Diffusion limits for networks of Markov-modulated infinite-server queues

Jansen, H. M.; Mandjes, Michel; Turck, Koen De; Wittevrongel, Sabine

doi:10.1016/j.peva.2019.102039

Cited by 14 publications

(28 citation statements)

References 26 publications

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“…the discussion above (25). This is likely to carry over to the modulated setting of Remark 5.4, when also speeding up the background process, in line with results found for networks of modulated infinite-server queues in [18,56]. In more general terms, it would also be interesting to further explore the relationship between linear stochastic fluid networks and infinite-server queueing networks; see already, for example, [49,66].…”

Section: Linear Stochastic Fluid Networksupporting

confidence: 71%

Shot-noise queueing models

Boxma

Mandjes

2021

Queueing Syst

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We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process.

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Section: Linear Stochastic Fluid Networksupporting

confidence: 71%

Shot-noise queueing models

Boxma

Mandjes

2021

Queueing Syst

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“…by the boundedness of f , [1,Proposition 3.2] and [17,Theorem 5.2]. Thus, we can replace 1(J n (s) = k) by π k for all k ∈ K in (4.9), when we let (n, T ) → ∞.…”

Section: 3mentioning

confidence: 99%

“…The system operates in the "averaged" Halfin-Whitt (H-W) regime, namely, it is critically loaded in an average sense, but it may be underloaded or overloaded for a given state of the environment. This situation is different from the standard H-W regime for many-server queues, which requires that the system is critically loaded as the arrival rates and number of servers get large; see, e.g., [2,7,16,17]. policies, i.e., work-conserving and non-preemptive polices.…”

Section: Introductionmentioning

confidence: 96%

“…Similarly, we have that E[τ n 2,i (t)] and E[R * ,i (τ n 2,i (t))] are finite. Thus, applying Lemma 3.2 in [22] and Theorem 8.7 on page 87 of [14], and using the decomposition in (2.3) and Lemma 3.1 in [17], we conclude thatX n is a semi-martingale with respect to the filtration F n := { F n t : t ≥ 0}, where F n t := σ{S n i (s), R n i (s), X n i (0) : i ∈ I, s ≤ t} ∨ σ{A n i (s), J n (s), Z n i (s) : i ∈ I, s ≥ 0} ∨ N , and N is a collection of P-null sets. Since the processes A n (t), J n (t) and Z n (t) are adapted to F n t , we can replace F n by the smaller filtration F n .…”

mentioning

confidence: 99%

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Optimal Control of Markov-Modulated Multiclass Many-Server Queues

et al. 2019

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We study multiclass many-server queues for which the arrival, service and abandonment rates are all modulated by a common finite-state Markov process. We assume that the system operates in the "averaged" Halfin-Whitt regime, which means that it is critically loaded in the average sense, although not necessarily in each state of the Markov process. We show that under any static priority policy, the Markov-modulated diffusion-scaled queueing process is geometrically ergodic. This is accomplished by employing a solution to an associated Poisson equation in order to construct a suitable Lyapunov function. We establish a functional central limit theorem for the diffusion-scaled queueing process and show that the limiting process is a controlled diffusion with piecewise linear drift and constant covariance matrix. We address the infinite-horizon discounted and long-run average (ergodic) optimal control problems and establish asymptotic optimality.We first establish a FCLT for the Markov-modulated diffusion-scaled queueing processes under any admissible scheduling policy (only considering work-conserving and preemptive policies). Proper scaling is needed in order to establish weak convergence of the queueing processes. In particular, since the arrival processes are of order n, and the switching rates of the background process are assumed to be of order n α for α > 0, the queueing processes are centered at the 'averaged' steady state, which is of order n, and are then scaled down by a factor of an n β , with β := max{ 1 /2, 1− α /2}, in the diffusion scale. Thus, when α ≥ 1, we have the usual diffusion scaling with β = 1 /2, which is due to the fact that the very fast switching of the environment results in an 'averaging' effect for the arrival, service and abandonment processes of the queueing dynamics. The limit queueing process is a piecewise Ornstein-Uhlenbeck diffusion process with a drift and covariance given by the corresponding 'averaged' quantities under the stationary distribution of the background process. When α = 1, both the variabilities of the queueing and background processes are captured in the covariance matrix, while when α > 1, only the variabilities of the queueing process is captured. On the other hand, when α < 1, the proper diffusion scaling requires β = 1 − α /2, for which we obtain a similar piecewise Ornstein-Uhlenbeck diffusion process with the covariance matrix capturing the variabilities of the background process only.The ergodic properties of this class of piecewise linear diffusions (and Lévy-driven stochastic differential equations) have been studied in [6,13], and these results can be applied directly to our model. The study of the ergodic properties of the diffusion-scaled processes, however, is challenging. Ergodicity of switching Markov processes has been an active research subject. For switching diffusions, stability has been studied in [18,19,21]. However, studies of ergodicity of switching Markov processes are scarce. Recently in [8,9], some kind of hypoellipticity criterion with H...

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“…. , n. As shown in [18,Section 3], this (·) can be considered the fluid limit [29] corresponding to M (N ) (·).…”

Section: Scaling Limit: Functional Central Limit Theoremmentioning

confidence: 99%

Queues on a Dynamically Evolving Graph

2018

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This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in parallel. The links that connect the queues have the special feature that they are unreliable, in the sense that their status alternates between 'up' and 'down'. If a link between two nodes is down, with a fixed probability each of the clients attempting to use that link is lost; otherwise the client remains at the origin node and reattempts using the link (and jumps to the destination node when it finds the link restored). For these networks we present the following results: (a) a system of coupled partial differential equations that describes the joint probability generating function corresponding to the queues' time-dependent behavior (and a system of ordinary differential equations for its stationary counterpart), (b) an algorithm to evaluate the (time-dependent and stationary) moments, and procedures to compute user-perceived performance measures which facilitate the quantification of the impact of the links' outages, (c) a diffusion limit for the joint queue length process. We include explicit results for a series relevant special cases, such as tandem networks and symmetric fully connected networks.

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Diffusion limits for networks of Markov-modulated infinite-server queues

Cited by 14 publications

References 26 publications

Shot-noise queueing models

Shot-noise queueing models

Optimal Control of Markov-Modulated Multiclass Many-Server Queues

Queues on a Dynamically Evolving Graph

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