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Jagerman D.L. — Difference equations with applications to queues
Jagerman D.L. — Difference equations with applications to queues

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Название: Difference equations with applications to queues

Автор: Jagerman D.L.


Jagerman (mathematics, Rutgers U.) explains techniques that he has found particularly useful in solving the many practical problems that have been put to him over the years from such fields as engineering, communication, physics, information theory, and astronomy. He presents a theory of difference and functional equations with continuous argument based on a generalization of the Riemann integral introduced by N. E. Norlund, allowing differentiation with respect to the independent variable and permitting greater flexibility in constructing solutions and approximations. He solves the nonlinear first-order equation by a variety of methods, including an adaptation of the Lie-Grobner theory. He also looks at the linear equation with constant coefficients and linear difference equations with polynomial coefficients.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 254

Добавлена в каталог: 04.06.2005

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Предметный указатель
Alternate representation of sum      89
Approximation method      176
Approximation method, approximate inversion      176
Approximation method, approximation to P(t,0)      177
Approximation to generators      161
Approximation to solution      114
Approximation to solution error estimate      114
Bandlimited functions      105
Bandlimited functions, samples on half line      105
Bernoulli polynomials, relation to Zeta function      55
Bernoulli's polynomials      50
Bernoulli's polynomials, Bernoulli numbers      51
Bernoulli's polynomials, complementary argument theorem      54
Bernoulli's polynomials, generating function      54
Bernoulli's polynomials, generating function for Bernoulli numbers      55
Bernoulli's polynomials, multiplication theorem      53
Bernoulli's polynomials, problem of Bernoulli      51
Bernoulli's polynomials, table of Bernoulli numbers      53
Bernoulli's polynomials, table of Bernoulli polynomials      52
beta function      7
Birth-death model      137
Birth-death model solution      138
Bound for remainder      83
Branching process approximation      145
Branching process approximation, critical case      146
Branching process approximation, nonnegative variable      146
Branching process approximation, subcritical case      146
Branching process approximation, supercritical case      145
Branching processes      136
Branching processes, generator for population      137
Casorati's determinant      202
Casorati's theorem      26
Casorati's theorem construction of a difference equation      29
Class of linear transformations      89
Class of linear transformations, associated function      90
Class of linear transformations, Borel transform      90
Class of linear transformations, exponential order of sequence      89
Class of linear transformations, maximum modulus      90
Class of linear transformations, vector space      90
Coefficients $A_{\nu} (\omega)$      127
Complete equation      222
Constant coefficients      172
Constant coefficients, characteristic equation      173
Constant coefficients, homogeneous equation      173
Constant coefficients, multiple roots      173
Constant coefficients, simple roots      173
Convergence of infinite products      113
Convolution integral representation      101
Convolution integral representation, derived error bound      102
Convolution integral representation, error estimate theorem      102
Critical points      143
Critical points, classification      144
Depression of order      201
Derivative      129
Derivative approximate mean work in M/G/l      131
Derivative approximation for Erlang loss function      130
Derivative approximation to u'(x)/u(x)      130
Differential-difference equations      122
Differential-difference equations examples of F-equation      124
Differential-difference equations method of Truesdell      122
Differential-difference equations reduction procedure      123
Differentiation      39
Differentiation formula      81
Differentiation formula asymptotic properties      82 97
Differentiation mth, derivative of sum      82
Equation of nth order      158
Equation of nth order examples      158
Equations reducible to constant coefficients      188
Erlang loss function      161
Error bound for transformation      101
Error bound for transformation, derived error bound      102
Error bound for transformation, error transformation      101
Euler equation      25
Euler summability      159
Euler summability, application of summability      161
Euler summability, circle of summability      160
Euler summability, consistency      160
