This paper considers Gaussian flows multiplexed in a queueing network. A single node being a useful but often incomplete setting, we examine more advanced models. We focus on a (two-node) tandem queue, fed by a large number of Gaussian inputs. With service rates and buffer sizes at both nodes scaled appropriately, Schilder's sample-path large-deviations theorem can be applied to calculate the asymptotics of the overflow probability of the second queue. More specifically, we derive a lower bound on the exponential decay rate of this overflow probability and present an explicit condition for the lower bound to match the exact decay rate. Examples show that this condition holds for a broad range of frequently used Gaussian inputs. The last part of the paper concentrates on a model for a single node, equipped with a priority scheduling policy. We show that the analysis of the tandem queue directly carries over to this priority queueing system.
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…
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…
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 consider a discrete-time preemptive priority queue with different service rates for two classes of customers, one with high-priority and the other with low-priority. This model corresponds to the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server. Due to the possibility of customers' arriving and departing at the same time in a discrete-time queue, the model considered in this paper is more complicated. In this model, we focus on the characterization of exact tail asymptotics for the joint stationary…
We study the asymptotics of the stationary sojourn time Z of a "typical customer" in a tandem of single-server queues. It is shown that, in a certain "intermediate" region of light-tailed service time distributions, Z may take a large value mostly due to a large value of a single service time of one of customers. Arguments used in the paper allow us to obtain also an elementary proof of the logarithmic asymptotics for the tail distribution of the stationary sojourn time in the whole class of light-tailed distributions.
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…
This paper presents a new method for deriving the waiting time distribution of the lowest class in an accumulating priority (AP) queue with positive Lévy input. The priority of a non-atomic customer (particle) is a function of their class and waiting time in the system, and the particles with the highest AP are the next to be processed. The method relies on the construction of a workload overtaking process and solving a first-passage time problem using an appropriate stopping time.
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…
We calculate asymptotics of the distribution of the number of customers in orbit in a two-class priority retrial $M/G/1$-type queueing model. In this model, priority customers wait in line while non-priority customers join an orbit and retry later. Although the generating function and moments of the number of customers in orbit has been analyzed before, asymptotics of the distribution have not been thoroughly investigated. We use singularity analysis of the probability generating function to do just that. Our results show that different regimes exist for these asymptotics in case of light-tail…
In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system consisting of three M/M/1 queues with threshold policy. It turns out that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues. In addition, the queue lengths of stable queues possess the same distributions as a priority polling system with N-policy vacation. Based on this fact, we provide the exact tail asymptotics of the vacation polling system to approximate the tail distribution of the queue le…
This paper considers a multichannel preemptive-resume priority queueing system with a Poisson input and an arbitrary service time distribution depending on the priority of job. Jobs of the same priority are serviced according to the LIFO rule. If, at moment of job arrival, all servers are busy, and at least one server is busy with the service of a job of a not higher than the priority of arriving job, then the service of a job is preempted such that the priority of preempted job is lowest from the priorities of the jobs in service. The service of a preempted job is resumed later. The paper pro…
In this paper, we deal with a two-queue polling system attended by a single server. The server visits the queues according to a Markovian routing mechnism. There are two-class customers in the first queue. Customers of each queue are served in the exhaustive discipline. For this model, we study the joint number of customers, the cycle time and the waiting times. we are also devoted to deriving the exact asymptotics for the scaled delay in the heavy-traffic scenario. In addition, the scaled delays with large switch-over times are discussed for the first time in the Markovian polling systems. Fi…
The problem of exact evaluation of the mean service cycle time in tandem systems of single-server queues with both infinite and finite buffers is considered. It is assumed that the interarrival and service times of customers form sequences of independent and identically distributed random variables with known mean values. We start with tandem queues with infinite buffers, and show that under the above assumptions, the mean cycle time exists. Furthermore, if the random variables which represent interarrival and service times have finite variance, the mean cycle time can be calculated as the max…
Stochastic networks with complex structures are key modelling tools for many important applications. In this paper, we consider a specific type of network: the retrial queueing systems with priority. This type of queueing system is important in various applications, including telecommunication and computer management networks with big data. For this type of system, we propose a detailed stochastic decomposition approach to study its asymptotic behaviour of the tail probability of the number of customers in the steady-state for retrial queues with two types (Type-1 and Type-2) of customers, in …
Motivated by applications in computing and telecommunication systems, we investigate the problem of estimating p-quantile of steady-state sojourn times in a single-server multi-class queueing system with non-preemptive priorities for p close to 1. The main challenge in this problem lies in efficient sampling from the tail event. To address this issue, we develop a regenerative simulation algorithm with importance sampling. In addition, we establish a central limit theorem for the estimator to construct the confidence interval. Numerical experiments show that our algorithm outperforms benchmark…
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…
The tandem fluid queueing model is a useful tool for performance analysis and control design for a variety of transportation systems. In this article, we study the joint impact of stochastic capacity and spillback on the long-time properties of this model. Our analysis focuses on the system of two fluid queueing links in series. The upstream link has a constant capacity (saturation rate) and an infinite buffer size. The downstream link has a stochastic capacity and a finite buffer size. Queue spillback occurs when the the downstream link is full. We derive a necessary condition and a sufficien…
The performance of non-preemptive M/M/1 queueing system with two priority is analyzed. By using complementary variable method to make vector Markov process and analyzing the state-change equations of the queueing system, the generating function of two kinds of customers'length distribution are derived under non-preemptive priority .Through further discussion, the probability of the server that it is working or free and average length of two kinds of customers are also derived.
We provide the first analytical expressions for the expected waiting time of high-priority customers in the delayed APQ by exploiting a classical conservation law for work-conserving queues. Additionally, we describe an algorithm to compute the expected waiting times of both low-priority and high-priority customers, which requires only the truncation of sums that converge quickly in our experiments. These insights are used to demonstrate how the accumulation rate and delay level should be chosen by health care practitioners to optimize common key performance indicators (KPIs). In particular, w…
In this paper we consider a single-server cyclic polling system consisting of two queues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low priority customers. For this situation the following service disciplines are considered: gated, globally gated, and exhaustive. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution at polling epochs, and the steady-state marginal queue length distributions for each customer type.
We consider a Markovian many server queueing system in which customers are preemptively scheduled according to exogenously assigned priority levels. The priority levels are randomly assigned from a continuous probability measure rather than a discrete one and hence, the queue is modeled by an infinite dimensional stochastic process. We analyze the equilibrium behavior of the system and provide several results. We derive the Radon-Nikodym derivative (with respect to Lebesgue measure) of the measure that describes the average distribution of customer priority levels in the system; we provide a f…
The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the $M/SM/1$ queue, has been well-studied since its introduction by Neuts in 1966. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic. In particular, we use it in the context of gap acceptance models where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority. Traditionally, gap acceptance…
We consider a Markovian single server queue in which customers are preemptively scheduled by exogenously assigned priority levels. The novelty in our model is that the priority levels are randomly assigned from a continuous probability measure rather than a discrete one. Because the priority levels are drawn from a continuum, the queue is modeled by a measure-valued stochastic process. We analyze the steady state behavior of this process and provide several results. We derive a measure that describes the average distribution of customer priority levels in the system; we provide a formula for t…
Quasi-birth-and-death (QBD) processes with infinite ``phase spaces'' can exhibit unusual and interesting behavior. One of the simplest examples of such a process is the two-node tandem Jackson network, with the ``phase'' giving the state of the first queue and the ``level'' giving the state of the second queue. In this paper, we undertake an extensive analysis of the properties of this QBD. In particular, we investigate the spectral properties of Neuts's R-matrix and show that the decay rate of the stationary distribution of the ``level'' process is not always equal to the convergence norm of …
We analyze the latency or sojourn time L(m,n) for the last customer in a batch of n customers to exit from the m-th queue in a tandem of m queues in the setting where the queues are in equilibrium before the batch of customers arrives at the first queue. We first characterize the distribution of L(m,n) exactly for every m and n, under the assumption that the queues have unlimited buffers and that each server has customer independent, exponentially distributed service times with an arbitrary, known rate. We then evaluate the first two leading order terms of the distributions in the large m and …
In this paper we consider a single-server cyclic polling system. Between visits to successive queues, the server is delayed by a random switch-over time. The order in which customers are served in each queue is determined by a priority level that is assigned to each customer at his arrival. For this situation the following service disciplines are considered: gated, exhaustive, and globally gated. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state margin…
In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service …
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.
Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a two-node tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we…
The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element $ζ$ of a certain class of càdlàg paths that take values in the space of signed measures on the half-line to a càdlàg path that takes values in the space of non-negative measures on $[0,\infty)$ in such a way that for each $x > 0$, the path $t \mapsto ζ_t[0,x]$ is transformed via a Skorokhod map on the half-line, and the regulating functions for different $x > 0$ are coupled. We establ…
One of the basic properties of a queueing network is stability. Roughly speaking, it is the property that the total number of jobs in the network remains bounded as a function of time. One of the key questions related to the stability issue is how to determine the exact conditions under which a given queueing network operating under a given scheduling policy remains stable. While there was much initial progress in addressing this question, most of the results obtained were partial at best and so the complete characterization of stable queueing networks is still lacking. In this paper, we resol…
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 priority queueing system where a single processor serves k classes of packets that are generated randomly following Poisson processes. Our objective is to compute the expected Peak Age of Information (PAoI) under various scenarios. In particular, we consider two situations where the buffer size at each queue is one and infinite, and in the infinite buffer size case we consider First Come First Serve (FCFS) and Last Come First Serve (LCFS) as service disciplines. For the system with buffer size one at each queue, we derive PAoI exactly for the case of exponential service time and …
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…
The presented study elaborates a multi-server priority queueing model considering the pre-emptive repeat policy and phase-type distribution (PH) for retrial process. The incoming heterogeneous calls are categorized as handoff calls and new calls. The arrival and service processes of both types of calls follow marked Markoian arrival process (MMAP) and PH distribution with distinct parameters, re-spectively. An arriving new call will be blocked when all the channels are occupied, and consequently will join the orbit (virtual space) of infinite capacity to retry following PH distribution. When a…
The presented study elaborates a multi-server catastrophic retrial queueing model considering preemptive repeat priority policy with phase-type (PH) distributed retrial times. For the sake of comprehension, the scenario of model operation prior and later to the occurrence of the disaster is referred to as the normal scenario and as the catastrophic scenario, respectively. In the normal scenario, the incoming heterogeneous calls are categorized as handoff calls and new calls. Handoff calls are provided controllable preemptive priority over new calls. In the catastrophic scenario, when a disaste…
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…
We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT-results to compare SRPT with FIFO from a large-deviations point of view.
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…
In this paper, we present counter-intuitive examples for the multiclass queueing network system. In the system, each station may serve more than one job class with differentiated service priority, and each job may require service sequentially by more than one service station. In our examples, the network performance is improved even when more workloads are admitted for service.
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…
