New bounds for expected delay in FIFO M/G/c queues

Most bounds for expected delay, E[D], in GI/GI/c queues are modifications of bounds for the GI/GI/1 case. In this paper we exploit a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two. (S AND T denote generic service and interarrival times, respectively.) We derive two different families of new bounds for expected delay, both in terms of moments of S AND T. Our first bound is applicable when E[S₂] < ∞. Our second bound for the first time does not require finite variance of S; it only involves terms of the form E[S^β], where 1 < β < 2. We conclude by comparing our bounds to the best known bound of this type, as well as values obtained from simulation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

Further delay moment results for FIFO multiserver queues

Using a new family of service disciplines, we provide weaker sufficient conditions for finite stationary delay moments in FIFO multiserver queues. This extends the work in Sigman and Scheller-Wolf [6] to GI/GI/s queues with = E[S]/E[T] s-1 are the familiar Kiefer and Wolfowitz conditions actually known to be necessary. For the case when < 1, we provide sufficient conditions for finite mean stationary delay, expressed as a function of the number of servers in the system. The limit of these conditions as s is the requirement that E[S] < , which is the condition for finite mean stationary delay in a FIFO GI/GI/ queue. Both of these results highlight the interplay between traffic intensity and service time distribution in determining the behavior of delay moments in multiserver queues.     

On Some Properties of Randomly Indexed Sequences of Random Elements

Let be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {N _n ,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {F _k ,k≥1} be a family of probability laws and let X be some random element defined on and taking values in (S,ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {X _n,k ,n,k≥1} defined on the same probability space and taking values in (S,ρ), and such that , and moreover as  n→∞. Furthermore, we consider the speed of convergence to X as n→∞.      

Moment convergence rates in the law of the logarithm for dependent sequences

Let {X _n ; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set S _n = Σ _k=1 ⁿ X _k , M _n = max _k≤n |S _k |, n ≥ 1. Suppose σ ² = EX ₁² + 2Σ _k=2^∞ EX ₁ X _k (0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{M _n −σɛ√n log n}₊ and E{|S _n | − σɛ√n log n}₊ as ɛ ↘ 0 and E{σɛ√π ² π/8logn − M _n }₊ as ɛ ↗ ∞ are obtained.     

Convergence in law of partial sum processes in p-variation norm

Let X ₁, X ₂, … be a sequence of independent identically distributed real-valued random variables, S _n be the nth partial sum process S _n (t) ≔ X ₁ + ⋯ X _⌊tn⌋, t ∈ [0, 1], W be the standard Wiener process on [0, 1], and 2 < p < ∞. It is proved that n ^−1/2 S _n converges in law to σW as n → ∞ in p-variation norm if and only if EX ₁ = 0 and σ ² = EX ₁² < ∞. The result is applied to test the stability of a regression model. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-21/07     

On a Furstenberg-Katznelson-Weiss Type Theorem over Finite Fields

Using Fourier analysis, Covert, Hart, Iosevich, and Uriarte-Tuero (2008) showed that if the cardinality of a subset of the 2-dimensional vector space over a finite field with q elements is ≥ ρq ², with q^-1/2 << r ￡ 1{q^{-1/2} \ll \rho \leq 1} then it contains an isometric copy of ≥ cρq ³ triangles. In this note, we give a graph theoretic proof of this result.     

On the Complete Convergence of Moving Average Process with Banach Space Valued Random Elements

Let {Y _i ;−∞<i<∞} be a doubly infinite sequence of independent random elements taking values in a separable real Banach space and stochastically dominated by a random variable X. Let {a _i ;−∞<i<∞} be an absolutely summable sequence of real numbers and set V _i =∑ _k=−∞^∞ a _i+k Y _i ,i≥1. In this paper, we derive that if and E|X| ^μ log  ^ρ |X|<0, for some μ (0<μ<2, μ≠1) and ρ>0 then for all ε>0. This work was partially supported by the Korean Research Foundation Grant funded by the Korean Government (KRF-2006-353-C00006, KRF-2006-251-C00026).     

Hypergraphs with independent neighborhoods

For each k ≥ 2, let ρ _k ∈ (0, 1) be the largest number such that there exist k-uniform hypergraphs on n vertices with independent neighborhoods and (ρ _k + o(1))( _k ⁿ ) edges as n → ∞. We prove that ρ _k = 1 − 2logk/k + Θ(log log k/k) as k → ∞. This disproves a conjecture of Füredi and the last two authors.     

Kneading sequences for unimodal expanding maps of the interval

LetP andAC be two primary sequences with min{P, AC}≥RLR ^∞,ρ(P) and ρ(AC) be the eigenvalues ofP andAC, respectively. Letf∈C ⁰ (I, I) be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved thatf has the kneading sequenceK(f)≥(RC) ^*m *P if λ≥(ρ(P))^1/2m, andK(f)>(RC) ^*m*AC*E for any shift maximal sequenceE if λ>(ρ(AC))^1/2m. The value of (ρ(P))^1/2m or (ρ(AC))^1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one. Project supported by the National Natural Science Foundation of China     

Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

We consider oriented bond or site percolation on ℤ ^d ₊. In the case of bond percolation we denote by P _p the probability measure on configurations of open and closed bonds which makes all bonds of ℤ ^d ₊ independent, and for which P _p {e is open} = 1 −P _p e {is closed} = p for each fixed edge e of ℤ ^d ₊. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v ₀ = 0, v ₁, v ₂, …, starting at the origin, such that lim inf _{n →∞} (1/n) ∑ _i=1 ⁿ X(e _i ) ≥ρ, where e _i is the edge {v _i−1 , v _i }. [MZ92] showed that there exists a critical probability p _c = p _c (ρ, d) = p _c (ρ, d, bond) such that there is a.s. no ρ-percolation for p < p _c and that P _p {ρ-percolation occurs} > 0 for p > p _c . Here we find lim _{d →∞} d ^1/ρ p _c (ρd, bond) = D ₁ , say. We also find the limit for the analogous quantity for site percolation, that is D ₂ = lim _{d →∞} d ^1/ρ p _c (ρ, d, site). It turns out that for ρ < 1, D ₁ < D ₂ , and neither of these limits equals the analogous limit for the regular d-ary trees. Received: 7 January 1999 / Published online: 14 June 2000     

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ ^d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ ^d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.     

Range of Brownian Motion with Drift

Let (B _δ (t)) _{t ≥ 0} be a Brownian motion starting at 0 with drift δ > 0. Define by induction S ₁=− inf _{t ≥ 0} B _δ (t), ρ₁ the last time such that B _δ (ρ₁)=−S ₁, S ₂=sup_{0≤ t ≤ρ 1} B _δ (t), ρ₂ the last time such that B _δ (ρ₂)=S ₂ and so on. Setting A _k =S _k +S _k+1; k ≥ 1, we compute the law of (A ₁,...,A _k ) and the distribution of (B _δ (t+ρ l) − B _δ (ρ _l ); 0 ≤ t ≤ ρ _l-1 − ρ _l )_{2 ≤ l ≤ k} for any k ≥ 2, conditionally on (A ₁,...,A _k ). We determine the law of the range R _δ (t) of (B _δ (s)) _{s≥ 0} at time t, and the first range time θ_δ (a) (i.e. θ_δ (a)=inf{t > 0; R _δ (t) > a}). We also investigate the asymptotic behaviour of θ _δ (a) (resp. R _δ (t)) as a → ∞ (resp. t → ∞).     

Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions

We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t ^−ν with 1 < ν ⩽ 2 , so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a → 1, then W, multiplied by an appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a → 1, then W, multiplied by another appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the negative exponential distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the limit-classifications of even and odd-order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(T _max)/D(T _min) associated withM[.] in terms of the behaviour of the determinants det [[f _rg_s](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.     

Wavelet characterization of Hörmander symbol classS ρ,δ m and applications

In this paper, we characterize the symbol in Hormander symbol classS _ρ ^m ,δ (m ∈ R, ρ, δ ≥ 0) by its wavelet coefficients. Consequently, we analyse the kerneldistribution property for the symbol in the symbol classS _ρ ^m ,δ (m ∈R, ρ > 0, δ≥ 0) which is more general than known results ; for non-regular symbol operators, we establish sharp L²-continuity which is better than Calderón and Vaillancourt’s result, and establishL ^p (1 ≤p ≤∞) continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator’s continuity on the basis of the wavelets coefficients in phase space.     

Homotopy Types of Stabilizers and Orbits of Morse Functions on Surfaces

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ ¹ or the circle S ¹, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h ⁻¹ for h ∊ D(M) and f ∊ C^∞ (M,P). Let f: M → P be an arbitrary Morse mapping, Σ _f the set of critical points of f, D(M,Σ _f ) the subgroup of D(M) preserving Σ _f , and S(f), S (f,Σ _f ), O(f), and O(f,Σ _f ) the stabilizers and the orbits of f with respect to D(M) and D(M,Σ _f ). In fact S(f) = S(f,Σ _f ).In this paper we calculate the homotopy types of S(f), O(f) and O(f,Σ _f ). It is proved that except for few cases the connected components of S(f) and O(f,Σ _f ) are contractible, π _k O(f) = π _k M for k ≥ 3, π₂ O(f) = 0, and π₁ O(f) is an extension of π₁ D(M) ⊕ Z ^k (for some k ≥ 0) with a (finite) subgroup of the group of automorphisms of the Kronrod-Reeb graph of f.We also generalize the methods of F. Sergeraert to give conditions for a finite codimension orbit of a tame smooth action of a tame Lie group on a tame Fréchet manifold to be a tame Fréchet manifold itself. In particular, we obtain that O(f) and O(f, Σ _f ) are tame Fréchet manifolds. Communicated by Peter Michor Vienna Mathematics Subject Classifications (2000): 37C05, 57S05, 57R45.     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

负相依随机变量的自正则化中心极限定理与部分和的方差估计

§ 1　 IntroductionWe firstintroduce some concepts.Random variables X and Y are called negative dependent ( ND) if for any pair ofmonotonically non-decresing functions f and g,Cov{ f( X) ,g( Y) }≤ 0 .Clearly itis equivalenttoP( X≤ x,Y≤ y)≤ P( X≤ x) P( Y≤ y)for all x,y∈R.A random sequence{ Xi,i≥ 1 } is said to be negative quadrant dependent( NQD) if any pairof variables Xi,Xj( i≠j) are ND.A sequence of random variables{ Xi,i≥ 1 } is said to be linear negative quadrand depend…