Анализ системы обслуживания с повторными вызовами, неоднородными приборами и марковским процессом поступления

Multi-server retrial queueing system with heterogeneous servers is analyzed. Requests arrive to the system according to the Markovian arrival process. Arriving primary requests and requests retrying from orbit occupy an available server with the highest service rate, if there is any available server. Otherwise, the requests move to the orbit having an infinite capacity. The total retrial rate infinitely increases when the number of requests in orbit increases. Service periods have exponential distribution. Behavior of the system is described by multi-dimensional continuous-time Markov chain which belongs to the class of asymptotically quasi-toeplitz Markov chains. This allows to derive simple and transparent ergodicity condition and compute the stationary probabilities distribution of chain states. Presented numerical results illustrate the dynamics of some system effectiveness indicators and the importance of considering of correlation in the requests arrival process.Анализируется многолинейная система массового обслуживания с повторными попытками и разнородными приборами. Запросы поступают в систему в соответствии с марковским процессом прибытия. Прибывающие первичные запросы и запросы, которые повторяют попытки попасть на обслуживание с орбиты, занимают свободный прибор с самой высокой скоростью обслуживания, если таковой имеется. В противном случае, если все приборы заняты,  запросы переходят на орбиту  бесконечной емкости, с которой осуществляют повторные попытки попасть на обслуживание. Общая интенсивность потока повторных попыток бесконечно возрастает с увеличением числа запросов на орбите. Время обслуживания запроса имеет экспоненциальное распределение с интенсивностью, зависящей от номера прибора. Поведение системы описывается многомерной цепью Маркова с непрерывным временем, которая принадлежит классу асимптотически квазитеплицевых цепей Маркова. Это позволяет вывести простое и прозрачное условие эргодичности и вычислить стационарное распределение вероятностей состояний цепи. Представленные численные результаты иллюстрируют динамику некоторых показателей эффективности системы и важность учета корреляции в процессе поступления запросов

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"

MSF-Model: Modeling Metastable Failures in Replicated Storage Systems

Metastable failure is a recent abstraction of a pattern of failures that occurs frequently in real-world distributed storage systems. In this paper, we propose a formal analysis and modeling of metastable failures in replicated storage systems. We focus on a foundational problem in distributed systems -- the problem of consensus -- to have an impact on a large class of systems. Our main contribution is the development of a queuing-based analytical model, MSF-Model, that can be used to characterize and predict metastable failures. MSF-Model integrates novel modeling concepts that allow modeling metastable failures which was interactable to model prior to our work. We also perform real experiments to reproduce and validate our model. Our real experiments show that MSF-Model predicts metastable failures with high accuracy by comparing the real experiment with the predictions from the queuing-based model

Large deviations analysis for the M/H2/n+MM/H_2/n + M queue in the Halfin-Whitt regime

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems

Essays on Service Information, Retrials and Global Supply Chain Sourcing

In many service settings, customers have to join the queue without being fully aware of the parameters of the service provider (for e.g., customers at check-out counters may not know the true service rate prior to joining). In such blind queues\u27\u27, customers typically make their decisions based on the limited information about the service provider\u27s operational parameters from past experiences, reviews, etc. In the first essay, we analyze a firm serving customers who make decisions under arbitrary beliefs about the service parameters. We show, while revealing the service information to customers improves revenues under certain customer beliefs, it may however destroy consumer welfare or social welfare. When consumers can self-organize the timing of service visits, they may avoid long queues and choose to retry later. In the second essay, we study an observable queue in which consumers make rational join, balk and (costly) retry decisions. Retrial attempts could be costly due to factors such as transportation costs, retrial hassle and visit fees. We characterize the equilibrium under such retrial behavior, and study its welfare effects. With the additional option to retry, consumer welfare could worsen compared to the welfare in a system without retrials. Surprisingly, self-interested consumers retry too little (in equilibrium compared to the socially optimal policy) when the retrial cost is low, and retry too much when the retrial cost is high. We also explore the impact of myopic consumers who may not have the flexibility to retry. In the third essay, we propose a comprehensive model framework for global sourcing location decision process. For decades, off-shoring of manufacturing to China and other low-cost countries was a no-brainer decision for many U.S. companies. In recent years, however, this trend is being challenged by some companies to re-shore manufacturing back to the U.S., or to near-shore manufacturing to Mexico. Our model framework incorporates perspectives over the entire life cycle of a product, i.e., product design, manufacturing and delivering, and after-sale service support, and we use it to test the validity of various competing theories on global sourcing. We also provide numerical examples to support our findings from the model

