Abstract
Our model is a constrained homogeneous random walk in Z+d. The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed in [11]. In this paper we show that computing exactly the stationary probability for this type of random walks is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems.
- G. Fayolle, V. A. Malyshev, and M. V. Menshikov. Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, 1995.Google ScholarCross Ref
- D. Gamarnik. On deciding stability of constrained homogeneous random walks and queueing systems. Mathematics of Operations Research, 27(2):272-293, 2002. Google ScholarDigital Library
- I. A. Ignatyuk and V. A. Malyshev. Classification of random walks in Z+d. Selecta Mathematica, 12:129-194, 1993.Google Scholar
- I. A. Ignatyuk, V. A. Malyshev, and V. V. Scherbakov. Boundary effects in large deviation problems. Russ. Math. Surv., 49:41-99, 1994.Google ScholarCross Ref
- I. A. Kurkova and Yu. M. Suhov. Malyshev's theory and JS-queues. Asymptotics of stationary probabilities. Submitted, 2001.Google Scholar
- V. A. Malyshev. Analytic method in the theoy of two-dimensional random walks. Sib.Math.J., 13(6):1314-1327, 1972.Google Scholar
- V. A. Malyshev. Classification of two-dimensional positive random walks and almost linear semimartingales. Dokl. Akad. Nauk SSSR, 202:526-528, 1972.Google Scholar
- V. A. Malyshev. Asymptotic behaviour of stationary probabilities for two-dimensional positive random walks. Sib.Math.J., 14(1):156-169, 1973.Google ScholarCross Ref
- V. A. Malyshev. Networks and dynamical systems. Adv.Appl.Prob., 25:140-175, 1993.Google ScholarCross Ref
- M. V. Menshikov. Ergodicity and transience conditions for random walks in the positive octant of space. Soviet.Math.Dokl., 217:755-758, 1974.Google Scholar
- S. P. Meyn and R. L. Tweedie. Computable bounds for geometric convergence rates of Markov chains. Ann. of Appl. Prob., 4:981-1011, 1994.Google ScholarCross Ref
- A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman and Hall, 1995.Google Scholar
Recommendations
On the Undecidability of Computing Stationary Distributions and Large Deviation Rates for Constrained Random Walks
We consider a constrained homogeneous random walk in Z+d. Such random walks are used to model various stochastic processes, most importantly multiclass Markovian queueing networks operating under state-dependent scheduling policies. These applications ...
Resolution-stationary random number generators
Besides speed and period length, the quality of uniform random number generators (RNGs) is usually assessed by measuring the uniformity of their point sets, formed by taking vectors of successive output values over their entire period length. For F"2-...
Stationary distribution and cover time of random walks on random digraphs
We study properties of a simple random walk on the random digraph D"n","p when np=dlogn, d>1. We prove that whp the value @p"v of the stationary distribution at vertex v is asymptotic to deg^-(v)/m where deg^-(v) is the in-degree of v and m=n(n-1)p is ...
Comments