On a new formula for the transient state probabilities for M/M/1 queues and computational implications

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.

Download Full-text

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100002061 ◽

1963 ◽

Vol 59 (1) ◽

pp. 117-124 ◽

Cited By ~ 4

Author(s):

A. Wragg

Keyword(s):

Difference Equations ◽

Queueing Theory ◽

Bessel Functions ◽

Formal Solution ◽

Monomer Concentration ◽

Arrival Rate ◽

Time Dependent ◽

Type Integral ◽

Dependent Parameter ◽

Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

Download Full-text

Using the discrete time modelling approach to evaluate the time-dependent behaviour of queueing systems

Journal of the Operational Research Society ◽

10.1057/palgrave.jors.2600783 ◽

1999 ◽

Vol 50 (8) ◽

pp. 777-788 ◽

Cited By ~ 14

Author(s):

Dave Worthington ◽

A Wall

Keyword(s):

Discrete Time ◽

Queueing Systems ◽

Time Dependent ◽

Time Dependent Behaviour ◽

Dependent Behaviour ◽

Modelling Approach

Download Full-text

Higher monotonicity properties of normalized Bessel functions

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691314610074 ◽

2014 ◽

Vol 12 (05) ◽

pp. 1461007

Author(s):

Yongping Liu

Keyword(s):

Bessel Function ◽

Bessel Functions ◽

Positive Zero ◽

Local Maxima ◽

Monotonicity Properties

Denote by Jν the Bessel function of the first kind of order ν and μν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szegö states that the sequence [Formula: see text] is decreasing, another theorem of theirs states that the sequence [Formula: see text] has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence [Formula: see text] has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x-ν+1|Jν-1(x)|, x ∈ (0, ∞), i.e. the sequence [Formula: see text] has higher monotonicity properties.

Download Full-text

Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Due to a Time-Dependent Couple

Zeitschrift für Naturforschung A ◽

10.1515/zna-2011-1-207 ◽

2011 ◽

Vol 66 (1-2) ◽

pp. 40-46 ◽

Cited By ~ 1

Author(s):

Corina Fetecau ◽

Muhammad Imran ◽

Constantin Fetecau

Keyword(s):

Exact Solutions ◽

Couette Flow ◽

Bessel Functions ◽

Time Dependent ◽

Second Grade ◽

Governing Equations ◽

Material Parameters ◽

Taylor Couette ◽

On the Points of Inflection of Bessel Functions of Positive Order, I

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-049-x ◽

1990 ◽

Vol 42 (5) ◽

pp. 933-948 ◽

Cited By ~ 5

Author(s):

Lee Lorch ◽

Peter Szego

Keyword(s):

Bessel Function ◽

Bessel Functions ◽

Primary Concern ◽

Second Derivative ◽

Positive Order ◽

Fixed Rank

The primary concern addressed here is the variation with respect to the order v > 0 of the zeros jʺvk of fixed rank of the second derivative of the Bessel function Jv(x) of the first kind. It is shown that jʺv1 increases 0 < v < ∞ (Theorem 4.1) and that jʺvk increases in 0 < v ≤ 3838 for fixed k = 2, 3,… (Theorem 10.1).

Download Full-text

Higher order exponential split operator method for solving time-dependent Schrödinger equations

Canadian Journal of Chemistry ◽

10.1139/v92-078 ◽

1992 ◽

Vol 70 (2) ◽

pp. 555-559 ◽

Cited By ~ 24

Author(s):

André D. Bandrauk ◽

Hai Shen

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Operator Method ◽

Computation Time ◽

Time Dependent ◽

Schrodinger Equations ◽

Schrödinger Equations ◽

Exponential Form ◽

Operator Solution ◽

Split Operator

A new method of splitting exponential operators is proposed for the exponential form of the operator solution to the time-dependent Schrödinger equation. The method is shown to hold for any desired accuracy in the time increment. A comparison of different algorithms is made as a function of accuracy and computation time. Keywords: splitting operator, Fast Fourier Transform (FFT), Schrödinger equations.

Download Full-text

The zeros of bessel functions.

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1918.0006 ◽

1918 ◽

Vol 94 (659) ◽

pp. 190-206 ◽

Cited By ~ 5

Keyword(s):

Bessel Function ◽

Bessel Functions ◽

Practical Importance ◽

Zeros Of Bessel Functions ◽

Order Of Magnitude ◽

Low Order

1. Although many results are known concerning the zeros of Bessel functions,* the greater number of these results are of practical importance only in the case of functions of comparatively low order. For example, McMahon has given a formula† for calculating the zeros of the Bessel function J n ( x ), namely that, if k 1 , k 2 , k 3 , ..., are the positive zeros arranged in ascending order of magnitude, then ks = β- 4 n 2 -1/8β - 4(4 n 2 -1)(28 n 2 -31)/3.(8β) 3 -..., where β = 1/4 π (2 n +4 s —1).

Download Full-text

K-Bessel functions associated to a 3-rank Jordan algebra

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.2863 ◽

2005 ◽

Vol 2005 (18) ◽

pp. 2863-2870

Author(s):

Hacen Dib

Keyword(s):

Bessel Function ◽

Linear Combination ◽

Jordan Algebra ◽

Bessel Functions

Using the Bessel-Muirhead system, we can express theK-Bessel function defined on a Jordan algebra as a linear combination of the J-solutions. We determine explicitly the coefficients when the rank of this Jordan algebra is three after a reduction to the rank two. The main tools are some algebraic identities developed for this occasion.

Download Full-text

Unified integrals involving product of multivariable polynomials and generalized Bessel functions

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v38i6.39291 ◽

2019 ◽

Vol 38 (6) ◽

pp. 73-83

Author(s):

K. S. Nisar ◽

D. L. Suthar ◽

Sunil Dutt Purohit ◽

Hafte Amsalu

Keyword(s):

Bessel Function ◽

Bessel Functions ◽

General Character ◽

Polynomial Functions ◽

Finite Product ◽

Generalized Bessel Function ◽

Integral Formulas ◽

Generalized Bessel Functions ◽

Multivariable Polynomial ◽

Integral Formulae

The aim of this paper is to evaluate two integral formulas involving a finite product of the generalized Bessel function of the first kind and multivariable polynomial functions which results are expressed in terms of the generalized Lauricella functions. The major results presented here are of general character and easily reducible to unique and well-known integral formulae.

Download Full-text