Index

Absorbing state Average-cost criterion, 326, 360 in Markov decision process, 330 Back-substitution, 182-183 Absorption, 466 Bailey's , 209, 247-248 Accelerating convergence, 273 Balance equations, 412 Acceleration, 268, 270 global, 130, 411-412, 414 Action space, 326 job-class, 423 Adjoint, 215, 230 local, 130, 422 Agarwal, 289, 298 partial, 411-414, 418, 422 Aggregate station, 437 station, 411, 413-414, 418, 422 Aggregation, 57 and blocking, 427 Aggregation matrix, 102 failure of, 425 Aggregation step, 103 restored, 427, 429, 435 Aliased sequence, 277 Barthez, 272 Aliasing, 266 BASTA, 373 See also Error, aliasing Benes, 282, 284, 287 Alternating series, 268 Bernoulli arrivals, 373 Analytic, 206, 208, 210, 216, 235, 238 Bernoulli process, 387 Analyticity condition, 294 Bertozzi, 304 And gate, 462 Binomial average, 270 Approximation, 375, 377-378, 380-381, 388, Binomial distribution, 270 392-393, 396 Block elimination, 175, 179, 183 of transition matrix, 180, 357 and paradigms of Neuts, 187-189 Arbitrary-epoch tail probabilities, 381 Block iterative methods, 93 Arbitrary service time, 381 Block Jacobi, 94 Argument principle, 207, 239 Block SOR, 94 Arnoldi's method, 53, 83, 97 Block-splitting, 93 Arrival-first, 366 Blocking, 158, 425-427 Asmussen, 293, 299, 301 Blocking probability, 410 Assembly line, 415 ,283 Asymptotic behavior, 243 time dependent, 280, 282 Asymptotic formulas, 282 Borovkov, 281 Asymptotic parameter, 294 Boundary probabilities, 206, 210, 218 Asymptotics, 303 Bounding methodology, 428-429 Automata: see Stochastic automata Bounds, 351, 429, 432, 434-435, 437 Availability, 46, 465 and optimal design, 436 instantaneous, 447 for waiting time, 396 interval, 447 intuitively obvious, 430 limiting interval, 448 on loss probability, 429, 432, 434 point availability, 69 on minimal costs, 436 steady-state, 448 on throughput, 429, 432, 439 482 COMPUTATIONAL PROBABILITY

on utilization, 429 rate of, 89, 358 Branch point, 238 Convex, 243-244 Breakdown, 426, 434-437, 439 Convolution algorithm, 304, 423, 428 Bremmer, 260 Convolution, 206, 259, 285, 315 Bromwich inversion integral, 263 smoothing, 272 Buffer, 411, 437-438 n-fold, 261 Busy cycle, 291, 367 Conway, 304 Busy period, 261, 367, 387 Costs, 325, 328 in MIGll queue, 280 terminal, 329 time-homogeneous, 335 Call center, 194, 433 uniformly bounded, 357 bilingual, 199 Coupling matrix, 102 Capacity constraint, 411, 427 Covariance function, 281, 287 Cauchy contour integral, 276 Cramer-Lundberg approximation, 294 Censored chain, 214, 221, 230 Crout method, 184, 191 Censoring, 173, 175, 177, 180, 182, 189-190 CTMC, 154, 412 Central limit theorem, 259 See also Continuous-time Markov chain Characteristic function, 258 Cumulants, 299 Chaudhry, 289, 298 Cumulative distribution function, 262 QPACK software package, 381, 397 Cumulative reward, 45, 48, 60-61 QROOT software package, 376, 381, 390 cumulative impulse reward, 46 Chebyshev-Subspace iteration, 97 cumulative operational time, 46, 69 Closed queueing network: see Queueing cumulative rate reward, 45, 64, 71 network Curse of dimensionality, 7, 410 Coefficient of variation, 457 Cutset, 141 Cohen, 257, 290, 301 Combinatorial explosion, 424 DA (Delayed Access), 367 Communication systems, 365 Damped sequence, 277 Compact storage schemes, 82, 85 Damping, 266 Companion matrix, 222 Darling, 286 Complementary cdf, 262 Davies, 272 Complementation Davis, 263, 265 of transition matrix, 180 Decision, 325 Complex-conjugate, 