A conditional dichotomy theorem for stochastic processes with independent increments

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M_t and N_t, respectively, and let the “time-jump” measures M and N be defined over [0, T] × $\text{R}$?{0} by M((t₁, t₂] × A) = M_t2(A) ? M_t1(A) and N((T₁, t₂] × A) = N_t2(A) ? N_t1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.     

M/M/c Retrial Queue with Multiclass of Customers

We consider the M/M/c retrial queues with multiclass of customers. We show that the stationary joint distribution for the number of customers in service facility and orbit converges to those of the ordinary M/M/c with discriminatory random order service (DROS) policy as retrial rate tends to infinity. Approximation formulae for the distributions of the number of customers in service facility, the mean number of customers in orbit and the sojourn time distribution of a customer are presented. The approximations are compared with exact and simulation results.     

The genus of a module

Let R be a Dedekind domain satisfying the Jordan-Zassenhaus theorem (e.g., the ring of integers in a number field) and Λ a module finite R-algebra. We extend classical results of Jacobinski, Roiter, and Drozd on orders and lattices. In particular, it is shown that the genus of a finitely generated Λ-module M is finite. Moreover, given M, there exist a positive integer t and a finite extension S of R such that a Λ-module N is the genus of M if and only if M^(t) ? N^(t) if and only if M ? S ? N ? S.     

A Discrete-Time Geo /G/1 Retrial Queue with General Retrial Times

We consider a discrete-time Geo/G/1 retrial queue in which the retrial time has a general distribution and the server, after each service completion, begins a process of search in order to find the following customer to be served. We study the Markov chain underlying the considered queueing system and its ergodicity condition. We find the generating function of the number of customers in the orbit and in the system. We derive the stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions for our queueing system and its corresponding standard system. Also, we develop recursive formulae for calculating the steady-state distribution of the orbit and system sizes. Besides, we prove that the M/G/1 retrial queue with general retrial times can be approximated by our corresponding discrete-time system. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics.     

The parametrized family of metric Mahler measures

Let M(α) denote the (logarithmic) Mahler measure of the algebraic number α. Dubickas and Smyth, and later Fili and the author, examined metric versions of M. The author generalized these constructions in order to associate, to each point in t∈(0,∞], a metric version Mt of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions Mt, using them to present an equivalent form of Lehmer?s conjecture. We show that the function t?Mtt(α) is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph t?Mt(α) for rational α.     

Approximation of M/M/s/K retrial queue with nonpersistent customers

We consider the M/M/s/K retrial queues in which a customer who is blocked to enter the service facility may leave the system with a probability that depends on the number of attempts of the customer to enter the service facility. Approximation formulae for the distributions of the number of customers in service facility, waiting time in the system and the number of retrials made by a customer during its waiting time are derived. Approximation results are compared with the simulation.     

A G/M/1 retrial queue with constant retrial rate

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.     

Splitting trees with neutral Poissonian mutations I: Small families

We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree [9], and the population counting process (N_t;t≥0) is a homogeneous, binary Crump-Mode-Jagers process.We assume that individuals independently experience mutations at constant rate θ during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called an allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,…,N_t.We mainly use two classes of tools: coalescent point processes, as defined in [15], and branching processes counted by random characteristics, as defined in [11] and [13]. We provide explicit formulae for the expectation of A(k,t) conditional on population size in a coalescent point process, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics, in the same vein as in [19].Last, we separately compute the expected homozygosity by applying a method introduced in [14], characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.     

Single server retrial queues with two way communication

The main aim of this paper is to study the steady state behavior of an M/G/1-type retrial queue in which there are two flows of arrivals namely ingoing calls made by regular customers and outgoing calls made by the server when it is idle. We carry out an extensive stationary analysis of the system, including stability condition, embedded Markov chain, steady state joint distribution of the server state and the number of customers in the orbit (i.e., the retrial group) and calculation of the first moments. We also obtain light-tailed asymptotic results for the number of customers in the orbit. We further formulate a more complicate but realistic model where the arrivals and the service time distributions are modeled in terms of the Markovian arrival process (MAP) and the phase (PH) type distribution.     

Some structural properties of a Markovian storage/production system

We consider a storage/production system with state-dependent production rate and state-dependent demand arrival rate. Every arriving demand gives rise to a 'peak' in the trajectory of the content process. We characterize the processes N ₀(x)and N₁(x), defined as the number of peaks and the number of record peaks, respectively, before the content reaches the level x. The results are applied to the virtual waiting time process W(t) of a M/G/1 queue. Assuming that W(0)= x₀, M(x) is defined to be the number of arrivals before the virtual waiting time drops from x₀ to x₀-x(0⩽ x ⩽ x₀) ; in particular, Mx₀)is the number of customers arriving during the first busy period. It is shown that (M(x))(0⩽ x ⩽ x₀)is a compound Poisson process, and its jump size distribution is derived in closed form. This revised version was published online in June 2006 with corrections to the Cover Date.     

