IJPAM: Volume 31, No. 2 (2006)

THE $BMAP/PH/\infty$ QUEUE
AS AN INFINITE SERVER NETWORK

Dieter Baum
Subdepartment of Computer Science
Department IV
University of Trier
Trier, D-54286, GERMANY
e-mail: baum@uni-trier.de


Abstract.We consider the $BMAP/PH/\infty$ queue in its most general form as an open network of $w$ infinite-server stations with exponential service time distributions and Markov routing. Customers arrive from outside the network according to a spatial BMAP as defined in [5]. As a consequence, the starting vector of any phase type service time distribution in the $BMAP/PH/\infty$ station is customer specific and can depend on the batch size as well as the phase transition of the underlying BMAP phase process. We present results for the transient as well as the stationary behavior of the network by deriving expressions for the joint distributions and generating functions of customer numbers at arbitrary network stations. The approach is based on former studies on the $BMAP/G/\infty$ queue [2, 5]. It supplements the detailed analysis of the $BMAP/PH/\infty$ queue given by Masuyama and Takine [12]. The network oriented analysis may be relevant for telecommunications performance analyses.

Received: June 29, 2006

AMS Subject Classification: 60K25, 90B22

Key Words and Phrases: $BMAP/PH/\infty$ station, queueing networks in random environment, spatial BMAPs

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 31
Issue: 2