arXiv subjects
Publications and source records attributed to Vyacheslav M. Abramov.
The paper disproves a basic theorem on quasi-birth-and-death processes given in [M. F. Neuts (1995). Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover, New York].
In the present paper we define conservative and semiconservative random walks in $\mathbb{Z}^d$ and study different families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (Pólya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in $\mathbb{Z}^d$ by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in $\mathbb{Z}^d$ for any $d\geq3$ and transient random walks in $\mathbb{Z}^2$.
This paper studies a discrete model of a large dam where the difference between lower and upper levels, $L$, is assumed to be large. Passage across the levels leads to damage, and the damage costs of crossing the lower or upper level are proportional to the large parameter $L$. Input stream of water is described by compound Poisson process, and the water cost depends upon current level of water in the dam. The aim of the paper is to choose the parameters of output stream (specifically defined in the paper) minimizing the long-run expenses that include the damage costs and water costs. The present paper addresses the important question \emph{How does the structure of water costs affect the optimal solution?} We prove the existence and uniqueness of a solution. A special attention is attracted to the case of linear structure of the costs. As well, the paper contributes to the theory of state-dependent queueing systems. The inter-relations between important characteristics of a state-dependent queueing system are established, and their asymptotic analysis that involves analytic techniques of Tauberian theory and heavy traffic approximations is provided.
Conservative and semiconservative random walks in $\mathbb{Z}^d$ were introduced and studied in [V.M. Abramov, J. Theor. Probab. (2017). https://doi.org/10.1007/s10959-017-0747-3]. In the present paper, we extend these concepts for random walks in $\mathbb{R}^d$ introducing semiconservative random walks in weak sense and construct such a family of random walks in $\mathbb{R}^d$.
In this paper, we establish two different results. The first result is a characterization theorem saying that if the stationary state probabilities for originally described Markovian discriminatory processor sharing (DPS) system have a closed product geometric form (the exact definition is given in the paper), then the system must only be Egalitarian, i.e. all flows in this system must have equal priorities. The second result is the tail asymptotics for the stationary probabilities. We provide a detailed asymptotic analysis of the system, and obtain the exact asymptotic form of the stationary probabilities in DPS systems when the number of flows in the system is large.
In this paper, we prove a characterization theorem on the number of losses during a busy period in $GI^X/GI^Y/1/n$ queueing systems, in which interarrival time distribution belongs to the class NWUE.
Let $X_1$, $X_2$, $...$ be a sequence of independently and identically distributed random variables with $\mathsf{E}X_1=0$, and let $S_0=0$ and $S_t=S_{t-1}+X_t$, $t=1,2,...$, be a random walk. Denote $τ={cases}\inf\{t>1: S_t\leq0\}, &\text{if} \ X_1>0, 1, &\text{otherwise}. {cases}$ Let $α$ denote a positive number, and let $L_α$ denote the number of level-crossings from the below (or above) across the level $α$ during the interval $[0, τ]$. Under quite general assumption, an inequality for the expected number of level-crossings is established. Under some special assumptions, it is proved that there exists an infinitely increasing sequence $α_n$ such that the equality $\mathsf{E}L_{α_n}=c\mathsf{P}\{X_1>0\}$ is satisfied, where $c$ is a specified constant that does not depend on $n$. The result is illustrated for a number of special random walks. We also give non-trivial examples from queuing theory where the results of this theory are applied.
This paper studies large dam models where the difference between lower and upper levels, $L$, is assumed to be large. Passage across the levels leads to damage, and the damage costs of crossing the lower or upper level are proportional to the large parameter $L$. Input stream of water is described by compound Poisson process, and the water cost depends upon current level of water in the dam. The aim of the paper is to choose the parameters of output stream (specifically defined in the paper) minimizing the long-run expenses. The particular problem, where input stream is ordinary Poisson and water costs are not taken into account, has been studied in [Abramov, \emph{J. Appl. Prob.}, 44 (2007), 249-258]. The present paper addresses the question \textit{How does the structure of water costs affect the optimal solution?} Under natural assumptions we prove an existence and uniqueness of a solution and study the case of linear structure of the costs.
The $M/GI/m/n$ queueing system with $m$ homogeneous servers and the finite number $n$ of waiting spaces is studied. Let $λ$ be the customers arrival rate, and let $μ$ be the reciprocal of the expected service time of a customer. Under the assumption $λ=mμ$ it is proved that the expected number of losses during a busy period is the same value for all $n\geq1$, while in the particular case of the Markovian system $M/M/m/n$ the expected number of losses during a busy period is $\frac{m^m}{m!}$ for all $n\geq0$. Under the additional assumption that the probability distribution function of a service time belongs to the class NBU or NWU, the paper establishes simple inequalities for those expected numbers of losses in $M/GI/m/n$ queueing systems.
This paper studies a closed queueing network containing a hub (a state dependent queueing system with service depending on the number of units residing here) and $k$ satellite stations, which are $GI/M/1$ queueing systems. The number of units in the system, $N$, is assumed to be large. After service completion in the hub, a unit visits a satellite station $j$, $1\leq j\leq k$, with probability $p_j$, and, after the service completion there, returns to the hub. The parameters of service times in the satellite stations and in the hub are proportional to $\frac{1}{N}$. One of the satellite stations is assumed to be a bottleneck station, while others are non-bottleneck. The paper establishes the continuity of the queue-length processes in non-bottleneck satellite stations of the network when the service times in the hub are close in certain sense (exactly defined in the paper) to the exponential distribution.
