Operational queueing system in the scheme of diffusion approximation with evolution averaging

We consider an operational queuing system of the type [SM|M|∞]^N in the scheme of diffusion approximation. The queueing system is described by a semi-Markov random evolution. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 708–714, May, 2006.     

Limit theorems for systems of the type <Emphasis Type="Italic">M</Emphasis><Superscript>θ</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">b</Emphasis> with resume level of input stream

We consider a queuing system of the type M ^θ/G/1/b in which the input stream is regulated by a certain threshold level. The asymptotic properties of the first busy period and the number of calls served during this period are studied. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 884–889, July, 2007.     

Maslov probability spaces related to the Bose—Einstein and Fermi—Dirac statistics

For systems of indistinguishable particles, we describe probability spaces factored by the equivalence relations identifying configurations which differ by permutation of particles, under the condition that identical states are forbidden (Fermi—Dirac statistics) or admissible (Bose—Einstein statistics). It is assumed that the states of particles have different probabilities; these correspond either to the presence of an external potential, or to a pair interaction potential, or to a collective interaction. The spaces constructed in the paper are related to specific queuing models. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 746–759, May, 1999.     

On the Full Idle Time in Multiphase Queueing Systems

The modern queueing theory is a powerful tool for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems arising in the network theory and communications theory (such as the so-called multiphase queueing systems, tandem queues, or series of queueing systems). We present heavy traffic limit theorems for the full idle time in multiphase queueing systems. We prove functional limit theorems for values of the full idle time of a queueing system, which is its important probability characteristic. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 367–386, July–September, 2005.     

A limit theorem for mixing processes subject to rarefaction. II

The limit theorem proved in the first part of this paper is applied to the well-known schemes of processes subject to rarefaction arising in queuing theory, mathematical biology, and in problems for counters. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 603–612, May, 1998.     

Rate of convergence to ergodic distribution for queue length in systems of the type <Emphasis Type="Italic">M</Emphasis><Superscript>θ</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis>

For finite-capacity queuing systems of the type M ^θ/G/1, convenient formulas for the ergodic distribution of the queue length are found, an estimate for the rate of convergence of the distribution of the queue length in the transient mode to the ergodic distribution is obtained, and computational algorithms for finding the rate of convergence are presented. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1169–1178, September, 2007.     

Compound Transient Phenomena in Multiphase Queuing Systems. III

The author continues the work on functional limit theorems in multiphase queuing systems (QS) under heavy traffic. In this paper there are proved theorems for the waiting time of a job when at phases of a system various conditions of heavy traffic are satisfied (compound transient phenomena).     

About the Sojourn Time Process in Multiphase Queueing Systems

Multiphase queueing systems (MQS) (tandem queues, queues in series) are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message switching systems, retransmission of video images, asembly lines, etc.). In this paper, we deal with approximations of MQS and present a heavy traffic limit theorems for the sojourn time of a customer in MQS. Functional limit theorems are proved for the customer sojourn time – an important probability characteristic of the queueing system under conditions of heavy traffic.      

Simulation modelling of abnormal activities in computer networks

The state of a network can be described by a set of configuration parameters. The suggested method consists in singling out some parameters in order to detect the on-going attack traffic. The results of the modelling are given in the paper. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 352, 2008, pp. 120–128.     

Asymptotic optimality of the Round–Robin policy in multipath routing with resequencing

We consider a model of a multipath routing system where arriving customers are routed to a set of identical, parallel, single server queues according to balancing policies operating without state information. After completion of service, customers are required to leave the system in their order of arrival, thus incurring an additional resequencing delay. We are interested in minimizing the end-to-end delay (including time at the resequencing buffer) experienced by arriving customers. To that end we establish the optimality of the Round–Robin routing assignment in two asymptotic regimes, namely heavy and light traffic: In heavy traffic, the Round–Robin customer assignment is shown to achieve the smallest (in the increasing convex stochastic ordering) end-to-end delay amongst all routing policies operating without queue state information. In light traffic, and for the special case of Poisson arrivals, we show that Round–Robin is again an optimal (in the strong stochastic ordering) routing policy. We illustrate the stochastic comparison results by several simulation examples. The work of the first author was supported through an ARCHIMEDES grant by the Greek Ministry of Education. The work of the second author was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.     

Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Formal asymptotics of eigenmodes for oscillating elastic spatial bodies with concentrated masses

Limit spectral problems are derived for the problem on oscillations of a solid with small heavy (or light) inclusions. The asymptotic ansatzs for eigenvalues and eigenvectors, as well as the limit problems, are crucially dependent on both the relation between the geometric and physical parameters and the disposition of the inclusions. It is established that, for heavy inclusions, the limit problems are united into a more complex resultant problem describing the “far action” in the set of inclusions. Bibliography: 39 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 31–76.     

The discrete-time queue with autoregressive inputs revisited

In this paper, we present an exact queuing analysis of a discrete-time queue whose arrival process is correlated and consists of a discrete autoregressive model of order 1 (DAR(1)). The functional equation describing this DAR(1)/D/1 queuing model, originally derived in Hwang and Sohraby (Queuing Systems 43 (2003)29–41), is manipulated and transformed into a mathematical tractable form. By using simple analytical transform techniques, we show how our proposed approach allows us to derive an equivalent (yet simpler) expression for the steady-state probability generating function (pgf) of the queue length, as originally derived in Hwang and Sohraby (Queuing Systems 43 (2003)29–41). From this pgf, we characterize the distribution of the packet delay. New numerical results related to packet loss ratio and mean delay of the DAR(1)/D/1 queue are also presented. The proposed approach outlines an alternate solution technique and a general framework under which more complex time-series based queuing models can be analyzed. AMS Subject Classifications 60K25     

Green’s model of cylindrical vortex decay

Modeling vortex decay is essential for estimating the risk that during landing an aircraft will hit the wake vortex from a preceding aircraft. The relevance of this problem has increased due to the growing traffic in airports and the development of vortex forecasting systems. Green’s model, one of the models describing the decay of point vortices, is generalized in this article to cylindrical vortices. __________ Translated from Prikladnaya Matematika i Informatika, No. 25, pp. 70–80, 2007.     

On the intrinsic normalizations of the seminonholonomic complexesSN Gr(m, n, (m+1)(n−m)−ϱ). I

Some intrinsic normalizations of seminonholonomic complexes of then-dimensional projective space are considered. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 517–538, October–December, 1999. Translated by R. Lapinskas     

Large deviations for integer centered poisson approximation

Large deviations in a centered Poisson approximation are examined. For lattice distributions, the centered Poisson approximation is more universal than the normal and standard Poisson laws. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 9–23, January–March, 1999. Translated by V. Čekanavičius     

Heavy traffic analysis of controlled multiplexing systems

The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The basic control problem for the physical system is hard, and the heavy traffic approach provides much simplification. Owing to the presence of the control, as well as to the fact that the cost function of main interest is “ergodic”, the problem cannot be fully treated with “classical” methods of heavy traffic analysis for queueing networks. A basic result is that the optimal average costs per unit time for the physical problem converge to the optimal cost per unit time for the limit stationary process as the number of sources and the time interval goes to infinity. This convergence is both in the mean and pathwise senses. Furthermore, a “nice” nearly optimal control for the limit system provides nearly optimal values for the physical system, under heavy traffic, in both a mean and pathwise sense. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

Standardization of distributions of the chord length and distance within oval domains

The distributions of the chord length and distance within oval domains are expressed in terms of the distributions of these values in a circle and of some parameters of oval domains. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 4, pp. 469–474, October–December, 1999. Translated by E. Gečiauskas     

On the Analysis of the Virtual Waiting Time in Open Queueing Networks

The model of an open queueing network in heavy traffic has been developed. These models are mathematical models of computer networks in heavy traffic. A limit theorem has been presented for the virtual waiting time of a customer in heavy traffic in open queueing networks. Finally, we present an application of the theorem—a reliability model from computer network practice.