Analysis of Two Queueing Models with Explicit Rate Operators and Stationary Distributions

10.6186/IJIMS.2011.22.2.5

Hong-Bo Zhang；Ding-Hua Shi

QBD process with countable phases ； rate operator ； stationary distribution ； the T-SPH/M/1 queue ； the M/T-SPH/1 queue

International Journal of Information and Management Sciences

22：2（2011 / 06 / 01）

177 - 188

英文

For QBD(quasi-birth-and-death) processes with countably many phases, it is well known that stationary distributions have operator-geometric forms. However, it is a challenging problem to determine the closed-form of both the rate operator (an infinite matrix) and stationary distribution for a given QBD process. In this paper, we will derive explicit rate operators and the stationary distributions for two special models: the T-SPH/M/1 queue and the M/T-SPH/1 queue, where T-SPH denotes the phase type distribution defined on the birth-and death process with countably many states.

1. Zhang, H. B.,Shi, D. H.(2010).Explicit Solution for M/M/1 Preemptive Queue.International Journal of Information and Management Sciences,21,197-208.
   連結：
2. Alfa, A. S.(2000).Discrete time queues and matrix-analytic methods.TOP,10,147-210.
3. Breuer, L.,Baum, D.(2005).An introductin to queueing theory and matrix-analytic methods.Springer.
4. Brualdi, R. A.(2004).Introductory Combinatorics.Prentice Hall.
5. Li, H.,Miyazawa, M.,Zhao, Y. Q.(2007).Geometric decay in a QBD process with countable background states with applications to a join-the-shortest-queue model.Stochastic Models,23,413-438.
6. Miller, D. R.(1981).Computation of steady-state probabilities for M/M/1 priority queues.Operations Research,29,945-958.
7. Neuts, M. F.(1981).Matrix-geometric solutions in stochastic models: an algorithmic Approach.Baltimore:The Johns Hopkins University Press.
8. Ramaswami, V.,Taylor, P. G.(1996).Some properties of the rate operators in level dependent quasi birth and death processes with a countable number of phases.Stochastic Models,12,143-164.
9. Sakuma, Y.,Miyazama, M.(2005).On the effect of finite buffer truncation in a two-node Jackson network.Journal of Applied Probability,42,199-222.
10. Shi, D. H.,Guo, J. L.,Liu, L. M.(1997).SPH-distributions and the rectangle-iterative algorithm.Matrix-analysis methods in stochastic models,New York:
11. Shi, D. H.,Guo, J. L.,Liu, L. M.(2005).On the SPH-distribution class.Acta Mathematica Scientia,25B,201-214.
12. Takahashi, Y.,Fujimoto, K.,Makimoto, N.(2001).Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with countable number of phases.Stochastic Models,1,1-24.
13. Tian, N. S.,Zhao, X. Q.(2008).The M/M/1 queue with single working vacation.International Journal of Information and Management Sciences,19,621-634.
14. Tweedie, R. L.(1982).Operator-geometric stationary distributions of Markov chains with application to queueing models.Advances in Applied Probability,14,368-391.
15. Zhang, H. B.,Shi, D. H.(2009).The M/M/1 qaueue with Bernoulli-schedule-controlled vacation and vacation interruption.International Journal of Information and Management Sciences,20,579-587.
16. Zhang, H. B.、Shi, D. H.(2008)。Tail analysis of the stationary queue length distributions for two T-SPH queues。OR Transactions，12，60-70。

1. Poongothai, V.,Arivudainambi, D.(2012).Solving Integer Programming Problems with a Variable Number of Switching Costs, Balking and Feedback.International Journal of Information and Management Sciences,23(4),395-407.