We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is routed to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We prove that stability in this model is achieved if and only if there exists

The idea behind the recently introduced “age-of-information” performance measure of a network message processing system is that it indicates our knowledge regarding the “freshness” of the most recent piece of information that can be used as a criterion for real-time control. In this foundational paper, we examine two such measures, one that has been extensively studied in the recent literature and

Donsker’s theorem is perhaps the most famous invariance principle result for Markov processes. It states that, when properly normalized, a random walk behaves asymptotically like a Brownian motion. This approach can be extended to general Markov processes whose driving parameters are taken to a limit, which can lead to insightful results in contexts like large distributed systems or queueing networks

We consider a model for transitory queues in which only a finite number of customers can join. The queue thus operates over a finite time horizon. In this system, also known as the \(\Delta _{(i)}/G/1\) queue, the customers decide independently when to join the queue by sampling their arrival time from a common distribution. We prove that, when the queue satisfies a certain heavy-traffic condition

We study a model of wireless networks where users move at speed \(\theta \ge 0\), which has the original feature of being defined through a fixed-point equation. Namely, we start from a two-class processor-sharing queue to model one representative cell of this network: class 1 users are patient (non-moving) and class 2 users are impatient (moving). This model has five parameters, and we study the case

This paper considers a multi-type fluid queue with priority service. The input fluid rates are modulated by a Markov chain, which is common for all fluid types. The service rate of the queue is constant. Various performance measures are derived, including the Laplace–Stieltjes transform and the moments of the stationary waiting time of the fluid drops and the queue length distributions. An Erlangization-based

It has long been conjectured that the tight upper bound for the mean steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions is attained asymptotically by two-point distributions. The two-point distribution for the interarrival time has one mass point at 0, but the service-time distribution involves a limit; there is one mass

We investigate the stability condition of a multiserver queueing system. Each customer needs simultaneously a random number of servers to complete the service. The times taken by each server are independent. The input flow is assumed to be a regenerative one. The service time has an exponential, phase-type or hyper-exponential distribution. The stability criteria for the models are established. It

We study a Markovian single-server ticket queue where, upon arrival, each customer can draw a number from a take-a-number machine, while the number of the customer currently being served is displayed on a panel. The difference between the above two numbers is called the “virtual queue length.” We consider a nonhomogeneous population of customers comprised of two types: “regular” and “strategic.” Upon

We consider a general k-dimensional discounted infinite server queueing process (alternatively, an incurred but not reported claim process) where the multivariate inputs (claims) are given by a k-dimensional finite-state Markov chain and the arrivals follow a renewal process. After deriving a multidimensional integral equation for the moment-generating function jointly to the state of the input at

We analyze the time-dependent behavior of various types of infinite-server queueing systems, where, within each system we consider, jobs interact with one another in ways that induce batch departures from the system. One example of such a queue was introduced in the recent paper of Frolkova and Mandjes (Stochastic Models, 2019) in order to model a type of one-sided communication between two users in

A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depend on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump diffusion. In both cases, the joint dynamics are constructed so

This paper studies stationary customer flows in an open queueing network. The flows are the processes counting customers flowing from one queue to another or out of the network. We establish the existence of unique stationary flows in generalized Jackson networks and convergence to the stationary flows as time increases. We establish heavy-traffic limits for the stationary flows, allowing an arbitrary

Men and jobs alike are characterized by a single trait, which may take on categorical values according to given population frequencies. Men arrive to the system following a Poisson process and wait till jobs are assigned to them. Jobs arrive to the system following another, independent, Poisson process. An arriving job must be assigned to a waiting man immediately, or be discarded, ensuing no gain

A result of Ward and Glynn (Queueing Syst 50(4):371–400, 2005) asserts that the sequence of scaled offered waiting time processes of the \(GI/GI/1+GI\) queue converges weakly to a reflected Ornstein–Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate

Polling systems are systems consisting of multiple queues served by a single server. In this paper, we analyze polling systems with a server that is self-ruling, i.e., the server can decide to leave a queue, independent of the queue length and the number of served customers, or stay longer at a queue even if there is no customer waiting in the queue. The server decides during a service whether this

Motivated by applications in biological systems, we show for certain multiclass queueing networks that time-dependent distributions for the multiclass queue-lengths can have a factorized form which reduces the problem of computing such distributions to a similar problem for related single-class queueing networks. We give an example of the application of this result to an enzymatic processing network

Arrival processes to service systems often display fluctuations that are larger than anticipated under the Poisson assumption, a phenomenon that is referred to as overdispersion. Motivated by this, we analyze a class of discrete-time stochastic models for which we derive heavy-traffic approximations that are scalable in the system size. Subsequently, we show how this leads to novel capacity sizing

The aim of this work concerns the performance analysis of systems with interacting queues under the join the shortest queue policy. The case of two interacting queues results in a two-dimensional random walk with bounded transitions to non-neighboring states, which in turn results in complicated boundary behavior. Although this system violates the conditions of the compensation approach due to the

We consider a multi-class multi-server retrial queueing system. Customers of each class arrive from outside the system according to a Poisson process. The service times for each class of customers are assumed to be exponentially distributed with service rates depending on both the customers’ class and the servers. We define the offered load and provide stability and instability conditions for this

We consider a decentralised multi-access algorithm, motivated primarily by the control of transmissions in a wireless network. For a finite single-hop network with arbitrary interference constraints, we prove stochastic stability under the natural conditions. For infinite and finite single-hop networks, we obtain broad rate-stability conditions. We also consider symmetric finite multi-hop networks

We analyse the evolution of a system designed for reservation of some items in advance (for example, seats in aircrafts or trains or bus) by customers arriving at random moments. The reservation has to be done by the server in one of the K time frames. At the beginning of the pth time frame, the inventoried items in it (as well as those sold from it earlier) have life time distribution which is a p-fold

A mean-field extension of the queueing system (GI/GI/1) is considered. The process is constructed as a Markov solution of a martingale problem. Uniqueness in distribution is also established under a slightly different set of assumptions on intensities in comparison with those required for existence.

Cloud computing task management has a critical role in the efficient operation of the cloud resources, i.e., the servers. The task management handles critical and complicated decisions, overcoming the inherent dynamic nature of cloud computing systems and the additional complexity due to the large magnitude of resources in such systems (tens of thousands of servers). Due to the fact that servers may

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably

Motivated by service systems, such as telephone call centers and emergency departments, we consider admission control for a two-class multi-server loss system with periodically varying parameters and customers who may abandon from service. Assuming mild conditions for the parameters, a dynamic programming formulation is developed. We show that under the infinite horizon discounted problem, there exists