arXiv subjects

Masakiyo Miyazawa

Publications and source records attributed to Masakiyo Miyazawa.

At least 19 recordsLinked to original sources

arXiv2026

Heavy-traffic limit of stationary distributions of a state-dependent queue

Inspired by the work of Atar and Miyazawa [1] (2026) as well as applications to energy-saving problems, we are interested in the heavy-traffic limit of the stationary queue length distribution, which is not addressed in [1]. In this paper, we consider this heavy-traffic limit for the single server queue which has the most general possible state-dependence. Namely, arrival and service speeds may take any values depending on the queue length. Here, the terminology, heavy-traffic limit, stands for a diffusion-scaled limit in heavy-traffic for processes, distributions and modeling primitives. This general model is referred to as a state-dependent queue. There are two motivations for this generalization. One is interest in the state-dependent queue itself because it allows finer control of service speed in application. Another is making it clear how the heavy-traffic limit is obtained under what conditions for the state-dependent queue. Thus, we start to study basic properties of this state-dependent queue, including its stability. We then take the sequence of the stationary distributions of its diffusion scaled queue-length processes. We have three main results for this sequence. We first show that it is tight if the heavy-traffic limit of their drifts exists and is negative as the queue length goes to infinity, where a drift is the arrival speed minus the service speed. We next assume the condition that the limit of every vaguely convergent subsequence has a density, which is referred to as a density condition, and show that the heavy-traffic limit of the stationary distributions is obtained in a closed form if and only if that negative drift condition holds. We then show that the density condition is always satisfied for the multi-level queue, so the problem is nicely solved for the multi-level queue.

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arXiv2026

Diffusion limit for the stationary distribution of a history-dependent two-level M/M/1 queue

Recently, Atar and Miyazawa [2] introduced a multi-level GI/G/1 queue with a finite number of levels, where both the arrival and service rates depend on the level corresponding to the current queue length. For this model, they proved that the diffusion limit of its queue length process in heavy traffic is the level-dependent reflected Brownian motion of [6]. In a subsequent study, Kobayashi et al. [4] derived the corresponding diffusion limit of the stationary distribution. These studies are motivated by the control of service capacity depending on the queue length. We are interested in the more general case where this control may also depend on the history of the queue length. As the first step toward such a generalization, we specialize the multi-level GI/G/1 queue to a two-level M/M/1 queue. We then extend the dynamics of this model so that its arrival and service rates depend not only on the current queue length but also on the recent history of queue lengths. Under the stability condition for this model, we first compute its stationary distribution in closed form, then derive its diffusion limit in heavy traffic. Finally, using this diffusion limit, we derive approximation formulas for the stationary distribution and then numerically assess their accuracy.

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arXiv2025

Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition

This paper studies a single server queue in heavy traffic, with general inter-arrival and service time distributions, where arrival and service rates vary discontinuously as a function of the (diffusively scaled) queue length. It is proved that the weak limit is given by the unique-in-law solution to a stochastic differential equation in $[0,\infty)$ with discontinuous drift and diffusion coefficients. The main tool is a semimartingale decomposition for point processes introduced in \cite{dal-miy}, which is distinct from the Doob-Meyer decomposition of a counting process. Whereas the use of this tool is demonstrated here for a particular model, we believe it may be useful for investigating the scaling limits of queueing models very broadly.

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arXiv2025

Tight matrices and heavy traffic steady state convergence in queueing networks

We are interested to prove that the stationary distribution of a multiclass queueing network converges to the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in heavy traffic. A key condition for this convergence is that the sequence of the pre-limit stationary distributions under appropriate scaling is tight. In Braverman et al.(2025), a sufficient condition for this tightness is introduced in the term of the reflection matrix $R$ of the SRBM, which is coined for $R$ to be ``tight''. In this paper, we study how we can verify this tightness of $R$ of an SRBM. For a $2$-dimensional SRBM, we give necessary and sufficient conditions for $R$ to be tight, while, for a general dimension, we only give sufficient conditions. We then apply these results to the SRBMs arising from the diffusion approximations of multiclass queueing networks with static buffer priority service disciplines that are studied in Braverman et al.(2025). It is shown that $R$ is always tight for this network with two stations if $R$ is completely-$\sr{S}$. For the case of more than two stations, it is shown that $R$ is tight for reentrant lines with last-buffer-first-service (LBFS) discipline, but it is not always tight for reentrant line with first-buffer-first-service (FBFS) discipline.

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arXiv2025

Heavy-tail asymptotics for the length of a busy period in a Generalised Jackson Network

We consider a Generalised Jackson Network with finitely many servers, a renewal input and $i.i.d.$ service times at each queue. We assume the network to be stable and, in addition, the distribution of the inter-arrival times to have unbounded support. This implies that the length of a typical busy period $B$, which is the time between two successive idle periods, is finite a.s. and has a finite mean. We assume that the distributions of the service times with the heaviest tails belong to the class of so-called intermediate regularly varying distributions. We obtain the exact asymptotics for the probability ${\mathbb P} (B>x)$, as $x\to\infty$. For that, we show that the Principle of a Single Big Jump holds: $B$ takes a large value mainly due to a single unusually large service time.

