Wyniki wyszukiwania - BazTech

1

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Zeifman A., Razumchik R., Satin Y., Kiseleva K., Korotysheva A., Korolev V.

International Journal of Applied Mathematics and Computer Science

|

2018

|

Vol. 28, no. 1

141--154

EN

In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

2

Estimation of Network Disordering Effects by In-depth Analysis of the Resequencing Buffer Contents in Steady-state

Pechinkin A., Razumchik R.

Journal of Telecommunications and Information Technology

|

2016

|

nr 1

53--41

EN

The paper is devoted to the analytic analysis of resequencing issue, which is common in packet networks, using queueing-theoretic approach. The authors propose the mathematical model, which describes the simplest setting of packet resequencing, but which allows one to make the first step in the in-depth-analysis of the queues dynamics in the resequencing buffer. Specifically consideration is given to N-server queueing system (N > 3) with single infinite capacity buffer and resequencing, which may serve as a model of packet reordering in packet networks. Customers arrive at the system according to Poisson flow, occupy one place in the buffer and receive service from one of the servers, which is exponentially distributed with the same parameter. The order of customers upon arrival has to be preserved upon departure. Customers, which violated the order are kept in resequencing buffer which also has infinite capacity. It is shown that the resequencing buffer can be considered as consisting of n, 1 ≤ n ≤ N −1, interconnected queues, depending on the number of busy servers, with i-th queue containing customers, which have to wait for i service completions before they can leave the system. Recursive algorithm for computation of the joint stationary distribution of the number of customers in the buffer and servers, and each queue in resequencing buffer are being obtained. Numerical examples, which show the dynamics of the characteristics of the queues in resequencing buffer are given.

3

Iterative Algorithm for Threshold Calculation in the Problem of Routing Fixed Size Jobs to Two Parallel Servers

Konovalov M., Razumchik R.

Journal of Telecommunications and Information Technology

|

2015

|

nr 3

32--38

EN

At present, solutions of many practical problems require signicant computational resources and systems (grids, clouds, clusters etc.), which provide appropriate means are constantly evolving. The capability of the systems to full quality of service requirements pose new challenges for the developers. One of the well-known approaches to increase system performance is the use of optimal scheduling (dispatching) policies. In this paper the special case of the general problem of nding optimal allocation policy in the heterogeneous n-server system processing xed size jobs is considered. There are two servers working independently at constant but di erent speeds. Each of them has a dedicated queue (of innite capacity) in front of it. Jobs of equal size arrive at the system. Inter-arrival times are i.i.d. random variables with general distribution with nite mean. Each job upon arrival must be immediately dispatched to one of the two queues wherefrom it will be served in FCFS manner (no pre-emption). The objective is the minimization of mean job sojourn time in the system. It is known that under this objective the optimal policy is of threshold type. The authors propose scalable fast iterative non-simulation algorithm for approximate calculation of the policy parameter (threshold). Numerical results are given.

4

Ergodicity and perturbation bounds for inhomogeneous birth and death processes with additional transitions from and to the origin

Zeifman A., Korotysheva A., Satin Y., Korolev V., Shorgin S., Razumchik R.

International Journal of Applied Mathematics and Computer Science

|

2015

|

Vol. 25, no. 4

787--802

EN

Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, time-dependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided.

5

Statistical Analysis of Message Delay in SIP Proxy Server

Abaev P., Razumchik R., Uglov I.

Journal of Telecommunications and Information Technology

|

2014

|

nr 4

79--87

EN

Single hop delay of SIP message going through SIP proxy server operating in carriers backbone network is being analyzed. Results indicate that message sojourn times inside SIP server in most cases do not exceed order of tens of milliseconds (99% of all SIP-I messages experience less than 21 ms of sojourn delay) but there were observed very large delays which can hardly be attributed to message specic processing procedures. It is observed that delays are very variable. Delay components distribution that is to identied are not exponentially distributed or nearly constant even per message type or size. The authors show that measured waiting time and minimum transit time through SIP server can be approximated by acyclic phase-type distributions but accuracy of approximation at very high values of quantiles depends on the number outliers in the data. This nding suggests that modeling of SIP server with queueing system of GjPHjc type may server as an adequate solution.

6

Analysis of an M|G|1|R queue with batch arrivals and two hysteretic overload control policies

Gaidamaka Y., Pechinkin A., Razumchik R., Samouylov K., Sopin E.

International Journal of Applied Mathematics and Computer Science

|

2014

|

Vol. 24, no. 3

519--534

EN

Hysteretic control of arrivals is one of the most easy-to-implement and effective solutions of overload problems occurring in SIP-servers. A mathematical model of an SIP server based on the queueing system [...] with batch arrivals and two hysteretic loops is being analyzed. This paper proposes two analytical methods for studying performance characteristics related to the number of customers in the system. Two control policies defined by instants when it is decided to change the system’s mode are considered. The expression for an important performance characteristic of each policy (the mean time between changes in the system mode) is presented. Numerical examples that allow comparison of the efficiency of both policies are given.

7

Statistical analysis and modeling of SIP traffic for parameter estimation of server hysteretic overload control

Abaev P., Razumchik R., Uglov I.

Journal of Telecommunications and Information Technology

|

2013

|

nr 4

22--31

EN

The problem of overload control in Session Initiation Protocol (SIP) signaling networks gives rise to many questions which attract researchers from theoretical and practical point of view. Any mechanism that is claimed to settle this problem down demands estimation of local (control) parameters on which its performance is greatly dependent. In hysteretic mechanism these parameters are those which define hysteretic loops. In order to find appropriate values for parameters one needs adequate model of SIP traffic flow circulating in the network under consideration. In this paper the attempt is made to address this issue. Analysis of SIP traffic collected from telecommunication operator’s network is presented. Traffic profile is built. It is shown that fitting with Markov Modulated Poisson Process with more than 2 phases is accurate. Estimated values of its parameters are given.