Analyzing <Emphasis Type="Italic">M/M/</Emphasis>1 Queues with Perturbations in the Arrival Process

This paper investigates when the M/M/1 model can be used to predict accurately the operating characteristics of queues with arrival processes that are slightly different from the Poisson process assumed in the model. The arrival processes considered here are perturbed Poisson processes. The perturbations are deviations from the exponential distribution of the inter-arrival times or from the assumption of independence between successive inter-arrival times. An estimate is derived for the difference between the expected numbers in perturbed and M/M/1 queueing systems with the same traffic intensity. The results, for example, indicate that the M/M/1 model can predict the performance of the queue when the arrival process is perturbed by inserting a few short inter-arrival times, an occasional batch arrival or small dependencies between successive inter-arrival times. In contrast, the M/M/1 is not a good model when the arrival process is perturbed by inserting a few long inter-arrival times.     

Almost Sure Approximation of the Superposition of the Random Processes

This article presents sufficient conditions, which provide almost sure (a.s.) approximation of the superposition of the random processes S(N(t)), when càd-làg random processes S(t) and N(t) themselves admit a.s. approximation by a Wiener or stable Lévy processes. Such results serve as a source of numerous strong limit theorems for the random sums under various assumptions on counting process N(t) and summands. As a consequence we obtain a number of results concerning the a.s. approximation of the Kesten–Spitzer random walk, accumulated workload input into queuing system, risk processes in the classical and renewal risk models with small and large claims and use such results for investigation the growth rate and fluctuations of the mentioned processes.     

Hitting Times of Points and Intervals for Symmetric Lévy Processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function P ^x (T _B >t), where T _B is the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove sharp two-sided estimates of the transition density of the process killed after hitting B.     

Compound Geometric Distribution of Order <Emphasis Type="Italic">k</Emphasis>

The distribution of the number of trials until the first k consecutive successes in a sequence of Bernoulli trials with success probability p is known as geometric distribution of order k. Let T _k be a random variable that follows a geometric distribution of order k, and Y ₁,Y ₂,… a sequence of independent and identically distributed discrete random variables which are independent of T _k . In the present article we develop some results on the distribution of the compound random variable \(S_{k} =\sum_{t=1}^{T_{k}}Y_{t}\).     

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup _t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.     

Adjoining batch Markov arrival processes of a Markov chain

A batch Markov arrival process (BMAP) X* = (N, J) is a 2-dimensional Markov process with two components, one is the counting process N and the other one is the phase process J. It is proved that the phase process is a time-homogeneous Markov chain with a finite state-space, or for short, Markov chain. In this paper, a new and inverse problem is proposed firstly: given a Markov chain J, can we deploy a process N such that the 2-dimensional process X* = (N, J) is a BMAP? The process X* = (N, J) is said to be an adjoining BMAP for the Markov chain J. For a given Markov chain the adjoining processes exist and they are not unique. Two kinds of adjoining BMAPs have been constructed. One is the BMAPs with fixed constant batches, the other one is the BMAPs with independent and identically distributed (i.i.d) random batches. The method we used in this paper is not the usual matrix-analytic method of studying BMAP, it is a path-analytic method. We constructed directly sample paths of adjoining BMAPs. The expressions of characteristic (D _k , k = 0, 1, 2 · · ·) and transition probabilities of the adjoining BMAP are obtained by the density matrix Q of the given Markov chain J. Moreover, we obtained two frontal Theorems. We present these expressions in the first time.     

On λ-Fold Equipartite Oberwolfach Problem with Uniform Table Sizes

We consider the following generalization of the Oberwolfach problem:”At a gathering there are n delegations each having m people. Is it possible to arrange a seating of mn people present at s round tables T ₁, T ₂, . . . , T _s (where each T _i can accommodate \( t_i\geq3 \) people and \( \sum t_i = mn \)) for k different meals so that each person has every other person not in the same delegation for a neighbor exactly λ times?” For λ= 1, Liu has obtained the complete solution to the problem when all tables accommodate the same number t of people. In this paper, we give the completesolution to the problem for \( \lambda\geq2 \) when all tables have uniform sizes t.     

Functional weak convergence of partial maxima processes

For a strictly stationary sequence of nonnegative regularly varying random variables (X _n ) we study functional weak convergence of partial maxima processes \(M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]\) in the space D[0, 1] with the Skorohod J ₁ topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J ₁ and M ₁ functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition \(\mathcal {A}(a_{n})\) with the time component.     

The realization problem for tail correlation functions

For a stochastic process {X _t } _t∈T with identical one-dimensional margins and upper endpoint τ _up its tail correlation function (TCF) is defined through \(\chi ^{(X)}(s,t) = \lim _{\tau \to \tau _{\text {up}}} P(X_{s} > \tau \,\mid \, X_{t} > \tau )\). It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on T×T coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of χ are derived. If T is finite, the set of TCFs on T×T forms a convex polytope of \(\lvert T \rvert \times \lvert T \rvert \) matrices. Several general results reveal its complex geometric structure. Up to \(\lvert T \rvert = 6\) a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as \(\lvert T \rvert \geq 3\) grows.     

