We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of next queue n+1. The question addressed is how steady-state queues scale as N goes to infinity. We show that the answer depends on whether lambda is below or above the critical value 1/4: in the former case queues remain uniformly stochastically bounded, while otherwise they grow to infinity.
The problem is essentially reduced to the behavior of the system with infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.
We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain.
Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering in the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between the Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. Th…
It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, that is, Burke's theorem in this context. In this short note we prove convergence to this invariant measure: if we have an arbitrary continuous process satisfying some mild conditions as initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.
We consider a two-node tandem queueing network in which the upstream queue is GI/GI/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. To investigate the evolution of workload in the second queue, we introduce and study a process M, called the plateau process, which encodes most of the information in the workload process. We focus on the case of infinite-variance service times and show that under appropriate scaling, workload in the first queue converges, and although the workload in the second queue do…
The semimartingale reflecting Brownian motion (SRBM) can be a heavy traffic limit for many server queueing networks. Asymptotic properties for stationary probabilities of the SRBM have attracted a lot of attention recently. However, many results are obtained only for the two-dimensional SRBM. There is only little work related to higher dimensional ($\geq 3$) SRBMs. In this paper, we consider a three dimensional SRBM: A three dimensional Brownian-driven tandem queue with intermediate inputs. We are interested in tail asymptotics for stationary distributions. By generalizing the kernel method an…
In this paper, we study a controllable tandem queueing system consisting of two nodes and a controller, in which customers arrive according to a Poisson process and must receive service at both nodes before leaving the system. A decision maker dynamically allocates the number of service resource to each node facility according to the number of customers in each node. In the model, the objective is to minimize the long-run average costs. We cast these problems as Markov decision problems by dynamic programming approach and derive the monotonicity of the optimal allocation policy and the relatio…
We consider a tandem queue with coupled processors, which is subject to global breakdowns. When the network is in the operating mode and both queues are non empty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the non-empty station. Moreover, arrival rates depend on the state of the network. The system is described by a Markov modulated random walk in the quarter plane representing the number of jobs in the two stations and the state of the network. By applying the generatin…
Tandem queues with finite buffer capacity commonly exist in practical applications. By viewing a tandem queue as an integrated system, an innovative approach has been developed to analyze its performance through the insight from reduction method. In our approach, the starvation at the bottleneck caused by service time randomness is modeled and captured by interruptions. Fundamental properties of tandem queues with finite buffer capacity are examined. We show that in general system service rate of a dual tandem queue with finite buffer capacity is equal or smaller than its bottleneck service ra…
Tandem queueing systems are widely-used stochastic models that arise from many real-life service operations systems. Motivated by the desire to understand the trade-off between the performance and complexity of policies for capacity-constrained tandem queueing systems, we investigate the long-run expected time-average revenue, the gain, of the service provider for various pricing policies. The gain-maximization problem is formulated as a Markov decision process model but the optimal policy, which dynamically adjusts service prices, is hard to obtain due to the curse of dimensionality. For gene…
We consider an infinite-server queue into which customers arrive according to a Cox process and have independent service times with a general distribution. We prove a functional large deviations principle for the equilibrium queue length process. The model is motivated by a linear feed-forward gene regulatory network, in which the rate of protein synthesis is modulated by the number of RNA molecules present in a cell. The system can be modelled as a tandem of infinite-server queues, in which the number of customers present in a queue modulates the arrival rate into the next queue in the tandem…
In a two node tandem network, customers decide to join or balk by maximizing a given profit function whose costs are proportional to the sojourn time they spend at each queue. Assuming that their choices are taken without knowing the complete state of the system, we show that a pure threshold equilibrium policy exists. In particular we analyze the case when the partial information consists in informing the arrival customers of the total number of users in the network.
