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Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit. (English) Zbl 1136.60369
Summary: We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ${ℝ}_{+}^{n}$ with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes.
##### MSC:
 60K30 Applications of queueing theory 60J60 Diffusion processes 60J75 Jump processes 60K25 Queueing theory
##### References:
 [1] Reiman, M.I.: Open queueing networks in heavy traffic. Math. Oper. Res. 9, 441–458 (1984) · Zbl 0549.90043 · doi:10.1287/moor.9.3.441 [2] Williams, R.J.: Reflecting diffusions and queueing networks. In: Proceedings of the International Congress of Mathematicians, vol. III, Berlin, 1998, number Extra vol. III, pp. 321–330 (electronic) (1998) [3] Harrison, J.M., Williams, R.J.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115 (1987) [4] Kushner, H.J.: Heavy Traffic Analysis of Controlled Queueing and Communication Networks. Applications of Mathematics, vol. 47. Springer, New York (2001) [5] Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Applications of Mathematics, vol. 46. Springer, New York (2001) [6] Robert, P.: Stochastic Networks and Queues. Applications of Mathematics, vol. 52. Springer, New York (2003) [7] Mandelbaum, A., Pats, G.: State-dependent stochastic networks. Part I: Approximations and applications with continuous diffusion limits. Ann. Appl. Probab. 8(2), 569–646 (1998) · Zbl 0945.60025 · doi:10.1214/aoap/1028903539 [8] Ramasubramanian, S.: A subsidy-surplus model and the Skorokhod problem in an orthant. Math. Oper. Res. 25(3), 509–538 (2000) · Zbl 1073.91610 · doi:10.1287/moor.25.3.509.12215 [9] Konstantopoulos, T., Last, G., Lin, S.-J.: On a class of Lévy stochastic networks. Queueing Syst. 46, 409–437 (2004) · Zbl 1061.90012 · doi:10.1023/B:QUES.0000027993.51077.f2 [10] Bardhan, I.: Further applications of a general rate conservation law. Stochastic Process. Appl. 60, 113–130 (1995) · Zbl 0846.60047 · doi:10.1016/0304-4149(95)00052-6 [11] Kella, O.: Non-product form of two-dimensional fluid networks with dependent Lévy inputs. J. Appl. Probab. 37(4), 1117–1122 (2000) · Zbl 0982.60047 · doi:10.1239/jap/1014843090 [12] Kella, O., Whitt, W.: Diffusion approximations for queues with server vacations. Adv. Appl. Probab. 22, 706–729 (1990) · Zbl 0713.60101 · doi:10.2307/1427465 [13] Whitt, W.: The reflection map with discontinuities. Math. Oper. Res. 26, 447–484 (2001) · Zbl 1073.90504 · doi:10.1287/moor.26.3.447.10588 [14] Piera, F.J., Mazumdar, R.R., Guillemin, F.M.: On product-form stationary distributions for reflected diffusions with jumps in the positive orthant. Adv. Appl. Probab. 37(1), 212–228 (2005) · Zbl 1064.60168 · doi:10.1239/aap/1113402406 [15] Piera, F.J., Mazumdar, R.R., Guillemin, F.M.: On the local times and boundary properties of reflected diffusions with jumps in the positive orthant. Markov Process. Relat. Fields 12(3), 561–582 (2006) [16] Mazumdar, R.R., Guillemin, F.M.: Forward equations for reflected diffusions with jumps. Appl. Math. Opt. 33(1), 81–102 (1996) · Zbl 0841.60062 · doi:10.1007/BF01187963 [17] Piera, F.J., Mazumdar, R.R., Guillemin, F.M.: On diffusions with random reflections and jumps in the orthant: boundary behavior and product-form stationary distributions. Submitted, available at http://www.cec.uchile.cl/$\sim$fpiera (2005) [18] Atar, R., Budhiraja, A.: Stability properties of constrained jump-diffusion processes. Electron. J. Probab. 7(22), 1–31 (2002) [19] Shashiashvili, M.: A lemma of variational distance between maximal functions with application to the Skorokhod problem in a nonnegative orthant with state-dependent reflection directions. Stoch. Stoch. Rep. 48, 161–194 (1994) [20] Whitt, W.: Stochastic-Process Limits. Springer Series in Operations Research. Springer, New York (2002) [21] Dupuis, P., Ishii, H.: SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21(1), 554–580 (1993) · Zbl 0787.60099 · doi:10.1214/aop/1176989415 [22] Reiman, M.I., Williams, R.J.: A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Relat. Fields 77(1), 87–97 (1988) · Zbl 0617.60081 · doi:10.1007/BF01848132 [23] Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics, vol. 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994) [24] Protter, P.: Stochastic Integration and Differential Equations: A New Approach. Applications of Mathematics, vol. 21. Springer, New York (1990) [25] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. A Series of Comprehensive Studies in Mathematics, vol. 293. Springer, Berlin (2005) [26] Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. A Series of Comprehensive Studies in Mathematics, vol. 288. Springer, Berlin (2003) [27] Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Probability and its Applications. Springer, New York (2002) [28] Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics. Wiley, New York (1999)