Stability of Queueing Systems with Impatience, Balking and Non-Persistence of Customers

Abstract

The operation of many queueing systems is adequately described by the structured multidimensional continuous-time Markov chains. The most well-studied classes of such chains are level-independent Quasi-Birth-and-Death processes, $GI/M/1$ type and $M/G/1$ type Markov chains, generators of which have the block tri-diagonal, lower- and upper-Hessenberg structure, respectively. All these classes assume that the matrices of transition rates are quasi-Toeplitz. This property greatly simplifies their analysis but makes them inappropriate for the study of many important systems, e.g., retrial queues with a retrial rate depending on the number of customers in orbit, queues with impatient customers, etc. The importance of such systems attracts significant interest to their analysis. However, in the literature, there is a methodological gap relating to the ergodicity condition of the corresponding Markov chains. To fulfill this gap and facilitate the analysis of a wide range of such systems, we show that under non-restrictive assumptions, the following hold true: (i) if the customers can balk or are impatient or non-persistent, then the Markov chain describing the behavior of the system belongs to the class of asymptotically quasi-Toeplitz Markov chains; (ii) this chain is ergodic; (iii) known algorithms can be applied for the calculation of the stationary distribution of the corresponding queueing system.

1. Introduction

The impatience (reneging, abandonment, etc.) of customers is a typical feature of many (if not all) real-world service systems. If the processing of a customer that waits in the queue does not start during a certain period of time (patience time), the customer leaves the system, independently of other customers, without service. Impatience is related to psychological reasons if the customers are humans, the obsolescence of information, perishing of the products, departure of the waiting mobile user from the cell, etc. As early works on queues with customers’ impatience, the papers [1,2,3,4,5] can be mentioned. Now, the literature devoted to the analysis of queues with impatient customers is very extensive. Results of research in the field of queues with customer impatience are presented, e.g., in [6,7,8,9].

If the total rate of customer departure due to impatience increases with growth in the number of customers in the buffer, it is intuitively clear that the number of customers in the buffer never becomes infinite. This means that such a system is always stable. However, to the best of our knowledge, these intuitive reasonings are not supported by formal statements proven in the existing literature. Here, we present such a proof for a rather wide class of queueing models.

The motivation for writing this paper is twofold. On one hand, in recent times, we have reviewed a lot of papers where the authors consider queues with customer impatience and completely ignore a discussion of ergodicity conditions or prove it in a non-correct way. On the other hand, we have done our own intensive research on queues with customer impatience, and usually, the reviewers of our manuscripts require the full formal proof of ergodicity of the considered Markov chain. Therefore, we believe that the results presented in this paper will be helpful to researchers. These results remove the problem of the proof of ergodicity and also present the recommendation of numerically stable algorithms for the computation of the stationary distribution of the system states.

The structure of the text is the following. Section 2 contains more detailed motivation for this paper’s preparation and some necessary preliminary information from the theory of structured multidimensional Markov chains ($MC$s). In Section 3, the ergodicity of the queueing system with impatient customers is proved for the system, the counterpart of which with patient customers is described by the Quasi-Birth-and-Death ($QBD$) process or $M/G/1$ type $MC$. The ergodicity in the case of customers balking with the probability of joining the queue approaching zero when the queue length increases is also proven there. In Section 4, the ergodicity of the multi-server retrial queueing system with impatient customers is stated. In Section 5, the ergodicity of the multi-server retrial queueing system with non-persistent customers is stated. Section 6 concludes the paper.