377, 392 Decision rule, 326 Complex Fourier series, 266 greedy, 353 Complex-valued functions, 262 Decomposable, 157, 174 Complex variables, 263 Decomposition Computation time, 114 for dimension reduction, 309-311 Computational complexity, 7, 139, 309 of a network, 439 Computer networking, 410 Decompositional methods, 97 Computer systems, 365 Decreasing failure rate, 27 Conditional decomposition, 309 Delay, 410 Congestion, 432 average, 197-198 Conservation of probability, 206, 209-210, Delayed access, 367 212, 218, 226, 228, 231, 249 Departure-first, 366 Continuous, 210, 216, 238, 243 Dependability, 65, 445-446, 469-473 Continuous-time control, 359 Depth-first search, 105 Continuous-time Markov chain, 154, 219, DES: see Discrete event system 232, 412, 453 Descriptor, 123 Continuous-time Markov decision chain, Determinant, 216-218, 238, 240, 282 358-359 Deterministic arrivals, 397 Contour integral, 263, 289 Deterministic service time, 381, 422 Contour, 276 Diagonally dominant, 240 Contraction mapping, 342, 349, 358 irreducibly, 240 Control strictly, 240 continuous-time, 359 Difference equations, 225 Convergence Difference operator, 264 pointwise, 358 Diffusion process, 281 INDEX 483

Dimension reduction truncation, 265, 270, 274 by decomposition, 309-311 Euler summation, 268, 270, 273, 278, Directed graph, 105 314-315 Direct methods: see Solution method Euler's constant, 301 Disaggregation step, 103 Event, 3, 16 Discount factor, 330 immediate, 15, 29-33 Discounted-cost problem, 330 schedule of, 3, 14 Discrete event Exchangeability, 60, 64 simulation, 3 Expected level, 193 system, 3, 14 Exponential damping, 300 Discrete-time Markov chain, 12, 14, Exponential polynomial, 457 154-155,207 Exponential service time, 412 Discretization error, 265-266, 268 Distribution Fault tree, 451, 462-463 binomial, 270 with repeated events, 451 gamma, 260, 297-298 Feldmann, 301 geometric, 397 Feller, 259, 268 infinitely divisible, 247 Fill-in, 82, 85, 92 marginal, 420-421, 424 Finite differences, 264 normal,281 Finite-horizon problem, 329, 332, 334 phase-type, 5, 27 Finite-state approximations, 357 stationary, 81, 206, 214, 219, 229 First passage time, 47, 284, 387 steady-state, 281 Fixed point, 341, 349 Divide and conquer, 99 iteration, 91, 471 Doetsch, 260, 262 Flannery, 296 Drift, 212, 219 Flow equations, 412 DTMC, 12 Fluid flow models, 71 See also Discrete-time Markov chain Fork and join queue, 427 Dubner,272 Fourier Duffield, 280 coefficients, 266 Durbin, 272 series, 265-266 Dynamic programming, 332, 336 method,266 transform, 168, 258 EAS (early arrival system), 367-368, 397 discrete, 277 Eigenvalues, 51, 207, 217, 222, 234, 238, 243, FT,451 282 See also Fault tree distinct, 239 FTRE,451 multiplicity of, 239 See also Fault tree with repeating Eigenvectors, 207, 217, 243, 282 events Elimination of non-optimal actions, 355 Functional dependency cycle, 140 Emptiness Functional rates, 135 probability of, 29 Functional transitions, 118, 123 Entropy, 412 Erlang, 365 G matrix, 221 Erlang blocking probability, 281, 283 Gamma distribution, 260, 297-298 Erlang loss model, 279 GASTA,373 Error, 265 Gaussian elimination, 86-87, 340 absolute, 293 Gauss-Seidel, 91 aliasing, 266-267, 269-272, 277-279, 303, backward, 82, 89 311-312 forward, 82, 89 bound, 265, 272, 275 ordering of states, 90 discretization, 265-268, 276, 312 Gaver, 258, 272, 301 balancing with roundoff error, 271 Gaver-Stehfest procedure, 264 estimate, 270 Generalized eigenvector, 222 in Euler summation, 272, 274 Generalized Erlang, 292 relative, 293 Generalized tensor product, 135, 137-139 roundoff, 265, 268, 271-272, 278-279, 312 Generating function, 206, 208, 275, 303, 307 control of, 268, 279 Generator, 219, 232 484 COMPUTATIONAL PROBABILITY

