arXiv subjects

Tetsuya Takine

Publications and source records attributed to Tetsuya Takine.

7 recordsLinked to original sources

arXiv2026

Computing transient probabilities in Markovian queues with balking conditional on a fixed number of joined customers

We analyze the transient behavior of a Markovian queue with balking, conditional on exactly $K$ customers joining the system in a finite time interval $[0, T]$. A key quantity of interest is the cumulative number of balking customers, for which we consider the probability generating function (PGF) and derive an equation it satisfies. This approach circumvents the computational burden arising from dealing with high-dimensional Markovian system, which arises when attempting to directly compute the joint distribution of the cumulative number of balking customers together with the total number of joining customers and the number of customers in the system. To compute state transitions in $(t, T]$ efficiently, we examine the conditional state probability at time $T$ given the system state at time $T - u$, and show that it satisfies a linear differential equation in $u$. For piecewise-constant arrival rates, we develop a numerical procedure for evaluating the moment of the total number of balking customers, jointly with the cumulative number of joining customers and the number of customers in the system, under the condition that $K$ customers joined. We also present numerical examples that highlight counterintuitive behaviors arising from this conditioning, along with explanations of the underlying mechanisms.

math.PR↗

arXiv2026

Time-dependent queue length distribution in queues fed by $K$ customers in a finite interval

We consider queueing models, where customers arrive according to a continuous-time binomial process on a finite interval. In this arrival process, a total of $K$ customers arrive in the finite time interval $[0,T]$, where arrival times of those $K$ customers are independent and identically distributed according to an absolutely continuous distribution defined by its probability density function $f(t)$ on $(0,T]$. To analyze the time-dependent queue length distribution of this model, we introduce the auxiliary model with non-homogeneous Poisson arrivals and show that the time-dependent queue length distribution in the original model is given in terms of the time-dependent joint distribution of the numbers of arrivals and departures in the auxiliary model. Next, we consider a numerical procedure for computing the time-dependent queue length distribution in Markovian models with piecewise constant $f(t)$. A particular feature of our computational procedure is that the truncation error bound can be set as the input. Some numerical examples are also provided.

math.PR↗

arXiv2026

Mean Waiting Times in Discrete-Time Priority Queues with Geometrically Distributed Idle Periods

This paper considers the mean waiting times in discrete-time preemptive-resume and nonpreemptive priority single-server queues fed by K independent batch Markovian arrival streams with geometrically distributed idle periods. While being active, the k-th (k = 1, 2, ..., K) arrival stream feeds at least one customer to the queue, where the number of arriving customers depends on the state of the underlying Markov chain. Service times of class $k$ customers are independent and identically distributed according to a general distribution. For these queues, we derive explicit formulae for the mean waiting times of customers in each class.

math.PR↗

arXiv2026

The Gated-Service M/GI/1 Queue with Single Vacations and Its Application to Batch-Service Queues

In this paper, we consider the M/GI/1 queue with single vacations under the gated service discipline. We obtain the probability generating function of the stationary queue length, the Laplace-Stieltjes transform of the system delay distribution in steady state, and the joint transform of the busy cycle length and the number of customers served in the busy cycle. Furthermore, as an application, we consider a batch-service M/G/1 queue, where service times depend on the number of customers in batch.

math.PR↗

arXiv2024

Exact Analysis of the Age of Information in the Multi-Source M/GI/1 Queueing System

We consider a situation that multiple monitoring applications (each with a different sensor-monitor pair) compete for a common service resource such as a communication link. Each sensor reports the latest state of its own time-varying information source to its corresponding monitor, incurring queueing and processing delays at the shared resource. The primary performance metric of interest is the age of information (AoI) of each sensor-monitor pair, which is defined as the elapsed time from the generation of the information currently displayed on the monitor. Although the multi-source first-come first-served (FCFS) M/GI/1 queue is one of the most fundamental model to describe such competing sensors, its exact analysis has been an open problem for years. In this paper, we show that the Laplace-Stieltjes transform (LST) of the stationary distribution of the AoI in this model, as well as the mean AoI, is given by a simple explicit formula, utilizing the double Laplace transform of the transient workload in the M/GI/1 queue.

cs.NI↗

arXiv2019

A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues

This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribution of the AoI is given in terms of the stationary distributions of the system delay and the peak AoI. To demonstrate its applicability and usefulness, we analyze the AoI in single-server queues with four different service disciplines: first-come first-served (FCFS), preemptive last-come first-served (LCFS), and two variants of non-preemptive LCFS service disciplines. For the FCFS and the preemptive LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and GI/M/1 queues are considered. With these results, we further show comparison results for the mean AoI's in the M/GI/1 and GI/M/1 queues under those service disciplines.

cs.PF↗

arXiv2014

Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.

math.PR↗