Transient solutions for multi-server queues with finite buffers

Abstract

Transient solutions for M/M/c queues are important for staffing call centers, police stations, hospitals and similar institutions. In this paper we show how to find transient solutions for M/M/c queues with finite buffers by using eigenvalues and eigenvectors. To find the eigenvalues, we create a system of difference equations where the coefficients depend on a parameter x. These difference equations allow us to search for all eigenvalues by changing x. To facilitate the search, we use Sturm sequences for locating the eigenvalues. We also show that the resulting method is numerically stable.

Closed Form Transient Solution of Continuous Time Markov Chains Through Uniformization

“…Note also that the vector B given by (18) has exactly the same form than in the DTMC (equation (12)). Thus, the remark 2 regarding the computation of the vector B for the DTMC, is applicable now changing the matrix P by Q.…”

“…This property only holds in some special cases, as in M/M/c queues. This fact has been exploited in [18] to obtain a close form transient solution for these class of queues.…”