2. Preliminary Information

2.1. Basic Information about Multidimensional Continuous-Time Markov Chains under Study

The operation of many queueing systems can be described by a suitably constructed multidimensional continuous-time $MC$ $\{i_t,n_t\},t\ge 0.$ The first component of the $MC$ $i_t$ has a countable state space, $i_t\ge 0,$ and corresponds to the current number of customers in the queueing system, buffer, orbit or network. The process $n_t$ describes the transitions of a finite component of the $MC$. The process $n_t$ having a finite state space indeed may be a whole finite set of finite components representing various auxiliary processes, e.g., the number of busy or broken servers in multi-server systems, the state of the underlying process of arrivals (if the arrivals occur in the Markov arrival process ($MAP$) or batch Markov arrival process ($BMAP$) or marked Markov arrival process ($MMAP$), etc.), the state of the underlying process of service (if the service time has a phase type, $PH$ distribution, or service is defined by the Markov service process, $MSP$), the random environment, which has an impact on the system operation or the number of customers at other stages of a tandem system with finite intermediate buffers, etc. An account of the physical meaning of these components is very important for writing down the generator of the $MC$. However, for the purposes of this paper, we assume that the values of these finite components are enumerated in some order. Thus, without the loss of generality, for the simplicity of denotations, in Section 2, we consider the case of only one finite component $n_t.$ In Section 3 and Section 4 devoted to retrial queues, we consider the case where the finite component $n_t$ is two-dimensional.

The set of states of the $MC$ having the fixed value, say, i of the first, countable, component is called level i of the $MC,i\ge 0.$ We suppose that there exists a finite number $i^0,$ $i^0\ge 0,$ such that the cardinalities of all levels i such that $i>i^0$ are equal. In particular, in Section 2, we will assume that there exists an integer number N such that the component $n_t$ of the $MC$ admits, for any $t,t\ge 0,$ and all $i_t\ge i^0$, the values in the set $\{0,1,2,\dots ,N\}.$ The cardinality of the levels i for $i\ge i^0$ is equal to $N+1.$ The levels having numbers $0,1,\dots ,i_0-1$ can have various dimensions, and we do not impose any specific assumptions about the behavior of the $MC$ for these levels except the obvious requirement of the boundedness of the transition rates from these levels.

Let $\mathcal{Q}$ be the generator of the $MC$ $\{i_t,n_t\},t\ge 0.$ Within this paper, we assume that this generator has the upper-Hessenberg structure

$$\mathcal{Q}=\left(\begin{array}{cccccc}{\mathcal{Q}}_{0,0}& {\mathcal{Q}}_{0,1}& {\mathcal{Q}}_{0,2}& {\mathcal{Q}}_{0,3}& {\mathcal{Q}}_{0,4}& \dots \\ {\mathcal{Q}}_{1,0}& {\mathcal{Q}}_{1,1}& {\mathcal{Q}}_{1,2}& {\mathcal{Q}}_{1,3}& {\mathcal{Q}}_{1,4}& \dots \\ O& {\mathcal{Q}}_{2,1}& {\mathcal{Q}}_{2,2}& {\mathcal{Q}}_{2,3}& {\mathcal{Q}}_{2,4}& \dots \\ O& O& {\mathcal{Q}}_{3,2}& {\mathcal{Q}}_{3,3}& {\mathcal{Q}}_{3,4}& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right)$$

(1)

where the matrices ${\mathcal{Q}}_{i,j}$ consist of the entries ${\mathcal{Q}}_{i,j}^{(n,n^{\prime })}$ defining, except the diagonal entries ${\mathcal{Q}}_{i,i}^{(n,n)}$, transition rates from the state $(i,n)$ to the state $(j,n^{\prime }).$ The mentioned diagonal entries are negative. Their moduli define the rates of the $MC$ $\eta$ exit from the corresponding state. For $i>i^0,j\ge i-1,$ ${\mathcal{Q}}_{i,j}$ are square matrices of size $N+1.$ Here, O denotes zero matrix. By I, we will denote the identity matrix. If it is necessary, the size of the matrix can be indicated by the subscript.

The popular partial case suggests that the matrix $\mathcal{Q}$ is block tri-diagonal. The $MC$s having a generator of a block tri-diagonal structure are called Quasi-Birth-and-Death processes ($QBD$).

