Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now

Scheduling control for queueing systems with many servers: asymptotic optimality in heavy traffic

A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system's state. We examine two versions of the problem: ``nonpreemptive,'' where service is uninterruptible, and ``preemptive,'' where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to $V$, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value $V$. The construction of these policies is based on a Hamilton--Jacobi--Bellman equation associated with $V$.Comment: Published at http://dx.doi.org/10.1214/105051605000000601 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Greedy walk on the real line

We consider a self-interacting process described in terms of a single-server system with service stations at each point of the real line. The customer arrivals are given by a Poisson point processes on the space-time half plane. The server adopts a greedy routing mechanism, traveling toward the nearest customer, and ignoring new arrivals while in transit. We study the trajectories of the server and show that its asymptotic position diverges logarithmically in time.Comment: Published at http://dx.doi.org/10.1214/13-AOP898 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Earliest-deadline-first service in heavy-traffic acyclic networks

This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process

On the Convergence of Multiclass Queueing Networks in Heavy Traffic

The subject of this paper is the heavy traffic behavior of a general class of queueing networks with first-in-first-out (FIFO) service discipline. For special cases which require various assumptions on the network structure, several authors have proved heavy traffic limit theorems to justify the approximation of queueing networks by reflected Brownian motions (RBM's). Based on these theorems, some have conjectured that the Brownian approximation may in fact be valid for a more general class of queueing networks. In this paper, we prove that the Brownian approximation does not hold for such a general class of networks. Our findings suggest that studying Brownian models of non-FIFO queueing networks may perhaps be more fruitful

The Trouble with Diversity: Fork-Join Networks with Heterogenous Customer Population

Consider a feedforward network of single-server stations populated by multiple job types. Each job requires the completion of a number of tasks whose order of execution is determined by a set of deterministic precedence constraints. The precedence requirements allow some tasks to be done in parallel (in which case tasks would "fork") and require that others be processed sequentially (where tasks may "join"). Jobs of a. given type share the same precedence constraints, interarrival time distributions, and service time distributions, but these characteristics may vary across different job types. We show that the heavy traffic limit of certain processes associated with heterogeneous fork-join networks can be expressed as a semimartingale reflected Brownian motion with polyhedral state space. The polyhedral region typically has many more faces than its dimension, and the description of the state space becomes quite complicated in this setting. One can interpret the proliferation of additional faces in heterogeneous fork-join networks as (i) articulations of the fork and join constraints, and (ii) results of the disordering effects that occur when jobs fork and join in their sojourns through the network