Strong Truncation Approximation in Tandem Queues with Blocking

Abstract

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.

1. 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. 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], 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.

2. 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: 𝜇𝑃_(𝑘,𝑙)=^∞𝑁_𝑖=0𝑗=0𝜇_(𝑖,𝑗)𝑃_{(𝑖,𝑗);(𝑘,𝑙)},()=(2.1)𝑃𝑓𝑘,𝑙^∞𝑁_𝑖=0𝑗=0𝑓𝑖,𝑗𝑃_{(𝑘,𝑙);(𝑖,𝑗)}.(2.2) Introduce to 𝔐 the class of norms of the form: 𝜇_𝜐=^∞𝑁_𝑖=0𝑗=0𝜐𝑖,𝑗𝜇_(𝑖,𝑗),(2.3) 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 𝑓_𝜐=sup_𝑘sup_𝑙𝑓(𝑘,𝑙)𝜐(𝑘,𝑙),𝑘,𝑙∈ℤ₊×{0,…,𝑁}.(2.4) Let us consider 𝔅, the space of bounded linear operators on the space {𝜇∈𝔐∶‖𝜇‖_𝜐<∞}, with norm ‖𝑄‖_𝜐=sup_𝑘sup_𝑙1𝜐(𝑘,𝑙)^∞𝑁_𝑖=0𝑗=0𝜐𝑖,𝑗𝑄_{(𝑘,𝑙);(𝑖,𝑗)},𝑘,𝑙∈ℤ₊×{0,…,𝑁}.(2.5) Let 𝜈 and 𝜇 be two invariant measures and suppose that these measures have finite 𝜐-norm. Then 𝜈𝑓−𝜇𝑓≤𝜈−𝜇_𝜐𝑓_𝜐inf_𝑘inf_𝑙𝜐(𝑘,𝑙),𝑘,𝑙∈ℤ₊×{0,…,𝑁}.(2.6)

For our analysis, we will assume that 𝑣(𝑘,𝑙) is of a particular form 𝜐(𝑘,𝑙)=𝛼^𝑘𝛽^𝑙, for 𝛼>1 and 𝛽>1, which implies inf_𝑘inf_𝑙𝜐(𝑘,𝑙)=1,𝑘,𝑙∈ℤ₊×{0,…,𝑁}.(2.7) Hence, the bound 6 becomes 𝜈𝑓−𝜇𝑓≤𝜈−𝜇_𝜐sup_𝑘sup_𝑙𝑓(𝑘,𝑙)𝜐(𝑘,𝑙),𝑘,𝑙∈ℤ₊×{0,…,𝑁}.(2.8)

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<𝜖} 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: (a)𝜋ℎ>0, 𝜎𝟏=1, 𝜎ℎ>0, (b)the kernel 𝑇=𝑃−ℎ∘𝜎 is nonnegative, (c)the 𝜐-norm of the kernel 𝑇 is strictly less than one, that is, ‖𝑇‖_𝜐<1, (d)‖𝑃‖_𝜐<∞, 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 𝑃−𝑃_𝜐<1−‖𝑇‖_𝜐‖𝐼−Π‖_𝜐,(2.9) then, the following bound holds ‖𝜋−𝜋‖_𝜐≤‖𝜋‖_𝜐‖𝐼−Π‖_𝜐𝑃−𝑃_𝜐1−‖𝑇‖_𝜐−‖𝐼−Π‖_𝜐𝑃−𝑃_𝜐,(2.10) 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 ‖𝐼−Π‖_𝜐≤1+‖𝟏‖_𝜐‖𝜋‖_𝜐.(2.11) In this case, we can also bound ‖𝜋‖_𝜐 by (𝜎𝜐)(𝜋ℎ).1−𝜌(2.12)

3. Analysis of the Model

3.1. 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)→0as𝑖→∞,and𝜆(𝑖)∶nonincreasingin𝑖.(3.1)

