Markov models are frequently used for performance modeling. However most models do not have closed form solutions, and numerical solutions are often not feasible due to the large or even infinite state space of models of practical interest. For that, the statespace truncation is often demanded for computation of this kind of models. In this paper, we use the strong stability approach to establish analytic error bounds for the truncation of a tandem queue with blocking. Numerical examples are carried out to illustrate the quality of the obtained error bounds.
Queueing networks consisting of several service stations are more suitable for representing the structure of many systems with a large number of resources than models consisting of a singleservice station. Particularly, the queueing networks are used for the performance and reliability evaluation of computer, communication, and manufacturing systems [
The determination of the steadystate probabilities of all possible states of the network can be regarded as the central problem of queueing theory. The mean values of all other important performance measures of the network can be calculated from these. Several efficient algorithms for the exact solution of queueing networks are introduced. However, the memory requirements and computation time of these algorithms grow exponentially with the number of job classes in the system. For computationally difficult problems of networks with a large number of job classes, we resort to approximation methods [
Many approximation methods for nonproductform networks are discussed in the literature (see [
The study of approximating the stationary probabilities of an infinite Markov chain by using finite Markov chains was initiated by Seneta [
A different though related line of research is that of perturbed Markov chains. General results on perturbation bounds for Markov chains are summarized by Heidergott and Hordijk [
In this paper we are interested in computing the error bound of the stationary queue length distributions of queueing networks through finite truncation of some buffers, provided their stability holds. So, it is natural to approximate the stationary distribution of queueing networks through truncating some buffers. We may expect that such a truncation well approximates the original model as the truncation level (or size) becomes large. Therefore, we extend the applicability of the strong stability approach [
The paper is organized as follows. Section
The main tool for our analysis is the weighted supremum norm, also called
Let us note that
Let
For our analysis, we will assume that
We say that the Markov chain
Thereby, the Markov chain
the kernel
the
It has been shown in [
Let
Note that the term
Consider two stations in series: a tandem queue of M/M/1/
The steadystate joint queue size distribution of this tandem queue system does not exhibit a closed product form expression [
Let
In order to apply the strong stability approach, we consider the same truncation considered by Van Dijk [
Equation (
For our bounds, we require bounds on the basic input entities such as
For ease of reference, we introduce the following condition:
Essential for our numerical bound on the deviation between stationary distributions
If condition
By definition, we have
If
If
If
From (
if
Let
Provided that condition
We have
if
If
From (
if
If
If
From (
if
If
If
From (
In order to obtain
For
For all
In the following lemma we will identify the range for
For that, we choose the measurable function
Provided that
We have
Let
Provided that
We have
We verify that
Then,
By Theorem
Let
Note that
Following the line of thought put forward in Section
Under the conditions put forward in Theorem
Note that the bound (
In this section we will apply our bound put forward in Theorem
Compute the stationary distribution
Compute the stationary distribution
Calculate
In order to compare the both errors (the real and that obtained by the strong stability approach), we calculated the real values of the error with the same norm that we have calculated the approximation.
For the first numerical example we set
Comparison of our bounds.




5  9.3416  0.0381 
10  2.0045  0.0304 
15  0.8600  0.0296 
20  0.4372  0.0250 
25  0.2386  0.0224 
30  0.1349  0.0210 
35  0.0777  0.0201 
40  0.0452  0.0196 
45  0.0265  0.0193 
50  0.0156  0.0192 
Comparison of our bounds.




5 

0.0015 
10 


15 


20 


25 


30 


35 


40 


45 


50  2.8747 
3.2874 
Define the inputs:
the service rate of the first station (
the service rate of the second station (
the arrival mean rate
the function
the number of buffers in the second station (
the truncation level (
the step
Verify the intensity trafic condition:
if
else (*the system is unstable*), go to Step
Determine
if
else, go to Step
For each value of
Calculate
else
Determine
End.
We compared our expected approximation error (
Errors, comparative curves.
Errors, comparative curves.
An alternative method for computing bounds on perturbations of Markov chains is the series expansion approach to Markov chains (SEMC). The general approach of SEMC has been introduced in [