233 research outputs found
Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results
Our model is a constrained homogeneous random walk in a nonnegative orthant
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 by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly 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
Right-convergence of sparse random graphs
The paper is devoted to the problem of establishing right-convergence of
sparse random graphs. This concerns the convergence of the logarithm of number
of homomorphisms from graphs or hyper-graphs \G_N, N\ge 1 to some target
graph . The theory of dense graph convergence, including random dense
graphs, is now well understood, but its counterpart for sparse random graphs
presents some fundamental difficulties. Phrased in the statistical physics
terminology, the issue is the existence of the log-partition function limits,
also known as free energy limits, appropriately normalized for the Gibbs
distribution associated with . In this paper we prove that the sequence of
sparse \ER graphs is right-converging when the tensor product associated with
the target graph satisfies certain convexity property. We treat the case of
discrete and continuous target graphs . The latter case allows us to prove a
special case of Talagrand's recent conjecture (more accurately stated as level
III Research Problem 6.7.2 in his recent book), concerning the existence of the
limit of the measure of a set obtained from by intersecting it with
linearly in many subsets, generated according to some common probability
law.
Our proof is based on the interpolation technique, introduced first by Guerra
and Toninelli and developed further in a series of papers. Specifically, Bayati
et al establish the right-convergence property for Erdos-Renyi graphs for some
special cases of . In this paper most of the results in this paper follow as
a special case of our main theorem.Comment: 22 page
On the rate of convergence to stationarity of the M/M/N queue in the Halfin-Whitt regime
We prove several results about the rate of convergence to stationarity, that
is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We
identify the limiting rate of convergence to steady-state, and discover an
asymptotic phase transition that occurs w.r.t. this rate. In particular, we
demonstrate the existence of a constant s.t. when a certain
excess parameter , the error in the steady-state approximation
converges exponentially fast to zero at rate . For , the
error in the steady-state approximation converges exponentially fast to zero at
a different rate, which is the solution to an explicit equation given in terms
of special functions. This result may be interpreted as an asymptotic version
of a phase transition proven to occur for any fixed n by van Doorn [Stochastic
Monotonicity and Queueing Applications of Birth-death Processes (1981)
Springer]. We also prove explicit bounds on the distance to stationarity for
the M/M/n queue in the Halfin-Whitt regime, when . Our bounds scale
independently of in the Halfin-Whitt regime, and do not follow from the
weak-convergence theory.Comment: Published in at http://dx.doi.org/10.1214/12-AAP889 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On deciding stability of multiclass queueing networks under buffer priority scheduling policies
One of the basic properties of a queueing network is stability. Roughly
speaking, it is the property that the total number of jobs in the network
remains bounded as a function of time. One of the key questions related to the
stability issue is how to determine the exact conditions under which a given
queueing network operating under a given scheduling policy remains stable.
While there was much initial progress in addressing this question, most of the
results obtained were partial at best and so the complete characterization of
stable queueing networks is still lacking. In this paper, we resolve this open
problem, albeit in a somewhat unexpected way. We show that characterizing
stable queueing networks is an algorithmically undecidable problem for the case
of nonpreemptive static buffer priority scheduling policies and deterministic
interarrival and service times. Thus, no constructive characterization of
stable queueing networks operating under this class of policies is possible.
The result is established for queueing networks with finite and infinite buffer
sizes and possibly zero service times, although we conjecture that it also
holds in the case of models with only infinite buffers and nonzero service
times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002)
272--293] and uses the so-called counter machine device as a reduction tool.Comment: Published in at http://dx.doi.org/10.1214/09-AAP597 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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