Spectral expansion solution

{{Use American English|date = January 2019}}

{{Short description|Means of solving M/M/c queue models in queueing theory}}

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.{{Cite journal | last1 = Chakka | first1 = R. | journal = Annals of Operations Research | volume = 79 | pages = 27–44 | doi = 10.1023/A:1018974722301 | year = 1998 | title = Spectral expansion solution for some finite capacity queues}} 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.{{Cite journal | last1 = Mitrani | first1 = I. | last2 = Chakka | first2 = R. | doi = 10.1016/0166-5316(94)00025-F | title = Spectral expansion solution for a class of Markov models: Application and comparison with the matrix-geometric method | journal = Performance Evaluation| volume = 23 | issue = 3 | pages = 241 | year = 1995 }}{{cite book |first1=J.|last1= Daigle|first2= D. | last2=Lucantoni | chapter=Queueing systems having phase-dependent arrival and service rates | title=Numerical Solutions of Markov Chains | editor-first1=William J. | editor-last1=Stewart| year= 1991 | pages= 161–202 | isbn = 9780824784058}}