A tight bound on the throughput of queueing networks with blocking

In this paper, we present a bounding methodology that allows to compute a tight lower bound on the cycle time of fork--join queueing networks with blocking and with general service time distributions. The methodology relies on two ideas. First, probability masses fitting (PMF) discretizes the service time distributions so that the evolution of the modified network can be modelled by a Markov chain. The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the orresponding interval. Second, we take advantage of the concept of critical path, i.e. the sequence of jobs that covers a sample run. We show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a lower bound on the original cycle time. The tightness of the bound is shown on computational experiments. Finally, we discuss the extension to split--and--merge networks and approximate estimations of the cycle time.queueing networks, blocking, throughput, bound, probability masses fitting, critical path.

Simple and explicit bounds for multi-server queues with 1/(1−ρ)1/(1 - \rho) (and sometimes better) scaling

We consider the FCFS $GI/GI/n$ queue, and prove the first simple and explicit bounds that scale as $\frac{1}{1-\rho}$ (and sometimes better). Here $\rho$ denotes the corresponding traffic intensity. Conceptually, our results can be viewed as a multi-server analogue of Kingman's bound. Our main results are bounds for the tail of the steady-state queue length and the steady-state probability of delay. The strength of our bounds (e.g. in the form of tail decay rate) is a function of how many moments of the inter-arrival and service distributions are assumed finite. More formally, suppose that the inter-arrival and service times (distributed as random variables $A$ and $S$ respectively) have finite $r$th moment for some $r > 2.$ Let $\mu_A$ (respectively $\mu_S$) denote $\frac{1}{\mathbb{E}[A]}$ (respectively $\frac{1}{\mathbb{E}[S]}$). Then our bounds (also for higher moments) are simple and explicit functions of $\mathbb{E}\big[(A \mu_A)^r\big], \mathbb{E}\big[(S \mu_S)^r\big], r$, and $\frac{1}{1-\rho}$ only. Our bounds scale gracefully even when the number of servers grows large and the traffic intensity converges to unity simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale better than $\frac{1}{1-\rho}$ in certain asymptotic regimes. More precisely, they scale as $\frac{1}{1-\rho}$ multiplied by an inverse polynomial in $n(1 - \rho)^2.$ These results formalize the intuition that bounds should be tighter in light traffic as well as certain heavy-traffic regimes (e.g. with $\rho$ fixed and $n$ large). In these same asymptotic regimes we also prove bounds for the tail of the steady-state number in service. Our main proofs proceed by explicitly analyzing the bounding process which arises in the stochastic comparison bounds of amarnik and Goldberg for multi-server queues. Along the way we derive several novel results for suprema of random walks and pooled renewal processes which may be of independent interest. We also prove several additional bounds using drift arguments (which have much smaller pre-factors), and make several conjectures which would imply further related bounds and generalizations

Validity of heavy traffic steady-state approximations in generalized Jackson Networks

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavy-traffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called ``interchange-of-limits'' for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Towards a System Theoretic Approach to Wireless Network Capacity in Finite Time and Space

In asymptotic regimes, both in time and space (network size), the derivation of network capacity results is grossly simplified by brushing aside queueing behavior in non-Jackson networks. This simplifying double-limit model, however, lends itself to conservative numerical results in finite regimes. To properly account for queueing behavior beyond a simple calculus based on average rates, we advocate a system theoretic methodology for the capacity problem in finite time and space regimes. This methodology also accounts for spatial correlations arising in networks with CSMA/CA scheduling and it delivers rigorous closed-form capacity results in terms of probability distributions. Unlike numerous existing asymptotic results, subject to anecdotal practical concerns, our transient one can be used in practical settings: for example, to compute the time scales at which multi-hop routing is more advantageous than single-hop routing

On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference