A stochastic network with mobile users in heavy traffic

We consider a stochastic network with mobile users in a heavy traffic regime. We derive the scaling limit of the multidimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos (preprint, 2012), which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times.     

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded \(G{/}GI{/}n+GI\) queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.     

Heavy-Traffic Limits for Loss Proportions in Single-Server Queues

We establish heavy-traffic stochastic-process limits for the queue-length and overflow stochastic processes in the standard single-server queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related single-server models with finite waiting room, such as the finite dam, satisfy the same heavy-traffic stochastic-process limits. As a consequence, we obtain heavy-traffic limits for the proportion of customers or input lost over an initial interval. Except for an interchange of the order of two limits, we thus obtain heavy-traffic limits for the steady-state loss proportions. We justify the interchange of limits in M/GI/1/K and GI/M/1/K special cases of the standard GI/GI/1/K model by directly establishing local heavy-traffic limits for the steady-state blocking probabilities.     

An overview of Brownian and non-Brownian FCLTs for the single-server queue

We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated. This revised version was published online in June 2006 with corrections to the Cover Date.     

A heavy-traffic analysis of a closed queueing system with a GI/∞ service center

This paper studies the heavy-traffic behavior of a closed system consisting of two service stations. The first station is an infinite server and the second is a single server whose service rate depends on the size of the queue at the station. We consider the regime when both the number of customers, n, and the service rate at the single-server station go to infinity while the service rate at the infinite-server station is held fixed. We show that, as n→∞, the process of the number of customers at the infinite-server station normalized by n converges in probability to a deterministic function satisfying a Volterra integral equation. The deviations of the normalized queue from its deterministic limit multiplied by √n converge in distribution to the solution of a stochastic Volterra equation. The proof uses a new approach to studying infinite-server queues in heavy traffic whose main novelty is to express the number of customers at the infinite server as a time-space integral with respect to a time-changed sequential empirical process. This gives a new insight into the structure of the limit processes and makes the end results easy to interpret. Also the approach allows us to give a version of the classical heavy-traffic limit theorem for the G/GI/∞ queue which, in particular, reconciles the limits obtained earlier by Iglehart and Borovkov. This revised version was published online in June 2006 with corrections to the Cover Date.     

Diffusion approximations for double-ended queues with reneging in heavy traffic

We study a double-ended queue consisting of two classes of customers. Whenever there is a pair of customers from both classes, they are matched and leave the system. The matching is instantaneous following the first-come–first-match principle. If a customer cannot be matched immediately, he/she will stay in a queue. We also assume customers are impatient with generally distributed patience times. Under suitable heavy traffic conditions, we establish simple linear asymptotic relationships between the diffusion-scaled queue length process and the diffusion-scaled offered waiting time processes and show that the diffusion-scaled queue length process converges weakly to a diffusion process that admits a unique stationary distribution.

     

State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic

Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56–90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45–56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs).     

On the fluid limit of the M/G/∞ queue

The paper deals with the fluid limits of some generalized M/G/∞ queues under heavy-traffic scaling. The target application is the modeling of Internet traffic at the flow level. Our paper first gives a simplified approach in the case of Poisson arrivals. Expressing the state process as a functional of some Poisson point process, an elementary proof for limit theorems based on martingales techniques and weak convergence results is given. The paper illustrates in the special Poisson arrivals case the classical heavy-traffic limit theorems for the G/G/∞ queue developed earlier by Borovkov and by Iglehart, and later by Krichagina and Puhalskii. In addition, asymptotics for the covariance of the limit Gaussian processes are obtained for some classes of service time distributions, which are useful to derive in practice the key parameters of these distributions.     

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has a clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes processes have wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.     

Comparison conjectures about the M/G/s queue

We conjecture that the equilibrium waiting-time distribution in an M/G/s queue increases stochastically when the service-time distribution becomes more variable. We discuss evidence in support of this conjecture and others based partly on light-traffic and heavy-traffic limits. We also establish an insensitivity property for the case of many servers in light traffic.     

Asymptotically tight steady-state queue length bounds implied by drift conditions

The Foster–Lyapunov theorem and its variants serve as the primary tools for studying the stability of queueing systems. In addition, it is well known that setting the drift of the Lyapunov function equal to zero in steady state provides bounds on the expected queue lengths. However, such bounds are often very loose due to the fact that they fail to capture resource pooling effects. The main contribution of this paper is to show that the approach of “setting the drift of a Lyapunov function equal to zero” can be used to obtain bounds on the steady-state queue lengths which are tight in the heavy-traffic limit. The key is to establish an appropriate notion of state-space collapse in terms of steady-state moments of weighted queue length differences and use this state-space collapse result when setting the Lyapunov drift equal to zero. As an application of the methodology, we prove the steady-state equivalent of the heavy-traffic optimality result of Stolyar for wireless networks operating under the MaxWeight scheduling policy.     

Infinite-server systems with Coxian arrivals

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.

     

A network of priority queues in heavy traffic: One bottleneck station

In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them. Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.     

Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit

We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ℝ₊ ⁿ with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes. F.J. Piera’s research supported in part by CONICYT, Chile, FONDECYT Project 1070797. R.R. Mazumdar’s research supported in part by NSF, USA, Grant 0087404 through Networking Research Program, and a Discovery Grant from NSERC, Canada.     

Loss bounds for a finite-capacity queue based on interval-wise traffic observation

We study a single-server finite-capacity queue with batch fluid inputs. We assume that the input traffic is observed in an interval-wise basis; that is, along the time axis divided into fixed-length intervals, with the amount of work brought into the queue during each interval sequentially observed. Since the sequence of the amounts of work during respective intervals, which is referred to as the interval-wise input process in this paper, does not reveal complete information about an input process, we cannot be precisely aware of the performance of the queue. Thus, in this paper, we focus on knowing the performance limit of the queue based on the interval-wise input process. In particular, we establish an upper limit of the long-run loss ratio of the workload in the queue. Our results would reveal useful tools to evaluate the performance of queuing systems when information about input processes in a fine timescale is not available.     

Heavy-traffic asymptotics for the single-server queue with random order of service

We consider the GI/GI/1 queue with customers served in random order, and derive the heavy-traffic limit of the waiting-time distribution. Our proof is probabilistic, requires no finite-variance assumptions, and makes the intuition provided by Kingman (Math. Oper. Res. 7 (1982) 262) rigorous.     

Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

We consider \(GI/Ph/n+M\) parallel-server systems with a renewal arrival process, a phase-type service time distribution, \(n\) homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin–Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy-traffic and steady-state limits for \(GI/Ph/n+M\) queues.     

Queues with random back-offs

We consider a broad class of queueing models with random state-dependent vacation periods, which arise in the analysis of queue-based back-off algorithms in wireless random-access networks. In contrast to conventional models, the vacation periods may be initiated after each service completion, and can be randomly terminated with certain probabilities that depend on the queue length. We first present exact queue-length and delay results for some specific cases and we derive stochastic bounds for a much richer set of scenarios. Using these, together with stochastic relations between systems with different vacation disciplines, we examine the scaled queue length and delay in a heavy-traffic regime, and demonstrate a sharp trichotomy, depending on how the activation rate and vacation probability behave as function of the queue length. In particular, the effect of the vacation periods may either (i) completely vanish in heavy-traffic conditions, (ii) contribute an additional term to the queue lengths and delays of similar magnitude, or even (iii) give rise to an order-of-magnitude increase. The heavy-traffic trichotomy provides valuable insight into the impact of the back-off algorithms on the delay performance in wireless random-access networks.