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arXiv2025

Heavy traffic limit of stationary distribution of the multi-level single server queue

Atar and Miyazawa recently introduced a single server queue with queue length dependent arrival and service processes, and name it a multi-level queue. They prove that the heavy traffic limit of its queue length process weakly converges to a reflected diffusion with discontinuously state-dependent drift and deviations. We derive the heavy traffic limit of the stationary queue length distribution of this multi-level queue in a closed form, which agrees with the stationary distribution of the reflected diffusion obtained by Miyazawa (2024, Journal of the Indian Society for Probability and Statistics). Thus, those results show the limit interchange of process and stationary distribution in heavy traffic. The multi-level queue is a simpler version of the 2-level GI/G/1 queue of Miyazawa (2025, Advances in Applied Probability, to appear) and its extension for multi-levels. For this 2-level queue in heavy traffic, the process limit is unknown, and the distributional limit is obtained for limited cases under extra conditions. Nevertheless, it is shown that the method developed in Miyazawa (2025) perfectly works for the present multi-level queue.

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arXiv2025

Diffusion approximation of the stationary distribution of a two-level single server queue

We consider a single server queue which has a threshold to change its arrival process and service speed by its queue length, which is referred to as a two-level single server queue. This model is motivated by an energy saving problem for a single server queue whose arrival process and service speed are controlled. To get its performance in tractable form, we study the limit of the stationary distribution of the queue length in this two-level queue under scaling in heavy traffic. Except for a special case, this limit corresponds to its diffusion approximation. It is shown that this limiting distribution is truncated exponential (or uniform if the drift is null) below the threshold level and exponential above it under suitably chosen system parameters and generally distributed inter-arrival times and workloads brought by customers. This result is proved under a mild limitation on arrival parameters using the so called BAR approach studied in Braverman, Dai and Miyazawa (2017, 2023) and Miyazawa (2017, 2023). We also intuitively discuss about a diffusion process corresponding to the limit of the stationary distribution under scaling.

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arXiv2024

The stationary distributions of state-dependent diffusions reflected at one and two sides

Consider a one-dimensional diffusion process which has state-dependent drift and deviation and is reflected at the origin, which is called a one-side reflected diffusion or simply reflected diffusion. We are particularly interested in the case that its drift and deviation are discontinuous. We define this reflected diffusion as the solution of a stochastic integral equation, and find conditions for its positive recurrence, We then derive its stationary distribution under these conditions. As a related problem, we also consider the case that it is reflected at two sides, which is called a two-sides reflected diffusion. Its existence, positive recurrence and stationary distribution are similarly studied. In the literature, these problems are studied through a state-dependent diffusion on the whole line particularly when the drift and deviation are discontinuous. However, the reflected process itself is not defined in such a study. Thus, the stationary distribution has not been fully studied for a general state-dependent reflected diffusion. We aim to fills this insufficiency and to make the stationary distributions of reflected diffusions widely available in application.

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arXiv2024

Multi-level reflecting Brownian motion on the half line and its stationary distribution

A semi-martingale reflecting Brownian motion is a popular process for diffusion approximations of queueing models including their networks. In this paper, we are concerned with the case that it lives on the nonnegative half-line, but the drift and variance of its Brownian component discontinuously change at its finitely many states. This reflecting diffusion process naturally arises from a state-dependent single server queue, studied by the author (2024). Our main interest is in its stationary distribution, which is important for application. We define this reflecting diffusion process as the solution of a stochastic integral equation, and show that it uniquely exists in the weak sense. This result is also proved in a different way by Atar, Castiel and Reiman (2022,2023). In this paper, we consider its Harris irreducibility and stability, that is, positive recurrence, and derive its stationary distribution under this stability condition. The stationary distribution has a simple analytic expression, likely extendable to a more general state-dependent SRBM. Our proofs rely on the generalized Ito formula for a convex function and local time.

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arXiv2024

Palm problems arising in BAR approach and its applications

We consider Palm distributions arising in a Markov process with time homogeneous transitions which is jointly stationary with multiple point processes. Motivated by a BAR approach studied in the recent paper Braverman, Dai and Miyazawa (2023}, we are interested in two problems; when this Markov process inherits the same Markov structure under the Palm distributions, and how the state changes at counting instants of the point processes can be handled to derive stationary equations when there are simultaneous counts and each of them influences the state changes. We affirmatively answer to the first problem, and propose a framework for resolving the second problem. We also discuss how those results can be applied in deriving BAR's for the diffusion approximation of queueing models in heavy traffic. In particular, as their new application, the heavy traffic limit of the stationary distribution is derived for a single server queue with a finite waiting room. Those examples show how the BAR approach is useful under the proposed framework.