2.2. Level-Independent Quasi-Birth-and-Death Processes and $M/G/1$ Type Markov Chains

The most well studied in the literature classes of $MC$s having the structure of form (1) are the level-independent $QBD$ and $M/G/1$ type $MC$s. These classes assume that the matrices of transition rates are block quasi-Toeplitz, i.e., the value of the block ${\mathcal{Q}}_{i,j}$ for $i>i^0,j\ge i-1,$ depends on the difference $j-i$ but does not depend on i and j separately. The name quasi-Toeplitz goes back to the concept of the Toeplitz matrix. The prefix “quasi” reflects the possibility of a violation of the Toeplitz property of the generator for some low levels.

For $M/G/1$ type $MC$s, there exist matrices ${\mathcal{Q}}^{-},{\mathcal{Q}}^0,{\mathcal{Q}}^{+k},k\ge 1,$ such that

$${\mathcal{Q}}_{i,i-1}={\mathcal{Q}}^{-},{\mathcal{Q}}_{i,i}={\mathcal{Q}}^0,{\mathcal{Q}}_{i,i+k}={\mathcal{Q}}^{+k},k\ge 1,i>i^0.$$

For level-independent $QBD$s, ${\mathcal{Q}}^{+k}=O$ for $k>1.$ For brevity, we will denote ${\mathcal{Q}}^{+1}={\mathcal{Q}}^{+}.$

The classes of the level-independent $QBD$ and $M/G/1$ type $MC$s were investigated in detail by M. Neuts; see, e.g., his seminal books [10,11]. Following M. Neuts, we assume that the matrix $\mathbf{Q}={\mathcal{Q}}^{-}+{\mathcal{Q}}^0+{\displaystyle \sum _{k=1}^{\infty }}{\mathcal{Q}}^{+k}$ is irreducible.

The necessary and sufficient condition for the ergodicity of $M/G/1$ type $MC$s is given by M. Neuts in the following form.

Lemma 1.

$M/G/1$ type $MC$ is ergodic if and only if the inequality

$$\mathbf{q}{\mathcal{Q}}^{-}\mathbf{e}>\mathbf{q}\sum _{k=1}^{\infty }k{\mathcal{Q}}^{+k}\mathbf{e}$$

(2)

holds good where the row vector $\mathbf{q}$ is the unique solution to the system

$$\mathbf{q}\mathbf{Q}=\mathbf{0},\mathbf{q}\mathbf{e}=1,$$

where $\mathbf{0}$ is the zero row vector and the column vector $\mathbf{e}$ has all entries equal to 1.

The following statement immediately follows from this lemma.

Corollary 1.

The level-independent $QBD$ is ergodic if and only if the inequality

$$\mathbf{q}{\mathcal{Q}}^{-}\mathbf{e}>\mathbf{q}{\mathcal{Q}}^{+}\mathbf{e}$$

(3)

holds good.

Sometimes, in application to a concrete queueing system, it is possible to analytically obtain the vector $\mathbf{q}$ and reduce inequalities (2) or (3) to a simple scalar form.

2.3. Level-Dependent Quasi-Birth-and-Death Processes and $M/G/1$ Type Markov Chains, Asymptotically Quasi-Toeplitz Markov Chains

The classes of the level-independent $QBD$ and $M/G/1$ type $MC$s are extremely useful for the analysis of a variety of queueing models, in particular, various queueing models with an infinite buffer. However, the inherent feature of many important queueing systems is that the $MC$ describing the behavior of the system is level-dependent. This implies that the blocks of the corresponding generator describing transition rates from the level i to the level j depend not only on the difference $j-i$ but also on i and j separately. It is mentioned in [12] that level-dependent $QBD$s are often more realistic and, while efficient and stable numerical solution techniques are available for level-independent $QBD$s, there are only a few approaches that try to exploit the block structure in the level-dependent case.

