Calculating exit times for series Jackson networks

1987 ◽  
Vol 24 (1) ◽  
pp. 226-234 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Bessel Functions ◽  
Transition Probabilities ◽  
Special Functions ◽  
Exit Time ◽  
Modified Bessel Functions ◽  
Exit Times ◽  
Jackson Network ◽  
Jackson Networks ◽  
Positive Orthant ◽  
New Family

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

Calculating exit times for series Jackson networks

1987 ◽  
Vol 24 (01) ◽  
pp. 226-234 ◽  
Author(s):  
William A. Massey
Keyword(s):  
Bessel Functions ◽  
Transition Probabilities ◽  
Special Functions ◽  
Exit Time ◽  
Modified Bessel Functions ◽  
Exit Times ◽  
Jackson Network ◽  
Jackson Networks ◽  
Positive Orthant ◽  
New Family

We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.

scholarly journals Analytic approximations for special functions, applied to the modified Bessel functions I2(x) and I2/3(x)

2018 ◽  
Vol 11 ◽  
pp. 1028-1033 ◽  
Author(s):  
Pablo Martin ◽  
Jorge Olivares ◽  
Fernando Maass ◽  
Elvis Valero
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Modified Bessel Functions ◽  
Analytic Approximations

Special functions, infinite divisibility and transcendental equations

1979 ◽  
Vol 85 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
C. Ping May
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Infinite Divisibility ◽  
Probability Density Functions ◽  
Integral Representations ◽  
Density Functions ◽  
Modified Bessel Functions ◽  
Explicit Formulas ◽  
Linear Combinations ◽  
Transcendental Equations

AbstractWe establish integral representations for quotients of Tricomi ψ functions and of several quotients of modified Bessel functions and of linear combinations of them. These integral representations are used to prove the complete monotonicity of the functions considered and to prove the infinite divisibility of a three parameter probability distribution. Limiting cases of this distribution are the hitting time distributions considered recently by Kent and Wendel. We also derive explicit formulas for the Kent–Wendel probability density functions.

scholarly journals On Applications of Some Special Functions in Statistics

2020 ◽  
Vol 8 (6) ◽  
pp. 1902-1908
Keyword(s):  
Characteristic Function ◽  
Laplace Transform ◽  
Bessel Functions ◽  
Special Functions ◽  
Probability Distributions ◽  
Laguerre Polynomials ◽  
Real Life ◽  
Moment Generating Function ◽  
Cumulative Density Function ◽  
Modified Bessel Functions

In this paper we will introduce some probability distributions with help of some special functions like Gamma, kGamma functions, Beta, k-Beta functions, Bessel, modified Bessel functions and Laguerre polynomials and in mathematical analysis used Laplace transform. We will also obtain their cumulative density function, expected value, variance, Moment generating function and Characteristic function. Some characteristics and real life applications will be computed in tabulated for these distributions

Analytical Solution of the Transient Response of Gas-to-Gas Crossflow Heat Exchanger With Both Fluids Unmixed

1986 ◽  
Vol 108 (4) ◽  
pp. 722-727 ◽  
Author(s):  
D. D. Gvozdenac
Keyword(s):  
Heat Exchanger ◽  
Bessel Functions ◽  
Special Functions ◽  
Time Varying ◽  
Modified Bessel Functions ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Temperature Distributions ◽  
The Mean ◽  
Mixed Fluid

The dynamic response of a single-pass crossflow heat exchanger with both fluids unmixed to arbitrary time varying inlet temperatures of fluids is investigated analytically. The initial spatial temperature distribution of the heat exchanger core is arbitrary as well. Analytical solutions for temperature distributions of both fluids and the wall as well as the mean mixed fluid temperatures at the exit are presented. The solutions are found by using Laplace transform method and special functions in the form of series of modified Bessel functions.

scholarly journals INEQUALITIES FOR SOME SPECIAL FUNCTIONS-II

1995 ◽  
Vol 26 (3) ◽  
pp. 235-242
Author(s):  
S. K. BISSU ◽  
C. M. JOSHI
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Upper Bounds ◽  
Modified Bessel Functions ◽  
Lower And Upper Bounds ◽  
Confluent Hypergeometric Functions

Some inequalities for Bessel functions, modified Bessel functions of the first kind and of their ratios involving both lower and upper bounds are given. The inequalities improve the results of earlier authours. Also incorporated in the discussion are some inequalities for the ratios of confluent hypergeometric functions of one variable.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

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