Discrete-time queues with zero-regenerative arrivals: moments and examples

In this paper we investigate a single-server discrete-time queueing system with single-slot service times. The stationary ergodic arrival process this queueing system is subject to, satisfies a regeneration property when there are no arrivals during a slot. Expressions for the mean and the variance of the queue content in steady state are obtained for this broad class which includes among others autoregressive arrival processes and M/G/infinity-input or train arrival processes. To illustrate our results, we then consider a number of numerical examples

Rejoinder on: queueing models for the analysis of communication systems

In this rejoinder, we respond to the comments and questions of three discussants of our paper on queueing models for the analysis of communication systems. Our responses are structured around two main topics: discrete-time modeling and further extensions of the presented queueing analysis

An Introduction to the RESearch Queueing Package for Modeling Computer Systems and Communication Networks

A queueing network is an important tool for modeling systems where performance is principally affected by contention for resources. Such systems include computer systems, communication networks and manufacturing lines. In order to effectively use queuing networks as performance models, appropriate software is necessary for definition ofthe networks to be solved, for solution ofthe networks and for examination of the performance measures obtained. The RESearch Queueing Package (RESQ) and the RESearch Queueing Package Modeling Environment (RESQME) form a system for constructing, solving and analyzing extended queueing network models. We refer to the class of RESQ networks as extended because of characteristics which allow effective representation of system detail. RESQ incorporates a high level language to concisely describe the structure of the model and to specify constraints on the solution. A main feature of the language is the capability to describe models in a hierarchical fashion, allowing an analyst to define submodels to be used analogously to use of macros in programming languages. RESQ also provides a variety of methods for estimating the accuracy of simulation results and for determining simulation run lengths. RESQME is a graphical interface for RESQ. In this introduction, we limit our examples to computer systems and communication networks. Acknowledgement: The authors wish to thank their co-developers of RESQME: Jim Kurose and Kurt Gordon. We also want to thank Ben Antanaitis, Howard Jachter, Jack Servier, Daniel Souday and Peter Welch for their many suggestions which helped improve the RESQME package and Anil Aggarwal, Al Blum, Gary Burkland, Rocky Chang, Janet Chen, Diana Coles, Prakash Deka, Paul Lnewner, and Geoff Parker for their work in implementing RESQME. We would also like to thank our users for their ideas and feedback that we tried to incorporate in RESQ and RESQME. We remain indebted to Charlie Sauer for his design, guidance, inspiration, and development ofthe RESQ languag

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"

Stochastic approximation of symmetric Nash equilibria in queueing games

We suggest a novel stochastic-approximation algorithm to compute a symmetric Nash-equilibrium strategy in a general queueing game with a finite action space. The algorithm involves a single simulation of the queueing process with dynamic updating of the strategy at regeneration times. Under mild assumptions on the utility function and on the regenerative structure of the queueing process, the algorithm converges to a symmetric equilibrium strategy almost surely. This yields a powerful tool that can be used to approximate equilibrium strategies in a broad range of strategic queueing models in which direct analysis is impracticable

Unreliable Retrial Queues in a Random Environment

This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints

Response time distribution in a tandem pair of queues with batch processing

Response time density is obtained in a tandem pair of Markovian queues with both batch arrivals and batch departures. The method uses conditional forward and reversed node sojourn times and derives the Laplace transform of the response time probability density function in the case that batch sizes are finite. The result is derived by a generating function method that takes into account that the path is not overtake-free in the sense that the tagged task being tracked is affected by later arrivals at the second queue. A novel aspect of the method is that a vector of generating functions is solved for, rather than a single scalar-valued function, which requires investigation of the singularities of a certain matrix. A recurrence formula is derived to obtain arbitrary moments of response time by differentiation of the Laplace transform at the origin, and these can be computed rapidly by iteration. Numerical results for the first four moments of response time are displayed for some sample networks that have product-form solutions for their equilibrium queue length probabilities, along with the densities themselves by numerical inversion of the Laplace transform. Corresponding approximations are also obtained for (non-product-form) pairs of “raw” batch-queues – with no special arrivals – and validated against regenerative simulation, which indicates good accuracy. The methods are appropriate for modeling bursty internet and cloud traffic and a possible role in energy-saving is considered

Performance analysis of a polling model with BMAP and across-queue state-dependent service discipline

As various video services become popular, video streaming will dominate the mobile data traffic. The H.264 standard has been widely used for video compression. As the successor to H.264, H.265 can compress video streaming better, hence it is gradually gaining market share. However, in the short term H.264 will not be completely replaced, and will co-exist with H.265. Using H.264 and H.265 standards, three types of frames are generated, and among different types of frames exist dependencies. Since the radio resources are limited, using dependencies and quantities of frames in buffers, an appropriate time division transmission policy can be applied to transmit different types of frames sequentially, in order to avoid the occurrence of video carton or decoding failure. Polling models with batch Markovian arrival process (BMAP) and across-queue state-dependent service discipline are considered to be effective means in the design and optimization of appropriate time division transmission policies. However, the BMAP and across-queue state-dependent service discipline of the polling models lead to the large state space and several coupled state transition processes, which complicate the performance analysis. There have been very few researches in this regard. In this paper, a polling model of this type is analyzed. By constructing a supplementary embedded Markov chain and applying the matrix-analytic method based on the semi-regenerative process, the expressions of important performance measures including the joint queue length distribution, the customer blocking probability and the customer mean waiting time are obtained. The analysis will provide inspiration for analyzing the polling models with BMAP and across-queue state-dependent service discipline, to guide the design and optimization of time division transmission policies for transmitting the video compressed by H.264 and H.265