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.

     

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.     

Heavy-traffic analysis of the sojourn time in a three node Jackson network with overtaking

We consider a Jackson network consisting of three first-in-first-out (FIFO)M/M/1 queues. When customers leave the first queue they can be routed to either the second or third queue. Thus, a customer that traverses the network by going from the first to the second to the third queue, can be overtaken by another customer that is routed from the first queue directly to the third. We study the distribution of the sojourn time of a customer through the three node network, in the heavy traffic limit. A three term heavy traffic asymptotic approximation to the sojourn time density is derived. The leading term shows that the nodes decouple in the heavy traffic limit. The next two terms, however, do show the dependence of the sojourn times at the individual nodes and give quantitative measures of the effects of overtaking.     

Fractional Brownian Heavy Traffic Approximations of Multiclass Feedforward Queueing Networks

We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.AMS subject classification: primary 90B15, secondary 60K25, 68M20     

Steady state approximations of limited processor sharing queues in heavy traffic

We investigate steady state properties of limited processor sharing queues in heavy traffic. Our analysis builds on previously obtained process limit theorems, and requires the interchange of steady state and heavy traffic limits, which are established by a coupling argument. The limit theorems yield explicit approximations of the steady state queue length and response time distribution in heavy traffic, of which the quality is supported by simulation experiments.     

Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A fork-join queueing model: Diffusion approximation,integral representations and asymptotics

We consider two parallel M/M/1 queues which are fed by a single Poisson arrival stream. An arrival splits into two parts, with each part joining a different queue. This is the simplest example of a fork-join model. After the individual parts receive service, they may be joined back together, though we do not consider the join part here. We study this model in the heavy traffic limit, where the service rate in either queue is only slightly larger than the arrival rate. In this limit we obtain asymptotically the joint steady-state queue length distribution. In the symmetric case, where the two servers are identical, this distribution has a very simple form. In the non-symmetric case we derive several integral representations for the distribution. We then evaluate these integrals asymptotically, which leads to simple formulas which show the basic qualitative structure of the joint distribution function.     

Asymptotic Analysis of Traffic Lights Performance Under Heavy Traffic Assumption

The main drawback of Markov models for traffic lights performance considered in our previous investigations is exponential distribution of intervals between lights switchings. To analyze the impact of this assumption we introduce a model with arbitrary distribution of interswitching intervals. An algorithm is proposed to calculate imbedded Markov chain stationary probabilities and mean length of a queue at crossroads. Although the difference between two models (exponentially distributed and constant intervals) is slight for traffic intensity ρ?≈?0.5, it is significant for ρ close to 1. We investigate the queue length behaviour as ρ → 1. Weak convergence of normalized characteristics (waiting time, queue length etc.) to exponential ones can be established under heavy traffic assumption. To prove one uses the asymptotic equivalence of these characteristics to supremum of a random walk with zero reflecting boundary.     

Improved bounds for queues with delayed arrivals

Improved bounds are developed for a queue where arrivals are delayed by a fixed time. For moderate to heavy traffic, a simple improved upper bound is obtained which only uses the first two moments of the service time distribution. We show that our approach can be extended to obtain bounds for other types of delayed arrival queues. For very light traffic, asymptotically tight bounds can be obtained using more information about the service time distribution. While an improved upper bound can be obtained for light to moderate traffic it is not particularly easy to apply. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the stability of open networks: A unified approach by stochastic dominance

Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrivai times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the pth moment of the queue length process is bounded for allt if the p+lth moment of the service times at all queues are finite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for allt. When the interarrivai times are unbounded and non-lattice (resp. spreadout), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied.     

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.     

基于Stackelberg博弈的时变双层交通分配模型

提出一个时变双层交通分配模型,其中上层网络管理者设立了一个路段的最大排队长度,其目标是使由网络流和排队长度定义的总出行时间最小.目标函数在离散时段内以路段流量和排队长度作为决策变量,同时考虑不同类型的信号交叉口延误的影响.下层网络用户的反应依赖于上层管理者的决策,其选择是使自身感知阻抗最小的路径,服从一个基于成对组合Logit的路径选择模型,构成一个成对组合Logit的均衡分配问题.结合了交通分配和流传播方法,将其表示为一个均衡约束下的双层数学规划问题,形成了一个Stackelberg非合作博弈.使用遗传算法求解该双层规划问题,并采用实证分析来表现模型的特征和算法的计算表现.结果表明路径重叠、路段流量、路段排队长度等因素对网络均衡流分布均有显著影响.     

