Light traffic for workload in queues

General exact light traffic limit theorems are given for the distribution of steadystate workloadV, in open queueing networks having as input a general stationary ergodic marked point process {(t _n ,K _n )n0 (where t_n denotes the arrival time and K_n the routing and service times of the nth customer). No independence assumptions of any kind are required of the input. As the light traffic regime, it is only required that the Palm distribution for the exogenous interarrival time converges weakly to infinity (while the service mechanism is not allowed to change much). As is already known in the context of a single-server queue, work is much easier to deal with mathematically in light traffic than is customer delayD, and consequently, our results are far more general than existing results forD. We obtain analogous results for multi-channel and infinite-channel queues. In the context of open queueing networks, we handle both the total workload in the network as well as the workload at isolated nodes.Research supported in part by the Japan Society for the Promotion of Science during the author's fellowship in Tokyo, and by NSF Grant DDM 895 7825.     

具有可变到达率的多重休假Geo~(λ_1,λ_2)/G/1排队分析

本文考虑顾客到达与服务员休假相关的多重休假离散时间排队系统,用更新过程及u-变换分析了系统的队长性质.分别得到系统在三种时点(n~-,n~+,n)处的队长分布的递推解,进而揭示了在不同到达率条件下系统队长分布不再具有随机分解特性,得到了系统在四种时点(n~-,n~+,n,离去时点D_n)处稳态队长分布的重要关系(不同于连续时间排队系统).     

A fluid system with coupled input and output,and its application to bottlenecks in ad hoc networks

This paper studies a fluid queue with coupled input and output. Flows arrive according to a Poisson process, and when n flows are present, each of them transmits traffic into the queue at a rate c/(n+1), where the remaining c/(n+1) is used to serve the queue. We assume exponentially distributed flow sizes, so that the queue under consideration can be regarded as a system with Markov fluid input. The rationale behind studying this queue lies in ad hoc networks: bottleneck links have roughly this type of sharing policy. We consider four performance metrics: (i) the stationary workload of the queue, (ii) the queueing delay, i.e., the delay of a ‘packet’ (a fluid particle) that arrives at the queue at an arbitrary point in time, (iii) the flow transfer delay, i.e., the time elapsed between arrival of a flow and the epoch that all its traffic has been put into the queue, and (iv) the sojourn time, i.e., the flow transfer time increased by the time it takes before the last fluid particle of the flow is served. For each of these random variables we compute the Laplace transform. The corresponding tail probabilities decay exponentially, as is shown by a large-deviations analysis. F. Roijers’ work has been carried out partly in the SENTER-NOVEM funded project Easy Wireless.     

Large deviations for a simple closed queueing model

We consider a simple queueing model with one service station. The arrival and service processes have intensitiesa(N–Q _t) andNf(N ^–1 Q _t), where Q_t is the queue length,N is a large integer,a>0 andf(x) is a positive continuous function. We establish the large deviation principle for the sequence of the normalized queue length processq _N ^t =N ^–1Q_t,N1 for both light (a<f(0)) and heavy (af(0)) traffic and use this result for an investigation of ergodic properties ofq _N ^t ,N 1.     

The Recursive Solution for Geom/G/1（E,SV） queue with Feedback and Single Server Vacation

Using recursive method,this paper studies the queue size properties at any epoch n + in Geom/G/1(E,SV) queueing model with feedback under LASDA (late arrival system with delayed access) setup.Some new results about the recursive expressions of queue size distribution at different epoch (n+,n,n-) are obtained.Furthermore the important relations between stationary queue size distribution at different epochs are discovered.The results are different from the relations given in M/G/1 queueing system.The model discussed in this paper can be widely applied in many kinds of communications and computer network.     

Diffusion approximations for open queueing networks with service interruptions

This paper establishes functional central limit theorems describing the heavy-traffic behavior of open single-class queueing networks with service interruptions. In particular, each station has a single server which is alternatively up and down. There are two treatments of the up and down times. The first treatment corresponds to fixed up and down times and leads to a reflected Brownian motion, just as when there are no service interruptions, but with different parameters. To represent long rare interruptions, the second treatment has growing up and down times with the up and down times being of ordern andn ^1/2, respectively, when the traffic intensities are of order 1-n^–1/2. In this case we establish convergence in the SkorohodM ₁ topology to a multidimensional reflection of multidimensional Brownian motion plus a multidimensional jump process.     

Birth control for giants

The standard Erdős-Rényi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n/2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v ₁, v ₂}, {v ₃, v ₄} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time t _c so that S(t) → ∞ as t approaches t _c from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < t _c . Employing classic results of Cramér on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t > t _c we show the existence of a giant component, so that t = t _c may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component. Research supported by the Australian Research Council, the Canada Research Chairs Program and NSERC. Research partly carried out while the author was at the Department of Mathematics and Statistics, University of Melbourne.     

