Difference Equations with Applications to Queues

Preface......Page 14
Contents......Page 18
1.  OPERATORS......Page 24
2.  FACTORIAL FUNCTION–—STIRLING NUMBERS......Page 27
3.  BETA FUNCTION–—FACTORIAL SERIES......Page 30
4.  [psi]-FUNCTION AND PRIMITIVES......Page 33
5.  LAPLACE AND MELLIN TRANSFORMATIONS......Page 35
PROBLEMS......Page 40
1.  GENESIS OF DIFFERENCE EQUATIONS......Page 44
3.  THE M/M/1 DELAY MODEL......Page 47
5.  THE EULER EQUATION......Page 48
PROBLEMS......Page 54
1.  INTRODUCTION......Page 55
2.  PRINCIPAL SOLUTION......Page 56
3.  SOME PROPERTIES OF THE SUM......Page 57
4.  SUMMATION OF SERIES......Page 60
5.  SUMMATION BY PARTS......Page 61
7.  EXTENSION OF DEFINITION OF SUM......Page 62
8.  REPEATED SUMMATION......Page 63
9.  SUM OF LAPLACE TRANSFORMS......Page 64
10.  HOMOGENEOUS FORM AND BOUNDS......Page 67
11.  BERNOULLI\'S POLYNOMIALS......Page 73
12.  COMPUTATIONAL FORMULAE......Page 79
PROBLEMS......Page 91
2.  THE EULER-MACLAURIN EXPANSION......Page 98
3.  EXISTENCE OF THE PRINCIPAL SUM......Page 100
4.  TRIGONOMETRIC EXPANSIONS......Page 107
5.  A CLASS OF LINEAR TRANSFORMATIONS......Page 112
6.  APPLICATIONS TO EXPANSIONS AND FUNCTIONAL EQUATIONS......Page 119
7.  APPLICATION TO THE NÖRLUND SUM......Page 122
8.  BOUND, ERROR ESTIMATE, AND CONVOLUTION FORM......Page 124
9.  CONSIDERATION OF SOME INTEGRAL EQUATIONS......Page 125
PROBLEMS......Page 128
1.  INTRODUCTION......Page 132
2.  THE LINEAR HOMOGENEOUS EQUATION......Page 133
3.  THE INHOMOGENEOUS EQUATION......Page 138
4.  THE DIFFERENTIAL–DIFFERENCE EQUATION......Page 145
5.  DERIVATIVE......Page 152
6.  FUNCTIONAL EQUATIONS......Page 155
7.  U-OPERATOR SOLUTION OF [Delta over h]Z = [theta](Z)......Page 162
8.  CRITICAL POINTS......Page 166
9.  A BRANCHING PROCESS APPROXIMATION......Page 168
10.  A PERTURBATION SOLUTION OF [Delta over h]Z = [theta](Z)......Page 170
11.  HALDANE\'S METHOD FOR [Delta over h]Z = [theta](Z)......Page 172
12.  SOLUTION OF G([phi](z)) – I(z)G(z) = m(z).......Page 174
13.  SIMULTANEOUS FIRST-ORDER EQUATIONS......Page 177
PROBLEMS......Page 185
1.  INTRODUCTION......Page 195
2.  THE HOMOGENEOUS EQUATION......Page 196
3.  THE INHOMOGENEOUS EQUATION......Page 202
4.  EQUATIONS REDUCIBLE TO CONSTANT COEFFICIENTS......Page 211
5.  PARTIAL DIFFERENCE EQUATIONS......Page 212
PROBLEMS......Page 221
1.  INTRODUCTION......Page 223
2.  DEPRESSION OF ORDER......Page 224
3.  THE OPERATORS [pi] AND [rho]......Page 226
4.  GENERAL OPERATIONAL SOLUTION......Page 233
5.  EXCEPTIONAL CASES......Page 240
6.  THE COMPLETE EQUATION......Page 245
7.  THE LCFS M/M/C QUEUE WITH RENEGING—INTRODUCTION......Page 249
8.  FORMULATION AND SOLUTION......Page 251
9.  AN M/M/1 PROCESSOR-SHARING QUEUE......Page 257
PROBLEMS......Page 259
References......Page 260
C......Page 264
F......Page 265
L......Page 266
N......Page 267
S......Page 268
V......Page 269