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Kapodistria S, Palmowski Z. Matrix geometric approach for random walks: Stability condition and equilibrium distribution. STOCH MODELS 2017. [DOI: 10.1080/15326349.2017.1359096] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Stella Kapodistria
- Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
| | - Zbigniew Palmowski
- Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wroclaw, Poland
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Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps. ADV APPL PROBAB 2016. [DOI: 10.1017/s0001867800007138] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
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.
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Kobayashi M, Miyazawa M. Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps. ADV APPL PROBAB 2016. [DOI: 10.1239/aap/1401369699] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
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.
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Khanchi A, Lamothe G. Simulating tail asymptotics of a Markov chain. Stat Probab Lett 2011. [DOI: 10.1016/j.spl.2011.04.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Kimura T, Daikoku K, Masuyama H, Takahashi Y. Light-Tailed Asymptotics of Stationary Tail Probability Vectors of Markov Chains of M/G/1 Type. STOCH MODELS 2010. [DOI: 10.1080/15326349.2010.519661] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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