MARKOV CHAINS WITH QUASITOEPLITZ TRANSITION MATRIX : APPLICATIONS "

Application problems are investigated for the Markov chains with quasitoeplitz transition matrix. Generating functions of transient and steady state probabilities, first zero hitting probabilities and mean times are found for various particular cases, corresponding to some known patterns of feedback ( “warm-up,” “switch at threshold” etc.), Level depending dams and queue-depending queueing systems of both M/G/1 and MI/G/1 types with arbitrary random sizes of arriving and departing groups are studied.

INTRODUCTION in this paper we investigate the application problems for the general results of our preceding papers [1] and [2] on the Markov chains with quasitoeplitz transition matrix, elements of which depend on the difference of indices except for the first column and some first rows.
We consider here some particular cases of this structure with various types of relationships between the toeplitz part of the transition matrix and its first nontoeplitz rows.These relationships correspond to some types of state-dependence known in literature: "warm-up," with one row in the nontoeplitz part; "Bailey type," with equal first rows; "switch at a threshold," with the first rows of another toeplitz structure.We refer, for example, to Abolnikov and Dukhovny [3,4] and Neuts [5] for results on transient and steady state probabilities and the ergodicity criterion in the case where toeplitz rows have left or fight zero parts.
Here we proceed from the general results of [1] and [2] to specify formulas for the generating functions (GF's) of the transient and steady state probabilities, first zero hitting (FZH) probabilities and mean times, and the ergodicity criterion for all mentioned patterns of state-dependence, with no significant restriction put on the toeplitz part.
We introduce here the model of a dam with additional irregular loss ("vaporization"), being significant only over some dam level, and we treat it as a particular case of Markov chains with quasitoeplitz transition matrix with a switch at a threshold.
Bulk queueing systems M/G/1 and GI/M/1 are considered with no restriction put on the size distributions of arriving and departing groups.They are also treated as a particular case of the general quasitoeplitz model.

GENERAL RESULTS TABLE
To make this paper self-contained, we list here assumptions, denotations and results of[l] and [2].
In this section we consider some simple particular cases of the general model.a) n = 0 homogeneous random walk with absorption in zero, no state- dependence.
To obtain formulas for Uo(x)and P0, we have used normalizing equalities W(x,1) = (1 x) "1 and p(1) = 1, respectively, and one simple identity for functions with absolutely summable Laurent coefficients: if a(z) = _ajzi L , then (28)   [T+a(z)]lz=l = ., j = Oza(z)e(z).j= Hence, equality (21) also holds here.In some applications, for example queueing systems with waiting server, the system's being empty may cause an additional input with GF, say, b(z)= o bjz., so Ao(z) = Tb(z)H(z), and we can use the formulas above.
Let b(z) = z, then (29) The latter is a generalization of the Pollyatchek-Khinchin formula, that follows from here when H(z For example, Ai(z) = A(z), i = 0, 1, n 1, is the generalization of the case that takes place in Bailey's bulk queue model.
Two obvious identifies directly follow from the definitions of operators T/, T', T .
(33) and (34) are obtained by the same method.
Considering FZH problems we can see that for the Bailey type transition matrix, (12) looks like n-1 qi : x.qj(z) Dn" 1A(z)R'(x,z)T+zn'I'jR'(X,z)" j=l + (x-1)[1 + xR/(x,z)Dn'A(z)e(z)R'(x,z)].Since the right side of this doesn't depend on i, then all qi(x), < n, are equal, so (35) and (36) follow immediately, as n-1 Z n-j = (Z n. 1)e(z), T/e(z)R'(x,z) "1 = O. j=l Formulas (38) and (39) are obtained in the same manner.b) Ai(z) = zi[21(z), i < n,---a switch at a threshold n.To provide the conditions of descent and ascent we assume that the Laurent expansion of /(z) contains positive and negative powers of z with nonzero coefficients.Formulas (6) to ( 14) give all the GF's that we need if we replace Ai(z) by zit(z), i < n.

CALCULATION PROBLEMS
The efficiency of calculations based on the aforementioned formulas strongly depends on the possibility of calculating effectively the functions R+/-(x,z) and R+/-(z) and the effect of the operators Di, T+, T-, and T .
With all the models in applications being approximate, one can always design a model in which H(z) is approximated with a function that is meromorphically extensible to either F + or .In fact, it is sufficient to replace either the negative-indexes part of hj} (for ,r'+) or the positive-indexes part of it (for ) with respective parts of some suitable function, or even simply to truncate these parts at some level.It can be done with all necessary precision.(To retain the ergodicity criterion and provide the necessary proximity between R+-(z).and+/-(z), corresponding to approximate (z), we must estimate such approximation in terms of .**j h i j jl ) It must also be mentioned that this analytical property of H(z) arises in applications in the most natural manner.In fact, no other cases were in study to date.Lemma 1.Let the ergodicity criterion (2) be true and {h j} be an aperiodic sequence.
Proof.It was proved in [1] that Indrof the functions 1 xH(z) and z[ 1 H(z)]e(z) is zero, provided x (0,1) or H'(1) < 0, respectively.For the functions meromorphic in /"+ or F', Ind r is the difference between the numbers of roots and poles in/"+ (F-), so these numbers are equal.
The other parts of Lemma 1 have a similar proof, but in case 2) we introduce normalizing coefficients k(x) and k to ensure R'(x,,) = 1 and R'(,,,,) = 1.
Given the aforementioned analytical properties of H(z), the following lemma makes it possible for us to calculate the effect of operators T / and T-.
Denote Lm. (Z) as the Lagrange polynomial, interpolating the functionf(z) values at some points e,...,e m.
Let's introduce, for example, the model with "vaporization" that becomes significant at some level n.This means that besides the regular release m there is also additional irregular loss fit,.The data available may contain information sufficient to estimate only the order of decline for probabilities of the higher values of fly, not the upper bound of it.This makes it reasonable to simulate flu as not limited, with a GF bt,(z) that is a rational function with the prescribed decline of coefficients.
As by(z-1) is a rational function with the poles in F+, we can calculate R+-(x,z), R+-(z) and the effects of the operators T/, T', and T O with the help of Lemmas 1 and 2.
7. QUEUEING SYSTEMS WITH A LIMITED QUEUE LENGTH DEPENDENCE In this section we consider the queueing systems of M/G/1 and .GI/M/1 types with operating modes depending on queue length at some special control moments.
Namely, let t: be the moments immediately following service completions in M/G/1 or preceding arrivals in GI/M/1, and let be the queue length at these moments.The operating mode of the system results in 7'= ct-fl--the difference between the numbers of customers admitted to and taken from the waiting line between tg and tg+.Obviously (56) + = max(0, + ).Denote Hi(z) = E{zT'; = i}.We will refer to this dependence on as limited if Hi(z) = H(z), i > n.
The definition of tg ensures that {g} defined by (56) is a Markov chain for both M/G/1 and GI/M/1 types (imbedded chain); the limited dependence pattern provides a quasitoeplitz structure of its transition matrix.
it is natural for queueing systems that ),take on both positive and negative values, no matter how long the queue is, which provides the conditions of descent and ascent.The existence of averages is also the common feature of all the models, so the "natural properties" of the transition matrix are really natural.
Hence, no matter whether we have a M/G/1 or GI/M/1 type, or no matter what the group size distributions might be, all the results of sections 2, 3, and 4 are valid here for series K(z); H(z) .**hz. i=1