In this article statistical bounds for certain output characteristics of the $M/GI/1/n$ and $GI/M/1/n$ loss queueing systems are derived on the basis of large samples of an input characteristic of these systems, such as service time in the $M/GI/1/n$ queueing system or interarrival time in the $GI/M/1/n$ queueing system. The analysis of this article is based on application of Kolmogorov's statistics for empirical probability distribution functions.
Let $\mathcal{G}(\frak{g}_1,\frak{g}_2)$ be the class of all probability distribution functions of positive random variables having the given first two moments $\frak{g}_1$ and $\frak{g}_2$. Let $G_1(x)$ and $G_2(x)$ be two probability distribution functions of this class satisfying the condition $|G_1(x)-G_2(x)|<ε$ for some small positive value $ε$ and let $\widehat{G}_1(s)$ and, respectively, $\widehat{G}_2(s)$ denote their Laplace-Stieltjes transforms. For real $μ$ satisfying $μ\frak{g}_1>1$ let us denote by $γ_{G_1}$ and $γ_{G_2}$ the least positive roots of the equations $z=\widehat{G}_1(μ-μz)$ and $z=\widehat{G}_2(μ-μz)$ respectively. In the paper, the upper bound for $|γ_{G_1}-γ_{G_2}|$ is derived. This upper bound is then used to find lower and upper bounds for the loss probabilities in different large loss queueing systems.
Under the assumption that the initial population size of a Galton-Watson branching process increases to infinity, the paper studies asymptotic behavior of the population size before extinction. More specifically, we establish asymptotic properties of the conditional moments (which are exactly defined in the paper).
The book of Lajos Takács \emph{Combinatorial Methods in the Theory of Stochastic Processes} has been published in 1967. It discusses various problems associated with $$ P_{k,i}=\mathrm{P}{\sup_{1\leq n\leqρ(i)}(N_n-n) 0$, and $ρ(i)$ is the smallest $n$ such that $N_n=n-i$, $i\geq1$. (If there is no such $n$, then $ρ(i)=\infty$.) (*) is a discrete generalization of the classic ruin probability, and its value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence $\{Q_k\}_{k\geq0}$ satisfies the recurrence relation of convolution type: $Q_0\neq0$ and $Q_k=\sum_{j=0}^kπ_jQ_{k-j+1}$. Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey concerns only with one of the areas of application associated with asymptotic behavior of $Q_k$ as $k\to\infty$. The theorem on asymptotic behavior of $Q_k$ as $k\to\infty$ and further properties of that limiting sequence are given on pages 22-23 of the aforementioned book by Takács. In the present survey we discuss applications of Takács' asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Many of the results presented in this survey have appeared recently, and some of them are new. In addition, further applications of Takács' theorem are discussed.
The paper studies a single-server queueing system with autonomous service and $\ell$ priority classes. Arrival and departure processes are governed by marked point processes. There are $\ell$ buffers corresponding to priority classes, and upon arrival a unit of the $k$th priority class occupies a place in the $k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total $k$th buffer content. The values $N^{(k)}$ are assumed to be large, and queueing systems both with finite and infinite buffers are studied. In the case of a system with finite buffers, the values $N^{(k)}$ characterize buffer capacities. The paper discusses a circle of problems related to optimization of performance measures associated with overflowing the quota of buffer contents in particular buffers models. Our approach to this problem is new, and the presentation of our results is simple and clear for real applications.
In this paper continuity theorems are established for the number of losses during a busy period of the $M/M/1/n$ queue. We consider an $M/GI/1/n$ queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution.
Consider a dam model, $L^{upper}$ and $L^{lower}$ are upper and, respectively, lower levels, $L = L^{upper}-L^{lower}$ is large and if the level of water is between these bounds, then the dam is said to be in a normal state. Passage across lower or upper levels leads to damage. Let $J_1=j_1L$ and $J_2=j_2L$ denote the damage costs per time unit of crossing the lower and, correspondingly, upper level where $j_1$ and $j_2$ are given real constants. It is assumed that input stream of water is described by a Poisson process, while the output stream is state dependent. Let $L_t$ denote the level of water in time $t$, and $c_{L_t}$ denote the water cost at level $L_t$ ($L^{lower} L^{upper}\}$ and $q_i=\lim_{t\to\infty}\mathbf{P}\{L_t=i\}$ ($L^{lower}<i\leq L^{upper}$) exist, the aim of the paper is to choose the parameters of an output stream (specifically defined in the paper) minimizing the long-run expenses $$J=p_1J_1+p_2J_2+\sum_{i=L^{lower}+1}^{L^{upper}}q_ic_i.$$
The paper studies asymptotic behavior of the loss probability for the $GI/M/m/n$ queueing system as $n$ increases to infinity. The approach of the paper is based on applications of classic results of Takács (1967) and the Tauberian theorem with remainder of Postnikov (1979-1980) associated with the recurrence relation of convolution type. The main result of the paper is associated with asymptotic behavior of the loss probability. Specifically it is shown that in some cases (precisely described in the paper) where the load of the system approaches 1 from the left and $n$ increases to infinity, the loss probability of the $GI/M/m/n$ queue becomes asymptotically independent of the parameter $m$.