Performance Evaluation 70 (2013) 663–681 Contents lists available at ScienceDirect Performance Evaluation journal homepage: www.elsevier.com/locate/peva On the numerical solution of Kronecker-based infinite level-dependent QBD processes H. Baumann a , T. Dayar b,∗ , M.C. Orhan b , W. Sandmann c a Department of Applied Stochastics and Operations Research, Clausthal University of Technology, D-38678 Clausthal–Zellerfeld, Germany b Department of Computer Engineering, Bilkent University, TR-06800 Bilkent, Ankara, Turkey c Campus E1 3, Room 325, Saarland University, D-66123 Saarbrücken, Germany article info Article history: Available online 11 June 2013 Keywords: Markov chain Level-dependent QBD process Kronecker product Matrix analytic method Steady-state expectation Call center abstract Infinite level-dependent quasi-birth-and-death (LDQBD) processes can be used to model Markovian systems with countably infinite multidimensional state spaces. Recently it has been shown that sums of Kronecker products can be used to represent the nonzero blocks of the transition rate matrix underlying an LDQBD process for models from stochastic chemical kinetics. This paper extends the form of the transition rates used recently so that a larger class of models including those of call centers can be analyzed for their steady- state. The challenge in the matrix analytic solution then is to compute conditional expected sojourn time matrices of the LDQBD model under low memory and time requirements after truncating its countably infinite state space judiciously. Results of numerical experiments are presented using a Kronecker-based matrix-analytic solution on models with two or more countably infinite dimensions and rules of thumb regarding better implementations are derived. In doing this, a more recent approach that reduces memory requirements further by enabling the computation of steady-state expectations without having to obtain the steady-state distribution is also considered. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Continuous-time infinite level-dependent quasi-birth-and-death (LDQBD) processes [1–3] are continuous-time Markov chain (CTMC) processes that have block tridiagonal transition rate matrices of the form Q =        Q 0,0 Q 0,1 Q 1,0 Q 1,1 Q 1,2 . . . . . . . . . Q l,l−1 Q l,l Q l,l+1 . . . . . . . . .        when their states are appropriately ordered. This study is about their numerical steady-state analyses, and hereafter, we shall be omitting the terms continuous-time and infinite in designating the LDQBD models we consider since they are all continuous-time and infinite. ∗ Corresponding author. Tel.: +90 312 290 1981; fax: +90 312 266 4047. E-mail addresses: hendrik.baumann@tu-clausthal.de (H. Baumann), tugrul@cs.bilkent.edu.tr (T. Dayar), morhan@cs.bilkent.edu.tr (M.C. Orhan), sandmann@cs.uni-saarland.de (W. Sandmann). 0166-5316/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.peva.2013.05.001