Reduced-Load Equivalence and Induced Burstiness in GPS Queues with Long-Tailed Traffic Flows

We analyze the queueing behavior of long-tailed traffic flows under the Generalized Processor Sharing (GPS) discipline. We show a sharp dichotomy in qualitative behavior, depending on the relative values of the weight parameters. For certain weight combinations, an individual flow with long-tailed traffic characteristics is effectively served at a constant rate. The effective service rate may be interpreted as the maximum average traffic rate for the flow to be stable, which is only influenced by the traffic characteristics of the other flows through their average rates. In particular, the flow is essentially immune from excessive activity of flows with heavier-tailed traffic characteristics. In many situations, the effective service rate is simply the link rate reduced by the aggregate average rate of the other flows. This confirms that GPS-based scheduling algorithms provide a potential mechanism for extracting significant multiplexing gains, while isolating individual flows. For other weight combinations however, a flow may be strongly affected by the activity of heavier-tailed flows, and may inherit their traffic characteristics, causing induced burstiness. The stark contrast in qualitative behavior illustrates the crucial importance of the weight parameters.     

Boundary Modified Contact Processes

We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate _e (where e is for external). Other births occur at rate _i (where i is for internal). Deaths occur at rate 1. The case _e= _i is the well known basic contact process for which there is a critical value _c>1 such that if the birth rate is larger than _c the process has a positive probability of surviving. Our main motivation here is to understand the relative importance of the external birth rates. We show that if _e1 then the process always dies out while if _e>1 and if _i is large enough then the process may survive. We also show that if _i< _c the process dies out for all _e. To extend this notion to d>1 we introduce a second process that has an epidemiological interpretation. For this process each site can be in one of three states: infected, a susceptible that has never been infected, or a susceptible that has been infected previously. Furthermore, the rates at which the two types of susceptible become infected are different. We obtain some information about the phase diagram about this case as well.     

Large Deviations of Queues Sharing a Randomly Time-Varying Server

We consider a discrete-time model where multiple queues, each with its own exogenous arrival process, are served by a server whose capacity varies randomly and asynchronously with respect to different queues. This model is primarily motivated by the problem of efficient scheduling of transmissions of multiple data flows sharing a wireless channel. We address the following problem of controlling large deviations of the queues: find a scheduling rule, which is optimal in the sense of maximizing

A Load-Balanced Network with Two Servers

A load-balanced network with two queues Q ₁ and Q ₂ is considered. Each queue receives a Poisson stream of customers at rate _i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p _i ^*, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system.     

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.     

Asymptotic behavior of general M-estimates for regression and scale with random carriers

Summary Let (xini, y _i be a sequence of independent identically distributed random variables, where x _i R ^p and y _i R, and let R ^p be an unknown vector such that y _i =x _i +u _i (^*), where u _i is independent of x _i and has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, ( ^*,^*), defined as solutions of the system: , where r= (y _i –x _i ^1*/)^*, with R ^p ×RR and RR. This class contains estimators of (, ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on and and assuming the joint distribution of (x _i , y _i ) to fulfill the model (^*) only approximately.     

Large Buffer Asymptotics for Fluid Queues with Heterogeneous M/G/∞ Weibullian Inputs

In this paper we consider a generalization of the so called M/G/ model where M types of sessions enter a buffer. The instantaneous rates of the sessions are functions of the occupancy of an M/G/ system with Weibullian G distributions. In particular we assume that a session of type i transmits r _i cells per unit time and lasts for a random time with a Weibull distribution given by x) \sim {\text{e}}^{{\text{ - }}\gamma ix^{\alpha i} } $$ " align="middle" border="0"> , where 0< _i <1, _i >0. We show that the complementary buffer occupancy distribution for large buffer size is Weibullian whose parameters can be determined as the solution of a deterministic nonlinear knapsack problem. For _i <0.5, upper and lower bound factors are determined. When specialized to the homogeneous case, i.e., when all the sessions are identical, the result coincides with a lower bound reported in the literature.     

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.     

Geodesic Equivalence via Integrability

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor g ⁱg¯_j are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Liouville integrable. This gives us topological obstacles to geodesic equivalence.     

Componentwise estimation of ordered parameters ofk (≥2) exponential populations

We consider the estimation of ordered parameters ofk ( 2) exponential distributions by improving upon the usual estimators. TheBrewsterzidek technique is used to find sufficient conditions for an estimator of _i and/or _i (i=1,...,k), to be inadmissible with respect to the MSE criterion where _i and _i are the location and scale parameters respectively of thei-th exponential population. Using these sufficient conditions improved estimators of _i and/or _i (i=1,...,k) are obtained.     

Simultaneous Extensions of Topogenities

Let (X,) be a topological space, X _i X for i I, and < _i be a topogenity on X _i . We look for a topogenity < on X such that is the topology induced by < and <|X _i =< _i for i I.     

Sign-Preserving Approximation of Periodic Functions

We prove the Jackson theorem for a zero-preserving approximation of periodic functions (i.e., in the case where the approximating polynomial has the same zeros y _i) and for a sign-preserving approximation [i.e., in the case where the approximating polynomial is of the same sign as a function f on each interval (y _i, y _{i – 1})]. Here, y _i are the points obtained from the initial points – y _{2s y 2s}–1 <...< y₁ < using the equality yi = y_{i + 2s} + 2 furthermore, these points are zeros of a 2-periodic continuous function f.     