Let (𝑖,𝑗) denote the number of customers at stations 1(𝑖) and 2(𝑗), 𝑀=𝜆+𝜇₁+𝜇₂, and consider the discrete time Markov chain with one-step transition probabilities given by [4]: 𝑃_{(𝑖,𝑗);(𝑖+1,𝑗)}=𝜆(𝑖)𝑀,𝑃_{(𝑖,𝑗);(𝑖,𝑗−1)}=𝜇₂𝑀𝟏_{𝑗>0},𝑃_{(𝑖,𝑗);(𝑖−1,𝑗+1)}=𝜇₁𝑀𝟏_{𝑖>0}𝟏_{𝑗<𝑁},𝑃_{(𝑖,𝑗);(𝑖,𝑗)}𝜆(𝑖)=1−+𝜇₁𝟏_{𝑖>0}𝟏_{𝑗<𝑁}+𝜇₂𝟏_{𝑗>0}𝑀.(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: 𝑃_{(𝑄,𝑗);(𝑄,𝑗)}𝜇=1−¹𝟏_{𝑖>0}𝟏_{𝑗<𝑁}+𝜇₂𝟏_{𝑗>0}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.

3.2. 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 𝜐(𝑘,𝑙)=𝛼^𝑘𝛽^𝑙,(𝑘,𝑙)∈𝐒,(3.4) as our norm-defining mapping.

For ease of reference, we introduce the following condition: 𝜆min𝜇₁,𝜇₂<1.(𝐂) 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 𝑃−𝑃_𝜐≤()𝜆(𝑄)𝛼+1𝑀=Δ𝛼,𝛽.(3.5)

Proof. By definition, we have 𝑃−𝑃_𝜐=sup_{0≤𝑖≤𝑄}sup_{0≤𝑗≤𝑁}1𝜐𝑖,𝑗^𝑄𝑁_𝑚=0𝑛=0𝜐(𝑚,𝑛)𝑃_{(𝑖,𝑗);(𝑚,𝑛)}−𝑃_{(𝑖,𝑗);(𝑚,𝑛)}.=sup_{0≤𝑖≤𝑄}sup_{0≤𝑗≤𝑁}𝑆𝑖,𝑗,(3.6) where 𝑆=1𝑖,𝑗𝜐𝑖,𝑗^𝑄𝑁_𝑚=0𝑛=0𝜐(𝑚,𝑛)𝑃_{(𝑖,𝑗);(𝑚,𝑛)−𝑃(𝑖,𝑗);(𝑚,𝑛)}.(3.7) For 0≤𝑖≤𝑄−1: 𝑆𝑖,𝑗=0.(3.8) For 𝑖=𝑄: 𝑆=1𝑄,𝑗𝛼^𝑄sup_{0≤𝑗≤𝑁}1𝛽^𝑗𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑄,𝑗);(𝑚,𝑛)}−𝑃_{(𝑄,𝑗);(𝑚,𝑛)}.(3.9) If 𝑗=0: 𝑆()=1𝑄,0𝛼^𝑄1𝛽⁰𝛼^𝑄−1𝛽¹𝜇₁𝑀−𝜇₁𝑀+𝛼^𝑄𝛽⁰𝜆(𝑄)1−𝑀−𝜇₁𝑀𝜇−1+¹𝑀+𝛼^𝑄+1𝛽⁰𝜆(𝑄)𝑀=()𝜆(𝑄)𝛼+1𝑀.(3.10) If 0<𝑗<𝑁: 𝑆=1𝑄,𝑗𝛼^𝑄sup_{0≤𝑗≤𝑁}1𝛽^𝑗𝛼^𝑄−1𝛽¹𝜇₁𝑀−𝜇₁𝑀+𝛼^𝑄𝛽^𝑗−1𝜇₂𝑀−𝜇₂𝑀+𝛽^𝑗𝜆(𝑄)1−𝑀−𝜇₁𝑀−𝜇₂𝑀𝜇−1+¹𝑀+𝜇₂𝑀+𝛼^𝑄+1𝛽^𝑗𝜆(𝑄)𝑀=()𝜆(𝑄)𝛼+1𝑀.(3.11) If 𝑗=𝑁: 𝑆()=1𝑄,𝑁𝛼^𝑄1𝛽^𝑁𝛼^𝑄𝛽^𝑁−1𝜇₂𝑀−𝜇₂𝑀+𝛽^𝑁𝜆(𝑄)1−𝑀−𝜇₂𝑀𝜇−1+²𝑀=𝜆(𝑄)𝑀.(3.12)
From (3.10), (3.11), and (3.12) we have 𝑃−𝑃_𝜐=()𝜆(𝑄)𝛼+1𝑀.(3.13) For 𝑖>𝑄:
if 0≤𝑗≤𝑁, then we have 𝑃_{(𝑖,𝑗);(𝑚,𝑛)}=𝑆1,if𝑚=0and𝑛=0,0,otherwise,(3.14)𝑖,𝑗=sup_𝑖>𝑄sup_{0≤𝑗≤𝑁}1𝑣=𝑖,𝑗1+𝛼^𝑖𝛽^𝑗𝜆(𝑖)1−𝑀−𝜇₁𝑀1_𝑗<𝑁−𝜇₂𝑀1_𝑗>0+𝛼^𝑖+1𝛽^𝑗𝜆(𝑖)𝑀+𝛼^𝑖−1𝛽^𝑗+1𝜇₁𝑀1_𝑗<𝑁+𝛼^𝑖𝛽^𝑗−1𝜇₂𝑀1_𝑗>0≤1𝛼^𝑄+1+()𝜆()𝛼−1𝑄+1𝑀−𝛽1−𝛼𝜇₁𝑀1_𝑗<𝑁−11−𝛽𝜇₂𝑀1_𝑗>0.(3.15) From (3.8), (3.13), and (3.15) we have 𝑃−𝑃_𝜐≤()𝜆(𝑄)𝛼+1𝑀.(3.16)

