The input/output process of a queue (Q2756664)

The authors discuss various descriptions of the stationary input/output processes of a node in a queueing system. Let \(D(0), D(1)\) and \(D(2)\) be \(m \times m\) matrices. The matrix \(D(0)\) has negative diagonal elements. Its off-diagonal elements and all elements of \(D(1)\) and \(D(2)\) are nonnegative. The sum \(D = D(0) + D(1) +D(2)\) is an irreducible generator. The stationary probability vector of \(D\) is denoted by \(\theta.\) The Markovian arrival process (MAP) with two types of events is the point process defined by the transition epochs in a \(2m\)-state Markov renewal process with the transition probability matrix \(F(x) = {1\over 2} (F_{ij}(x))_{i=1,2}^{j=1,2}\), where NEWLINE\[NEWLINE F_{11}(x) =F_{21}(x) = \int_0^x \exp [D(0)u] D(1) du;\;\;F_{12}(x) =F_{22}(x) = \int_0^x \exp [D(0)u] D(2) du. NEWLINE\]NEWLINE For a Markovian queueing node the input/output process is a MAP with two types of events, the admission and the departures. The matrices \(D(1)\) and \(D(2)\) contain the transition rates of all events by which the member of jobs in the node increases or decreases. Let \(c(1)= \theta D(1)(e \theta - D)^{-1}\), \(c(2)= \theta D(2)(e \theta - D)^{-1}.\) The authors prove that under some assumptions the following equation holds true: \(c(1) D(1) e = c(2) D(2) e.\) The essence of this assertion is the global conservation of the indices of dispersion for counts of a second-order description of the input stream through a finite Markovian node.