Stability of Earliest-Due-Date,First-Served Queueing Networks

We study multiclass queueing networks with the earliest-due-date, first-served (EDDFS) discipline. For these networks, the service priority of a customer is determined, upon its arrival in the network, by an assigned random due date. First-in-system, first-out queueing networks, where a customer's priority is given by its arrival time in the network, are a special case. Using fluid models, we show that EDDFS queueing networks, without preemption, are stable whenever the traffic intensity satisfies _j <1 for each station j.     

Asymptotic Analysis of Two Coupled Queues with Vastly Different Arrival Rates and Finite Customer Capacities

We consider two coupled queues, with each having a finite capacity of customers. When both queues are nonempty they evolve independently, but when one becomes empty the service rate in the other changes. Such a model corresponds to a generalized processor sharing (GPS) discipline. We study the joint distribution p(m, n) of finding (m, n) customers in the (first, second) queue, in the steady state. We study the problem in an asymptotic limit of “heavy traffic,” where also the arrival rate to the second queue is assumed to be small relative to that of the first. The capacity of the first queue is scaled to be large, while that of the second queue is held constant. We consider several different scalings, and in each case obtain limiting differential and/or difference equation for p(m, n), and these we explicitly solve. We show that our asymptotic approximations are quite accurate numerically. This work supplements previous investigations into this GPS model, which assumed infinite capacities/buffers. The present model corresponds to a random walk in a lattice rectangle, where p(m, n) satisfies a different boundary condition on each edge.     

Quasi-reversibility of a discrete-time queue and related models

We consider a discrete-time queueing system and its application to related models. The model is defined byX _n+1=X_n+A_n-D_n+1 with discrete states, whereX _n is the queue-length at the nth time epoch,A _n is the number of arrivals at the start of the nth slot andD _n+1 is the number of outputs at the end of the nth slot. In this model, the arrival process {A _n} is described as a sequence of independently and identically distributed random variables. The departureD _n+1 depends only on the system sizeX _n+A_n at the beginning of the time slot.We study the reversibility for the model. The departure discipline in which the system has quasi-reversibility is determined. Models with special arrival processes were studied by Walrand [8] and sawa [7]. In this paper, we generalize their results. Moreover, we consider discrete-time queueing networks with some reversible nodes. We then obtain the product-form solution for these networks.     

Distribution sensitivity in a highway flow model

We examine a model of traffic flow on a highway segment, where traffic can be impaired by random incidents (usually, collisions). Using analytical and numerical methods, we show the degree of sensitivity that the model exhibits to the distributions of service times (in the queueing model) and incident clearance times. Its sensitivity to the distribution of time until an incident is much less pronounced. Our analytical methods include an M/G_t/∞ analysis (G_t denotes a service process whose distribution changes with time) and a fluid approximation for an M/M/c queue with general distributions for the incident clearance times. Our numerical methods include M/PH₂/c/K models with many servers and with phase‐type distributions for the time until an incident occurs or is cleared. We also investigate different time scalings for the rate of incident occurrence and clearance. Copyright © 2009 John Wiley & Sons, Ltd.     

On the distribution of the number of packets in the fluid flow approximation of packet arrival streams

Fluid flow approximations are widely used for approximating models of communication systems where packet arrival streams are generated in a regular manner over certain intervals (constant rate). The appropriate mathematical model for describing those bursty arrival streams in the fluid flow framework are the well-known Markov modulated rate processes (MMRP). The paper deals with the distribution of the numberN(t) of packets in the interval [0,t] of MMRP. For two-state MMRPs and their superpositions we derive formulas for the distribution ofN(t) and its density. Further we give asymptotic results. The presented numerical results and simulation studies illustrate the goodness of the fluid flow approximation and show that the proposed numerical algorithms work well even in the case of multiplexing a large number of burst silence sources.This work was partially supported by a grant from the Deutsche Bundespost TELEKOM.     

Queueing systems fed by many exponential on-off sources: an infinite-intersection approach

In queueing theory, an important class of events can be written as ‘infinite intersections’. For instance, in a queue with constant service rate c, busy periods starting at 0 and exceeding L > 0 are determined by the intersection of the events , i.e., queue Q _t is empty at 0 and for all t∊ [0, L] the amount of traffic A _t arriving in [0,t) exceeds the server capacity. Also the event of exceeding some predefined threshold in a tandem queue, or a priority queue, can be written in terms of this kind of infinite intersections. This paper studies the probability of such infinite intersections in queueing systems fed by a large number n of i.i.d. traffic sources (the so-called ‘many-sources regime’). If the sources are of the exponential on-off type, and the queueing resources are scaled proportional to n, the probabilities under consideration decay exponentially; we explicitly characterize the corresponding decay rate. The techniques used stem from large deviations theory (particularly sample-path large deviations). M. Mandjes is also with Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, the Netherlands, and EURANDOM, Eindhoven, the Netherlands. Work done while P. Mannersalo was on leave at CWI. MSC 2000: 60F10 (Large deviations), 60K25 (Queueing theory)     

Structure of subsemigroups of factor-powers of finite symmetric groups

For the factor-powerFP(S _n ) of the symmetric groupS _n , we describe regular elements, maximal subgroups, isolated and fully isolated subsemigroups, and also maximal nilpotent subsemigroups whose zero elements coincide with the zero element of the semigroupFP(S _n ). Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 341–354, September, 1995. This research was partially supported by the Foundation for Fundamental Research of the State Committee for Science and Engineering of the Ukraine.     