Approximation of M/M/c retrial queue with PH-retrial times

We consider the M/M/c retrial queues with PH-retrial times. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. Some numerical results are presented.     

Distribution and characteristic functions for correlated complex Wishart matrices

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t₁, t₂, where t₁<t₂. In particular, we derive the following results: the joint density of the elements of A(t₁), A(t₂), the joint density of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂), the characteristic function of the elements of A(t₁), A(t₂), the characteristic function of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂). In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.     

Random evolutions in discrete and continuous time

This paper points out a connection between random evolutions and products of random matrices. This connection is useful in predicting the long-run growth rate of a single-type, continuously changing population in randomly varying environments using only observations at discrete points in time. A scalar Markov random evolution is specified by the n×n irreducible intensity matrix or infinitesimal generator Q = (q_ij) of a time-homogeneous Markov chain and by n finite real growth rates (scalars) s_i. The scalar Markov random evolution is the quantity M^C(t) = exp(Σⁿ_j=1s_jg^C_j (t)), where g^C_j(t) is the occupancy times in state j up to time t. The scalar Markov product of random matrices induced by this scalar Markov random evolution is the quantity M^D(t) = exp(Σⁿ_j=1s_jg^D_j (t)), where g^D_j(t) is the occupancy time in state j up to and including t of the discrete-time Markov chain with stochastic one-step transition matrix P = e^Q. We show that lim_t→∞(1/t)E(logM^D(t))=lim_t→∞(1/t)E(logM^C(t)) but that in general lim_t→∞(1/t)logE(M^C(t)) ≠ lim_t→∞(1/t)logE(M^D(t)). Thus the mean Malthusian parameter of population biologists is invariant with respect to the choice of continuous or discrete time, but the rate of growth of average population size is not. By contrast with a single-type population, in multitype populations whose growth is governed by non-commuting operators, the mean Malthusian parameter may be destined for a less prominent role as a measure of long-run growth.     

On products of matroids

We show, by means of counterexamples, that products with rank rk(M)rk(N) of a matroid M by a matroid N do not exist in general, and that there is no free-est product of M by N. We prove that a canonical product of M by N (having rank rk(M)+rk(N)?1) is a free-est product in a certain (weaker) sense.     

Betti and Tachibana numbers

The Tachibana numbers t _r (M), the Killing numbers k _r (M), and the planarity numbers p _r (M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n ? 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers b _r (M) is investigated. In particular, it is proved that if b _r (M) = 0, then the corresponding Tachibana number has the form t _r (M) = k _r (M) + p _r (M) for t _r (M) > k _r (M) > 0. In the special case where b ₁(M) = 0 and t ₁(M) > k ₁(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.     

Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

In this paper we set up a representation theorem for tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky Fan norms. Examples of tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property include unitarily invariant norms on finite factors (type II₁ factors and Mn(C)) and symmetric gauge norms on L^∞[0,1] and Cn. As the first application, we obtain that the class of unitarily invariant norms on a type II₁ factor coincides with the class of symmetric gauge norms on L^∞[0,1] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286-300] on unitarily invariant norms on Mn(C). As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766] is obtained for finite von Neumann algebras satisfying the weak Dixmier property. As the third application, some classical results in non-commutative Lp-theory (e.g., non-commutative Hölder's inequality, duality and reflexivity of non-commutative Lp-spaces) are obtained for general unitarily invariant norms on finite factors. We also investigate the extreme points of N(M), the convex compact set (in the pointwise weak topology) of normalized unitarily invariant norms (the norm of the identity operator is 1) on a finite factor M. We obtain all extreme points of N(M₂(C)) and some extreme points of N(Mn(C)) (n?3). For a type II₁ factor M, we prove that if t (0?t?1) is a rational number then the Ky Fan tth norm is an extreme point of N(M).     

Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule

We consider a single server queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under the so called linear retrial policy. This model extends both the classical M/G/1 retrial queue with linear retrial policy as well as the M/G/1 queue with two phases of service and Bernoulli vacation model. We carry out an extensive analysis of the model.     

Existence of a double random integral with respect to stable measures

In this paper, the authors study a double random integral of the form ∫₀¹∫₀¹f(s,t) M(ds) M(dt), where M(0,t) is a stable process with independent increments. Basically, the Wiener approach is used, and the existence of the above integral is established for a wide class of functions f.     

Maximum and minimum of one-dimensional diffusions

Let M_t be the maximum of a recurrent one-dimensional diffusion up till time t. Under appropriate conditions, there exists a distribution function F such that |P(M_t?x) ? F^t(x)|→0as t and x go to infinity. This reduces the asymptotic behavior of the maximum to that of the maximum of independent and identically distributed random variables with distribution function F. A new proof of this fact is given which is based on a time change of the Ornstein-Uhlenbeck process. Using this technique, the asymptotic independence of the maximum and minimum is also established. Moreover, this method allows one to construct stationary processes in which the limiting behavior of M_t is essentially unaffected by the stationary distribution. That is, there may be no relationship between the distribution F above and the marginal distribution of the process.