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1.
This paper studies the queue-length process in a closed Jackson-type queueing network with the large number N of homogeneous customers by methods of the theory of martingales and by the up- and down-crossing method. The network considered here consists of a central node (hub), being an infinite-server queueing system with exponentially distributed service times, and k single-server satellite stations (nodes) with generally distributed service times with rates depending on the value N. The service mechanism of these k satellite stations is autonomous, i.e., every satellite server j serves the customers only at random instants that form a strictly stationary and ergodic sequence of random variables. Assuming that the first k-1 satellite stations operate in light usage regime the paper considers the cases where the kth satellite station is a bottleneck node. The approach of the paper is based both on development of the method from the paper by Kogan and Liptser [16], where a Markovian version of this model has been studied, and on development of the up- and down-crossing method. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

2.
Bounds on the number of row sums in ann×n, non-singular (0,1)-matrixA sarisfyingA tA=diag (k 11,…,k nn),k jj>0,λ1=…=λee+1=…=λn are obtained which extend previous results for such matrices.  相似文献   

3.
We consider the random variable ζ = ξ1ρ+ξ2ρ2+…, where ξ1, ξ2, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξi = 0) = p0, P(ξi = 1) = p1, 0 < p0 < 1. Let β = 1/ρ be the golden number. The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η1ρ + η2ρ2 + … where the random variables ηk are {0, 1}-valued and ηkηk+1 = 0. The infinite random word η = η1η2 … ηn … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous. We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic properties of the invariant Erdős measure. We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable when ξ1, ξ2, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47.  相似文献   

4.
Summary Let {X n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS n=Xn,1+…+X n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f n(x)∼ defined on a stationary sequence {X j∼, whereX n.f=fn(Xj). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.  相似文献   

5.
The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (X k : k = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The results are also extended for random fields (X k: k, ℓ = 1, 2, …).  相似文献   

6.
The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σ k of arbitrary degree k in random variables X i (i=1,2,…,m) defined on special subsets of commutative rings ℛ m with identity of finite characteristic m. It is shown that the probability distributions of the random elements σ k (X 1,…,X m ) tend to a limit when m→∞ if X 1,…,X m form a Markov chain of finite degree μ over a finite set of states V, V⊂ℛ m , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain.   相似文献   

7.
Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the array. We define monotone (d,m)-trapezoids as monotone triangles with m rows where the d−1 top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row (k 1,…,k n ) is given by a polynomial α(n;k 1,…,k n ) in the k i ’s. The main purpose of this paper is to show that the number of monotone (d,m)-trapezoids with prescribed top and bottom row appears as a coefficient in the expansion of a specialisation of α(n;k 1,…,k n ) with respect to a certain polynomial basis. This settles a generalisation of a recent conjecture of Romik et al. (Adv. Math. 222:2004–2035, 2009). Among other things, the result is used to express the number of monotone triangles with bottom row (1,2,…,i−1,i+1,…,j−1,j+1,…,n) (which is, by the standard bijection, also the number of n×n alternating sign matrices with given top two rows) in terms of the number of n×n alternating sign matrices with prescribed top and bottom row, and, by a formula of Stroganov for the latter numbers, to provide an explicit formula for the first numbers. (A formula of this type was first derived by Karklinsky and Romik using the relation of alternating sign matrices to the six-vertex model.)  相似文献   

8.
Regard an element of the set of ranked discrete distributions Δ := {(x 1, x 2,…):x 1x 2≥…≥ 0, ∑ i x i = 1} as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the Δ-valued Markov process in which pairs of clusters of masses {x i , x j } merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time −∞ with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991, 1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees. Received: 6 October 1998 / Revised version: 16 May 1999 / Published online: 20 October 2000  相似文献   

9.
Bramson  Maury 《Queueing Systems》2001,39(1):79-102
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.  相似文献   

10.
For a natural number k, define an oriented site percolation on ℤ2 as follows. Let x i , y j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ2 closed if x i = y j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i 0, j 0)(i 1, j 1) · · · such that 0 = i 0i 1≤· · ·, 0 = j 0j 1≤· · ·, and each site (i n , j n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x i ) and (y j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000  相似文献   

11.
A graph G is a k-sphere graph if there are k-dimensional real vectors v 1,…,v n such that ijE(G) if and only if the distance between v i and v j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v 1,…,v n such that ijE(G) if and only if the dot product of v i and v j is at least 1.  相似文献   

12.
The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S j for j=1,…,K at unknown spot locations x j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S j and x j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.   相似文献   

