1,020 research outputs found
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
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
Stationary analysis of a single queue with remaining service time dependent arrivals
We study a generalization of the system (denoted by ) with
independent and identically distributed (iid) service times and with an arrival
process whose arrival rate depends on the remaining service
time of the current customer being served. We derive a natural stability
condition and provide a stationary analysis under it both at service completion
times (of the queue length process) and in continuous time (of the queue length
and the residual service time). In particular, we show that the stationary
measure of queue length at service completion times is equal to that of a
corresponding system. For we show that the continuous time
stationary measure of the system is linked to the system via a
time change. As opposed to the queue, the stationary measure of queue
length of the system at service completions differs from its marginal
distribution under the continuous time stationary measure. Thus, in general,
arrivals of the system do not see time averages. We derive formulas
for the average queue length, probability of an empty system and average
waiting time under the continuous time stationary measure. We provide examples
showing the effect of changing the reshaping function on the average waiting
time.Comment: 31 pages, 3 Figure
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