Lattice path counting and the theory of queues

In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract)Series: Research Report Series / Department of Statistics and Mathematic

Delay performance in random-access grid networks

We examine the impact of torpid mixing and meta-stability issues on the delay performance in wireless random-access networks. Focusing on regular meshes as prototypical scenarios, we show that the mean delays in an $L\times L$ toric grid with normalized load $\rho$ are of the order $(\frac{1}{1-\rho})^L$. This superlinear delay scaling is to be contrasted with the usual linear growth of the order $\frac{1}{1-\rho}$ in conventional queueing networks. The intuitive explanation for the poor delay characteristics is that (i) high load requires a high activity factor, (ii) a high activity factor implies extremely slow transitions between dominant activity states, and (iii) slow transitions cause starvation and hence excessively long queues and delays. Our proof method combines both renewal and conductance arguments. A critical ingredient in quantifying the long transition times is the derivation of the communication height of the uniformized Markov chain associated with the activity process. We also discuss connections with Glauber dynamics, conductance and mixing times. Our proof framework can be applied to other topologies as well, and is also relevant for the hard-core model in statistical physics and the sampling from independent sets using single-site update Markov chains

Analysis of Buffer Starvation with Application to Objective QoE Optimization of Streaming Services

Our purpose in this paper is to characterize buffer starvations for streaming services. The buffer is modeled as an M/M/1 queue, plus the consideration of bursty arrivals. When the buffer is empty, the service restarts after a certain amount of packets are \emph{prefetched}. With this goal, we propose two approaches to obtain the \emph{exact distribution} of the number of buffer starvations, one of which is based on \emph{Ballot theorem}, and the other uses recursive equations. The Ballot theorem approach gives an explicit result. We extend this approach to the scenario with a constant playback rate using T\`{a}kacs Ballot theorem. The recursive approach, though not offering an explicit result, can obtain the distribution of starvations with non-independent and identically distributed (i.i.d.) arrival process in which an ON/OFF bursty arrival process is considered in this work. We further compute the starvation probability as a function of the amount of prefetched packets for a large number of files via a fluid analysis. Among many potential applications of starvation analysis, we show how to apply it to optimize the objective quality of experience (QoE) of media streaming, by exploiting the tradeoff between startup/rebuffering delay and starvations.Comment: 9 pages, 7 figures; IEEE Infocom 201