The MAP/PH/1/N queue with flows of customers as a model for traffic control in telecommunication networks

A single-server queueing model with finite buffer and flows of customers is considered. Flow means a group of customers which should be sequentially processed in the system. In contrast to the standard batch arrival when a whole group of customers arrives into the system at one epoch, we assume that the customers of an accepted flow arrive one by one in exponentially distributed times. Service time has Phase type (PH) distribution. Generation of flows is described by the Markov Arrival Process (MAP). A flow consists of a random number of customers. This number is geometrically distributed and is not known at a flow arrival epoch. The number of flows, which can be admitted into the system simultaneously, is subject to control. Accepted flow can be lost, with a given probability, in the case of any customer from this flow rejection. Analysis of the joint distribution of the number of flows and customers in the system, flow loss probability and sojourn time distribution is implemented by means of the matrix technique and method of catastrophes. The effect of control on the main performance measures of the system is demonstrated numerically. The influence of correlation in the arrival process of flows, variation of service time and probability of a flow loss in case of any customer from this flow rejection is illustrated.

[1]  Christoph Lindemann,et al.  Modeling IP traffic using the batch Markovian arrival process , 2003, Perform. Evaluation.

[2]  H. Kesten,et al.  Priority in Waiting Line Problems 1). II , 1957 .

[3]  Moon Ho Lee,et al.  Queueing Model with Time-Phased Batch Arrivals , 2007, International Teletraffic Congress.

[4]  Daniel P. Heyman,et al.  Modeling multiple IP traffic streams with rate limits , 2003, TNET.

[5]  Alexander N. Dudin,et al.  Mobile Networks Modeling the access to a wireless network at hot spots , 2005, Eur. Trans. Telecommun..

[6]  Alexander N. Dudin,et al.  A Single-Server Queueing System with Batch Markov Arrivals, Semi-Markov Service, and Finite Buffer: Its Characteristics , 2002 .

[7]  Srinivas R. Chakravarthy.  The Batch Markovian Arrival Process: A Review and Future Work , 2001 .

[8]  David M. Lucantoni,et al.  New results for the single server queue with a batch Markovian arrival process , 1991 .

[9]  Alexander N. Dudin,et al.  Optimal hysteretic control for a BMAP/SM/1/N queue with two operation modes , 2000 .

[10]  Richard J. Harris,et al.  A simple IP flow blocking model , 2005 .

[11]  C. Blondia.  The n/g/l finite capacity queue , 1989 .

[12]  Alma Riska,et al.  Efficient fitting of long-tailed data sets into hyperexponential distributions , 2002, Global Telecommunications Conference, 2002. GLOBECOM '02. IEEE.

[13]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[14]  Che Soong Kim,et al.  Lack of Invariant Property of the Erlang Loss Model in Case of MAP Input , 2005, Queueing Syst. Theory Appl..

[15]  Wolfgang Fischer,et al.  The Markov-Modulated Poisson Process (MMPP) Cookbook , 1993, Perform. Evaluation.

[16]  Zhisheng Niu,et al.  A vacation queue with setup and close-down times and batch Markovian arrival processes , 2003, Perform. Evaluation.

[17]  D. van Dantzig.  Chaînes de Markof dans les ensembles abstraits et applications aux processus avec régions absorbantes et au problème des boucles , 1955 .