Euler summability, regularity      160
Euler — Maclaurin expansion      75
Euler — Maclaurin expansion, periodic Bernoullian functions      75
Euler — Maclaurin expansion, usual form of expansion      76
Exact expansion for P(t,0)      177
Exceptional cases      217
Exceptional cases multiple roots      217
Exceptional cases multiple roots — Newton series      219
Existence of principal sum      77
Expansion for g(z)      142
Expansion for g(z), approximation to g(z)      143
Expansions and functional equations      96
Expansions and functional equations, differential-difference equation      97
Expansions and functional equations, first order difference equation      98
Expansions and functional equations, Newton's expansion      96
Factorial function      4
Factorial series      7
Factorial series first kind      7
Factorial series second kind      7
Factorial series uniqueness      9
Finite source model      195
Functional equations      132
Gauss — Laguerre quadrature table      61
Gauss — Legendre quadrature table      161
General operational solution      210
General operational solution inverse factorial series      212
General operational solution reduction to canonical form      211
General operational solution use of R      213
Genesis      21
Genesis Clairault      22
Genesis geometric distribution      22
Genesis Poisson distribution      22
Genesis Riccati      22
Genesis two parameter family      23
GI/M/1 queue      125
Gregory — Laplace quadrature      64
Haldane's method      149
Haldane's method expansion for l/g(z)      150
Heymann's theorem      29
Homogeneous equation of order n      25
Homogeneous equation of order n congruent points      26
Homogeneous equation of order n fundamental system      26
Homogeneous equation of order n general form of solution      29
Homogeneous form      44
Homogeneous form bounds for eigenvalue and sum      46
Homogeneous form commutability      45
Homogeneous form convexity of eigenvalue      47
Homogeneous form derivative      44
Homogeneous form difference      44
Homogeneous form eigenfunction of homogeneous sum      45
Homogeneous form extension of summability      45
Homogeneous form representation of Laplace transforms      45
Homogeneous sum exponential weight      100
Identity in $\pi$      207
Indicial equation, Inverse factorial series      213
inhomogeneous equation      179 114
Inhomogeneous equation Boole's operational method      179
Inhomogeneous equation concomitant boundary condition      115
Inhomogeneous equation construction of solution      114
Inhomogeneous equation method of Broggi      182
Inhomogeneous equation partial fraction method      182
Inhomogeneous equation specification of u(0)      116
Inhomogeneous equation useful form of solution      115
Inhomogeneous form      197
Integral equations      102
Integral equations inhomogeneous      104
Invariant function      157
Inverse of $\pi$+m      206
Lagrange's method      190
Lagrange's variation of parameters      224
Landau      7
Landau associated series      7
Landau convergence      7
Laplace      12
Laplace abscissa      13
Laplace bilateral      13
Laplace convergence      12
Laplace convolution      13
Laplace numbers      62
Laplace numbers generating function      63
Laplace numbers, Laplace numbers table      63
Laplace operational properties      14
Laplace transform      12
Laplace's method      183
Laplace's observation      193
Laplace's observation, Boole's example      194
LCFS M/M/C queue with reneging      226
LCFS M/M/C queue with reneging final solution      231
LCFS M/M/C queue with reneging, formulation and solution      228
LCFS M/M/C queue with reneging, method for binomial equations      229
LCFS M/M/C queue with reneging, perturbation solution for small alpha      231
LCFS M/M/C queue with reneging, second solution      229
LCFS M/M/C queue with reneging, test customer      226
Level crossings of work      120
Linear homogeneous equation      110
Linear homogeneous equation gamma function      111
Linear homogeneous equation, infinite product solution      113
Linear homogeneous equation, multiplication formula      111
Linear homogeneous equation, solution in terms of gamma function      112
Linear transformation      91
Linear transformation class A      91
Linear transformation eigenfunction      93
Linear transformation generating function      91
Linear transformation inverse      94
Linear transformation sum and product      93
Lower bound theorem      49
M/G/l queue with reneging      121
M/G/l queue with reneging difference equation      121
M/G/l queue with reneging M/M/l case      122
M/G/l queue with reneging Volterra integral equation      121
M/M/C Erlang blocking model      24
M/M/l delay model      24
M/M/l delay model, differential-difference equation      24
M/M/l delay model, semigroup type      24
M/M/l processor sharing queue      234
