Strong Truncation Approximation in Tandem Queues with Blocking

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 state-space 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.


Introduction
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 single-service station.Particularly, the queueing networks are used for the performance and reliability evaluation of computer, communication, and manufacturing systems 1 .
The determination of the steady-state 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 2 .
Many approximation methods for nonproduct-form networks are discussed in the literature see 3 and references therein .Especially, the well-known technique applicable for limiting model sizes is state truncation 4, 5 .Indeed, approximating a countable-state Markov chain by using finite-state Markov chains is an interesting and often a challenging topic, which has attracted many researchers' attention.Computationally, when we solve for the stationary distribution, when it exists, of a countable-state Markov chain, the transition probability matrix of the Markov chain has to be truncated in some way into a finite matrix as a first step.We then compute the stationary distribution of this finite-state Markov chain as an approximation to that of the countable-state one.We expect that as the truncation size increases to infinity, the solution for the finite Markov chain converges to that of the countable-state Markov chain.While for many application problems the justification of the convergence could be made by the physical meanings of the finite and the countable state Markov chains, it is not always easy to formally justify this claim.
The study of approximating the stationary probabilities of an infinite Markov chain by using finite Markov chains was initiated by Seneta 6 in 1967 Kartashov 23 , and Mitrophanov 24 .In these works, the bounds for general Markov chains are expressed in terms of ergodicity coefficients of the iterated transition kernel, which are difficult to compute for infinite state spaces.These results were obtained using operator-theoretic and probabilistic methods.Some of these methods allow us to obtain quantitative estimates in addition to the qualitative affirmation of the continuity.
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 23, 25 to the case of truncation problem for a tandem queue with blocking.As is well known, this network is a multidimensional, nonproduct form queueing network see, for example, Van Dijk 4 .So, our interest is to see what conditions guarantee that the steady-state joint queue length distribution of this tandem queue system is well approximated by the finite buffer truncation of another one.Such conditions allow us to obtain better quantitative estimates on the stationary characteristics of the tandem queue with blocking and infinite buffers.
The paper is organized as follows.Section 2 contains the necessary definitions and notation.In Section 3, we present the considered network queueing model and we give a new perturbation bounds corresponding to the considered truncation problem.Numerical example is presented in Section 4. Eventually, we will point out directions of further research.

Strong Stability Approach
The main tool for our analysis is the weighted supremum norm, also called υ-norm, denoted by • υ , where υ is some vector with elements υ k, l > 1, for all k, l ∈ S Z × {0, . . ., N}.
Let us note that B N , the Borel field of the natural numbers, that is equipped with the discrete topology and consider the measurable space N, B N .
Let M {μ i,j } be the space of finite measures on B N and let η {f i, j } be the space of bounded measurable functions.We associate with each transition operator P the linear mappings: Introduce to M the class of norms of the form: where υ is an arbitrary measurable function not necessary finite bounded from below by a positive constant.This norm induces in the space η the norm Let us consider B, the space of bounded linear operators on the space {μ ∈ M : μ υ < ∞}, with norm Let ν and μ be two invariant measures and suppose that these measures have finite υ-norm.
For our analysis, we will assume that v k, l is of a particular form υ k, l α k β l , for α > 1 and β > 1, which implies

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Hence, the bound 6 becomes We say that the Markov chain X with transition kernel P verifying P υ < ∞ and invariant measure π is strongly υ-stable, if every stochastic transition kernel P in some neighborhood { P : P − P υ < } admits a unique invariant measure π such that π − π υ tends to zero as P − P υ tends to zero uniformly in this neighborhood.The key criterion of strong stability of a Markov chain X is the existence of a deficient version of P defined in the following.
Thereby, the Markov chain X with the transition kernel P and invariant measure π is strongly υ-stable with respect to the norm • υ if and only if there exist a measure σ and a nonnegative measurable function h on S such that the following conditions hold: where • denotes the convolution between a measure and a function and 1 is the vector having all the components equal to 1.
It has been shown in 22 that a Markov chain X with the transition kernel P is strongly stable with respect to υ if and only if a residual for P with respect to υ exists.Although the strong stability approach originates from stability theory of Markov chains, the techniques developed for the strong stability approach allow to establish numerical algorithms for bounding π − π υ .A bound on π − π υ is established in the following theorem.
Theorem 2.1 see 26 .Let P be strongly stable.If then, the following bound holds where Π is the stationary projector of P and I is the identity matrix.
Note that the term I − Π υ in the bound provided in Theorem 2.1 can be bounded by