As the most important queueing models described by the level-dependent $QBD$s, retrial queueing models and queues with customer impatience have to be mentioned. More information about the retrial queues, real-world examples and known results can be found, e.g., in the books [13,14] and papers [15,16,17,18,19].

The importance of retrial queueing models stems from their suitability for modeling various real-world systems, including contact centers, delivery systems and the extremely popular wireless communication networks. The total intensity of retrials of customers staying in the orbit in the overwhelming majority of real systems and networks depends on the number of these customers. This makes the $MC$ describing the behavior of the system level-dependent.

The impatience (reneging, abandonment, etc.) of customers also makes the M describing system behavior level-dependent.

Due to the practical importance of the analysis of level-dependent $MC$s, the notion of asymptotically quasi-Toeplitz Markov chains ($AQTMC$s) was introduced in the paper [20]. The rough intuitive definition of $AQTMC$s is as follows. The $AQTMC$ is an $MC$ with the block upper-Hessenberg structure (1) of a generator that does not possess the quasi-Toeplitz property; however, in asymptotic ones, for very large values of the countable component of the chain, this property appears. A more exact and formal definition of $AQTMC$ is given below.

The main motivation for introducing $AQTMC$s is the necessity of considering retrial queues, with the retrial rate proportional to the number of customers in orbit, the $BMAP$ arrival process and the phase-type distribution of service times, see [21]. The significant difference between a system with an infinite buffer and a similar system with retrials is that in the former system, a new customer is immediately picked up for service from the buffer when some server is released. In an analogous situation in the latter system, there exists an interval during which the server (or servers) remains idle despite the customer’s presence in the orbit. This period finishes via a new primary customer arrival or the retrial of a customer from the orbit. When the number of customers staying in the orbit infinitely increases, such a period becomes shorter and completely disappears in the limit. Therefore, the $AQTMC$ behaves in limit exactly as the corresponding limiting quasi-Toeplitz $MC$.

In the case when the generator of the $AQTMC$ is block tri-diagonal, the $AQTMC$ is a special case of the level-dependent $QBD.$ The difference is that no assumptions about the blocks of a generator are made for the level-dependent $QBD$, while their asymptotic behavior is suggested for $AQTMC.$

It is worth noting that, due to the absence of a quasi-Toeplitz property of the $MC$s, which describes many queueing systems with customer retrials and (or) impatience, the problem of solving the infinite system of equilibrium equations for the stationary probabilities of the chain is quite difficult. Therefore, many researchers impose, from the early beginning, quite unrealistic assumptions about the considered system, like the orbit capacity is finite, and the rate of retrials from the orbit is constant, independent of the current number of customers in the orbit, etc.

Some other researchers solve the infinite system of equilibrium equations via its truncation, see, e.g., [22]. The worst case, from a mathematical point of view, is when this is direct (brute force) truncation. Some equations are cut, and the remaining finite system is solved with the use of a computer. The better case is when the researchers use so-called soft truncation. One of the possible ways for soft truncation was offered by M.F. Neuts and B.M. Rao in [23]. This soft truncation suggests that the blocks of the generator for levels exceeding some fixed threshold become constant, independent of the level. In application to the analysis of a multi-server retrial queue, this means that after the number of customers in the orbit reaches some threshold (and until it drops below this threshold), the retrial rate becomes constant. But this suggestion is definitely not realistic because, usually, the total retrial rate is proportional to the number of customers staying in the orbit. When soft truncation is implemented, the results from [10] can be applied. If the generator is block tri-diagonal, i.e., the $MC$ is the $QBD,$ the vectors of the stationary probabilities of the states that belong to high levels have a matrix geometric form. When the truncation threshold is chosen suitably, the algorithm from [23] can give satisfactory results.

