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Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

Part of: Special processes Limit theorems Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Masahiro Kobayashi  and
Masakiyo Miyazawa
Masahiro Kobayashi*
Affiliation:
Tokyo University of Science
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
*
∗ Postal address: Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan.
∗ Postal address: Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan.
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Abstract

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We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.

Keywords

Two-dimensional reflecting random walkstationary distributionlinear convex ordertail asymptoticsunbounded jumptwo-node queueing networkbatch arrival60K25

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 365 - 399
Copyright
© Applied Probability Trust 

References

Arjas, E. and Speed, T. P. (1973). Symmetric Wiener–Hopf factorisations in Markov additive processes. Z. Wahrscheinlichkeitsth. 26, 105–118.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. (New York) 51), 2nd edn. Springer, New York.Google Scholar
Borovkov, A. A. and Moguĺskiı˘, A. A. (1996). Large deviations for stationary Markov chains in a quarter plane. In Probability Theory and Mathematical Statistics (Tokyo, 1995), World Scientific, River Edge, NJ, pp. 12–19.Google Scholar
Borovkov, A. A. and Moguĺskiı˘, A. A. (2001). Integro-local limit theorems for sums of random vectors that include large deviations. II. Theory Prob. Appl. 45, 3–22.Google Scholar
Borovkov, A. A. and Moguĺskiı˘, A. A. (2001). Large deviations for Markov chains in the positive quadrant. Russian Math. Surveys 56, 803–916.Google Scholar
Dai, J. G. and Miyazawa, M. (2011). Reflecting Brownian motion in two dimensions: exact asymptotics for the stationary distribution. Stoch. Systems 1, 146–208.Google Scholar
Dai, J. G. and Miyazawa, M. (2013). Stationary distribution of a two-dimensional SRBM: geometric views and boundary measures. Queueing Systems 74, 181–217.Google Scholar
Doetsch, G. (1974). Introduction to the Theory and Application of the Laplace Transformation. Springer, New York.Google Scholar
Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. John Wiley, New York.Google Scholar
Fayolle, G. (1989). On random walks arising in queueing systems: ergodicity and transience via quadratic forms as Lyapounov functions. I. Queueing Systems Theory Appl. 5, 167–183.Google Scholar
Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999). Random Walks in the Quarter-Plane. Algebraic Methods, Boundary Value Problems and Applications. Springer, Berlin.Google Scholar
Fayolle, G., Malyshev, V. A. and Meńshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.Google Scholar
Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands. I. SIAM J. Appl. Math. 44, 1041–1053. (Erratum: 45 (1985), 168.)CrossRefGoogle Scholar
Foley, R. D. and McDonald, D. R. (2005). Large deviations of a modified Jackson network: stability and rough asymptotics. Ann. Appl. Prob. 15, 519–541.Google Scholar
Guillemin, F. and Pinchon, D. (2004). Analysis of generalized processor-sharing systems with two classes of customers and exponential services. J. Appl. Prob. 41, 832–858.Google Scholar
Guillemin, F. and van Leeuwaarden, J. S. H. (2011). Rare event asymptotics for a random walk in the quarter plane. Queueing Systems 67, 1–32.Google Scholar
Guillemin, F., Knessl, C. and van Leeuwaarden, J. S. H. (2013). Wireless three-hop networks with stealing II: exact solutions through boundary value problems. Queueing Systems 74, 235–272.Google Scholar
Kobayashi, M. and Miyazawa, M. (2013). Revisiting the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. In Matrix-Analytic Methods in Stochastic Models, Springer, New York, pp. 145–185.CrossRefGoogle Scholar
Kobayashi, M., Miyazawa, M. and Zhao, Y. Q. (2010). Tail asymptotics of the occupation measure for a Markov additive process with an M/G/1-type background process. Stoch. Models 26, 463–486.CrossRefGoogle Scholar
Koshevoy, G. and Mosler, K. (1998). Lift zonoids, random convex hulls and the variability of random vectors. Bernoulli 4, 377–399.Google Scholar
Li, H. and Zhao, Y. Q. (2011). Tail asymptotics for a generalized two-demand queueing model—a kernel method. Queueing Systems 69, 77–100.Google Scholar
Li, H., Miyazawa, M. and Zhao, Y. Q. (2007). Geometric decay in a QBD process with countable background states with applications to a Join-the-shortest-queue model. Stoch. Models 23, 413–438.CrossRefGoogle Scholar
Miyazawa, M. (2009). Tail decay rates in double QBD processes and related reflected random walks. Math. Operat. Res. 34, 547–575.CrossRefGoogle Scholar
Miyazawa, M. (2011). Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP 19, 233–299.Google Scholar
Miyazawa, M. and Taylor, P. G. (1997). A geometric product-form distribution for a queueing network with non-standard batch arrivals and batch transfers. Adv. Appl. Prob. 29, 523–544.Google Scholar
Miyazawa, M. and Zhao, Y. Q. (2004). The stationary tail asymptotics in the GI/G/1-type queue with countably many background states. Adv. Appl. Prob. 36, 1231–1251.Google Scholar
Miyazawa, M. and Zwart, B. (2012). Wiener–Hopf factorizations for a multidimensional Markov additive process and their applications to reflected processes. Stoch. Systems 2, 67–114.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Nakagawa, K. (2004). On the exponential decay rate of the tail of a discrete probability distribution. Stoch. Models 20, 31–42.Google Scholar
Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press.Google Scholar
Rockafellar, R. T. (1970). Convex Analysis (Princeton Math. Ser. 28). Princeton University Press.Google Scholar
Scarsini, M. (1998). Multivariate convex orderings, dependence, and stochastic equality. J. Appl. Prob. 35, 93–103.CrossRefGoogle Scholar