ABSTRACT

The matrix form of this equation is the first equation in (5). By the Markovian property of the input MAP, <l>k,J{Y) is independent of Ui.JX, y) for l'!£,i,

Now, as for an application of Theorem 1 , we consider the waiting times in the MAPIGll queue. This leads to a generalization of th� classical Pollaczek-Khinchin formula. We will use the results of the previous section to define the relationship between the virtual and actual waiting time distributions. For this purpose, we define (for l'!£,i, j�):

(10)

(12)

Now, consider the following function of x: (C(i) Jo J( { vt::;X, J(t) =) I J(t, O -)=i } )dt. (14)

This function is an almost everywhere continuous, nondecreasing and piecewise linear function and is differentiable for all x>O. By the definition of 0ilx), we have

(17)

THEOREM 3. In the MAP/Gil queue, we have:

(19)

(20)