Spectral expansion solution explained

In probability theory, the spectral expansion solution method is a technique for computing the stationary probability distribution of a continuous-time Markov chain whose state space is a semi-infinite lattice strip.^[1] For example, an 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]}

Notes and References

1. Chakka . R. . Annals of Operations Research . 79 . 27–44 . 10.1023/A:1018974722301 . 1998 . Spectral expansion solution for some finite capacity queues.
2. Mitrani . I. . Chakka . R. . 10.1016/0166-5316(94)00025-F . Spectral expansion solution for a class of Markov models: Application and comparison with the matrix-geometric method . Performance Evaluation. 23 . 3 . 241 . 1995 .
3. Book: J.. Daigle. D. . Lucantoni . Queueing systems having phase-dependent arrival and service rates . Numerical Solutions of Markov Chains . William J. . Stewart. 1991 . 161–202 . 9780824784058.