However, to apply this algorithm, it is necessary first to prove that, under the fixed set of transition rates, the considered $QBD$ is ergodic and that the computed distribution is indeed the stationary distribution of the $MC.$ Unfortunately, the paper [23] does not contain information about the conditions for ergodicity of the $MC.$ Some researchers derive the ergodicity condition via the use of the results from [10] for level-independent $QBD$. But the obtained condition is the ergodicity condition for the $MC$ with other dynamics of the $MC$ when the current level of the $MC$ is above the truncation threshold. Obviously, it is not the ergodicity condition for the initial level-dependent $QBD.$

The problem of the derivation of ergodicity conditions in the case of level-dependent $MC$s is very important but is not sufficiently addressed in the existing literature.

The condition given for the level-dependent $QBD$s in [24] is not a constructive one. It is given as a requirement for the convergence of some matrix series, the terms of which contain the infinite set of matrices (denoted as $R_i$ in [24]) that are formally computed recursively. In the case of level-independent $QBD,$ the recursion turns to the quadratic matrix equation. As it is known, see [10], the solution of this equation with the required properties exists only if the $QBD$ is ergodic. Therefore, in the more complicated case of level-dependent $QBD$ considered here, the situation is more difficult; the existence of a solution to the infinite recursion has to be justified, and at least the ergodicity of $QBD$ has to be postulated. Thus, there is an evident vicious circle. To check the ergodicity, it is required to compute the matrices $R_i$ which, in turn, may have a chance to be computed only if the $QBD$ is ergodic.

2.4. Conditions for Ergodicity and Non-Ergodicity of Asymptotically Quasi-Toeplitz Markov Chains

Constructive sufficient conditions for the ergodicity and non-ergodicity of $AQTMC$s, a special case of which is an important class of level-dependent $QBD,$ were presented in [20]. Here, we briefly reproduce the results relevant to our analysis from [20].

According to the definition of $AQTMC$s, an $MC$ ${\chi }_t=\{i_t,n_t\}$ belongs to the class of $AQTMC$s if

(A)

Its generator has the upper-Hessenberg structure (1);

(B)

The following matrices $Y_k,k\ge 0,$ exist:

$$Y_k=\underset{i\to \infty }{lim}{\mathcal{U}}_i^{-1}{\mathcal{Q}}_{i,i+k-1}+{\delta }_{k,1}I,k\ge 0,$$

where ${\delta }_{k,1}=1$ if $k=1$ and ${\delta }_{k,1}=0$; otherwise, ${\mathcal{U}}_i$ is the diagonal matrix with the diagonal entries defined by the moduli of the diagonal entries of the matrix $Q_{i,i}.$ In other words,

$${\mathcal{U}}_i=-I\circ {\mathcal{Q}}_{i,i},i\ge i^0,$$

where ∘ is the Hadamard product of matrices symbol, see, e.g., [25], and the matrix ${\displaystyle \sum _{k=0}^{\infty }}{Y}_k$ is the stochastic one;

(C)

Some technical assumptions related to the requirement of the finiteness of the average size of the jump-up of the level of $AQTMC$ (see Theorem 4 in [20]) are fulfilled. These assumptions are evidently implemented, e.g., under the suggestion that $Y_k=O$ for $k>K+1$ where K is a finite integer, $K\ge 1.$ Thus, below, we impose this suggestion.

A sufficient condition for the ergodicity of $AQTMC$s proven in [20] is given as follows.

Let us introduce the matrix generating function $Y\left(z\right)={\displaystyle \sum _{k=0}^{\infty }}{Y}_k{z}^k,\left|z\right|<1.$ Ergodicity conditions are different depending on the irreducibility or reducibility of the matrix $Y\left(1\right).$ It is worth noting that although we supposed above that the matrix $\mathbf{Q}={\mathcal{Q}}^{-}+{\mathcal{Q}}^0+{\displaystyle \sum _{k=1}^{\infty }}{\mathcal{Q}}^{+k}$ is irreducible, the matrix $Y\left(1\right)$ can be (and often is) reducible. Therefore, two variants of the ergodicity condition have to be analysed.

