Geodesic
Monotonicity and comparison results for nonnegative dynamic systems. Part I: Discrete-time case
Kybernetika, Tome 42 (2006) no. 1, pp. 37-56 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In two subsequent parts, Part I and II, monotonicity and comparison results will be studied, as generalization of the pure stochastic case, for arbitrary dynamic systems governed by nonnegative matrices. Part I covers the discrete-time and Part II the continuous-time case. The research has initially been motivated by a reliability application contained in Part II. In the present Part I it is shown that monotonicity and comparison results, as known for Markov chains, do carry over rather smoothly to the general nonnegative case for marginal, total and average reward structures. These results, though straightforward, are not only of theoretical interest by themselves, but also essential for the more practical continuous-time case in Part II (see [DijkSl2]). An instructive discrete-time random walk example is included.
In two subsequent parts, Part I and II, monotonicity and comparison results will be studied, as generalization of the pure stochastic case, for arbitrary dynamic systems governed by nonnegative matrices. Part I covers the discrete-time and Part II the continuous-time case. The research has initially been motivated by a reliability application contained in Part II. In the present Part I it is shown that monotonicity and comparison results, as known for Markov chains, do carry over rather smoothly to the general nonnegative case for marginal, total and average reward structures. These results, though straightforward, are not only of theoretical interest by themselves, but also essential for the more practical continuous-time case in Part II (see [DijkSl2]). An instructive discrete-time random walk example is included.
Classification : 37N40, 39A10, 60J10, 60J27, 90A16, 91B62
Keywords: Markov chains; monotonicity; nonnegative matrices 
@article{KYB_2006_42_1_a1,
     author = {Dijk, Nico M. van and Sladk\'y, Karel},
     title = {Monotonicity and comparison results for nonnegative dynamic systems. {Part} {I:} {Discrete-time} case},
     journal = {Kybernetika},
     pages = {37--56},
     year = {2006},
     volume = {42},
     number = {1},
     mrnumber = {2208519},
     zbl = {1249.60168},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2006_42_1_a1/}
}
TY  - JOUR
AU  - Dijk, Nico M. van
AU  - Sladký, Karel
TI  - Monotonicity and comparison results for nonnegative dynamic systems. Part I: Discrete-time case
JO  - Kybernetika
PY  - 2006
SP  - 37
EP  - 56
VL  - 42
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/KYB_2006_42_1_a1/
LA  - en
ID  - KYB_2006_42_1_a1
ER  - 
%0 Journal Article
%A Dijk, Nico M. van
%A Sladký, Karel
%T Monotonicity and comparison results for nonnegative dynamic systems. Part I: Discrete-time case
%J Kybernetika
%D 2006
%P 37-56
%V 42
%N 1
%U http://geodesic.mathdoc.fr/item/KYB_2006_42_1_a1/
%G en
%F KYB_2006_42_1_a1
Dijk, Nico M. van; Sladký, Karel. Monotonicity and comparison results for nonnegative dynamic systems. Part I: Discrete-time case. Kybernetika, Tome 42 (2006) no. 1, pp. 37-56. http://geodesic.mathdoc.fr/item/KYB_2006_42_1_a1/

[1] Adan I. J. B. F., Wal J. van der: Monotonicity of the throughput in single server production and assembly networks with respect to buffer sizes. In: Queueing Networks with Blocking, North Holland, Amsterdam 1989, pp. 345–356

[2] Adan I. J. B. F., Wal J. van der: Monotonicity of the throughput of a closed queueing network in the number of jobs. Oper. Res. 37 (1989), 953–957 | DOI | MR

[3] Dijk N. M. van, Puterman M.: Perturbation theory for Markov reward processes with applications to queueing systems. Adv. in Appl. Probab. 20 (1988), 79–87 | DOI | MR

[4] Dijk N. M. van, Wal J. van der: Simple bounds and monotonicity results for multi-server exponential tandem queues. Queueing Systems 4 (1989), 1–16 | DOI

[5] Dijk N. M. van: On the importance of bias-terms for error bounds and comparison results. In: Numerical Solution of Markov Chains (W. J. Stewart, ed.), Marcel Dekker, New York 1991, pp. 640–654 | MR

[6] Dijk N. M. van: Bounds and error bounds for queueing networks. Ann. Oper. Res. 36 (2003), 3027–3030

[7] Dijk N. M. van: Queuing Networks and Product Forms. Wiley, New York 1993 | MR

[8] Dijk N. M. van, Sladký K.: Error bounds for dynamic nonnegative systems. J. Optim. Theory Appl. 101 (1999), 449–474 | DOI | MR

[9] Dijk N. M. van, Sladký K.: Monotonicity and comparison results for nonnegative dynamic systems. Part II: Continuous-time case and reliability application. Kybernetika 42 (2006), No. 2 (to appear)

[10] Dijk N. M. van, Taylor P. G.: On strong stochastic comparison for steady state measures of Markov chains with a performability application. Oper. Res. 36 (2003), 3027–3030

[11] Gross D., Miller D. R.: The randomization technique as a modelling tool and solution procedure over discrete state Markov processes. Oper. Res. 32 (1984), 343–361 | DOI | MR

[12] Keilson J., Kester A.: Monotone matrices and monotone Markov processes. Stoch. Process. Appl. 5 (1977), 231–241 | DOI | MR | Zbl

[13] Massey W. A.: Stochastic ordering for Markov processes on partially ordered spaces. Math. Oper. Res. 12 (1987), 350–367 | DOI | MR

[14] Melamed B., Yadin N.: Randomization procedures in the computation of cumulative-time distributions over discrete state Markov processes. Oper. Res. 32 (1984), 926–943 | DOI | MR | Zbl

[15] Shantikumar J. G., Yao D. D.: Monotonicity proporties in cyclic queueing networks with finite buffers. In: Queueing Networks with Blocking, North Holland, Amsterdam 1989, pp. 345–356

[16] Sladký K.: Bounds on discrete dynamic programming recursions I. Kybernetika 16 (1980), 526–547 | MR | Zbl

[17] Stoyan D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, New York 1983 | MR | Zbl

[18] Tsoucas P., Walrand J.: Monotonicity of throughput in non-Markovian networks. J. Appl. Probab. 26 (1989), 134–141 | DOI | MR | Zbl

[19] Whitt W.: Comparing counting processes and queues. Adv. in Appl. Probab. 13 (1981), 207–220 | DOI | MR | Zbl

[20] Whitt W.: Stochastic comparison for non-Markov processes. Math. Oper. Res. 11 (1986), 608–618 | DOI | MR