and synchronizing events, 119 Infinitesimal generator, 81 Geometric arrivals, 373 Ir,itial approximation, 83 Geometric, 160, 366, 377-378, 381, 392, 394 Initial value theorem, 269 Geometric distribution, 397 Insensitivity, 422 Georganas, 304 of bounds, 430, 435 Giffin, 260 Instantaneous state, 31 GI/G/1 paradigm, 156, 191-193 preservation, 31 GI/G/1 queue: see Queue, GI/G/1 Interarrival time GI/G/1 type process, 156-158 rational transform, 290 GI/M/1 paradigm, 168, 187-189 Interdependence graph, 310 GI/M/1 type process, 155-157, 159, 164 Inverse of a matrix, 174-175, 185 GI/M/1 queue: see Queue, GI/M/l Inversion dimension, 310 Global balance, 130,411-412,418,422 Inversion of generating functions, 375 Global generator, 118 Irreducible, 207, 210, 217, 226 G/M/l type process, 206, 223, 228-229, Iterative aggregation/disaggregation, 103 232-233 Iterative methods, 82 See also GI/M/l type process Jacobs, 301 G/M/l queue: see Queue, GI/M/l Jackson, 421 GMRES, 98, 115 Jackson networks, 24, 417 Graph, 310 Jagerman, 264, 272, 281-282, 284 GSPN (Generalized Stochastic Petri Net) Jagerman-Stehfest procedure, 263 17-22 Jensen's method, 54, 233 colored 22 See also Randomization, Uniformization GTH advantage, 86, 182 Job, 305, 422-423 GTP: see Generalized tensor product class, 305 Harmonic, 217, 249 job-class balance, 423 Heavy-traffic, 281 job-local balance, 422 approximation, 293 priority, 427 Hessenberg matrix, 98 type, 423 lower, 155 Johnsonbaugh, 273 upper, 156 Jordan chain, 217, 222, 234, 246 High-level formalism, 14 Jordan form, 222, 234 History, 326 Jump chain, 220, 232 Homogeneous in time, 455-456 Kao,259 Homers's method, 169 Keilson, 283, 286 Hosono, 272 Kendall functional equation, 261 Hyperexponential, 291 Kernel of the MRS, 459 Hypoexponential,291 Kingman, 299 Kleinrock, 260 IA (Immediate Access), 367 Knessl,282 Ideal decomposition, 309 Kobayashi, 259, 304 Idle time, 291 Kolmogorov's equations, 44, 48, 52, 56 Ill-conditioned, 87-88 Krylov subspace, 98 ILU: see Incomplete LU factorization Krylov subspace method, 53, 55 ILU preconditioning, 90 Kwok,272 ILU(O), 92 ILUK,93 Laguerre ILUTH,92 coefficients, 264 Imbedded Markov chain, 159, 173, 180, 365 functions, 264 Immediate access, 367 generating functions, 264 Incomplete LU factorization, 92 polynomials, 264 Increasing failure rate, 27 series, 264 Indicator function, 6 series algorithm, 272, 279 Infinite state space, 58 series inversion formula, 264 Infinite-horizon discounted problem, 325 series representation, 272 Infinite-server center, 306 Lam, 310-311, 315 Infinitely divisible distribution, 247 Laplace curtain, 258 INDEX 485

Laplace transform, 52, 55, 66, 262, 289 equation, 467 Laplace-Stieltjes transform, 262 moments, 458 Largeness avoidance, 470 sequence, 457-458 LAS (Late Arrival System), 367 theory, 457 LAS-DA, 367, 372-374, 397 Markov reward modcls, 60-61, 454 LAS-lA,367-368 Martin, 272 Last-come first-served Master/slave, 118 preemptive-resume, 306 Matrix analytic methods, 154, 206, 213, 221, preemptive, 422 229 Late arrival system, 367 Matrix equation, 166 Latouche, 169-170, 178 Matrix exponential, 50, 55 Lavenberg, 304, 308 Matrix geometric, 161, 168, 179, 195, 197, LCFSPR,306 229,233 See also