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)}=𝑃0,if𝑖=𝑗=0,_{(𝑖,𝑗);(𝑚,𝑛)},otherwise.(3.17)

Lemma 3.2. Provided that condition (𝐂) holds, it holds that ‖𝑇‖_𝜐()𝜆=1+𝛼−1𝑀+𝑚𝛼,𝛽𝑀=𝜌𝛼,𝛽<1,(3.18) where 𝑚𝛽𝛼,𝛽=max𝛼−1𝜇₁,1𝛽−1𝜇₂.(3.19)

Proof. We have 𝑇𝜐𝑖,𝑗=^𝑄𝑁_{𝑚≥0𝑛=0}𝜐(𝑚,𝑛)𝑇_{(𝑖,𝑗);(𝑚,𝑛)}.(3.20) For 𝑖=0:  if 𝑗=0: 𝑇𝜐(0,0)=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛×0=0.(3.21) If 0<𝑗≤𝑁: 𝑇𝜐0,𝑗=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(0,𝑗);(𝑚,𝑛)}=𝛼⁰𝛽^𝑗−1𝜇₂𝑀+𝛽^𝑗𝜆(0)1−𝑀−𝜇₂𝑀+𝛼¹𝜆(0)𝑀=𝛼⁰𝛽^𝑗()𝜆(0)1+𝛼−1𝑀+1𝛽𝜇−1²𝑀.(3.22)
From (3.21) and (3.22) we have 𝜌𝟏_{𝑖=0}()𝜆(0)=1+𝛼−1𝑀+1𝛽𝜇−1²𝑀.(3.23) For 1≤𝑖≤𝑄−1:  if 𝑗=0: 𝑇𝜐(𝑖,0)=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑖,0);(𝑚,𝑛)}=𝛼^𝑖−1𝛽¹𝜇₁𝑀+𝛼^𝑖𝛽⁰𝜆(𝑖)1−𝑀−𝜇₁𝑀+𝛼^𝑖+1𝛽⁰𝜆(𝑖)𝑀=𝛼^𝑖𝛽⁰𝛽1+𝛼𝜇−1¹𝑀+()𝜆(𝑖)𝛼−1𝑀.(3.24) If 0<𝑗<𝑁: 𝑇𝜐𝑖,𝑗=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑖,𝑗);(𝑚,𝑛)}=𝛼^𝑖−1𝛽^𝑗+1𝜇₁𝑀+𝛼^𝑖𝛽^𝑗−1𝜇₂𝑀+𝛽^𝑗𝜆(𝑖)1−𝑀−𝜇₁𝑀−𝜇₂𝑀+𝛼^𝑖+1𝛽^𝑗𝜆(𝑖)𝑀=𝛼^𝑖𝛽^𝑗()𝜆(𝑖)1+𝛼−1𝑀+𝛽𝛼𝜇−1¹𝑀+1𝛽𝜇−1²𝑀.(3.25) If 𝑗=𝑁: 𝑇𝜐(𝑖,𝑁)=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑖,𝑁);(𝑚,𝑛)}=𝛼^𝑖𝛽^𝑁−1𝜇₂𝑀+𝛽^𝑁𝜆(𝑖)1−𝑀−𝜇₂𝑀+𝛼^𝑖+1𝛽^𝑁𝜆(𝑖)𝑀=𝛼^𝑖𝛽^𝑁()𝜆(𝑖)1+𝛼−1𝑀+1𝛽𝜇−1²𝑀.(3.26)
