1. Introduction and Preliminaries
Multidimensional birth-death processes (BDP) were objects of a number of studies in queueing theory and other applied fields. The authors of these papers studied different special classes of homogeneous multidimensional BDPs under some restrictions and considered fluid approximations [
1], simulations [
2,
3,
4,
5,
6], large deviations [
7], stability [
8,
9], and other features. The problem of the product from solutions for such models was considered, for instance, in [
10,
11] (also, see the references therein). If the process is inhomogeneous and the transition intensities have a more general form, then the problem of computation of any probabilistic characteristics of the queueing model is much more difficult.
In the general case, it is impossible to obtain explicit solutions and their characteristics, as well as to construct any significant characteristics of the processes, as can be seen from the above list of works. This paper fills this gap and proposes a method of research and evaluation allowing one to estimate the rate of convergence for a one-dimensional projection of the multidimensional birth-death process. The approach also makes it possible to evaluate the main characteristics of the projection, as is demonstrated by the simplest example of an inhomogeneous two-dimensional process.
The background of our approach is the method of investigation of inhomogeneous BDP, see the detailed discussion and some preliminary results in [
12,
13,
14,
15]. Estimates for the state probabilities of one-dimensional projections of a multidimensional BDP were studied in [
16,
17]. However, within that methodology, it was impossible to obtain estimates of the rate of convergence, since the logarithmic norm of the operator cannot be applied to the corresponding nonlinear systems.
Here, we substantially modify that approach so that it can be used for estimation and construction of some explicit bounds on the rate of convergence for one-dimensional projection of a multidimensional BDP. Namely, in
Section 2, we develop a simple but efficient method for bounding the rate of convergence for an arbitrary (which may be nonlinear, depending on the number of parameters and so on) differential equation in the space of sequences
, and in
Section 3, we apply this method to bounding the rate of convergence for one-dimensional projections of BDP.
Let be a d-dimensional BDP such that in the interval , the following transitions are possible with order h: birth of a particle of type j, death of a particle of type j.
Let be the corresponding birth rate (from the state to the state ) and let be the corresponding death intensity (from the state to the state ). Denote .
To consider the existence and uniqueness, we renumber the states (only in this section), transforming the process into a one-dimensional one. Now, let the (finite or countable) state space of the vector process under consideration be arranged in a special order, say
. Denote by
, the corresponding state probabilities, and by
, the corresponding column vector of state probabilities. Applying our standard approach (see details in [
12,
14,
15]), we suppose in addition that all intensities are nonnegative functions locally integrable on
, and, moreover, in new enumeration,
where
are the corresponding transition intensities and all
are
uniformly in
i, that is,
, for any
.
We suppose that for any j, and almost all .
The probabilistic dynamics of the process is represented by the forward Kolmogorov system:
where
is the corresponding infinitesimal (intensity) matrix.
Throughout the paper, we denote the -norm by , i.e., , and for .
Let
be the set all stochastic vectors, i.e.,
-vectors with nonnegative coordinates and unit norm. We have the inequality
for almost all
. Hence, the operator function
from
into itself is bounded for almost all
and is locally integrable on
. Therefore, we can consider (
2) as a differential equation in the space
with bounded operator.
It is well known, see [
18], that the Cauchy problem for differential Equation (
2) has unique solution for an arbitrary initial condition, and
implies
for
.
We recall that a Markov chain
is called null-ergodic, if all
as
for any initial condition, and it is called weakly ergodic, if
as
for any initial condition
, see for instance [
12,
14].
3. Bounds on the Rate of Convergence for a Projection of Multidimensional BDP
Again, consider the forward Kolmogorov system (
2) in the original vector form. Then, we have
for any
.
In this section, we consider the one-dimensional process
for a fixed
j. Denote
. Then,
. The process
has nonzero jump rates only for unit jumps (
), namely, if
, then for small positive
h only the jumps
are possible with positive intensities, say
and
, respectively. Moreover, (
7) implies the equalities
and hence
and
Then, is a (in general, non-Markovian) birth and death process with birth and death intensities and , respectively, (that is, it is a process with possible infinitesimal jumps , the intensities of which depend on t and on the initial condition for the original multidimensional process .)
For any fixed initial distribution
and any
, the probability distribution
is unique. Hence,
and
uniquely define the system
for the vector
of state probabilities of the projection
under the given initial condition. Obviously, different initial conditions specify different systems.
Here, is the corresponding three-diagonal “birth-death” transposed intensity matrix such that all off-diagonal elements are nonnegative and all column-wise sums are equal to zero.
Let for all
and any
Then, from (
10) and (
11), we obtain the two-sided bounds
for any
k, any
t, and any initial conditions.
1. Let the state space of
be countable and
Put , , and . Let be a diagonal matrix, .
Note that in this situation, , and as implying null ergodicity of , that is as for any k.
Then, we have
implying the estimate
and the following statement.
Theorem 1. Let (15) hold for some j. Then, is null-ergodic and the following bounds hold:and Hence,and as for any . Remark 2. It should be noted that the above requirements are imposed only on this one coordinate.
We have
for any
. Set
. Then, from (
12), we obtain the system
where
,
, and the corresponding matrix
, where
for the corresponding elements of the matrix
.
For the solutions of system (
23), the rate of convergence is determined by the system
where all elements of
depend on
t and the initial condition of the original process.
Now, let
in accordance with (
22). Let
,
. Denote by
D, the upper triangular matrix
Let
. Then, the following bound holds:
Note that
, see detailed discussion in [
15], therefore, if
as
, then
is weakly ergodic.
Thus, we obtain the following statement.
Theorem 2. Let (22) hold for some j. Then, is weakly ergodic and the following bound holds:for any and any corresponding initial conditions. 4. Example
Consider a simple two-dimensional BDP with finite state space , , and the following transition intensities:
- (i)
from to ;
- (ii)
from to if ;
- (iii)
from to ;
- (iv)
from to ;
- (v)
from to ;
where , , , .
Then,
, and Theorem 2 gives bound (
27) with
.
We computed some important characteristics for the original process and its projection , namely:
Figure 1,
Figure 2 and
Figure 3 show the behaviour of the state probabilities for
, namely
,
, and
under two initial conditions for the original BDP:
- (i)
, for any (blue); and
- (ii)
, , for any such that (green).
Note that the corresponding initial conditions for the projection are , and .
These Figures illustrate the rate of convergence in a weak ergodic situation.
Figure 4 and
Figure 5 show the ’birth intensities’
and
for
under the same initial conditions.
Note that all the quantities are found by numerically solving the Cauchy problem for the forward Kolmogorov system (
2) and the corresponding system (
12) for its projection on the corresponding interval.
As can be seen from Theorem 2 and the figures below, to construct all the characteristics of interest with good accuracy, it suffices to carry out the numerical solution on the interval .
As was already noted, the projection of the original process is not a Markov process, and all probabilistic characteristics depend on the initial conditions of the original process.
It can be seen that these characteristics present comprehensive information concerning the behavior of the projection of the original process.