Line search by curve fitting in minimization algorithms

Summary Consider an unconstrained minimization of an uniformly convex functionf(z). Let be an algorithm that, for solving it constructs a sequence {z _i} withz _i+1 =z _i + ⁽ⁱ⁾ h _i ,h _i R ⁿ , ⁽ⁱ⁾ R ⁼ and –h _i ^T f(z _i ) > 0. For any algorithm that converges linearly and that uses parabolic or cubic interpolations for the line search, upper bounds on the number of function evaluations needed to approximate the minimum off(z) within a given accuracy, are calculated. The obtained results allow to compare the line search procedure under investigation.     

Ergodic and light traffic properties of a complex repairable system

A repairable system is composed of components ofI types. A component can be loaded, put on standby, queued or repaired. The repair facility is here assumed to be a queueing system of a rather general structure though interruption of repairs is not allowed. Typei components possess a lifetime distributionA _i (t) and repair time distributionB _t (t). The lifetime of componentj is exhausted with a state-dependent rate _j (t). A Markov process Z(t) with supplementary variables is built to investigate the system behaviour. An ergodic result, Theorem 1, is established under a set of conditions convenient for light traffic analysis. In Theorems 2 to 6, a light traffic limit is derived for the joint steady state distribution of supplementary variables. Applying these results, Theorems 7 to 10 derive light traffic properties of a busy period-measured random variable. Essentially, the concepts of light traffic equivalence due to Daley and Rolski (1992) and Asmussen (1992) are used. The asymptotic (light traffic) insensitivity of busy period and steady state parameters to the form ofA _i(t) [given their means and (in some cases) values of density functions for smallt], is observed under some analytic conditions.Some abbreviations & notations LHS, RHS lefthand side, righthand side - w.r.t. with respect to - (i.i.d.) r.v. (independent identically distributed) random variables - A ^c complement event - d.f., p.d.f. distribution function, probability density function - m.g.f. moment generating function - I _A indicator function of eventA - (t) =1–A(t)     

Independence of Likelihood Ratio Criteria for Homogeneity of Several Populations

Let _i be an i-tb population with a probability density function f(· | _i ) with one dimensional unknown parameter _i = 1, 2, ... , k. Let n _i sample be drawn from each _i . The likelihood ratio criteria _j|(j–1) for testing hypothesis that the first j parameters are equal against alternative hypothesis that the first (j – 1) parameters are equal and the j-th parameter is different with the previous ones are defined, j = 2, 3, ... , k. The paper shows the asymptotic independence of _j|(j–1)'s up to the order 1/n under a hypothesis of equality of k parameters, where n is a number of total samples.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

Convergence criteria for generalized gradient methods of solving locally Lipschitz feasibility problems

We prove the convergence of a class of iterative algorithms for solving locally Lipschitz feasibility problems, that is, finite systems of inequalities f _i (x)0, (i I), where each f _i is a locally Lipschitz functional on ⁿ . We also obtain a new convergence criterion for the so-called block-iterative projection methods of finding common points of finite families of convex closed subsets of ⁿ as defined by Aharoni and Censor ([3]).The work of Dan Butnariu was done while visiting the Department of Mathematics of the University of Texas at Arlington.     

A statistical estimate of the structure of multi-extremal problems

The solution of ak-extremal problem is defined as the set of pairs (x _i ^* , _i),i = 1, ,k, where x _t ^* isi ^th local minimum and _i is the volume of the set of attraction of this minimum. A Bayesian estimate ofk and ( ₁ , , _k ) is constructed.This paper has been written while the author was a CNR visiting professor at the Institute of Mathematics of the Milano University.     

Maximum of the fourth diameter in the family of continua with prescribed capacity

We obtain a complete solution of the problem of the maximum of the fourth diameter in the family of continua with capacity 1. Let E(o, eⁱ, e^–i). 0<i, e^–i; H(=cap E(o, eⁱ, e^–i). Let C() be the common point of three analytic arcs which form E(o, eⁱ, e^–i). One shows that the indicated maximum is realized by the continuum ={z:H(₀)z ²E(o, eⁱ, e^–i)} where ₀, o<₀z eⁱ z+C ( is a real and C is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 60–79, 1976.     

Stochastic algorithms with armijo stepsizes for minimization of functions

Stochastic algorithms for optimization problems, where function evaluations are done by Monte Carlo simulations, are presented. At each iteratex _i, they draw a predetermined numbern(i) of sample points from an underlying probability space; based on these sample points, they compute a feasible-descent direction, an Armijo stepsize, and the next iteratex _i+1. For an appropriate optimality function , corresponding to an optimality condition, it is shown that, ifn(i) , then (x _i) 0, whereJ is a set of integers whose upper density is zero. First, convergence is shown for a general algorithm prototype: then, a steepest-descent algorithm for unconstrained problems and a feasible-direction algorithm for problems with inequality constraints are developed. A numerical example is supplied.