On a doubly dynamically controlled supermarket model with impatient customers

Li, Quan‐Lin; Du, Ye; Dai, Guirong; Wang, Meng

doi:10.1016/j.cor.2014.10.004

Cited by 14 publications

(15 citation statements)

References 56 publications

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“…Load balancing is an interesting research direction in queueing networks with simply observable information, e.g., see Down and Lewis [103], Chou and Abraham [79], e Silva and Gerla [288], Li and Kameda [217], Li et al [220,221], Li [219] and Li and Lui [224].…”

Section: Objective Six: Controlled Queues and Network With Useful Inmentioning

confidence: 99%

An Overview for Markov Decision Processes in Queues and Networks

Fan

et al. 2019

Stochastic Models in Reliability, Network Security and System Safety

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Markov decision processes (MDPs) in queues and networks have been an interesting topic in many practical areas since the 1960s. This paper Provides a detailed overview on this topic and tracks the evolution of many basic results. Also, this paper summarizes several interesting directions in the future research. We hope that this overview can shed light to MDPs in queues and networks, and also to their extensive applications in various practical areas.One main purpose of this paper is to provide an overview for research on MDPs in queues and networks in the last six decades. Also, such a survey is first related to several other basic studies, such as, Markov processes, queueing systems, queueing networks, Markov decision processes, sensitivity-based optimization, stochastic optimization, fluid and diffusion control. Therefore, our analysis begins from three simple introductions:1 Markov processes and Markov decision processes, queues and queueing networks, and queueing dynamic control. (a) Markov processes and Markov decision processesThe Markov processes, together with the Markov property, were first introduced by a Russian mathematician: Andrei Andreevich Markov in 1906. See Markov [238] for more details. From then on, as a basically mathematical tool, the Markov processes have extensively been discussed by many authors, e.g., see some excellent books by Doob [99], Karlin [175], Karlin and Taylor [176], Chung [80], Anderson [21], Kemeny et al. [181], Meyn and Tweedie [241], Chen [77], Ethier and Kurtz [110] and so on. In 1960, Howard [165] is the first to propose and discuss the MDP (or stochastic dynamic programming) in terms of his Ph.D thesis, which opened up a new and important field through an interesting intersection between Markov processes and dynamic programming (e.g., see Bellman and Kalaba [32]). From then on, not only are the MDPs an important branch in the area of Markov processes, but also it is a basic method in modern dynamic control theory. Crucially, the MDPs have been greatly motivated and widely applied in many practical areas in the past 60 years. Readers may refer to some excellent books, for example, the discrete-time MDPs by Puterman [261], Glasserman and Yao [143], Bertsekas [33], Bertsekas and Tsitsiklis [34], Hernádez-Lerma and Lasserre [155, 156], Altman [9], Koole [193] and Hu and Yue [166]; the continuous-time MDPs by Guo and Hernández-Lerma [145]; the partially observable MDPs by Cassandra [67] and Krishnamurthy [196]; the competitive MDPs (i.e., stochastic game) by [127]; the sensitivity-based optimization by Cao [58]; some applications of MDPs by Feinberg and Shwartz (Eds.) [122]and Boucherie and Van Dijk (Eds.) [44]; and so on. (b) Queues and queueing networksIn the early 20th century, a Danmark mathematician: Agner Krarup Erlang, published a pioneering work [109] of queueing theory in 1909, which started the study of queueing theory and traffic engineering. Over the past 100 years, queueing theory has been regarded as a key mathematical tool not only for analyzing practical s...

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Section: Objective Six: Controlled Queues and Network With Useful Inmentioning

confidence: 99%

An Overview for Markov Decision Processes in Queues and Networks

Fan

et al. 2019

Stochastic Models in Reliability, Network Security and System Safety

Self Cite

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“…Markov chains by Benaim and Le Boudec [9] and Budhiraja and Majumder [20], the Markov decision processes by Gast and Gaujal [38] and Gast at al. [39], the continuoustime Markov chains by Dupuis and Fischer [30] and Budhiraja et al [18,19], and some practical examples include Mitzenmacher [68], Bobbio et al [11], Li et al [58,59], and Li and Lui [60].…”

Section: Nonlinear Markov Processesmentioning

confidence: 99%

“…On the other hand, we provide a simple method to describe the weak interaction of the N objects through introduction to the matrix Γ (N ) µ N (t) where µ N (t) is the empirical measure of the system of the N weakly interacting objects. Although our method is simple to deal with the weak convergence Γ (N ) µ N (t) =⇒ Γ (p (t)) as N → ∞, it is useful and effective in the mean-field study of many continuous-time big networks, readers may refer to Li et al [58,59] and Li [60] for more details.…”

Section: (B) the Limiting Processesmentioning

confidence: 99%

“…For the supermarket models, readers may refer to, such as, Vvedenskaya et al [83] for operator semigroup, Mitzenmacher [68] for density-dependent jump Markov processes, Turner [80] for martingale limits. Subsequent papers have been published on this theme, important examples include Vvedenskaya and Suhov [84], Graham [41,42], Luczak and McDiarmid [63,64], Li et al [58,59] and Li [60]. As a key generalization of the supermarket models, the fast Jackson networks were investigated by Martin and Suhov [66], Martin [65], and Suhov and Vvedenskaya [77].…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Markov processes in big networks

2016

Special Matrices

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Big networks express multiple classes of large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-eld theory and the nonlinear Markov processes to constructing a broad class of nonlinear continuous-time block-structured Markov processes, which can be used to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of big networks with weak interactions, where each big network is described as a continuous-time block-structured Markov process. Secondly, some e ective algorithms are given for computing the xed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkho center, the locally stable xed points, the Lyapunov functions and the relative entropy are developed to analyze stability or metastability of the system of weakly interacting big networks, and several interesting open problems are proposed with detailed interpretation. We believe that the methodology and results given in this paper can be useful and e ective in the study of big networks.

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“…In the queueing literature, one can find an extensive body of research addressing impatience phenomena observed in single or multi-server systems serving a single class of customers (see Altman and Yechiali [4], Yechiali [30], Perel and Yechiali [20]). Recently, Li et al [13] studied a supermarket model with impatient customers. However, to the best of our knowledge, no work has been done on the M X /G/1 retrial queueing system with impatient customers.…”

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confidence: 99%

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

Peng¹,

Wu²

2021

JIMO

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We consider an M X /G/1 retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until N or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.2010 Mathematics Subject Classification. Primary: 60K25; Secondary: 90B22.

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On a doubly dynamically controlled supermarket model with impatient customers

Cited by 14 publications

References 56 publications

An Overview for Markov Decision Processes in Queues and Networks

An Overview for Markov Decision Processes in Queues and Networks

Nonlinear Markov processes in big networks

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

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