M/M/l processor sharing queue singular perturbation solution      235
M/M/l processor sharing queue tagged customers      234
M/M/l queue transient solution      174
M/M/l queue transient solution, exact inversion      176
M/M/l queue transient solution, Laplace transform of solution      175
M/M/l queue with feedback      132
mellin      14
Mellin convolution      14 15
Mellin inverse factorial series      14
Mellin operational properties      14 15
Mellin product of series      16
Milne — Thomson      30
Milne — Thomson asymptotic criterion      30
Monomial equations in $\pi$      207
Monomial equations in $\pi$ multiple roots      208
Monomial equations in $\rho$      204
Monomial equations in $\rho$, Erlang loss function      204
Monomial equations in $\rho$, use of factorial series      204
Newton's expansion      2
Newton's expansion uniqueness      8
Newton's expansion, defined      2
Nonlinear difference equation      133
Nonlinear difference equation, generating function g(z|h)      133
Nonlinear difference equation, ordinary differential equation      133
Nonlinear difference equation, partial differential equation      134
Norlund      8
Norlund principal sum      75
Norlund sum for complex arguments      99
Norlund sum for complex arguments complementary argument formula      100
Norlund sum for complex arguments complementary argument theorem      100
Norlund sum for complex arguments expansion for      99
Norlund sum for complex arguments integral representation      99
Norlund uniform convergence      8
Norlund's expansion      56
Norlund's expansion, asymptotic properties      57
Norlund's expansion, error estimate      57
Norlund's expansion, homogeneous sum      59
Norlund's expansion, representation of sum      59
Null function      13
Numerical differentiation      60
Operational formulae      17
Operational formulae shift formula      17
Operational method of Boole      179
Operator $\pi$      205
Operators      1
Operators $\pi$ and $\rho$      203
Partial difference equations      189
Partial difference equations Bernoulli trials      190
Partial difference equations Pascal's triangle      189
Perturbation solution      147
Perturbation solution Norlund sum expression      147
Perturbation solution perturbation solution      148
Principal solution      33
Principal solution limits and span      33
Principal solution Norlund definition      33
Principal solution sum      33
Properties of sum      34
Properties of sum limiting form      36
Properties of sum linear change of variable      34
Properties of sum multiplication theorem      35
Properties of sum span integral      35
Properties of sum sum from primitive      36
Psi-function      10
Quadrature formula for sum      62
Quadrature of homogeneous sum      65
Quadrature of homogeneous sum coefficients for exponential weight      66
Quadrature of homogeneous sum coefficients for linear weight      68
Queueing model M/M/n      118
Queueing model M/M/n asymptotic expansion      120
Queueing model M/M/n differences of 1/B(0,a)      120
Queueing model M/M/n Erlang loss function      118
Queueing model M/M/n expansion for 1/B(x,a)      119
Queueing model M/M/n extension to continuous argument      119
Queueing model M/M/n Fortet integral      120
Reduction of order      213
Repeated summation      40
Representation theorem      90
Riccati form      135
Riccati form inifinitesimal generators      136
Riccati form solution      135
Roots differing by an integer      220
Separation of variables      191
Sequence of transformations      111
Sequence of transformations convergence of generating functions      95
Sequence of transformations convergence theorem      95
Shift formula      205
Simultaneous first-order equations      154
Simultaneous first-order equations limiting form      157
Simultaneous first-order equations Newton's expansion      156
Simultaneous first-order equations U-operator method      155
Simultaneous first-order equations, equivalent differential equations      154
Simultaneous first-order equations, equivalent partial differential equations      154
Simultaneous first-order equations, partial differences      156
Solution in Newton series      215
Solution in Newton series, descending factorials      216
Solution in Newton series, example — integral form of solution      216
Solution in Newton series, indicial equation      215
Solution of functional equation      152
Solution of functional equation example      153
Solution of functional equation example of M/M/l queue with feedback      152
Solution of functional equation, solution of special case      153
Stirling equation      192
Stirling numbers      4
Stirling numbers first kind      5
Stirling numbers second kind      5
Sum of Laplace transforms      41
Sum of Laplace transforms, representation theorem      41
Summability      39
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