Exact asymptotics for the stationary distribution of a Markov chain: a production model
Queueing Systems 62, No. 4, (2009)
Ivo Adan , R. D. Foley, and D. R. McDonald,
Currently, this paper would be the best starting point for reading about our approach to deriving exact asymptotics for the stationary distribution of queueing networks. This paper does include jitter cases; more importantly, this paper is the first of our papers to consider situations where the fluid limit to the rare event of interest is not a straight line, which we call cascades. In cascades, the transition matrix of the Markovian part of a certain Markov additive process is transient and has a radius of convergence $R > 1$. Furthermore, this paper is the first of our papers that uses a time reversal that starts from the rare event of interest. With the time reversal, we are able to derive the asymptotics of $\pi$ for most any divergent sequence of states. Using some of the results in this paper, many of our results in earlier papers that derive asymptotics along a discontinuity (like along an axis) could be extended to other divergent sequence of states.

Large deviations of a modified Jackson network: stability and rough asymptotics
Annals of Applied Probability 15, No. 1B, (2005)
R. D. Foley, and D. R. McDonald,
This paper and the next paper are companion papers analyzing a modified Jackson network. The modification is that one of the servers when idle may help the other server. The paper above determines the stability region and uses the traditional large deviations theory to derive the rough asymptotics of a large deviation at one of the queues. The following paper determines the exact asymptotics.

Bridges and networks: Exact asymptotics
Annals of Applied Probability 15, No. 1B (2005)
R. D. Foley, and D. R. McDonald,
This paper and the previous paper are companion papers analyzing a modified Jackson network. In this paper, we first encounter the "bridge phenomenon" where the most likely approach to the rare event hovers above the axis but rarely touching the axis. For bridges, the Markovian part of a certain Markov additive process is either null recurrent or 1-transient. There is a ratio limit theorem for the Green's function of this Markov additive process; however, to use the ratio limit theorem, we need the exact asymptotics for a reference point. We obtain the an estimate when the Markovian part is a birth-death process, but it would be useful to have a similar result for a more general Markovian part. For bridges, we obtained asymptotic expressions as $\ell \to \infty$ for $\pi(\ell,y)$ that are of the form $c \ell^{-1/2} \rho^\ell$ and $c \ell^{-3/2}\rho^\ell$.

Join the Shortest Queue: Stability and Exact Asymptotics,
Annals of Applied Probability 11, No. 3, (2001)
R. D. Foley, and D. R. McDonald,
This paper analyzes a join-the-shortest-queue system. The paper derives stability conditions. Under certain conditions, the exact asymptotics of $\pi$ along the diagonal and along axes are obtained. These results could be extended to other sequences using some of the time reversal ideas mentioned above. This paper considers "jitter cases" where the Markovian part of a certain Markov additive process is positive recurrent, which keeps the Markovian part close to some discontinuity such as the diagonal or an axis. This paper obtains the asymptotics of the Green's function of a positive recurrent Markov additive process that is periodic. For jitter cases, we obtain asymptotic expressions as $\ell \to \infty$ for $\pi(\ell,y)$ that are of the form $c \rho^\ell$.

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