A Multi-server Markovian Queueing Model with Primary and Secondary Services

We study a multi-server queueing model in which the arrivals occur according to a Markovian arrival process. An arriving customer either (a) is lost due to all main servers being busy; or (b) enters into service with one of the main servers and leaves the system (as a satisfied primary customer); (c) enters into a service with one of the main servers, gets service in self-service mode, and is impatient to get a final service with one of the main servers, may leave the system (as a dissatisfied secondary customer); or (d) enters into service with one of the main servers, gets service in self-service mode, becomes successful in getting a final service from one of the main servers, and departs. This queueing model is studied in a steady state

Optimal Threshold Control by the Robots of Web Search Engines with Obsolescence of Documents

A typical web search engine consists of three principal parts: crawling engine, indexing engine, and searching engine. The present work aims to optimize the performance of the crawling engine. The crawling engine finds new web pages and updates web pages existing in the database of the web search engine. The crawling engine has several robots collecting information from the Internet. We first calculate various performance measures of the system (e.g., probability of arbitrary page loss due to the buffer overflow, probability of starvation of the system, the average time waiting in the buffer). Intuitively, we would like to avoid system starvation and at the same time to minimize the information loss. We formulate the problem as a multi-criteria optimization problem and attributing a weight to each criterion. We solve it in the class of threshold policies. We consider a very general web page arrival process modeled by Batch Marked Markov Arrival Process and a very general service time modeled by Phase-type distribution. The model has been applied to the performance evaluation and optimization of the crawler designed by INRIA Maestro team in the framework of the RIAM INRIA-Canon research project

Multi-dimensional quasitoeplitz Markov chains

This paper deals with multi-dimensional quasitoeplitz Markov chains. We establish a sufficient equilibrium condition and derive a functional matrix equation for the corresponding vector-generating function, whose solution is given algorithmically. The results are demonstrated in the form of examples and applications in queues with BMAP-input, which operate in synchronous random environment

A Retrial BMAP/PH/N System

A multi-server retrial queueing model with Batch Markovian Arrival Process and phase-type service time distribution is analyzed. The continuous-time multi-dimensional Markov chain describing the behavior of the system is investigated by means of reducing it to the corresponding discrete-time multi-dimensional Markov chain. The latter belongs to the class of multi-dimensional quasi-Toeplitz Markov chains in the case of a constant retrial rate and to the class of multi-dimensional asymptotically quasi-Toeplitz Markov chains in the case of an infinitely increasing retrial rate. It allows to obtain the existence conditions for the stationary distribution and to elaborate the algorithms for calculating the stationary state probabilities

Queueing system with passive servers

In this paper the authors introduce systems in which customers are served by one active server and a group of passive servers. The calculation of response time for such systems is rendered by analyzing a special kind of queueing system in a synchronized random environment. For an embedded Markov chain, sufficient conditions for the existence of a stationary distribution are proved. A formula for the corresponding vector generating function is obtained. It is a matrix analog of the Pollaczek-Khinchin formula and is simultaneously a matrix functional equation. A method for solving this equation is proposed

Priority Multi-Server Queueing System with Heterogeneous Customers

In this paper, we analyze a multi-server queueing system with heterogeneous customers that arrive according to a marked Markovian arrival process. Customers of two types differ in priorities and parameters of phase type distribution of their service time. The queue under consideration can be used to model the processes of information transmission in telecommunication networks in which often the flow of information is the superposition of several types of flows with correlation of inter-arrival times within each flow and cross-correlation. We define the process of information transmission as the multi-dimensional Markov chain, derive the generator of this chain and compute its stationary distribution. Expressions for computation of various performance measures of the system, including the probabilities of loss of customers of different types, are presented. Output flow from the system is characterized. The presented numerical results confirm the high importance of account of correlation in the arrival process. The values of important performance measures for the systems with the correlated arrival process are essentially different from the corresponding values for the systems with the stationary Poisson arrival process. Measurements in many real world systems show poor approximation of real flows by such an arrival process. However, this process is still popular among the telecommunication engineers due to the evident existing gap between the needs of adequately modeling the real life systems and the current state of the theory of algorithmic methods of queueing theory

Queuing System with Two Types of Customers and Dynamic Change of a Priority

The use of priorities allows us to improve the quality of service of inhomogeneous customers in telecommunication networks, inventory and health-care systems. An important modern direction of research is to analyze systems in which priority of a customer can be changed during his/her stay in the system. We considered a single-server queuing system with a finite buffer, where two types of customers arrive according to a batch marked Markov arrival process. Type 1 customers have non-preemptive priority over type 2 customers. Low priority customers are able to receive high priority after the random amount of time. For each non-priority customer accepted into the buffer, a timer, which counts a random time having a phase type distribution, is switched-on. When the timer expires, the customer with some probability leaves the system unserved and with the complimentary probability gains the high priority. Such a type of queues is typical in many health-care systems, contact centers, perishable inventory, etc. We describe the behavior of the system by a multi-dimensional continuous-time Markov chain and calculate a number of the stationary performance measures of the system including the various loss probabilities as well as the distribution function of the waiting time of priority customers. The illustrative numerical examples giving insights into the system behavior are presented

The Queueing Model MAP/PH/1/N with Feedback Operating in a Markovian Random Environment

Queueing systems with feedback are well suited for the description of message transmission and manufacturing processes where a repeated service is required. In the present paper we investigate a rather general single server queue with a Markovian Arrival Process (MAP), Phase-type (PH)service-time distribution, a finite buffer and feedback which operates in a random environment. A finite state Markovian random environment affects the parameters of the input and service processes and the feedback probability. The stationary distribution of the queue and of the sojourn times as well as the loss probability are calculated. Moreover, Little’s law is derived