Last-come first-served methods, 258 preemptive-resume solution, 161 Least squares, 98 Maximal connected subsets, 310 Leontief substitution system, 346 McKenna, 304 Leung, 303, 308 MDC, 326 Levcl, 155, 214, 218-219, 221, 229, 233 See also Markov decision chain Levy-Desplanques theorem, 240 Mean time to absorption, 466 Lien, 310-311, 315 Mean time to failure, 447, 462, 466 Lifetime, 47 Mean-value .analysis, 411 Linear independence, 248 Memory requirement, 114 Linear programming, 345 Meromorphic, 242 dual problem, 346 Method of phases, 5 in Markov decision process, 345 M/G/l paradigm, 164, 170, 190-191 Little's theorem, 369, 374-375, 420 M/G/1 queue: see Queue, M/G/1 Load dependent service speeds, 421 M/G/1 type process, 155-157, 206-207, 213, Local balance: see balance equations 218, 221 Local transition, 118-119, 121, 141 Mitra, 282 Long-run dynamics, 100 M/M/1 bulk service, 246 Long-tail service-time, 299 Modeling power of a formalism, 25 Loss of significance, 86 Moment generating function, 294 Loss probability Moments, 309 bound on, 434 Monotone convergence theorem, 169 LU decompo:sition, 83--'84 MRGP, 457 Lucantoni, 262, 272, 279-280 See also Markov regenerative process Lumping of automata, 114 MRM,454 See also Markov reward models Manufacturing, 410 MRS, 457-458 Marginal distribution, 420-421, 424 See also Markov renewal sequence Marking, 18 MTTA,466 tangible, 19 See also Mean time to absorption vanishing, 19 MTTF,462 Markov chain See also Mean time to failure continuous-time, 154, 219, 232, 412, 453 Multi-dimensional generating functions, 304 discrete-time, 12, 14, 154-155, 207 Multi-dimensional transforms, 262, 303 Markov decision Multichain closed queueing networks, chain, 326 305-316 model,325 Multiplicity process, 325 of factors in denominator, 307 Markov policy, 327 of roots, 246 Markov property, 455 of zeros, 210, 216, 225, 231 Markov regenerative epoch,458 NCO, 82,100 process, 449, 457, 460, 468 See also kernel of, 468 Nearly-completely-decomposable Markov renewal Nearly alternating, 269 486 COMPUTATIONAL PROBABILITY

Nearly alternating series, 265 See also ODE Nearly-completely-decomposable, 82, Ordinary tensor product, 135 10G-101 Ornstein-Uhlenbeck process, 281 Nearly decomposable system, 89 reflected, 281 Nearly separable, 100 Orthogonal projection process, 96 Nearly uncoupled, 100 OTP,135 Nerman,301 See also Ordinary tensor product Network: see Queueing network, Stochastic OU process, 281 automata network, GSPN See also Ornstein-Uhlenbeck process Neuts, 155-156, 159-160, 258, 298 Overflow, 425-426, 433-434 Newton-Raphson root finding procedure, Overrelaxation: see SOR 293-294, 313-314 Non-decomposable matrix, 418 Pareto mixture of exponentials, 301 Non-homogeneous Markov chain, 55, 59, 62 Partial balance, 411 Non-Markovian models, 71 Partial fractions expansion, 245 Non-product-form networks, 430 Partial fractions, 291 Non-skip-free, 209, 213, 222, 224, 229, 234 Partial generating function, 305, 308 Nonlinear equations, 313 Partition function, 304, 307 Nonsingular,240 Perform ability, 45, 62, 69, 449 Norm, 348 point performability, 60 Normal distribution, 281 Performance analysis tool Normal form, 218, 232, 242 Q2,273 Normalization constant, 303, 416, 419, 423, Performance analysis, 435 431 Performance measures, 410 Normalizing constant, 132, 160 Periodic function, 265 Norming constant, 193 Periodic sequence, 277 Null recurrent, 212, 218, 228, 232, 237 Perron root, 101 Null vector, 207, 217, 222 