Asymptotic Hitting Distribution for a Reflected Random Walk in the Positive Quadrant

作者：Khanchi Aziz^*

来源：Stochastic Models, 2011, 27(2): 169-201.

DOI：10.1080/15326349.2011.567927

• Decay rate
• Exact asymptotics
• Harmonic function
• h-transform
• Jackson network with cooperation
• Markov additive chain
• Reflected random walk

Consider a stable skip free reflected random walk in the two dimensional nonnegative quadrant, where the main state is referred to as level. We establish the asymptotic distribution of the second coordinate (phase) at the moment the process hits a large level for the first time. The limit is derived under two scenarios. In the first, the chain reaches the destination level for the first time before returning to the starting level and in the second it reaches the destination level for the first time before returning to the starting state. We will show that under the same conditions the limits are equal. The existence of the hitting distribution in limit is surprising in the bridge case where the large deviations path tries to avoid the level axis. We also show that the reciprocal of the mean time to reach a level for the first time decays at a rate which is the same as the decay rate of the stationary distribution. Moreover, the results are applied to a modified Jackson network with partially coupled servers.

• 出版日期2011