Masakiyo Miyazawa

• Tokyo University of Science

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 l...

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...

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. The aim of this paper is to define this reflected diffusio...

We consider a single server queue which has a threshold to change its arrival process and service rate 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 rate may be changed. To get its performance in tractable...

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...

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 et al. in the BAR approach for multi-class queueing networks with SBP service policies, 2023), we are interested in two problems; w...

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 nonnegative half-line but the variance and drift of its Brownian component discontinuously change depending on its position. This reflecting diffus...

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...

We consider a single server queue which has a threshold to change its arrival process and service rate 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 rate may be changed. To get its performance in tractable...

We consider Palm distributions arising in a Markov process with time homogeneous transitions which is jointly stationary with multiple counting processes. Motivated by a BAR approach studied in the recent paper, Braverman Dai and Miya (2023}, we are interested in two problems; when this Markov process inherits the same Markov structure under the Pa...

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 an arriving customer itself and a customer being served is greater than a given threshold level c1≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage...

We extend the BAR-approach to prove heavy traffic steady-state convergence of multiclass queueing networks operating under static-buffer-priority service disciplines. Previously, a restrictive version of the BAR-approach was introduced to study generalized Jackson networks in which each service station has a single job class. The extension makes a...

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 pro...

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...

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...

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotic 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 Dynkin'...

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...

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 uniquel...

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...

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...

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...

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: the distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, wi...

In the seminal paper of Gamarnik and Zeevi [17], 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 il...

A classical result for the steady-state queue-length distribution of single-class queueing systems is the following: the distribution of the queue length just before an arrival epoch equals the distribution of the queue length just after a departure epoch. The constraint for this result to be valid is that arrivals, and also service completions, wi...

We are interested in a large queue in a $GI/G/k$ queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two different weak limit approximations are considered. One is for heavy traffic, an...

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 inter-arrival and service times have phase...

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...

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...

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...

Markov modulation is versatile in generalization for making a simple stochastic model which is often analytically tractable to be more flexible in application. In this spirit, we modulate a two dimensional reflecting skip-free random walk in such a way that its state transitions in the boundary faces and interior of a nonnegative integer quadrant a...

Diffusion processes have been widely used for approximations in the queueing theory. There are different types of diffusion approximations. Among them, we are interested in those obtained through limits of a sequence of models which describe queueing networks. Such a limit is typically obtained by the weak convergence of either stochastic processes...

We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. It is assumed that this reflecting random walk has skip free transitions. We are concerned with its time reversed process assuming that the stationary distribution exists. In general, the time reversed process may not be a reflecting random walk. In this paper...

We present a geometric interpretation of a product form stationary distribution for a $d$-dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The $d$-dimensional SRBM data can be equivalently specified by $d+1$ geometric objects: an ellipse and $d$ rays. Using these geometric objects, we establish nec...

We call a multidimensional distribution to be decomposable with respect to a partition of two sets of coordinates if the original distribution is the product of the marginal distributions associated with these two sets. We focus on the stationary distribution of a multidimensional semimartingale reflecting Brownian motion (SRBM) on a nonnegative or...

We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional workload process. Tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounde...

We consider a two-dimensional skip-free reflecting random walk on the non-negative integers, which is referred to as a 2-d reflecting random walk. We give necessary and sufficient conditions for the stationary distribution to have a product-form. We also derive simpler sufficient conditions for the product-form for a restricted class of 2-d reflect...

We are concerned with an $M/M$-type join the shortest queue ($M/M$-JSQ for short) with $k$ parallel queues for an arbitrary positive integer $k$, where the servers may be heterogeneous. We are interested in the tail asymptotic of the stationary distribution of this queueing model, provided the system is stable. We prove that this asymptotic for the...

In stochastic models for queues and their networks, random events evolve in time. A process for their backward evolution is referred to as a time reversed process. It is often greatly helpful to view a stochastic model from two different time directions. In particular, if some property is unchanged under time reversal, we may better understand that...

We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary dis...

We extend the framework of Neuts’ matrix analytic approach to a reflected process generated by a discrete time multidimensional Markov additive process. This Markov additive process has a general background state space and a real vector valued additive component, and generates a multidimensional reflected process. Our major interest is to derive a...

This paper presents a benchmarking suite that measures the performance of using sockets and eXtensible Markup Language remote procedure calls (XML-RPC) to exchange intra-node messages between Java virtual machines (JVMs). The paper also reports on an ...

We consider a parallel queueing model which has k identical servers. Assume that customers arrive from outside according to a Poisson process and join the shortest queue. Their service times have an i.i.d. exponential distribution, which is referred to as an M/MJSQ with k parallel queues. We are interested in the asymptotic behavior of the stationa...

We consider a two dimensional skip-free reflecting random walk on a nonnegative integer quadrant. We are interested in the tail asymptotics of its stationary distribution, provided its existence is assumed. We derive exact tail asymptotics for the stationary probabilities on the coordinate axis. This refines the asymptotic results in the literature...

