping, Courier Corporation, 1975.

[34] H. Kober, Dictionary of conformal representations, Dover New York, Vol. 2, 1957.

[35] R. P. Agarwal, K. Perera and S. Pinelas, An introduction to complex analysis, Springer, 2011.

[36] P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge University press, 2009.

[37] E-s. El-Hady, W. Forg-Rob, H. Nassar, and M. A. W. Mahmoud, On a functional equation arising from a difference equation characterizing the dynamics of a network gateway, Submitted for publication, (2015).

[38] D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, Springer Science & Business Media, 2007.