Queueing networks are notoriously difficult to analyze sans both Markovian and stationarity assumptions. Much of the theoretical contribution towards performance analysis of time-inhomogeneous single class queueing networks has focused on Markovian networks, with the recent exception of work in Liu and Whitt (2011) and Mandelbaum and Ramanan (2010). In this paper, we introduce transitory queueing networks as a model of inhomogeneous queueing networks, where a large, but finite, number of jobs arrive at queues in the network over a fixed time horizon. The queues offer FIFO service, and we assum…
We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for th…
In this paper we study the stationary workload distribution of a fluid tandem queue in heavy traffic. We consider different types of Lévy input, covering compound Poisson, $α$-stable Lévy motion (with $1<α<2$), and Brownian motion. In our analysis we separately deal with Lévy input processes with increments that have finite and infinite variance. A distinguishing feature of this paper is that we do not only consider the usual heavy-traffic regime, in which the load at one of the nodes goes to unity, but also a regime in which we simultaneously let the load of both servers tend to one, which, a…
Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ taking the steps $(1,0)$, $(-1,1)$ and $(0,-1)$ with probabilities $λ< (μ_1\neq μ_2)$; in particular, $X$ is assumed stable. Let $τ_n$ be the first time $X$ hits $\partial A_n = \{x:x(1)+x(2) = n \}$ For $x \in {\mathbb Z}_+^2, x(1) + x(2) < n$, the probability $p_n(x)= P_x( τ_n < τ_0)$ is a key performance measure for the queueing system represented by $X$. Let $Y$ be the constrained random walk on ${\mathbb Z} \times {\mathbb Z}_+$ with increments $(-1,0)$, $(1,1)$ and $(0,-1)$. Let $τ$ be the first time that the components of $Y$ e…
We study a queueing network with a single shared server, that serves the queues in a cyclic order according to the gated service discipline. 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 and tandem queues. We derive exact limits of the mean delays under both …
The paper analyzes the performance of tandem network of polling queue with setups. For a system with two-products and two-stations, we propose a new approach based on a partially-collapsible state-space characterization to reduce state-space complexity. In this approach, the size of the state-space is varied depending on the information needed to determine buffer levels and waiting times. We evaluate system performance under different system setting and comment on the numerical accuracy of the approach as well as provide managerial insights. Numerical results show that approach yields reliable…
In this paper we provide convergence analysis for a class of Brownian queues in tandem by establishing an exponential drift condition. A consequence is the uniform exponential ergodicity for these multidimensional diffusions, including the O'Connell-Yor process. A list of open problems are also presented.
This paper studies an infinite-server queue in a random environment, meaning that the arrival rate, the service requirements and the server work rate are modulated by a general càdlàg stochastic background process. To prove a large deviations principle, the concept of attainable parameters is introduced. Scaling both the arrival rates and the background process, a large deviations principle for the number of jobs in the system is derived using attainable parameters. Finally, some known results about Markov-modulated infinite-server queues are generalized and new results for several background …
We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.
This paper focuses on an infinite-server queue modulated by an independently evolving finite-state Markovian background process, with transition rate matrix $Q\equiv(q_{ij})_{i,j=1}^d$. Both arrival rates and service rates are depending on the state of the background process. The main contribution concerns the derivation of central limit theorems for the number of customers in the system at time $t\ge 0$, in the asymptotic regime in which the arrival rates $λ_i$ are scaled by a factor $N$, and the transition rates $q_{ij}$ by a factor $N^α$, with $α\in \mathbb R^+$. The specific value of $α$ h…
We provide the first algorithm that under minimal assumptions allows to simulate the stationary waiting-time sequence of a single-server queue backwards in time, jointly with the input processes of the queue (inter-arrival and service times). The single-server queue is useful in applications of DCFTP (Dominated Coupling From The Past), which is a well known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time assuming only finite mean of the inter-arrival and service times. In order to simulate the single-server queue in stationarity unt…
We study in this paper a variation of the acclaimed CSMA random access protocol. We will focus on the case where back-off rates at each node is polynomial in the size of the queue. Under a condition relating the exponent in the polynomial rates and the geometry of the interference graph, we prove convergence of the scaled process to a deterministic fluid limit up to the time a queue reaches $0$ on the fluid scale . We outline the difficulties arising at that time and solve them in the case of a complete interference graph. This paper relies on a new method to obtain a fully coupled stochastic …
Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ with increments $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$; $X$ represents, at arrivals and service completions, the lengths of two queues working in parallel whose service and interarrival times are exponentially distributed with arrival rates $λ_i$ and service rates $μ_i$, $i=1,2$; we assume $λ_i < μ_i$, $i=1,2$, i.e., $X$ is assumed stable. Without loss of generality we assume $ρ_1 =λ_1/μ_1 \ge ρ_2 = λ_2/μ_2$. Let $τ_n$ be the first time $X$ hits the line $\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$. Let $Y$ be the same ran…
In this paper we describe a perfect simulation algorithm for the stable $M/G/c$ queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how to build a dominated CFTP algorithm for perfect simulation of the super-stable $M/G/c$ queue operating under First Come First Served discipline, with dominating process provided by the corresponding $M/G/1$ queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service), and exploiting the fact that …
What determines the average length of a queue which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations induced delays are in high demand as queue structures appear in various natural and man-made systems. Here we demonstrate that a simple service resetting mecha…
