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Publications (164)
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...
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...
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...
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 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...
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...
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 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 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...
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...
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 eect of perturbations and inaccuracies of nonexponential service or arrival distributions in single- and multi-server queues. The general framework is worked out in detail for the three types of finite or infinite buer queues: GI/G/1/N, M/G/c/N, and GI/M/c/N. First, for the sta...
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...
We are concerned with a queueing network, described by a continuous-time Markov chain, in which each node is quasi-reversible. A new class of local balance equations is derived for the Markov chain with respect to a product-form distribution, which simultaneously provides an alternative and short proof for product form results of queueing networks...
Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint dist...
Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint dist...
This study concerns the equilibrium behavior of a general class of Markov network processes that includes a variety of queueing
networks and networks with interacting components or populations. The focus is on determining when these processes have product
form stationary distributions. The approach is to relate the marginal distributions of the pro...
A simple practical approximation is studied for a two-stage tandem queue with a finite first station. Explicit small error bounds are obtained for the mean queue length and the tail probabilities of the second queue. These error bounds are based on a new application of an existing error bound theorem for comparing Markov chains. The extension requi...
We consider characterizations of departure functions in Markovian queueing networks with batch movements and state-dependent routing in discrete-time and in continuous-time. For this purpose, the notion of structure-reversibility is introduced, which means that the time-reversed dynamics of a queueing network corresponds with the same type of queue...
We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers...
We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers...
This note studies the comparison of finite-buffer and nonexponential batch arrival systems of the form Gx/M/c/c + N with the corresponding systems, with N replaced by N', where N' can be smaller, larger, or infinite. If N' = ∞ the service times can be arbitrarily distributed. Both comparison and error bounds are obtained for performance measures su...
Formulas for level crossing probabilities, ladder height distributions and related characteristics of a general class of processes with stationary bounded variations and continuous decreasing components are derived under certain mild conditions. Results for a risk process with a constant premium rate and with a claim process generated by a stationa...
Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length d...
Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length d...
We introduce a discrete-time Jackson network with batch movements. Not more than one node simultaneously completes service, but arbitrary sizes of batch arrivals, departures and transfers are allowed under a Markovian routing of batches including changes of their sizes. This model corresponds with the continuous-time network with batch movements st...
A symmetric queue is known to have a nice property, the so-called insensitivity. In this paper, we generalize this for a single node queue with Poisson arrivals and background state, which changes at completion instants of lifetimes as well as at the arrival and departure instants. We study this problem by using the decomposability property of the...
Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group X, and relates integrations concerning them to each other. For X = R, we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was rec...
A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fix...
Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group , and relates integrations concerning them to each other. For , we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently...
A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fix...
We consider a discrete-time single-server queue with batch arrivals. Assuming that the arrival process of batches is renewal and using the point process approach, we present a relationship between characteristics of queue-length and waiting-time distributions. This relationship enables us to obtain a distributional form of Little's law (DFLL) for a...
Projects
Project (1)
Find suitable martingales for counting processes, queues and their networks. We apply them to study (a) asymptotic behaviors (of the moments of a renewal process and the the stationary distributions of queueing processes) and (b) diffusion approximations. We like to find what feature of martingales are useful in those studies.Of course, change of measure is one of them, but we like to find more.