We are concerned with the stationary distributions of reflecting processes on multidimensional nonnegative orthants and other related processes, provided they exist. Such stationary distributions arise in performance evaluation for various queueing systems and their networks. However, it is very hard to obtain them analytically, so our interest is...

We are concerned with the stationary distributions of reflecting processes on multidimensional nonnegative orthants and other related processes, provided they exist. Such stationary distributions arise in performance evaluation for various queueing systems and their networks. However, it is very hard to obtain them analytically, so our interest is...

We present three sets of results for the stationary distribution of a two-dimensional semimartingale-reflecting Brownian motion (SRBM) that lives in the non-negative quadrant. The SRBM data can equivalently be specified by three geometric objects, an ellipse and two lines, in the two-dimensional Euclidean space. First, we revisit the variational pr...

We are concerned with the stationary distribution of a d-dimensional semimartingale reflecting Brownian motion on a nonnegative orthant, provided it is stable, and conjecture about the tail decay rate of its marginal distribution in an arbitrary direction. Due to recent studies, the conjecture is true for d=2. We show its validity for the skew symm...

We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal station...

We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the non-negative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal statio...

This chapter discusses Palm calculus and its applications to various processes including queues and their networks. We aim to explain basic ideas behind this calculus. Since it differs from the classical approach using Markov processes, we scratch from very fundamental facts. The main target of Palm calculus is stationary processes, but we are also...

We are concerned with a discrete-time Markov additive process (MAP) generated by a Markov chain with transition probabilities similar to that for the M/G/1 queue. We are interested in its occupation measure before the additive component returns to the origin, and we study its asymptotic behavior as the additive component goes to infinity. This asym...

In this paper, we consider an M/M/s queue where customers may abandon waiting for service and renege the system without receiving their services. We assume that impatient time on waiting for each customer is independent and identically distributed non-negative random variable with a general distribution where the probability distribution is light-t...

We consider a Markov-modulated fluid queue with a finite buffer. It is assumed that the fluid flow is modulated by a background Markov chain which may have different transitions when the buffer content is empty or full. In Sakuma and Miyazawa (Asymptotic Behavior of Loss Rate for Feedback Finite Fluid Queue with Downward Jumps. Advances in Queueing...

We consider a Levy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the station- ary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper we consider only light-tailed inputs. Fo...

We extend the framework of Neuts' matrix analytic approach to a reflected process generated by a discrete time multidimensional Markov additive process. This Markov additive process has a general background state space and a real vector valued additive component, and generates a multidimensional reflected process. Our major interest is to derive a...

A double quasi-birth-and-death (QBD) process is the QBD process whose background process is a homogeneous birth-and-death process, which is a synonym of a skip free random walk in the two dimensional positive quadrant with homogeneous reflecting transitions at each boundary face. It is also a special case of a 0-partially homogenous chain introduce...

We consider a two dimensional reflected random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary, and referred to as a double M/G/1-type process. We are interested in tail asymptotic behavior of its stationary distribution, provided it exists. Assuming the arriving batch size dist...

We consider a feedback fluid queue with a finite buffer and downward jumps, where the net flow rate and the jump size for the buffer content are controlled by a background Markov chain with a finite state space. The feedback means that the transition structure of the background Markov chain may change when the buffer content becomes empty or full....

We are concerned with a two sided doubly quasi-birth-and-death process. Under a discrete time setting, this is a two dimensional skip free random walk on the half space whose second component is a nonnegative integer valued while its first component may take positive or negative integers. Our major interest is in the tail decay rate of the stationa...

Level-expanding quasi-birth-and-death (QBD) processes have been shown to be an efficient modeling tool for studying multi-dimensional systems, especially two- dimensional ones. Computationally, it changes the more challenging problem of dealing with algorithms for two-dimensional systems to a less challenging one for block-structured transition mat...

A geometric tail decay of the stationary distribution has been recently studied for the GI/G/1 type Markov chain with both countable level and background states. This method is essentially the matrix analytic approach, and simplicity is an obvious advantage of this method. However, so far it can be only applied to the α-positive case (or the jitter...

We are concerned with a finite buffer queue with arrival and service processes that are generated by a continuous time Markov chain with finitely many states. This model includes a many-server queue, and is described by a truncated QBD, where QBD stands for a quasi-birth-and-death process. It is shown that the loss probability of this queue geometr...

We consider congestion of traffics that are randomly produced in a bounded area. Those traffics start at local lines which are connected to a main line, and they have a common destination located at the end of the main line. We assume that a stream on the main line runs with a constant speed, while local traffics can join the main stream only if a...

In this paper, we consider a PH/M/2 queue in which each server has its own queue and arriving customers join the shortest queue. For this model, it has been conjectured that the decay rate of the tail probabilities for the shortest queue length in the steady state is equal to the square of the decay rate for the queue length in the corresponding PH...