13.
Let G n,k be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc n12,…, β k ), β12,…, βk = 0,1,2,…, β12 + … +β k n,c n(0,0,…, 0) = 1 and whenever β0n - (β1 + β2 + … + β k ) where Δc n12,…, β k ) =c n1 + 1, β2,…, β k )+c n12+1,…, β k )+…+c n12,…, β k +1) -c n12,…, β k ). Further, let Π n,k be the set of all symmetric probabilities on {0,1,2,…,k} n . We establish a one-to-one correspondence between the sets G n,k and Π n,k and use it to formulate and answer interesting questions about both. Assigning to G n,k the uniform probability measure, we show that, asn→∞, any fixed section {it{cn}(β12,…, β k ), 1 ≤ Σβ i m}, properly centered and normalized, is asymptotically multivariate normal. That is, converges weakly to MVN[0, Σ m ]; the centering constantsc 01, β2,…, β k ) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex inR k.  相似文献   

14.
Berger  Arthur  Bregman  Lev  Kogan  Yaakov 《Queueing Systems》1999,31(3-4):217-237
Asymptotic behavior of queues is studied for large closed multi-class queueing networks consisting of one infinite server station with K classes and M processor sharing (PS) stations. A simple numerical procedure is derived that allows us to identify all bottleneck PS stations. The bottleneck station is defined asymptotically as the station where the number of customers grows proportionally to the total number of customers in the network, as the latter increases simultaneously with service rates at PS stations. For the case when K=M=2, the set of network parameters is identified that corresponds to each of the three possible types of behavior in heavy traffic: both PS stations are bottlenecks, only one PS station is a bottleneck, and a group of two PS stations is a bottleneck while neither PS station forms a bottleneck by itself. In the last case both PS stations are equally loaded by each customer class and their individual queue lengths, normalized by the large parameter, converge to uniformly distributed random variables. These results are directly generalized for arbitrary K=M. Generalizations for KM are also indicated. The case of two bottlenecks is illustrated by its application to the problem of dimensioning bandwidth for different data sources in packet-switched communication networks. An engineering rule is provided for determining the link rates such that a service objective on a per-class throughput is satisfied. This revised version was published online in June 2006 with corrections to the Cover Date.  相似文献   

15.
Vladimir V. Anisimov 《TOP》1999,7(2):169-186
Some special classes of Switching Processes such as Recurrent Processes of a Semi-Markov type and Processes with Semi-Markov Switches are introduced. Limit theorems of Averaging Principle and Diffusion Approximation types are given. Applications to the asymptotic analysis of overloading state-dependent Markov and semi-Markov queueing modelsM SM,Q /M SM,Q /1/∞ and retrial queueing systemsM/G/1/w.r in transient conditions are studied. The paper was supported by INTAS Project 96-0828  相似文献   

16.
Recently, B. Y. Chen introduced a new intrinsic invariant of a manifold, and proved that everyn-dimensional submanifold of real space formsR m (ε) of constant sectional curvature ε satisfies a basic inequality δ(n 1,…,n k )≤c(n 1,…,n k )H 2+b(n 1,…,n k )ε, whereH is the mean curvature of the immersion, andc(n 1,…,n k ) andb(n 1,…,n k ) are constants depending only onn 1,…,n k ,n andk. The immersion is calledideal if it satisfies the equality case of the above inequality identically for somek-tuple (n 1,…,n k ). In this paper, we first prove that every ideal Einstein immersion satisfyingnn 1+…+n k +1 is totally geodesic, and that every ideal conformally flat immersion satisfyingnn 1+…+n k +2 andk≥2 is also totally geodesic. Secondly we completely classify all ideal semi-symmetric hypersurfaces in real space forms. The author was supported by the NSFC and RFDP.  相似文献   

17.
We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P 1, …, P k  ∈ Z[m] in one unknown m with P 1(0) = … = P k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with 1 \leqslant m \leqslant xe1 \leqslant m \leqslant x^\varepsilon, such that x + P 1(m), …, x + P k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P j  = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.  相似文献   

18.
Let X 1, X 2, … , X n be i.i.d. random variables with common distribution F, and let b 1, b 2, … , b n be real coefficients such that ∑ b j 2 = 1. We prove that F is close to the normal distribution in the Lévy metric whenever the distribution of the linear statistic ∑ b j X j is close to F.  相似文献   

19.
We consider the generalized convolution powers G α *u (x) of an arbitrary semistable distribution function G α (x) of exponent α∈(0,2), and prove that for all j, k∈{0,1,2,…} and u>0 the derivatives G α (k,j)(x;u)= k+j G α *u (x)/ x k u j , x∈ℝ, are of bounded variation on the whole real line ℝ. The proof, along with an integral recursion in j, is new even in the special case of stable laws, and the result provides a framework for possible asymptotic expansions in merge theorems from the domain of geometric partial attraction of semistable laws. An erratum to this article can be found at  相似文献   

20.
The paper studies the region of values Dm,1(T) of the system {ƒ(z1), ƒ(z2), …, ƒ(zm), ƒ(r)}, m e 1, where zj (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a2z2+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set Dm,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(zm) in the subclass of functions from the class T with prescribed values ƒ(zk) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.  相似文献   

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