Real class sizes

Following Wiener, we consider the zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We show that the variance of the number of zeroes in a long horizontal rectangle [?T,T] × [a, b] is asymptotically between cT and CT², with positive constants c and C. We also supply with conditions (in terms of the spectral measure) under which the variance grows asymptotically linearly with T, as a quadratic function of T, or has intermediate growth. The results are compared with known results for real stationary Gaussian processes and other models.     

Randomness of the square root of 2 and the giant leap,part 2

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic equation with integer coefficients; say, \(\alpha = \sqrt 2\). Given any rational number 0 < x < 1 (say, x = 1/2) and any positive integer n, we count the number of elements of the sequence α, 2α, 3α, ..., nα modulo 1 that fall into the subinterval [0, x]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” nx from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ n ≤ N. Depending on α and x, we may need an extra additive correction of constant times logarithm of N; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of N. If N is large, the distribution of this renormalized counting number, as n runs in 1 ≤ n ≤ N, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as N tends to infinity. This is the main result of the paper (see Theorem 1.1).     

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T?=?(X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper is to compare the results provided by these two methods: to which extent can they lead to different orders? We consider three cases: T is any tournament, T is strongly connected, T has only one Slater order. For each one of these three cases, we specify the maximum of the symmetric difference distance between Slater orders and Copeland orders. More precisely, thanks to a result dealing with arc-disjoint circuits in circular tournaments, we show that this maximum is equal to n(n???1)/2 if T is any tournament on an odd number n of vertices, to (n ²???3n?+?2)/2 if T is any tournament on an even number n of vertices, to n(n???1)/2 if T is strongly connected with an odd number n of vertices, to (n ²???3n???2)/2 if T is strongly connected with an even number n of vertices greater than or equal to 8, to (n ²???5n?+?6)/2 if T has an odd number n of vertices and only one Slater order, to (n ²???5n?+?8)/2 if T has an even number n of vertices and only one Slater order.     

Singularities of the wave trace for the Friedlander model

There has recently been renewed interest in the trace formula–in particular, that of the initial case of GL(2)–due to counting applications in the function field case. For these applications, one needs a very precise form of the trace formula, with all terms computed explicitly. Our aim in this work is to compute the trace formula for GL(2) over a number field in as full detail as was done for the function field case and to give an accessible exposition, being motivated by these applications to counting, but also by pure curiosity as to the optimal form of this plastic formula. We also explain a correction argument in our context here of GL(2). The idea is to introduce a global summand which does not change the formula globally but changes the local weighted orbital integrals at the hyperbolic terms, so that their limit at the identity becomes a unipotent contribution to the trace formula. This gives a harmonious and pleasing form to the formula. Finally, we put the trace formula in an invariant form; thus all its terms are distributions whose value at a test function f ^y (x) = f(y ^?1 xy) is independent of y in GL(2,A).     

A conditional limit theorem for a bivariate representation of a univariate random variable and conditional extreme values

We first consider a real random variable X represented through a random pair (R,T) and a deterministic function u as X = R?u(T). Under quite weak assumptions we prove a limit theorem for (R,T) given X>x, as x tends to infinity. The novelty of our paper is to show that this theorem for the representation of the univariate random variable X permits us to obtain in an elegant manner conditional limit theorems for random pairs (X,Y) = R?(u(T),v(T)) given that X is large. Our approach allows to deduce new results as well as to recover under considerably weaker assumptions results obtained previously in the literature. Consequently, it provides a better understanding and systematization of limit statements for the conditional extreme value models.     

Continuous and Discrete Age-Replacement Policies

This paper considers the combined continuous and discrete age-replacement policies when units deteriorate with age and use: a unit is replaced preventively before failure at time T of age or at number N of uses, whichever occurs first. The expected cost rate C (T,N) is derived, and both optimum time T* and number N* to minimize C (T,N) are discussed. There exist finite and unique T* and N* when the use occurs in a Poisson process under suitable conditions.     

Limit theorems for supremum of Gaussian processes over a random interval

Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) x) is considered, as x →∞.     

Optimal replacement policy based on maximum repair time for a random shock and wear model

We study a \(\delta \) shock and wear model in which the system can fail due to the frequency of the shocks caused by external conditions, or aging and accumulated wear caused by intrinsic factors. The external shocks occur according to a Bernoulli process, i.e., the inter-arrival times between two consecutive shocks follow a geometric distribution. Once the system fails, it can be repaired immediately. If the system is not repairable in a pre-specific time D, it can be replaced by a new one to avoid the unnecessary expanses on repair. On the other hand, the system can also be replaced whenever its number of repairs exceeds N. Given that infinite operating and repair times are not commonly encountered in practical situations, both of these two random variables are supposed to obey general discrete distribution with finite support. Replacing the finite support renewal distributions with appropriate phase-type (PH) distributions and using the closure property associated with PH distribution, we formulate the maximum repair time replacement policy and obtain analytically the long-run average cost rate. Meanwhile, the optimal replacement policy is also numerically determined by implementing a two-dimensional-search process.     

On a system of piecewise linear functions with vanishing integrals on <Emphasis Type="Italic">R</Emphasis>

For an admissible sequence T we define an orthonormal system consisting of piecewise linear functions with vanishing integrals on R. Necessary and sufficient conditions on T are found for the corresponding system to be a basis in H¹(R).     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.