Perturbation, 239 Numerical computations, 381 Petri net, 17, 114-115 Numerical differentiation, 276 See also GSPN Numerical example, 395 Phase, 155, 214, 219, 229 Numerical integration, 263, 265, 296 Phase-type distribution, 5, 27 Numerical inversion algorithms, 261 Phase-type expansion, 449, 456 Numerical performance, 377 Pivoting, 85 Numerical transform inversion, 257 Poisson arrivals, 412 Poisson-Charlier polynomials, 282 O'Cinneide, 273, 275 Poisson summation formula, 266-267, 277 ODE, 48, 55, 59, 70 Pole, 243, 245 Euler's method, 48 Policy, 326 multi-step method, 49 deterministic, 327 Runge-Kutta methods, 50 improvement, 344 single step method, 49 iteration, 340, 343 trapezoid method, 49 modified, 349-350 Ollson,301 Markov, 327 One-stage problem, 336 optimal, 329,336 Optimal stationary, 327 E-optimal, 329, 342 Pollaczek Contour Integral, 294 Optimal admission policy, 357 Pollaczek, 289, 294 Optimal control of queues, 326 Pollaczek-Khintchine formula, 290, 301 Optimal decision rule, 336 Polling, 72, 280 Optimal policy, 329, 336 Polynomial, 225, 245 Optimal stopping problem, 330, 332 Positive recurrent, 207, 209, 212, 218, 224, Optimal value function, 325, 329, 332, 228,232 334-336 Post-Widder formula Optimality equation, 325, 332, 334, 336-342, discrete analog of, 264 345, 355, 359-361 Post-Widder inversion formula, 264 Or gate, 462 Poularikas, 260 Ordinary differential equation, 282 Power method, 82, 86-87 INDEX 487

Power series coefficients, 211, 214, 235, 243 Mt/Gt/1, 280 Precision, 271, 279 MX /M~l, 158 Preconditioned power iterations, 82, 88-89 MX /M /1,158 Preconditioning, 91, 99 Queueing network, 113,409-441 matrix, 91 closed, 303-309, 311-316, 415-417 Prefactor, 268-269, 279 multichain, 305-316 Present value, 330 single server, 419 Press, 296 Jackson, 24, 417 Priority non-product form, 430 job class, 427 open, 415-417 shortest-job first, 427 with blocking, 124 Probability density function, 262 Queueing time, 374, 388 Probability generating function, 165, 210 Probability of emptiness, 291 R matrix, 233 , 306, 422 See also Rate matrix Product form, 115, 129-134,409,411,415, Rabinowitz, 263, 265 417,421 Radius of convergence, 236, 243-244, 276 as approximation, 428 Ramaswami, 170 generalized view, 421 Ramaswami method, 221 restored, 429 Random variable, 262 Production, 410 Randomization method, 54, 233 Project completion time, 259 See also Jensen's method, Projection, 206 U niformization method, 53, 147 Randomly changing environment, 158, 160 process, 95-96, 99 Rank,216 techniques, 82 Rate equations, 412 Prolog, 22 Rate matrix, 161, 179 Puterman, 325, 349 Rational generating function, 245 Rational transforms, 390-392 QBD process: see quasi birth and death RBD, 450, 461 process See also Reliability block diagram Quadrature, 263 Reachable state space Quasi birth and death process, 155-157, reduction of, 145 170-171,177, 189, 194 Real analytic, 243 Queue Recurrent, 157 BMAP/G/1,280 See also Positive recurrent DX /Dm /1, 397 Recursive relation, 283 Ek/Ek/1, 298 Reducible, 218, 232, 242 r a /r(3/1,297 Reference function, 358 Geom/G/1, 366, 394 Regular, 249 Geom(n)/G(n)/l/N, 382 Regular function, 217 Geomx /G/1, 396 Regular splitting, 348 Geom/GB /1, 397 Reif,304 GI/G/1, 156, 158, 366, 392 Reiser, 304 discrete-time, 388 Relative traffic intensities, 306 moment generating function, 294 Relative-cost function, 361 GI/Geom/1, 382, 394 Reliability, 47, 69, 435, 447 GI/Geom/1/N,382 Reliability block diagram, 461, 450 GI/Geom/c/c, 397 Reliability graph, 452 GI/M/1,159 Reliability models H2/G/1,293 composition, 471 K2/G/ 1,292 decomposition, 471 Km /G/1,290 non-Markovian, 466 M/D/c, 365 non-state space, 448 M/G/1, 164, 280 state space, 448 M/M/oo, 287 Remaining service time, 369 MMBP/G/1/N,387 Removal M/Mx /1,158 of functional terms, 144 488 COMPUTATIONAL PROBABILITY

