Regenerative Simulation for Queueing Networks with Exponential or Heavier Tail Arrival Distributions

Multiclass open queueing networks find wide applications in communication, computer and fabrication networks. Often one is interested in steady-state performance measures associated with these networks. Conceptually, under mild conditions, a regenerative structure exists in multiclass networks, making them amenable to regenerative simulation for estimating the steady-state performance measures. However, typically, identification of a regenerative structure in these networks is difficult. A well known exception is when all the interarrival times are exponentially distributed, where the instants corresponding to customer arrivals to an empty network constitute a regenerative structure. In this paper, we consider networks where the interarrival times are generally distributed but have exponential or heavier tails. We show that these distributions can be decomposed into a mixture of sums of independent random variables such that at least one of the components is exponentially distributed. This allows an easily implementable embedded regenerative structure in the Markov process. We show that under mild conditions on the network primitives, the regenerative mean and standard deviation estimators are consistent and satisfy a joint central limit theorem useful for constructing asymptotically valid confidence intervals. We also show that amongst all such interarrival time decompositions, the one with the largest mean exponential component minimizes the asymptotic variance of the standard deviation estimator.Comment: A preliminary version of this paper will appear in Proceedings of Winter Simulation Conference, Washington, DC, 201

Delay Performance of MISO Wireless Communications

Ultra-reliable, low latency communications (URLLC) are currently attracting significant attention due to the emergence of mission-critical applications and device-centric communication. URLLC will entail a fundamental paradigm shift from throughput-oriented system design towards holistic designs for guaranteed and reliable end-to-end latency. A deep understanding of the delay performance of wireless networks is essential for efficient URLLC systems. In this paper, we investigate the network layer performance of multiple-input, single-output (MISO) systems under statistical delay constraints. We provide closed-form expressions for MISO diversity-oriented service process and derive probabilistic delay bounds using tools from stochastic network calculus. In particular, we analyze transmit beamforming with perfect and imperfect channel knowledge and compare it with orthogonal space-time codes and antenna selection. The effect of transmit power, number of antennas, and finite blocklength channel coding on the delay distribution is also investigated. Our higher layer performance results reveal key insights of MISO channels and provide useful guidelines for the design of ultra-reliable communication systems that can guarantee the stringent URLLC latency requirements.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Continuous feedback fluid queues

We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves `as a Markov process' with generator $Q(y)$ at times when the buffer level is $y$, where the entries of $Q(y)$ are continuous functions of $y$. Moreover, the flow rates for the buffer may also depend continuously on the current buffer level. Such models are interesting in the context of closed-loop telecommunication networks, in which sources interact with network buffers, but may also be deployed in the study of certain production systems. \u

On level crossings for a general class of piecewise-deterministic Markov processes

We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. We study the point process of upcrossings of a level $b$ by the Markov process. Our main result shows that, under a suitable scaling $\nu(b)$, the point process converges, as $b$ tends to infinity, weakly to a geometrically compound Poisson process. We also prove a version of Rice's formula relating the stationary density of the process to level crossing intensities. This formula provides an interpretation of the scaling factor $\nu(b)$. While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite.Comment: 25 page

Some simple but challenging Markov processes

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are mathematically rich. Their math-ematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems