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This paper presents a comprehensive review of stochastic processes, with a particular focus on Markov chains and jump processes. The main results related to queuing systems are analyzed. Additionally, conditions that ensure the stability,…

Probability · Mathematics 2025-07-30 Carlos E. Martínez-Rodríguez

To ensure the effective and objective development of transportation networks, it is crucial to identify performance limitations across various subsystems. A timetable-independent assessment of infrastructure capacity at railway junctions is…

Systems and Control · Electrical Eng. & Systems 2025-05-27 Tamme Emunds , Nils Nießen

We consider $M/G/\infty$ queues with gated service and obtain results on the distribution of the stage length and the number of customers served in a stage when the system is stationary. The stage length density is expressed as an infinite…

Probability · Mathematics 2026-03-23 Dimitra Pinotsi , Michael A. Zazanis

For entropy driven balanced processes we obtain final states with Poisson, Bernoulli, negative binomial and P\'olya distributions. We apply this both for complex networks and particle production. For random networks we follow the evolution…

Statistical Mechanics · Physics 2017-03-08 Tamás S. Biró , Zoltán Néda

We study three non-equivalent queueing models in continuous time that each generalise the classical M/M/1 queue in a different way. Inter-event times in all models are Mittag-Leffler distributed, which is a heavy tail distribution with no…

Probability · Mathematics 2022-11-24 Jacob Butt , Nicos Georgiou , Enrico Scalas

Motivated by call center practice, we propose a tractable model for $\mbox{GI}/\mbox{GI}/n+\mbox{GI}$ queues in the efficiency-driven (ED) regime. We use a one-dimensional diffusion process to approximate the virtual waiting time process…

Probability · Mathematics 2015-09-17 Shuangchi He

This is an annotated bibliography on estimation and inference results for queues and related stochastic models. The purpose of this document is to collect and categorise works in the field, allowing for researchers and practitioners to…

Statistics Theory · Mathematics 2024-01-12 Azam Asanjarani , Yoni Nazarathy

We study a many-server queuing system with general service time distribution and state dependent service rates. The dynamics of the system are modeled using measure valued processes which keep track of the residual service times. Under…

Probability · Mathematics 2013-04-09 Anup Biswas

This paper studies load balancing for many-server ($N$ servers) systems. Each server has a buffer of size $b-1,$ and can have at most one job in service and $b-1$ jobs in the buffer. The service time of a job follows the Coxian-2…

Probability · Mathematics 2021-02-18 Xin Liu , Kang Gong , Lei Ying

We consider the following distributed service model: jobs with unit mean, general distribution, and independent processing times arrive as a renewal process of rate $\lambda n$, with $0<\lambda<1$, and are immediately dispatched to one of…

Probability · Mathematics 2018-07-10 David Gamarnik , John N. Tsitsiklis , Martin Zubeldia

The infinite-server queueing models with homogeneous and non-homogeneous arrivals of customers and catastrophes are considered. The probability generating functions of joint distributions of numbers of busy servers and served customers, as…

Optimization and Control · Mathematics 2018-07-24 Khanik Kerobyan

This contribution presents a derivation of the steady-state distribution of velocities and distances of driven particles on a onedimensional periodic ring. We will compare two different situations: (i) symmetrical interaction forces…

Data Analysis, Statistics and Probability · Physics 2009-11-13 Martin Treiber , Dirk Helbing

Explicit results are derived using simple and exact methods for the joint and marginal queue-length distributions for the M/M/c queue with two non-preemptive priority levels. Equal service rates are assumed. Two approaches are considered.…

Probability · Mathematics 2023-09-19 Josef Zuk , David Kirszenblat

In this paper, we consider a load balancing system under a general pull-based policy. In particular, each arrival is randomly dispatched to one of the servers whose queue lengths are below a threshold, if there are any; otherwise, this…

Performance · Computer Science 2018-08-22 Xingyu Zhou , Jian Tan , Ness Shroff

We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service…

Probability · Mathematics 2013-12-03 Ruslan Krenzler , Hans Daduna

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes…

Probability · Mathematics 2021-11-23 Sarat Babu Moka , Yoni Nazarathy , Werner Scheinhardt

The residual queue during a given study period (e.g., peak hour) is an important feature that should be considered when solving a traffic assignment problem under equilibrium for strategic traffic planning. Although studies have focused…

Systems and Control · Electrical Eng. & Systems 2025-01-14 Hao Fu , William H. K. Lam , Wei Ma , Yuxin Shi , Rui Jiang , Huijun Sun , Ziyou Gao

We introduce the {\Delta}(i)/GI/1 queue, a new queueing model. In this model, customers from a given population independently sample a time to arrive from some given distribution F. Thus, the arrival times are an ordered statistics, and the…

Probability · Mathematics 2014-12-09 Harsha Honnappa , Rahul Jain , Amy R. Ward

In this paper, we analyse a single server polling model with two queues. Customers arrive at the two queues according to two independent Poisson processes. There is a single server that serves both queues with generally distributed service…

Probability · Mathematics 2017-07-14 Mayank Saxena , Onno Boxma , Stella Kapodistria , Rudesindo Núñez Queija

By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions…

Probability · Mathematics 2010-07-12 Samuel N. Cohen , Robert J. Elliott