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different case...

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different case...

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates...

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates...

A general framework is provided to derive analytic error bounds for the effect of perturbations and inaccuracies of nonexponential service or arrival distributions in single- and multiserver queues. The general framework is worked out in detail for the three types of finite or infinite buffer queues: GI/G/1/N, M/G/c/N, and GI/M/c/N. First, for the...

Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that th...

We consider the stationary distribution of the M/GI/1 type queue when back- ground states are countable. We are interested in its tail behavior. To this end, we derive a Markov renewal equation for characterizing the stationary distribution using a Markov additive process that describes the number of customers in sys- tem when the system is not emp...

Asymptotic decay rates are considered for the stationary joint distributions of customer pop-ulations in a generalized Jackson network and a batch movement network. We first define an asymptotic decay rate for a multi-dimensional distribution concerning a tail set and a direction to decrease. Then, for the stationary joint distributions of customer...

In this paper, we introduce a new class of queueing networks called arrival first networks. We characterise its transition rates and derive the relationship between arrival rules, linear partial balance equations, and product form stationary distributions. This model is motivated by production systems operating under a kanban protocol. In contrast...

We consider a preemptive priority fluid queue with two buffers for continuous fluid and batch fluid inputs. Those two types of fluids are governed by a Markov chain with a finite state space, called a background process, and the continuous fluid is preemptively processed over the batch fluid. The stationary joint distribution of the two buffer cont...

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and...

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and...

This paper considers an M/G/1 queue in which service time distributions in each busy period change according to a finite state Markov chain, embedded at the arrival instants of customers. It is assumed that this Markov chain has an upper triangular transition matrix. Applying the regenerative cycle approach with respect to a busy period, we obtain...

It is well known that various characteristics in risk and queuing processes can be formulated as Markov renewal functions, which are determined by Markov renewal equations. However, those functions have not been utilized as they are expected. In this article, we show that they are useful for studying asymptotic decay in risk and queuing processes u...

We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(·) is subadditive. A natural definition of ‘traffic intensity under congestion’ in this setting is ρ := lim t→∞t-1Es (all tasks arriving during time [0,t]). We show that ρ > 1 and a finite mea...

We consider I fluid queues in parallel. Each fluid queue has a deterministic inflow with a constant rate. At a random instant subject to a Poisson process, random amounts of fluids are simultaneously reduced. The requested amounts for the reduction are subject to a general I -dimensional distribution. The queues with inventories that are smaller th...

We consider I fluid queues in parallel. Each fluid queue has a deterministic inflow with a constant rate. At a random instant subject to a Poisson process, random amounts of fluids are simultaneously reduced. The requested amounts for the reduction are subject to a general I-dimensional distribution. The queues with inventories that are smaller tha...

We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(Delta) is subadditive. A natural definition of "traffic intensity under congestion" in this setting is ae := lim t!1 t Gamma1 Es(all tasks arriving during time [0; t]): We show that ae ! 1, an...

This paper focuses on product form and related tractable stationary distributions in a general class of stochastic networks with finite numbers of nodes such that their network states are changed through signal transfers as well as internal transitions. Signals may be customers in traditional queueing applications, but we do not make any restrictio...

This special issue focuses on a stochastic network described by a multidimensional stochastic process with reflecting boundaries. This is a widely used model in queueing theory and operations research. We are particularly interested in its stationary characteristics. Although a variety of models have been studied for stationary properties, exact re...

Palm distributions are basic tools when studying stationarity in the context of point processes, queueing systems, fluid queues or random measures. The framework varies with the random phenomenon of interest, but usually a one-dimensional group of measure-preserving shifts is the starting point. In the present paper, by alternatively using a frame...

We are concerned with an M/G/1 queue in which service time distributions in each busy period may depend on the number of customers who have been served in the same busy period. This model is called an exceptional service model. Our major interest is to see a general structure of this model through the stationary waiting time distribution and some o...

We show that several truncation properties of queueing systems are consequences of a simple property of censored stochastic processes. We first consider a discrete-time stochastic process and show that its censored process has a truncated stationary distribution. When the stochastic process has continuous time, we present a similar result und...

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions,...

In this paper we extend the notion of quasi-reversibility and apply it to the study of queueing networks with instantaneous movements and signals. The signals treated here are considerably more general than those in the existing literature. The approach not only provides a unified view for queueing networks with tractable stationary distributions,...

We are concerned with an M/G/1 queue in which service time distributions in each busy period may depend on the number of customers who have been served in the same busy period. This model is called an exceptional service model. Our major interest is to see a general structure of this model through the stationary waiting time distribution and some o...

In a recent communication, François Beccelli suggested the importance of a general formulation in research when he commented that we see many beautiful flowers in a garden but cannot see ground. Julian Keilson is honored as one of the great pioneers in cultivating the ground of applied probability. I am pleased to have an opportunity to contribute...