of synchronizing events, 144 Series Rendez-vous, 118 alternating, 268 networks, 427 Fourier, 265-266 Renewal function, 261 See also Fourier, series Renewal process, 261, 457 complex Fourier series, 266 Repeating rows Laguerre method of, 213, 229 series, 264 Residual vector, 96 See also Laguerre series Resource sharing, 124 nearly alternating, 265 Response time, 410 power series coefficients, 211, 214, 235, 243 Reversible process, 283 Service centers, 305 Reward, 6, 328 Service-time distribution impulse reward, 26, 46, 63 arbitrary, 381 instantaneous reward, 60 deterministic, 381 rate and impulse based, 66 exponential, 412 rate functions, 454 geometric, 381 rate reward, 26, 45, 63 SHARPE package, 471 unbounded, 357 Short-run dynamics, 100 RG,452 Siegert, 286 See also Reliability graph Sigma function, 284 Riemann-Lebesgue lemma, 268 Significance Riordan, 282 loss of, 86 Romberg integration, 296 Similarity transformation, 51 Root vector, 206, 215, 223, 225 Simulation, 11, 15, 114 Roots, 160, 205, 229, 289, 366, 375-377, discrete event, 3 379-382, 388-392, 394-396, 398 Monte Carlo, 3 distinct, 375 Singularity, 207, 263, 295 multiple, 246 critical, 292 of unity, 237, 242 dominant, 292 repeated, 375 Skip-free Ross, 286 to the left, 208-209, 213 ROU process, 281 to the right, 223-224, 229 See also Ornstein-Uhlenbeck process, Slot, 365-366, 369 reflected boundaries, 369, 372 RouchE!'s theorem, 207, 235, 238, 375, 379, Smoothness, 268 389 SMP: see Semi-Markov process roundoff error: see Error, roundoff Sojourn time, 410 Routing Sojourn matrix, 306 above n, 161-163, 178 random, 417 in E', 174 state dependent, 425 Solution method Sample path, 183 Arnoldi, 83 SAN, 113, 117 combined direct-iterative, 90 See also Stochastic automata network decompositional, 82, 97 SAVE, 23 direct, 82-83 Scalar elimination, 181 iterative, 82 Scaled generating function, 312 SOR (Successive OverRelaxation), 82 Scaling, 269, 279, 281, 284 Spans, 351 in closed queueing networks, 311-314 Sparse matrix format, 34 Scheduled maintenance, 72 Sparseness, 310 Schur vector, 96 of routing chain, 310 Search algorithm, 296 Sparsity, 139 Semi-Markov decision chain, 358-359 Spectral expansion, 282 Semi-Markov decision process, 326 Spectral radius, 89, 218, 241, 243 Semi-Markov process, 280, 449, 457, 459, Splitting, 348 467 Squared coefficient of variation, 291 Sensitivity analysis, 70 Srikant, 281 INDEX 489

SSOR (Symmetric SOR), 90 Stochastic system, 1 Stable algorithm, 87 Stochastically smaller, 300 Stages, 326, 330 Stopping criterion, 341 method of, 456 Stroot, 272 State reduction, 181-184 Sturm sequence, 247 State space, 16, 326 Subdominant eigenvalue, 89-90 explosion, 114, 143 Sub-level, 155, 158, 160 generation of, 12, 33 Subnetwork, 437, 439 potential, 16 Subspace iteration, 96-97, 147 storage, 33 Substochastic, 179-180 State truncation, 470 matrix, 105, 174 State variables, 3 Successive approximation State-action frequencies, 347 in policy space, 340 States in function space, 340 recurrent, 157 Successive substitution, 169 tangible, 19 Summable, 224-225, 230, 234 transient, 157 Superharmonic, 250 vanishing, 