Traffic processes in a class of finite Markovian queues

This paper exposes the stochastic structure of traffic processes in a class of finite state queueing systems which are modeled in continuous time as Markov processes. The theory is presented for theM/E _k /φ/L class under a wide range of queue disciplines. Particular traffic processes of interest include the arrival, input, output, departure and overflow processes. Several examples are given which demonstrate that the theory unifies many earlier works, as well as providing some new results. Several extensions to the model are discussed.     

Convexity of single stage queueing systems with bulk arrival

In this paper we study the convexity of the waiting time, workload and the number of jobs in single stage queueing systems with respect to the bulk size of the arrival process. In particular we show that the number of jobs in a single server queueing systemG ^[x ]/GI/1 and in a multiple server queueing systemG ^[x]/M/c with bulk sizesx=(x ₁ ,x ₂ ,x ₃ ,...) is componentwise convex inx. This is in the sense of the sample path convexity introduced in Shaked and Shanthikumar [11]. These results have applications in the stochastic comparison of bulk arrival queueing systems.Research supported in part by NSF grant DDM-9113008.     

Heavy traffic limit theorems for a sequence of shortest queueing systems

A sequence of shortest queueing systems is considered in this paper. We give weak convergence theorems for the queue length and waiting time processes when the sequence of traffic intensities associated with the sequence of shortest queueing systems approaches the critical value (=1) at appropriate rates.Research supported by the National Natural Science Foundation of China.     

Heavy traffic resource pooling in parallel‐server systems

We consider a queueing system with r non‐identical servers working in parallel, exogenous arrivals into m different job classes, and linear holding costs for each class. Each arrival requires a single service, which may be provided by any of several different servers in our general formulation; the service time distribution depends on both the job class being processed and the server selected. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs onto available servers. A linear program involving only first‐moment data (average arrival rates and mean service times) is used to define heavy traffic for a system of this form, and also to articulate a condition of overlapping server capabilities which leads to resource pooling in the heavy traffic limit. Assuming that the latter condition holds, we rescale time and state space in standard fashion, then identify a Brownian control problem that is the formal heavy traffic limit of our rescaled scheduling problem. Because of the assumed overlap in server capabilities, the limiting Brownian control problem is effectively one‐dimensional, and it admits a pathwise optimal solution. That is, in the limiting Brownian control problem the multiple servers of our original model merge to form a single pool of service capacity, and there exists a dynamic control policy which minimizes cumulative cost incurred up to any time t with probability one. Interpreted in our original problem context, the Brownian solution suggests the following: virtually all backlogged work should be held in one particular job class, and all servers can and should be productively employed except when the total backlog is small. It is conjectured that such ideal system behavior can be approached using a family of relatively simple scheduling policies related to the cμ rule. This revised version was published online in June 2006 with corrections to the Cover Date.     

Insensitivity in discrete time queues with a moving server

Queues in which customers request service consisting of an integral number of segments and in which the server moves from service station to service station are of considerable interest to practitioners working on digital communications networks. In this paper, we present insensitivity theorems and thereby equilibrium distributions for two discrete time queueing models in which the server may change from one customer to another after completion of each segment of service. In the first model, exactly one segment of service is provided at each time point whether or not an arrival occurs, while in the second model, at most one arrival or service occurs at each time point. In each model, customers of typet request a service time which consists ofl segments in succession with probabilityb _t(l). Examples are given which illustrate the application of the theorems to round robin queues, to queues with a persistent server, and to queues in which server transition probabilities do not depend on the server's previous position. In addition, for models in which the probability that the server moves from one position to another depends only on the distance between the positions, an amalgamation procedure is proposed which gives an insensitive model on a coarse state space even though a queue may not be insensitive on the original state space. A model of Daduna and Schassberger is discussed in this context.This work was supported by the Australian Research Council.     

On the existence of harmonic morphisms from symmetric spaces of rank one

Summary In this paper we give a unified framework for constructing harmonic morphisms from the irreducible Riemannian symmetric spaces ℍH ⁿ, ℂH ⁿ, ℝH ² _t+1, ℍP ⁿ, ℂP ⁿ and ℝP ²ⁿ⁺¹ of rank one. Using this we give a positive answer to the global existence problem for the non-compact hyperbolic cases. This work was supported by The Swedish Natural Science Research Council. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.