# Spectral expansion solution

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In [probability theory](/source/Probability_theory), the **spectral expansion solution method** is a technique for computing the stationary probability distribution of a [continuous-time Markov chain](/source/Continuous-time_Markov_chain) whose state space is a semi-infinite lattice strip.[1] For example, an [M/M/c queue](/source/M/M/c_queue) where service nodes can breakdown and be repaired has a two-dimensional state space where one dimension has a finite limit and the other is unbounded. The stationary distribution vector is expressed directly (not as a transform) in terms of eigenvalues and eigenvectors of a matrix polynomial.[2][3]

## References

1. Chakka, R. (1998). "Spectral expansion solution for some finite capacity queues". *Annals of Operations Research*. **79**: 27–44. [doi:10.1023/A:1018974722301](https://doi.org/10.1023/A:1018974722301)

1. Mitrani, I. & Chakka, R. (1995). "Spectral expansion solution for a class of Markov models: Application and comparison with the matrix-geometric method". *[Performance Evaluation](/source/Performance_Evaluation)*. **23** (3): 241. [doi:10.1016/0166-5316(94)00025-F](https://doi.org/10.1016/0166-5316(94)00025-F)

1. Daigle, J. & Lucantoni, D. (1991). "Queueing systems having phase-dependent arrival and service rates". *Numerical Solutions of Markov Chains*. Stewart, William J. (ed.). pp. 161–202. ISBN 9780824784058.

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