Quasi-birth-and-death (QBD) processes with infinite “phase spaces” can exhibit unusual and interesting behavior. One of the simplest examples of such a process is the two-node tandem Jackson network, with the “phase” giving the state of the first queue and the “level” giving the state of the second queue.
In this paper, we undertake an extensive analysis of the properties of this QBD. In particular, we investigate the spectral properties of Neuts’s R-matrix and show that the decay rate of the stationary distribution of the “level” process is not always equal to the convergence norm of R. In fact, we show that we can obtain any decay rate from a certain range by controlling only the transition structure at level zero, which is independent of R.
We also consider the sequence of tandem queues that is constructed by restricting the waiting room of the first queue to some finite capacity, and then allowing this capacity to increase to infinity. We show that the decay rates for the finite truncations converge to a value, which is not necessarily the decay rate in the infinite waiting room case.
Finally, we show that the probability that the process hits level n before level 0 given that it starts in level 1 decays at a rate which is not necessarily the same as the decay rate for the stationary distribution.
References
Burke, P. J. (1956). The output of a queueing system. Oper. Res. 4 699--704.
Mathematical Reviews (MathSciNet):
MR83416
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
Mathematical Reviews (MathSciNet):
MR481884
Fujimoto, K., Takahashi, Y. and Makimoto, N. (1998). Asymptotic properties of stationary distributions in two-stage tandem queueing systems. J. Oper. Res. Soc. Japan 41 118--141.
Gail, H. R., Hantler, S. L. and Taylor, B. A. (1996). Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains. Adv. in Appl. Probab. 28 114--165.
Jackson, J. R. (1957). Networks of waiting lines. Oper. Res. 5 518--521.
Mathematical Reviews (MathSciNet):
MR93061
Kroese, D. P. and Nicola, V. F. (2002). Efficient simulation of a tandem Jackson network. ACM Transactions on Modeling and Computer Simulation (TOMACS) 12 119--141.
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix-Analytic Methods in Stochastic Modelling. ASA-SIAM, Philadelphia.
Latouche, G. and Taylor, P. G. (2000). Level-phase independence in processes of $GI/M/1$ type. J. Appl. Probab. 37 984--998.
Latouche, G. and Taylor, P. G. (2002). Truncation and augmentation of level-independent QBD processes. Stochastic Process. Appl. 99 53--80.
Latouche, G. and Taylor, P. G. (2003). Drift conditions for matrix-analytic models. Math. Oper. Res. 28 346--360.
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins Univ. Press.
Mathematical Reviews (MathSciNet):
MR618123
Neuts, M. F. (1989). Structured Stochastic Matrices of $M/G/1$ Type and Their Applications. Dekker, New York.
Ramaswami, V. (1997). Matrix analytic methods: A tutorial overview with some extensions and new results. In Matrix-Analytic Methods in Stochastic Models (S. R. Chakravarthy and A. S. Alfa, eds.) 261--296. Dekker, New York.
Ramaswami, V. and Taylor, P. G. (1996). Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stoch. Models 12 143--164.
Rudin, W. (1973). Functional Analysis. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet):
MR365062
Seneta, E. (1981). Nonnegative Matrices and Markov Chains. Springer, New York.
Mathematical Reviews (MathSciNet):
MR719544
Takahashi, Y., Fujimoto, K. and Makimoto, N. (2001). Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with a countable number of phases. Stoch. Models 17 1--24.
Tweedie, R. L. (1982). Operator-geometric stationary distributions for Markov chains with application to queueing models. Adv. in Appl. Probab. 14 368--391.
Mathematical Reviews (MathSciNet):
MR650129
