Many-Sources Large Deviations for Max-Weight Scheduling

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page

A linear programming approach to Markov reward error bounds for queueing networks

In this paper, we present a numerical framework for constructing bounds on stationary performance measures of random walks in the positive orthant using the Markov reward approach. These bounds are established in terms of stationary performance measures of a perturbed random walk whose stationary distribution is known explicitly. We consider random walks in an arbitrary number of dimensions and with a transition probability structure that is defined on an arbitrary partition of the positive orthant. Within each component of this partition the transition probabilities are homogeneous. This enables us to model queueing networks with, for instance, break-downs and finite buffers. The main contribution of this paper is that we generalize the linear programming approach of [1] to this class of models

Asymptotic analysis by the saddle point method of the Anick-Mitra-Sondhi model

We consider a fluid queue where the input process consists of N identical sources that turn on and off at exponential waiting times. The server works at the constant rate c and an on source generates fluid at unit rate. This model was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an alternate representation of the joint steady state distribution of the buffer content and the number of on sources. This is given as a contour integral that we then analyze for large N. We give detailed asymptotic results for the joint distribution, as well as the associated marginal and conditional distributions. In particular, simple conditional limits laws are obtained. These shows how the buffer content behaves conditioned on the number of active sources and vice versa. Numerical comparisons show that our asymptotic results are very accurate even for N=20

EUROPEAN CONFERENCE ON QUEUEING THEORY 2016

International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"

Combined analysis of transient delay characteristics and delay autocorrelation function in the Geo(X)/G/1 queue

We perform a discrete-time analysis of customer delay in a buffer with batch arrivals. The delay of the kth customer that enters the FIFO buffer is characterized under the assumption that the numbers of arrivals per slot are independent and identically distributed. By using supplementary variables and generating functions, z-transforms of the transient delays are calculated. Numerical inversion of these transforms lead to results for the moments of the delay of the kth customer. For computational reasons k cannot be too large. Therefore, these numerical inversion results are complemented by explicit analytic expressions for the asymptotics for large k. We further show how the results allow us to characterize jitter-related variables, such as the autocorrelation of the delay in steady state

Stability Region of a Slotted Aloha Network with K-Exponential Backoff

Stability region of random access wireless networks is known for only simple network scenarios. The main problem in this respect is due to interaction among queues. When transmission probabilities during successive transmissions change, e.g., when exponential backoff mechanism is exploited, the interactions in the network are stimulated. In this paper, we derive the stability region of a buffered slotted Aloha network with K-exponential backoff mechanism, approximately, when a finite number of nodes exist. To this end, we propose a new approach in modeling the interaction among wireless nodes. In this approach, we model the network with inter-related quasi-birth-death (QBD) processes such that at each QBD corresponding to each node, a finite number of phases consider the status of the other nodes. Then, by exploiting the available theorems on stability of QBDs, we find the stability region. We show that exponential backoff mechanism is able to increase the area of the stability region of a simple slotted Aloha network with two nodes, more than 40\%. We also show that a slotted Aloha network with exponential backoff may perform very near to ideal scheduling. The accuracy of our modeling approach is verified by simulation in different conditions.Comment: 30 pages, 6 figure

Large Deviations and Transient Multiplexing at a Buffered Resource

In this paper we discuss asymptotics associated with a large number of sources using a resource in a compact time interval. A large deviations condition is placed on the sum of the vectors that describe the stochastic behaviour of the sources and large deviations results deduced about the probability of exhaustion of the resource. This approach allows us to consider sources which are highly non-stationary in time. The examples in mind are a single server queue and a form of the Cramer-Lundburg model from risk theory. Connection is made with past work on stability of queues and effective bandwidths. A number of examples are presented to illustrate the strengths of this approach