2.11
In this case, we can also bound π υ by

Model Description and Assumptions
Consider two stations in series: a tandem queue of M/M/1/∞ and M/M/1/N.There is one server at each station, and customers arrive to station 1 in accordance to a Poisson process with a state-dependent rate λ i when i customers are present at station 1 Customer service times at station i are exponentially distributed with rate μ i , i 1, 2 The interarrival and service times are independent of one another.The size of the buffer at station 1 is infinite, whereas the buffer size at station 2 is N < ∞ .When the second station is saturated the servicing at the first station is stopped.Queueing is assumed to be first-come, first-served.
The steady-state joint queue size distribution of this tandem queue system does not exhibit a closed product form expression 27 .Numerical studies and approximation procedures have therefore been investigated widely see, for example, Boxma and Konheim 28 , Hillier and Boling 29 , and Latouche and Neuts 30 .Van Dijk 4 has analysed the same tandem queue system and obtained an explicit error bound for bias terms of reward structures.In this paper, like in 4 , we consider the truncation of the size of the buffer at station 1 to obtain another analytic error bound by using the strong stability approach 23 .Therefore, we assume that for some constant λ: λ i ≤ λ, λ i −→ 0 as i −→ ∞, and λ i : nonincreasing in i.

3.1
Let i, j denote the number of customers at stations 1 i and 2 j , M λ μ 1 μ 2 , and consider the discrete time Markov chain with one-step transition probabilities given by 4 :

3.2
In order to apply the strong stability approach, we consider the same truncation considered by Van Dijk 4 .Therefore, for a finite integer Q, we have the following truncation: M , P i,j ; 0,0 1 {i>Q,0≤j≤N} , P i,j ; m,n P i,j ; m,n , otherwise.

3.3
Equation 3.3 means that the queue size at station 1 is truncated at level Q by rejecting arrivals whenever i Q.We remark also that the two transition matrices P and P are stochastic.

Strong Stability Bounds
For our bounds, we require bounds on the basic input entities such as π and T .In order to establish those bounds, we have to specify υ.Specifically, for α > 1 and β > 1, we will choose as our norm-defining mapping.
For ease of reference, we introduce the following condition: This condition corresponds to the trafic intensity condition of the infinite system.Essential for our numerical bound on the deviation between stationary distributions π and π is a bound on the deviation of the transition kernel P from P .This bound is provided in the following lemma.Lemma 3.1.If condition C is satisfied, then where 3.9 If j 0:

3.11
If j N:

3.13
For i > Q: if 0 ≤ j ≤ N, then we have P i,j ; m,n 1, if m 0 and n 0, 0, otherwise, 3.14

3.15
Mathematical Problems in Engineering From 3.8 , 3.13 , and 3.15 we have Let T denote a deficient Markov kernel residual matrix for the transition matrix P that avoids jumps to state 0, 0 ; more specifically, for i, j , m, n let T i,j ; m,n 0, if i j 0, P i,j ; m,n , otherwise. 3.17

Lemma 3.2. Provided that condition C holds, it holds that
where

3.19
Proof.We have

3.22
From 3.21 and 3.22 we have

3.25
If j N:

3.27
For i Q:

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If 0 < j < N:

3.29
If j N:

3.31
In order to obtain ρ1 {i Q} < 1, we must impose that β < α.For i > Q: Tv i, j 1,
In the following lemma we will identify the range for α and β that leads to finite υnorm of π.
For that, we choose the measurable function h i, j 1 {i 0,j 0} 1, for i j 0, 0, otherwise, 3.35 and the probability measure σ i,j P 0,0 ; i,j .
Proof.We have

3.41
Theorem 3.4.Provided that C holds, then, for all α, β such that β < α < α 0 and 1 < β < β 0 , the discrete time Markov chain describing the tandem queue with blocking and finite buffers model is υ-strongly stable for the test function υ k, l α k β l .
T i,j ; m,n P i,j ; m,n − h i, j σ m,n 0, if i j 0, P i,j ; m,n , otherwise.

