arXiv subjects

Vyacheslav M. Abramov

Publications and source records attributed to Vyacheslav M. Abramov.

46 records · Page 3Linked to original sources

arXiv2007

Large closed queueing networks in semi-Markov environment and its application

The paper studies closed queueing networks containing a server station and $k$ client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is $N$. The expected times between departures in client stations are $(Nμ_j)^{-1}$. After a service completion in the server station, a unit is transmitted to the $j$th client station with probability $p_{j}$ $(j=1,2,...,k)$, and being processed in the $j$th client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities $p_{j}$ $(j=1,2,...,k)$ and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.

math.PR↗

arXiv2007

The stability of join-the-shortest-queue models with general input and output processes

The paper establishes necessary and sufficient conditions for stability of different join-the-shortest-queue models including load-balanced networks with general input and output processes. It is shown, that the necessary and sufficient condition for stability of load-balanced networks is associated with the solution of a linear programming problem precisely formulated in the paper. It is proved, that if the minimum of the objective function of that linear programming problem is less than 1, then the associated load-balanced network is stable.

math.PR↗

arXiv2007

Prediction of missing observations by a control method

Consider a time series with missing observations but a known final point. Using control theory ideas we estimate/predict these missing observations. We obtain recurrence equations which minimize sum of squares of a control sequence. An advantage of this method is in easily computable formulae and flexibility of its application to different structures of missing data.

math.ST↗

arXiv2006

Optimal control of a large dam

A large dam model is an object of study of this paper. The parameters $L^{lower}$ and $L^{upper}$ are its lower and upper levels, $L=L^{upper}-L^{lower}$ is large, and if a current level of water is between these bounds, then the dam is assumed to be in normal state. Passage one or other bound leads to damage. Let $J_1$ $(J_2)$ denote the damage cost of crossing the lower (upper) level. It is assumed that input stream of water is described by a Poisson process, while the output stream is state-dependent (the exact formulation of the problem is given in the paper). Let $L_t$ denote the dam level at time $t$, and let $p_1=\lim_{t\to\infty}\mathbf{P}\{L_t= L^{lower}\}$, $p_2=\lim_{t\to\infty}\mathbf{P}\{L_t> L^{upper}\}$ exist. The long-run average cost $J=p_1J_1+p_2J_2$ is a performance measure. The aim of the paper is to choose the parameter of output stream (exactly specified in the paper) minimizing $J$.

math.PR↗

arXiv2006

Stochastic inequalities for single-server loss queueing systems

The present paper provides some new stochastic inequalities for the characteristics of the $M/GI/1/n$ and $GI/M/1/n$ loss queueing systems. These stochastic inequalities are based on substantially deepen up- and down-crossings analysis, and they are stronger than the known stochastic inequalities obtained earlier. Specifically, for a class of $GI/M/1/n$ queueing system, two-side stochastic inequalities are obtained.

math.PR↗

arXiv2006

Multiserver queueing systems with retrials and losses

The interest to retrial queueing systems is due to their application to telephone systems. The paper studies multiserver retrial queueing systems with $n$ servers. Arrival process is a quite general point process. An arriving customer occupies one of free servers. If upon arrival all servers are busy, then the customer waits for his service in orbit, and after random time retries more and more to occupy a server. The orbit has one waiting space only, and arriving customer, who finds all servers busy and the waiting space occupied, losses from the system. Time intervals between possible retrials are assumed to have arbitrary distribution (the retrial scheme is exactly explained in the paper). The paper provides analysis of this system. Specifically the paper studies optimal number of servers to decrease the loss proportion to a given value. The representation obtained for loss proportion enables us to solve the problem numerically. The algorithm for numerical solution includes effective simulation, which meets the challenge of rare events problem in simulation.

math.PR↗

arXiv2005

Analysis of Multiserver Retrial Queueing System: A Martingale Approach and an Algorithm of Solution

The paper studies a multiserver retrial queueing system with $m$ servers. Arrival process is a point process with strictly stationary and ergodic increments. A customer arriving to the system occupies one of the free servers. If upon arrival all servers are busy, then the customer goes to the secondary queue, orbit, and after some random time retries more and more to occupy a server. A service time of each customer is exponentially distributed random variable with parameter $μ_1$. A time between retrials is exponentially distributed with parameter $μ_2$ for each customer. Using a martingale approach the paper provides an analysis of this system. The paper establishes the stability condition and studies a behavior of the limiting queue-length distributions as $μ_2$ increases to infinity. As $μ_2\to\infty$, the paper also proves the convergence of appropriate queue-length distributions to those of the associated `usual' multiserver queueing system without retrials. An algorithm for numerical solution of the equations, associated with the limiting queue-length distribution of retrial systems, is provided.

math.PR↗

arXiv2005

Asymptotic analysis of the GI/M/1/n loss system as n increases to infinity

This paper provides the asymptotic analysis of the loss probability in the $GI/M/1/n$ queueing system as $n$ increases to infinity. The approach of this paper is alternative to that of the recent papers of Choi and Kim [2000] and Choi et al [2000] and based on application of modern Tauberian theorems with remainder. This enables us to simplify the proofs of the results on asymptotic behavior of the loss probability of the abovementioned paper of Choi and Kim [2000] as well as to obtain some new results.

math.PR↗

arXiv2005

Asymptotic Behavior of the Number of Lost Messages

The goal of the paper is to study asymptotic behavior of the number of lost messages. Long messages are assumed to be divided into a random number of packets which are transmitted independently of one another. An error in transmission of a packet results in the loss of the entire message. Messages arrive to the $M/GI/1$ finite buffer model and can be lost in two cases as either at least one of its packets is corrupted or the buffer is overflowed. With the parameters of the system typical for models of information transmission in real networks, we obtain theorems on asymptotic behavior of the number of lost messages. We also study how the loss probability changes if redundant packets are added. Our asymptotic analysis approach is based on Tauberian theorems with remainder.

math.PR↗

arXiv2005

A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are $r$ server stations. At the initial time moment all units are distributed in the server stations, and the $i$th server station contains $N_i$ units, $i=1,2,...,r$, where all the values $N_i$ are large numbers of the same order. The total number of client stations is equal to $k$. The expected times between departures in the client stations are small values of the order $O(N^{-1})$ ~ $(N=N_1+N_2+...+N_r)$. After service completion in the $i$th server station a unit is transmitted to the $j$th client station with probability $p_{i,j}$ ~ ($j=1,2,...,k$), and being served in the $j$th client station the unit returns to the $i$th server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.

math.PR↗