Batch size distribution is taken arbitrary with

probability generating function G(z) = 0.1[z.sup.2] + 0.1[z.sup.3] + 0.8[z.sup.4] and mean batch size [bar.g] = 3.7.

[g.sup.(i).sub.n] (0) i--fold convolution of {[g.sub.n]} with itself and [g.sup.(0).sub.n] = [[delta].sub.n,0] [g.sup.(i).sub.n] = [n-1.summation over (k=1)] [g.sup.(i-1).sub.n-k] [g.sub.k] where [[delta].sub.m,n] is the Kroncker--Delta function X(z)

Probability generating function of X.

The probability that the system resides in each phase is calculated by both Matrix Geometric Method and

Probability Generating Function method and listed in Table 2.

Probability Generating Function. Let [p.sub.i] = [lim.sub.t[right arrow][infinity]]P{Q(t) = i}, i = 0, 1, ..., denote the steady-state probability of state i.

Furthermore, we describe some methodological aspects of queueing models with phase service by discussing the wel-established techniques, namely, supplementary variable technique (SVT),

probability generating function (PGF), embedded Markov chain (EMC), matrix geometric/matrix method, maximum entropy analysis (MEA) and many others as follows.

Note, these are not

probability generating functions as we still need to normalize the coefficients of [z.sup.2n+1] by dividing by the total probability of having alternating words of length 2n + 1.

of wards in which a caregiver works r Probability that a given caregiver works in a given ward Pk Probability that a caregiver works in k wards qk Probability that a ward has k caregivers working in it [f.sub.0] (x)

Probability generating function (pgf) for the degree distribution of caregivers [g.sub.0] (x) pgf for the degree distribution of wards [f.sub.1] (x) First select a random ward, and then select a random caregiver working there.

Probability generating function definitions for the number, weight and chromatographic chain distributions of radicals and polymer must be applied to the corresponding mass balances at each one of the cells in which the studied reactor is divided.

where M(s) is a nondefective

probability generating function and [m.sub.j] > 0 for all j [greater than or equal to] 1.

Finally, using a technical lemma on singularity analysis and composition of singular expansions, we are able to work out the asymptotics for the generating function [C.sup.[*](x, y, w) of rooted connected planar graphs, and from this the

probability generating function can be computed exactly.

On the other hand, the

probability generating function [psi](z) is defined in Eq 3.

The difference of two

probability generating function belongs to the following class.