From (3.24), (3.25), and (3.26) we have 𝜌𝟏_{{1≤𝑖≤𝑄−1}}()𝜆(𝑖)=1+𝛼−1𝑀+𝑚𝛼,𝛽𝑀.(3.27) For 𝑖=𝑄:  if 𝑗=0: 𝑇𝜐(𝑄,0)=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑄,0);(𝑚,𝑛)}=𝛼^𝑄−1𝛽¹𝜇₁𝑀+𝛼^𝑄𝛽⁰𝜇1−¹𝑀=𝛼^𝑄𝛽⁰𝛽1+𝛼𝜇−1¹𝑀.(3.28) If 0<𝑗<𝑁: 𝑇𝜐𝑄,𝑗=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑄,𝑗);(𝑚,𝑛)}=𝛼^𝑄−1𝛽^𝑗+1𝜇₁𝑀+𝛼^𝑄𝛽^𝑗−1𝜇₂𝑀+𝛽^𝑗𝜇1−¹𝑀−𝜇₂𝑀=𝛼^𝑄𝛽^𝑗𝛽1+𝛼𝜇−1¹𝑀+1𝛽𝜇−1²𝑀.(3.29) If 𝑗=𝑁: 𝑇𝜐(𝑄,𝑁)=^𝑄𝑁_{𝑚≥0𝑛=0}𝛼^𝑚𝛽^𝑛𝑃_{(𝑄,𝑁);(𝑚,𝑛)}=𝛼^𝑄𝛽^𝑁−1𝜇₂𝑀+𝛽^𝑁𝜇1−²𝑀=𝛼^𝑄𝛽^𝑁11+𝛽𝜇−1²𝑀.(3.30)
From (3.28), (3.29), and (3.30) we have 𝜌𝟏_{𝑖=𝑄}𝑚=max1+𝛼,𝛽𝑀𝛽,1+𝛼𝜇−1¹𝑀.(3.31)
In order to obtain 𝜌𝟏_{𝑖=𝑄}<1, we must impose that 𝛽<𝛼.
For 𝑖>𝑄: =𝛼𝑇𝑣𝑖,𝑗=1,_𝑖𝛽^𝑗𝛼^𝑖𝛽^𝑗,=𝛼^𝑖𝛽^𝑗𝛼^𝑄𝛽⁰,=𝛼^𝑖𝛽^𝑗1𝛼^𝑄,(3.32) From (3.23), (3.27), (3.31), and (3.32) we have 𝜌()𝜆𝛼,𝛽=max1+𝛼−1𝑀+𝑚𝛼,𝛽𝑀𝑚,1+𝛼,𝛽𝑀𝛽,1+𝛼𝜇−1¹𝑀.(3.33)
For all 𝛽 such that: 1<𝛽<min(𝜇₁,𝜇₂)/𝜆 and 𝛽<𝛼<1+(1−1/𝛽)(min(𝜇₁,𝜇₂)/𝜆), we will obtain 𝜌(𝛼,𝛽)<1, then, under the same condition, we finally obtain 𝑇𝜐𝑖,𝑗≤𝜌𝛼,𝛽𝜐𝑖,𝑗,∀𝑖,𝑗∈𝐒,(3.34) and it follows that the 𝜐-norm of 𝑇 is equal to 𝜌(𝛼,𝛽), which proves the claim.

In the following lemma we will identify the range for 𝛼 and 𝛽 that leads to finite 𝜐-norm of 𝜋.

For that, we choose the measurable function ℎ𝑖,𝑗=𝟏_{{𝑖=0,𝑗=0}}=1,for𝑖=𝑗=0,0,otherwise,(3.35) and the probability measure 𝜎_(𝑖,𝑗)=𝑃_{(0,0);(𝑖,𝑗)}.(3.36)

Lemma 3.3. Provided that (𝐂) holds, the 𝜐-norm of 𝜋 is bounded by ‖𝜋‖_𝜐=𝜋_(0,0)()𝜆(0)1−𝜌𝛼,𝛽1+𝛼−1𝑀=𝑐₀𝛼,𝛽<∞,(3.37) where 𝜌(𝛼,𝛽) was defined in (3.18).