Analysis of buffer allocations in time-dependent and stochastic flow lines

This thesis reviews and classifies the literature on the Buffer Allocation Problem under steady-state conditions and on performance evaluation approaches for queueing systems with time-dependent parameters. Subsequently, new performance evaluation approaches are developed. Finally, a local search algorithm for the derivation of time-dependent buffer allocations is proposed. The algorithm is based on numerically observed monotonicity properties of the system performance in the time-dependent buffer allocations. Numerical examples illustrate that time-dependent buffer allocations represent an adequate way of minimizing the average WIP in the flow line while achieving a desired service level

Transient Analysis of Large-scale Stochastic Service Systems

The transient analysis of large-scale systems is often difficult even when the systems belong to the simplest M/M/n type of queues. To address analytical difficulties, previous studies have been conducted under various asymptotic regimes by suitably accelerating parameters, thereby establishing some useful mathematical frameworks and giving insights into important characteristics and intuitions. However, some studies show significant limitations when used to approximate real service systems: (i) they are more relevant to steady-state analysis; (ii) they emphasize proofs of convergence results rather than numerical methods to obtain system performance; and (iii) they provide only one set of limit processes regardless of actual system size. Attempting to overcome the drawbacks of previous studies, this dissertation studies the transient analysis of large-scale service systems with time-dependent parameters. The research goal is to develop a methodology that provides accurate approximations based on a technique called uniform acceleration, utilizing the theory of strong approximations. We first investigate and discuss the possible inaccuracy of limit processes obtained from employing the technique. As a solution, we propose adjusted fluid and diffusion limits that are specifically designed to approximate large, finite-sized systems. We find that the adjusted limits significantly improve the quality of approximations and hold asymptotic exactness as well. Several numerical results provide evidence of the effectiveness of the adjusted limits. We study both a call center which is a canonical example of large-scale service systems and an emerging peer-based Internet multimedia service network known as P2P. Based on our findings, we introduce a possible extension to systems which show non-Markovian behavior that is unaddressed by the uniform acceleration technique. We incorporate the denseness of phase-type distributions into the derivation of limit processes. The proposed method offers great potential to accurately approximate performance measures of non-Markovian systems with less computational burden

Call centres with balking and abandonment: from queueing to queueing network models

The research on call centres has attracted many researchers from different disciplines recently. In this thesis, we focus on call centre modelling, analysis and design. In terms of modelling, traditionally call centres have been modelled as single-node queueing systems. Based on the Semiopen Queueing Network (SOQN) model proposed by Srinivasan et al. [42], we propose and study SOQN models with balking and abandonment (both exponential and general patience time distributions). In addition, we study the corresponding single-node queueing systems and obtain new results. For each model, we study the queue length distribution, waiting time distribution and the related performance measures. To facilitate the computation, we express the performance measures in terms of special functions. In terms of call centre design, we develop a design algorithm to determine the minimal number of CSRs (S) and trunk lines (N) to satisfy a given set of service level constraints. The explicit expressions for performance measures obtained allow for theoretical analysis of the performance measures. For example we prove monotonicity and convexity properties of performance measures for the M/M/S/N and M/M/S/N + M models. We also study the comparison of different patience time distributions for the M/M/S/N+G model. We provide numerical examples for each model and discuss numerical results such as monotonicity properties of performance measures. In particular, we illustrate the efficacy of our design algorithm for various models including patient, balking and abandonment models. The impact of model parameters on the design of call centres is also discussed based on the numerical examples. The results are computed using Matlab, where special functions are available