Lemma 2.

If the matrix $Y\left(1\right)$ is irreducible, the sufficient condition for the ergodicity of $AQTMC$s is the fulfillment of the inequality

$$\mathbf{y}\sum _{k=1}^{\infty }k{Y}_k\mathbf{e}<1$$

(4)

where the row vector $\mathbf{y}$ is the unique solution of the system

$$\mathbf{y}=\mathbf{y}\sum _{k=0}^{\infty }{Y}_k,\mathbf{y}\mathbf{e}=1.$$

(5)

If the matrix $Y\left(1\right)$ is reducible, then, by means of the coordinated permutation of rows and columns, the matrix $Y\left(1\right)$ can be represented in the normal form, see [26],

$$Y\left(1\right)=\left(\begin{array}{cccccccc}{Y}^{\{1,1\}}& O& O& \cdots & O& O& O& O\\ O& Y^{\{2,2\}}& O& \cdots & O& O& O& O\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮\\ O& O& O& \cdots & Y^{\{m,m\}}& O& O& O\\ Y_{m+1,1}& Y_{m+1,2}& Y_{m+1,3}& \cdots & Y_{m+1,m}& Y^{\{m+1,m+1\}}& O& O\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮\\ Y_{s,1}& Y_{s,2}& Y_{s,3}& \cdots & Y_{s,m}& Y_{s,m+1}& \cdots & Y^{\{s,s\}}\end{array}\right)$$

where $Y^{\{l,l\}},l=1,\dots ,m$ are irreducible stochastic matrices, the matrices $Y^{\{l,l\}},l=m+1,\dots ,s,$ are irreducible matrices, and for each $l,l=m+1,\dots ,s,$ at least one of the matrices $Y_{l,1},\dots ,Y_{l,l-1}$ is non-zero.

Correspondingly to the normal form of the matrix $Y\left(1\right)={\displaystyle \sum _{k=0}^{\infty }}{Y}_k,$ all matrices $Y_k,k\ge 0,$ can also be represented in a similar form. In particular, we denote by $Y_k^{\{l,l\}}$ the diagonal blocks of the matrix $Y_k,k\ge 0,l=1,\dots ,m.$

According to [20], the following statement is true.

Lemma 3.

In the case of the reducible matrix $Y\left(1\right)$, the sufficient condition for the ergodicity of $AQTMC$s is the fulfillment of all m inequalities

$${\mathbf{y}}^{\left\{l\right\}}\sum _{k=1}^{\infty }k{Y}_k^{\{l,l\}}\mathbf{e}<1,l=1,\dots ,m,$$

(6)

where the row vector ${\mathbf{y}}^{\left\{l\right\}}$ is the unique solution of the system

$${\mathbf{y}}^{\left\{l\right\}}={\mathbf{y}}^{\left\{l\right\}}\sum _{k=0}^{\infty }{Y}_k^{\{l,l\}},{\mathbf{y}}^{\left\{l\right\}}\mathbf{e}=1,l=1,\dots ,m.$$

(7)

The use of Lemmas 2 and 3 allows for the determination of ergodicity conditions for various queueing systems. Sometimes, these conditions can be easily verified numerically. Finite systems (5) or (7) of the linear algebraic equations are solved, and their solutions are substituted into inequalities (6) or (7). Sometimes, the inequalities can be reduced to a nice scalar form.

However, the application of these conditions to the analysis of concrete queueing models requires preliminary verification that the $MC$ describing a queueing model indeed belongs to the class of $AQTMC$s. To this end, a computation of the blocks $Y_k,k\ge 1,$ of the one-step transition probability matrix for the limiting discrete-time jump $MC$ is necessary. It may not be easy.

