Sojourn times in a processor sharing queue with service interruptions

We study the sojourn times of customers in an M/M/1 queue with the processor sharing service discipline and a server that is subject to breakdowns. The lengths of the breakdowns have a general distribution, whereas the on-periods are exponentially distributed. A branching process approach leads to a decomposition of the sojourn time, in which the components are independent of each other and can be investigated separately. We derive the Laplace–Stieltjes transform of the sojourn-time distribution in steady state, and show that the expected sojourn time is not proportional to the service requirement. In the heavy-traffic limit, the sojourn time conditioned on the service requirement and scaled by the traffic load is shown to be exponentially distributed. The results can be used for the performance analysis of elastic traffic in communication networks, in particular, the ABR service class in ATM networks, and best-effort services in IP networks.     

On the angular derivative and univalence

f . , , — , A f f(). , , f() 0 . , , ,A , f . , f() - f() . , , . (1976) ( ¦f(z)¦<1) . . (1969) ( ).     

Один Способ Замены Переменных В теории Параболических Уравнений и Его Применение

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Non-parametric Estimation for the M/G/∞ Queue

Given an M/G/ queue with input rate and service-time distribution G, we consider the problem of estimating and G from data on the queue-length process Q = (Q_t). Our motivation is to study departures of G from exponentiality, following recent work of Bingham and Dunham (1997, Ann. Inst. Statist. Math., 49, 667–679).     

Head of the line processor sharing for many symmetric queues with finite capacity

In this paper the steady-state behavior of many symmetric queues, under the head of the line processor-sharing discipline, is investigated. The arrival process to each of n queues is Poisson, with rateA, and each queue hasr waiting spaces. A job arriving at a full queue is lost. The queues are served by a single exponential server, which has a mean raten, and splits its capacity equally amongst the jobs at the head of each nonempty queue. The normal traffic casep=/< 1 is considered, and it is assumed thatn1 andr= 0(1). A 2-term asymptotic approximation to the loss probabilityL is derived, and it is found thatL = 0(n ^–r ), for fixedp. If6=(1–p)/p 1, then the approximation is valid if n² 1 and (r+ 1)²n, and in this caseL r!/(n)^r. Numerical values ofL are obtained forr = 1,2,3,4 and 5,n = 1000,500 and 200, and various values ofp< 1. Very small loss probabilities may be obtained with appropriate values of these parameters.     

Convergence of simultaneous Padé approximants to two dilogarithms

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Kanten-kritische graphen mit der zusammenhangszahl 2

A set of vertices of a graph G represents G if each edge of G is incident with at least one vertex of . A graph G is said to be edge-critical if the minimal number of vertices necessary to represent G decreases if any edge of G is omitted. Plummer [5] has given a method to construct an infinite family of edge-critical graphs with connectivity number 2. We use this method to construct a more extensive class of edge-critical graphs with connectivity number 2 and show that all edge-critical graphs with this connectivity number (K₂) can be constructed from smaller edge-critical graphs. Finally we give examples of edge-critical graphs not constructable from smaller ones by this method.     

An M/G/1 queue with second optional service

We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate (>0) all demand the first essential service, whereas only some of them demand the second optional service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/₂ (₂>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_k/1 and M/M/1 have been derived as particular cases.     

On approximation by Riesz means concerning orthogonal series

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Dedicated to Academician S. M. Nikol'skii on his 90th birthday

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This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Sharpening of Stečkin's theorem to strong approximation with exponentp

— , B _n , B p1., , p=1 , - . , , , p.     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run state behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both actual and virtual delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural explanations for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Bene's formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.     

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on R^d can be symmetrized to a function f^* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality f^*f, where f is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H¹,P then f^* belongs to H¹,P and f^*_pf_p.     

On optimality of spline interpolation for recovery of differentiable functions

W _p ^r (1p<) , . - r -1. =1 r-2.     

Transferring estimates for zonal convolution operators

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Unisecant curves on a general ruled surface

Let X be an irreducible algebraic curve of genus g smooth and proper over an algebraically closed field k, a locally free sheaf of rank 2 over X, F=P() the projective bundle associated to and :FX the canonical projection. Aunisecant curve on F is a curve (effective divisor) C on F such that the intersection number (C,–¹(x))=1, x X. Notice that a section of F over X or alternatively a sub-line bundle of means simply an irreducible unisecant curve. We give here some results on unisecant curves on F. In particular we are able to prove C.Segre's result regarding his general surfaces [8]. A more ample account including all the details will appear later.     

Investigation of Haar and Franklin series in Hardy spaces

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Жизнь и деятельность яноша Бояи

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k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.