Gagan Somashekar
Anshul Gandhi
Abstract
Continuous Time Markov chains (CTMCs) are widely used to model and analyze networked systems. A common analysis approach is to solve the system of balance equations governing the state transitions of a CTMC to obtain its steadystate probability distribution, and use the state probabilities to derive or compute various performance measures.
In many systems, the state space of the underlying CTMC is infinite and multi-dimensional with state-dependent transitions; exact analysis of such models is challenging. For example, the exact probability distribution of the number of jobs in the Discriminatory Processor Sharing (DPS) system, first proposed by Kleinrock in 1967 [4], is still an open challenge. Likewise, obtaining the exact state probabilities of quasi-birth-and-death (QBD) processes with leveldependent transitions is known to be challenging [1]; QBDs are infinite state space multi-dimensional Markov chains in which states are organized into levels and transitions are skip-free between the levels.
- Hendrik Baumann and Werner Sandmann. Numerical solution of level dependent quasi-birth-and-death processes. Procedia Computer Science, 1(1):1561--1569, 2010.Google Scholar
Cross Ref
- G. De Veciana and T. Konstantopoulos. Stability and performance analysis of networks supporting elastic services. IEEE/ACM Transactions on Networking, 9(1):2--14, 2001.Google ScholarDigital Library
- T. Dayar et al. Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics. Advances in Applied Probability, 43(4):1005--1026, 2011.Google ScholarCross Ref
- Leonard Kleinrock. Time-shared systems: A theoretical treatment. J. ACM, 14(2):242--261, 1967.Google ScholarDigital Library
- Kiran M. Rege and Bhaskar Sengupta. Queue-length distribution for the discriminatory processor-sharing queue. Operations Research, 44(4):653--657, 1996.Google ScholarDigital Library
- Petra Vis. Performance analysis of multi-class queueing models. PhD thesis, September 2017.Google Scholar
Index Terms
- Tighter Lyapunov Truncation for Multi-Dimensional Continuous Time Markov Chains with Known Moments
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