19 Superposition of automata, 115 Station balance, 411, 413-414, 418, 422 Superregular, 250 and blocking, 427 Supplementary variables, 4, 449 and failure of, 425 Supremum norm, 358 restored, 427, 429, 435 Symbolic mathematical software, 276 Stationary Symmetric Gauss Seidel, 90 distribution, 81, 206, 214, 219, 229 Synchronization of automata, 114, 123, 126 See also Stationary, probability vector Synchronized transitions, 118-119, 121, 123 policy, 327 functional, 119 probability vector, 207, 220, 224, 81 constant, 119 See also Stationary, distribution Synchronizing event, 118-123, 126-127, Stationary-excess cdf, 290 141-142 Steady-state owner of, 142 detection, 57 System dependencies, 453 analysis, 465 Tail probabilities, 377, 380-381, 388, 392 distribution, 281 arbitrary epoch, 381 probability, 81, 412 Takacs, 257 solutions, 5 Takacs theorem, 241 See also Solution method Tandem queue, 414-415, 425, 430-431 Stehfest, 272 Tangible state, 19 Step-by-step nested inversion, 305 TANGRAM, 22-23 Stiff models, 56, 67, 69, 471 Telecommunication, 410, 433 Stiffness, 472 Telephone operator, 194 Stochastic activity network, 20 bilingual, 194 Stochastic automata, 114, 119 Telephony, 365 grouping of, 143 Telescoping property, 308 ordering of, 143 Templeton, 289, 298 superposition, 115 Tensor, 115 synchronization, 114, 123, 126 addition, 116 Stochastic automata network, 113, 118-123 algebra, 115, 117 interacting, 118-119 product, 116-119, 121 non-interacting, 117-118 sum, 116-117, 123 Stochastic complement, 213, 221 Terminal cost, 329 Stochastic confusion, 30 Teukolsky, 296 Stochastic loss network, 303 Throughput, 307, 410, 420, 424,429,432, Stochastic matrix, 81 439 Stochastic Petri net, 114-115 Time to achieve a reward level, 46, 71 See also GSPN Time-dependent algebra, 114 blocking probability, 280 Stochastic reward net, 20 mean, 280 490 COMPUTATIONAL PROBABILITY

Toeplitz matrix, 168, 206-208, 215 adaptive uniformization, 57 Tolerance criterion, 83 Jensen's method, 54 Traffic equation, 417-418 randomization technique, 54 Traffic rate equation, 306 regenerative randomization, 58 Transform method, 206-207, 213, 248 uniformization power method, 56, 68 Transient, 212, 218, 228, 232 Uniformly bounded costs, 357 analysis Upper-bound extrapolation, 354 cumulative, 464 Utilization, 410, 420, 429 instantaneous, 464 Value determination, 344 state, 157 Value function, 328 Transient solutions, 5, 43-72, 280 of Value iteration, 340 Erlang loss model, 279-288 Vandermonde matrix, 248-249 Transition Van Der Pol, 260 enabled,18 Vanishing state, 19 immediate, 18 Vector subspaces, 95 priority, 19 Vector-descriptor multiplication, 134 timed,18 Vetterling, 296 Transition function, 44 Visit ratio, 306 Transition matrix, 5 Volterra integral equation, 468 approximation, 357 storage, 34 Waiting time, 158, 369, 374, 388, 410 Translate, 266 in GI/G/I queue, 288 Weakly coupled, 309 Trapezoidal rule, 265, 276 Weeks, 264 Tree algorithm, 311, 315 Weeks' algorithm, 264 Truncation Weighted supremum norms, 358 error, 265, 270, 274 Weiss, 272, 282 of state space, 131 Well-defined model, 30 of transition matrix, 180 Wiener-Hopf, 206, 223 Types of jobs, 423 Wimp, 268, 273 UDL factorization, 186-187 Winding number, 239 UL factorization, 193 Zero, 205, 238,242, 291 Uniform convergence, 358 See also Roots Uniformization, 54-55, 58, 66, 70, 221, 233 multiple, 246 See also Jensen's method, multiplicity of, 210, 216, 225, 231 Randomization of generating function, 375