Proof. We have ‖𝜋‖_𝜐≤(𝜎𝜐)(𝜋ℎ).1−𝜌(3.38) By definition, 𝜎𝜐=^∞𝑁_{𝑖≥0𝑗=0}𝜎_(𝑖,𝑗)ℎ()𝜆(0)𝑖,𝑗=1+𝛼−1𝑀,𝜋ℎ=^∞𝑁_{𝑖≥0𝑗=0}𝜋_(𝑖,𝑗)ℎ𝑖,𝑗=𝜋_(0,0).(3.39) Hence, ‖𝜋‖_𝜐=𝜋_(0,0)()𝜆(0)1−𝜌1+𝛼−1𝑀=𝑐₀𝛼,𝛽.(3.40)

Let 𝛼₀,𝛽=sup𝛼∶𝜌𝛼,𝛽<1₀=sup𝛽∶𝜌𝛼,𝛽<1.(3.41)

Theorem 3.4. Provided that (𝐂) holds, then, for all (𝛼,𝛽) such that 𝛽<𝛼<𝛼₀ and 1<𝛽<𝛽₀, 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^𝑘𝛽^𝑙.

Proof. We have 𝜋ℎ=𝜋_(0,0)>0, 𝜎𝟏=1, and 𝜎ℎ=𝜎_(0,0)=1−(𝜆(0)/𝑀)>0. 𝑇_{(𝑖,𝑗);(𝑚,𝑛)}=𝑃_{(𝑖,𝑗);(𝑚,𝑛)}−ℎ𝑖,𝑗𝜎_(𝑚,𝑛)=𝑃0,if𝑖=𝑗=0,_{(𝑖,𝑗);(𝑚,𝑛)},otherwise.(3.42) Hence, the kernel 𝑇 is nonnegative.
We verify that ‖𝑃‖_𝜐<∞. We have 𝑇=𝑃−ℎ∘𝜎⇒𝑃=𝑇+ℎ∘𝜎⇒‖𝑃‖_𝜐≤‖𝑇‖_𝜐+‖ℎ‖_𝜐‖𝜎‖_𝜐,(3.43) or, according to (2.5), ‖𝑇‖_𝜐=sup_0≤𝑖sup_{0≤𝑗≤𝑁}1𝜐𝑖,𝑗^∞𝑁_𝑚=0𝑛=0𝜐(𝑚,𝑛)𝑇_{(𝑖,𝑗);(𝑚,𝑛)}≤sup_0≤𝑖sup_{0≤𝑗≤𝑁}1𝜐𝜌𝜐𝑖,𝑗𝛼,𝛽𝑖,𝑗≤𝜌𝛼,𝛽<1.(3.44) According to (2.4) and (2.1), we have ‖ℎ‖_𝜐=sup_0≤𝑖sup_{0≤𝑗≤𝑁}1𝜐ℎ‖𝜎‖𝑖,𝑗𝑖,𝑗=1,_𝜐=^∞𝑁_𝑚=0𝑛=0𝜐(𝑚,𝑛)𝜎_(𝑚,𝑛)=^∞𝑁_𝑚=0𝑛=0𝛼^𝑚𝛽^𝑛𝑃_{(0,0);(𝑚,𝑛)}()𝜆(0)=1+𝛼−1𝑀≤1+𝛼₀𝜆−1𝑀<∞,(3.45) where 𝛼₀=sup𝛼∶𝜌𝛼,𝛽<1.(3.46)
Then, ‖𝑃‖_𝜐<∞.

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<𝛼₀ and 1<𝛽<𝛽₀, and under the condition: Δ𝛼,𝛽<1−𝜌𝛼,𝛽𝑐𝛼,𝛽,(3.47) we have the following estimate: ‖𝜋−𝜋‖_𝜐≤𝑐₀𝛼,𝛽𝑐𝛼,𝛽Δ𝛼,𝛽1−𝜌𝛼,𝛽−𝑐𝛼,𝛽Δ𝛼,𝛽=𝐒𝐒𝐁𝛼,𝛽,(3.48) where 𝑐(𝛼,𝛽)=1+𝑐₀(𝛼,𝛽) and Δ(𝛼,𝛽), 𝜌(𝛼,𝛽), and 𝑐₀(𝛼,𝛽) were already defined by the formulas (3.5), (3.18), and (3.37), respectively.