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arXiv2024

The BAR approach for multiclass queueing networks with SBP service policies

The basic adjoint relationship (BAR) approach is an analysis technique based on the stationary equation of a Markov process. This approach was introduced to study heavy-traffic, steady-state convergence of generalized Jackson networks in which each service station has a single job class. We extend it to multiclass queueing networks operating under static-buffer-priority (SBP) service disciplines. Our extension makes a connection with Palm distributions that allows one to attack a difficulty arising from queue-length truncation, which appears to be unavoidable in the multiclass setting. For multiclass queueing networks operating under SBP service disciplines, our BAR approach provides an alternative to the "interchange of limits" approach that has dominated the literature in the last twenty years. The BAR approach can produce sharp results and allows one to establish steady-state convergence under three additional conditions: stability, state space collapse (SSC) and a certain matrix being "tight." These three conditions do not appear to depend on the interarrival and service-time distributions beyond their means, and their verification can be studied as three separate modules. In particular, they can be studied in a simpler, continuous-time Markov chain setting when all distributions are exponential. As an example, these three conditions are shown to hold in reentrant lines operating under last-buffer-first-serve discipline. In a two-station, five-class reentrant line, under the heavy-traffic condition, the tight-matrix condition implies both the stability condition and the SSC condition. Whether such a relationship holds generally is an open problem.

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arXiv2023

Stability of a cascade system with two stations and its extension for multiple stations

We consider a two station cascade system in which waiting or externally arriving customers at station $1$ move to the station $2$ if the queue size of station $1$ including a customer being served is greater than a given threshold level $C_{1} \ge 1$ and if station $2$ is empty. Assuming that external arrivals are subject to independent renewal processes satisfying certain regularity conditions and service times are $i.i.d.$ at each station, we derive necessary and sufficient conditions for a Markov process describing this system to be positive recurrent in the sense of Harris. This result is extended to the cascade system with a general number $k$ of stations in series. This extension requires the actual traffic intensities of stations $2,3,\ldots, k-1$ for $k \ge 3$. We finally note that the modeling assumptions on the renewal arrivals and $i.i.d.$ service times are not essential if the notion of the stability is replaced by a certain sample path condition. This stability notion is identical with the standard stability if the whole system is described by the Markov process which is a Harris irreducible $T$-process.

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arXiv2020

Markov modulated fluid network process: Tail asymptotics of the stationary distribution

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotics behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on a well-known time-evolution formula of a Markov process, so-called Dynkin's formula, where a key ingredient is a suitable choice of test functions. Those results show how multidimensional tail asymptotics can be studied for the more than two-dimensional case, which is known as a hard problem.

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arXiv2018

A martingale view of Blackwell's renewal theorem and its extensions to a general counting process

Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and its extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell's renewal theorem holds.

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arXiv2018

Tails in a fixed-point problem for a branching process with state-independent immigration

We consider a fixed-point equation for a non-negative integer-valued random variable, that appears in branching processes with state-independent immigration. A similar equation appears in the analysis of a single-server queue with a homogeneous Poisson input, feedback and permanent customer(s). It is known that the solution to this equation uniquely exists under mild first and logarithmic moments conditions. We find further the tail asymptotics of the distribution of the solution when the immigration size and branch size distributions are heavy-tailed. We assume that the distributions of interest are dominantly varying and have a long tail. This class includes, in particular, (intermediate, extended) regularly varying distributions. We consider also a number of generalisations of the model.

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arXiv2018

Customer sojourn time in GI/G/1 feedback queue in the presence of heavy tails

We consider a single-server GI/GI/1 queueing system with feedback. We assume the service times distribution to be (intermediate) regularly varying. We find the tail asymptotics for a customer's sojourn time in two regimes: the customer arrives in an empty system, and the customer arrives in the system in the stationary regime. In particular, in the case of Poisson input we use the branching processes structure and provide more precise formulae. As auxiliary results, we find the tail asymptotics for the busy period distribution in a single-server queue with an intermediate varying service times distribution and establish the principle-of-a-single-big-jump equivalences that characterise the asymptotics.

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arXiv2017

Martingale approach for tail asymptotic problems in the generalized Jackson network

We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two station case, this problem has been recently solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probabilities of state sets of small volumes. Second, the interarrival and service times are generally distributed and light tailed, but of phase type in some cases. Third, we also study the case that there are more than two stations, although the asymptotic results are less complete. For them, we develop a martingale method, which has been recently applied to a single queue with many servers by the author.

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arXiv2017

Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach

In the seminal paper of Gamarnik and Zeevi (2006), the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steady-state approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks.

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