Impact of Bursty Traffic on Queues

The impact of bursty traffic on queues is investigated in this paper. We consider a discrete-time single server queue with an infinite storage room, that releases customers at the constant rate of c customers/slot. The queue is fed by an M/G/∞ process. The M/G/∞ process can be seen as a process resulting from the superposition of infinitely many ‘sessions’: sessions become active according to a Poisson process; a station stays active for a random time, with probability distribution G, after which it becomes inactive. The number of customers entering the queue in the time-interval [t, t + 1) is then defined as the number of active sessions at time t (t = 0,1, ...) or, equivalently, as the number of busy servers at time t in an M/G/∞ queue, thereby explaining the terminology. The M/G/∞ process enjoys several attractive features: First, it can display various forms of dependencies, the extent of which being governed by the service time distribution G. The heavier the tail of G, the more bursty the M/G/∞ process. Second, this process arises naturally in teletraffic as the limiting case for the aggregation of on/off sources [27]. Third, it has been shown to be a good model for various types of network traffic, including telnet/ftp connections [37] and variable-bit-rate (VBR) video traffic [24]. Last but not least, it is amenable to queueing analysis due to its very strong structural properties. In this paper, we compute an asymptotic lower bound for the tail distribution of the queue length. This bound suggests that the queueing delays will dramatically increase as the burstiness of the M/G/∞ input process increases. More specifically, if the tail of G is heavy, implying a bursty input process, then the tail of the queue length will also be heavy. This result is in sharp contrast with the exponential decay rate of the tail distribution of the queue length in presence of ‘non-bursty’ traffic (e.g. Poisson-like traffic). This revised version was published online in August 2006 with corrections to the Cover Date.     

Exact waiting time and queue size distributions for equilibrium <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 queues with Pareto service

This paper solves the problem of finding exact formulas for the waiting time cdf and queue length distribution of first-in-first-out M/G/1 queues in equilibrium with Pareto service. The formulas derived are new and are obtained by directly inverting the relevant Pollaczek-Khinchin formula and involve single integrals of non-oscillating real valued functions along the positive real line. Tables of waiting time and queue length probabilities are provided for certain parameter values under heavy traffic conditions.      

Optimal scheduling and power allocation in wireless networks with heavy traffic

This paper addresses the problem of joint transmit power allocation and time slot scheduling in a wireless communication system with time varying traffic. The system is handled by a single base station transmitting over time varying channels. This may be the case in practice of a hybrid TDMA-CDMA (Time Division Multiple Access-Code Division Multiple Access) system. The operating time horizon is divided into time slots; a fixed amount of power is available at each time slot. The users share each time slot and the power available at this time slot with the objective of minimizing the expected total queue length. The problem is reformulated, via a heavy traffic approximation, as the optimal control of a reflected diffusion in the positive orthant. We establish a closed form solution for the obtained control problem. The main feature that makes it possible is an astute choice of some auxiliary weighting matrices in the cost rate. The proposed solution relies also on the knowledge of the covariance matrix of the non-standard multi-dimensional Wiener process which is the driving process in the reflected diffusion. We then compute this covariance matrix given the stationary distribution of the multi-dimensional channel process. Further stochastic analysis is provided: the cost variance, and the Fokker–Planck equation for the distribution density of the queue length.     

On the Remaining Service Time upon Reaching a Given Level in M/G/1 Queues

The ability of effectively finding the distribution of the remaining service time upon reaching a target level in M/G/1 queueing systems is of great practical importance. Among other things, it is necessary for the estimation of the Quality-of-Service (QoS) provided by Asynchronous Transfer Mode (ATM) networks. The previous papers on this subject did not give a comprehensive solution to the problem. In this paper an explicit formula for this distribution is given. This formula is general as it includes any initial level of the length of the queue, any type of service distribution (heavy tails) and any traffic intensity ρ. Moreover, it is easy to use and fast in computation. To show this several numerical examples are presented. In addition, a solution of the similar problem in G/M/1 queues (which is the distribution of the remaining interarrival time) is given.     

A numerically efficient method for the MAP/D/1/K queue via rational approximations

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traffic model for use in the performance analysis of Asynchronous Transfer Mode (ATM) networks. In ATM networks, packets are of fixed length and the buffering memory in switching nodes is limited to a finite numberK of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Padé approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear differential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacityK. We show through numerical examples that, the idea of using Padé approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities.This work was done when the author was with the Bilkent University, Ankara, Turkey and the research was supported by TÜBITAK under Grant No. EEEAG-93.     

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.     

Sample path large deviations for a family of long-range dependent traffic and associated queue length processes

We consider the long-range dependent cumulative traffic generated by the superposition of constant rate fluid sources having exponentially distributed inter start times and Pareto distributed durations with finite mean and infinite variance. We prove a sample path large deviation principle when the session start time intensity is increased and the processes are centered and scaled appropriately. Properties of the rate function are investigated. We derive a sample path large deviation principle for a related family of stationary queue length processes. The large deviation approximation of the steady-state queue length distribution is compared with the corresponding empirical distribution obtained by a computer simulation. MSC 2000 Classifications: Primary 60F10; Secondary 60K25, 68M20, 90B22     

An interpolation approximation for the GI/G/1 queue based on multipoint Padé approximation

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy traffic conditions. In this paper, we propose a new approximation technique, which combines the light and heavy traffic characteristics. This interpolation approximation is based on the theory of multipoint Padé approximation which is applied at two points: light and heavy traffic. We show how this can be applied for estimating the waiting time moments of the ^GI/G/1 queue. The light traffic derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy traffic limits of the ^GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of traffic intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. This revised version was published online in June 2006 with corrections to the Cover Date.