Proof. Note that 𝛼∈]𝛽,𝛼₀[ and 𝛽∈]1,𝛽₀[ already implies 𝑐₀(𝛼,𝛽)<∞ and 𝜌(𝛼,𝛽)<1. Hence, Lemmas 3.2 and 3.3 apply.

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×𝐒𝐒𝐁𝛼,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 𝜚=𝜆/𝑀, this leads to the following optimization problem: min_𝛼,𝛽<𝐒𝐒𝐁𝛼,𝛽s.t.𝜌𝛼,𝛽<1,Δ𝛼,𝛽1−𝜌𝛼,𝛽𝑐.𝛼,𝛽(3.50)

4. Numerical Example

In this section we will apply our bound put forward in Theorem 3.5. For this, we implement an algorithm (the principal idea of this algorithm is the same as in [25]) on concrete cases. Indeed, we apply a computer program to determine the made error on stationary distribution due to the approximation, when the approximation is possible, as well as the norm from which the error is obtained. It is important, in this work, to give an idea about the performance of this approach. For this, we computed the real value of the error by enlarging the state space and derived the stationary distribution. For that, we elaborated a program in the Matlab environment according to the following steps.(1)Compute the stationary distribution 𝜋 of the tandem queue with blocking with infinite buffers (the original model) using the definition (𝜋=𝜇lim_𝑛→∞𝑃^𝑛), where 𝜇 is a probability measure.(2)Compute the stationary distribution 𝜋 of the tandem queue with blocking and finite buffers (the truncated model)sing the definition (𝜋=𝜇lim_𝑛→∞𝑃^𝑛), where 𝜇 is a probability measure.(3)Calculate ∑^𝑄_𝑖=0∑^𝑁_𝑚=0𝛼^𝑖𝛽^𝑚|𝜋_(𝑖,𝑚)−𝜋_(𝑖,𝑚)|.

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, 𝜇₁=5, 𝜇₂=2, and 𝑁=5, and for the second one we set 𝜆(𝑖)=(1−𝛾)𝛾^𝑖 with 𝛾=0.3, 𝜇₁=6, 𝜇₂=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.

Table 1 

Comparison of our bounds.

Table 2 

Comparison of our bounds.

4.1. Strong Stability Algorithm

Step 1. Define the inputs:(i)the service rate of the first station (𝜇₁);(ii)the service rate of the second station (𝜇₂);(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 2. Verify the intensity trafic condition:  if 𝜆/min(𝜇₁,𝜇₂)<1, go to Step 3, else (*the system is unstable*), go to Step 6.

Step 3. Determine  𝑀=𝜆+𝜇₁+𝜇₂, 𝛽₀=min(𝜇₁,𝜇₂)/𝜆 and put 𝛽=1+ℎ;  if 𝛽<𝛽₀, go to Step 4;  else, go to Step 6.

Step 4. For each value of 𝛽 determine the constant: 𝛼₀=sup{𝛼∶𝜌(𝛼,𝛽)<1}=1+(1−1/𝛽)(min(𝜇₁,𝜇₂)/𝜆); 𝛼=𝛽+ℎ, go to Step 5.

Step 5. Calculate 𝐒𝐒𝐁(𝛼,𝛽) and 𝛼=𝛼+ℎ, if 𝛼<𝛼₀ go to Step 5;
else 𝛽=𝛽+ℎ and go to Step 4.

Step 6. Determine (𝛼_opt,𝛽_opt) and 𝐒𝐒𝐁(𝛼_𝐨𝐩𝐭,𝛽_𝐨𝐩𝐭), where 𝐒𝐒𝐁𝛼_𝐨𝐩𝐭,𝛽_𝐨𝐩𝐭=min𝐒𝐒𝐁𝛼,𝛽.(4.1)

Step 7. End.

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.

Figure 1 

Errors, comparative curves.

Figure 2 

Errors, comparative curves.

5. 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.