3.42
Hence, the kernel T is nonnegative.We verify that P υ < ∞.We have or, according to 2.5 ,

3.44
According to 2.4 and 2.1 , we have
By Theorem 3.4, the general bound provided Theorem 2.1 can be applied to the kernels P and P for our tandem queues with blocking.Specifically, we will insert the individual bounds provided in Lemmas 3.1, 3.2, and 3.3, which yields the following result.Theorem 3.5.Let π and π be the steady-state joint queue size distributions of the discrete time Markov chains in the tandem queue with blocking and finite buffers and the tandem queue with blocking with infinite buffers respectively.Provided that C holds, then, for all β < α < α 0 and 1 < β < β 0 , and under the condition: we have the following estimate: where c α, β 1 c 0 α, β and Δ α, β , ρ α, β , and c 0 α, β were already defined by the formulas 3.5 , 3.18 , and 3.37 , respectively.
Following the line of thought put forward in Section 2, see 2.6 , we will translate the norm bound in Theorem 3.5 to bounds for individual performance measures f.Corollary 3.6.Under the conditions put forward in Theorem 3.5, it holds for any f such that f υ < ∞ that πf − πf ≤ f υ × SSB α, β .

3.49
Note that the bound 3.49 in Corollary 3.6 has α and β as free parameters.This give the opportunity to minimize the right hand side of the inequality 3.48 in Theorem 3.5 with respect to α, β .For given λ/M, this leads to the following optimization problem: 3.50

Numerical Example
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 λ i 1 − γ γ i with γ 0.1, μ 1 5, μ 2 2, and N 5, and for the second one we set λ i 1 − γ γ i with γ 0.3, μ 1 6, μ 2 4, and N 6.As a first step for applying our bound, in the both examples, we compute the values for α, β that minimize SSB α, β .Then, we can compute the bound put forward in Theorem 3.5 for various values for Q.The numerical results are presented in Tables 1 and 2.

Strong Stability Algorithm
Step 1. Define the inputs: i the service rate of the first station μ 1 ; ii the service rate of the second station μ 2 ; iii the arrival mean rate λ; iv the function λ i ; v the number of buffers in the second station N ; vi the truncation level Q ; vii the step h.
Step 4. For each value of β determine the constant: Step 5. Calculate SSB α, β and α α h, if α < α 0 go to Step 5; else β β h and go to Step 4.
We compared our expected approximation error SSB against numerical results RE and we easily observed that the real error on the stationary distribution is significantly smaller than the strong stability approach one.Furthermore, if we compare our expected approximation error against the numerical results, it is easy to see that the error decreases as the truncation level Q increases.We can notice the remarkable sensibility of the strong stability approach in the variation of the truncation level Q with regard to the real error.This means that the numerical error is really the point of the error which we can do when switching from the tandem queue with blocking and infinite buffers to the another one with finite buffers.Graphic comparison is illustrated in Figures 1 and 2.

Further Research
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 13 .SEMC for discrete time finite Markov chains is discussed in 15 , and SEMC for continuous time Markov chains is developed in 14 .The key feature of SEMC is that a bound for the precision of the approximation can be given.Unfortunately, SEMC requires numerical computation of the deviation matrix, which limits the approach in essence to Markov chains with finite state space.Perturbation analysis via the strong stability approach overcomes this drawback, however, in contrast to SEMC, no measure on the quality of the approximation can be given.
n P i,j ; m,n − P i,j ; m,n .
. Many up-to-date results were obtained by him and several collaborators.Most of their results are included in a paper by Gibson and Seneta 7 .Other references may be found therein and/or in another paper 8 published in the same year by the same authors.Other researchers, including Wolf 9 , used different approaches to those of Seneta et al.For instance, Heyman 10 provided a probabilistic treatment of the problem.Later, Grassmann and Heyman 11 justified the convergence for infinite-state Markov chains with repeating rows.All the above results are for approximating stationary distributions.Regarding more general issues of approximating a countable-state Markov chain, see the book by Freedman 12 .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 13 .One group of results concerns the sensitivity of the stationary distribution of a finite, homogeneous Markov chain see Heidergott et al. 14, 15 , and the bounds are derived using methods of matrix analysis; see the review of Cho and Meyer 16 and recent papers of Kirkland 17, 18 , and Neumann and Xu 19 .Another group includes perturbation bounds for finite-time and invariant distributions of Markov chains with general state space; see Anisimov 20 , Rachev 21 , Aïssani and Kartashov 22 ,

Table 1 :
Comparison of our bounds.

Table 2 :
Comparison of our bounds.