It appears that if the customers arriving at the system can balk (abandon) the system with the probability tending to 1 when queue length upon arrival infinitely increases, or if the customers waiting in the system are impatient or non-persistent, sometimes, this verification and the other steps for the proof that the considered $MC$ is ergodic are not necessary. It can be stated that the considered $MC$ belongs to the class of $AQTMC$s and is ergodic (the corresponding queueing system has a stationary regime of operation) for any set of the system parameters.

Here, we present the results, which allow us to skip, under quite non-restrictive assumptions, the necessity of proving the affiliation of the considered $MC$ to the class of $AQTMC$s, including the computation of the matrices $Y_k,k\ge 0.$ This justifies the direct use of the algorithms developed for the computation of the stationary distribution of $AQTMC$s in [20,27,28,29] to compute the stationary distribution of the considered queueing system. The use of the algorithms from [20,27] requires certain analytical derivations (calculation of the limits of some matrices) to obtain the explicit form of the blocks $Y_k$ of the one-step transition probability matrix of the limiting discrete-time $MC$ for the $AQTMC$. The algorithms proposed in [28,29] do not need such derivations because they operate directly only with the blocks of the generator of the $AQTMC$. It should also be stressed that the results of this paper render unnecessary the derivation and control of the fulfillment of an ergodicity condition for $MC$s describing various queueing systems because it is shown here that these $MC$s are always ergodic due to customers’ balking, impatience or non-persistence.

3. Impact of Customers’ Impatience in the Systems Described, in the Absence of Impatience, by the Level-Independent $\mathit{QBD}$ and $\mathit{M}/\mathit{G}/\mathit{1}$ Type $\mathit{MC}$

3.1. Problem Statement

Let us consider a queueing system, the behavior of which, in the absence of customer impatience and non-persistence, is described by a regular irreducible continuous-time $MC$ ${\tilde{\zeta }}_t=\{i_t,n_t\}$, where the countable component $i_t$ defines the number of customers in the system, $i_t\ge 0.$ The process $n_t$ describes the transitions of a finite component, defining, along with the component $i_t$, the dynamics of the system.

Let us now assume that the customers who arrive to receive service in the system are impatient. Impatience means the ability of a customer to depart (renege) from the system while waiting in a buffer. We denote the $MC$ describing this system as ${\zeta }_t=\{i_t,n_t\}.$ Note that the $MC$s ${\tilde{\zeta }}_t=\{i_t,n_t\}$ and ${\zeta }_t=\{i_t,n_t\}$ have the same state space but different dynamics.

We assume that if the current state of the queueing system belongs to level i, then, during an interval of very small length $\Delta ,$ with the probability ${\alpha }_i\Delta +o(\Delta )$, one customer permanently departs from the system, and the finite component $n_t$ makes the transition possible. The matrix of the corresponding transition probabilities is denoted by $\Psi .$ It is most likely that $\Psi =I.$ However, for generality, we admit any stochastic matrix $\Psi .$ We suppose that the impatience rates ${\alpha }_i$ tend to infinity when i goes to infinity. Note that the most popular and reasonable dependence of ${\alpha }_i$ on i is ${\alpha }_i=i\alpha$ or ${\alpha }_i=max\{0,i-J\}\alpha .$ Here, $\alpha$ is the impatience rate (the parameter of exponentially distributed patience time) of individual customers, and it is assumed that customers renege from the system due to impatience independently of each other; J is the number of customers that cannot renege from the system. For example, if the component $i_t$ is the number of customers in the system, and only customers staying in the buffer can renege, then J corresponds to the current number of customers that receive service.

Our aims are to prove that the $MC$ ${\zeta }_t$ belongs to the class of $AQTMC$s and is ergodic for any values of the system parameters.

3.2. Problem Solution

Firstly, we consider the partial case when the $MC$ ${\tilde{\zeta }}_t=\{i_t,n_t\}$ describing the dynamic of the system with patient customers is the level-independent $QBD$. This means that the generator $\tilde{\mathcal{Q}}$ of this $MC$ is the block tri-diagonal matrix having the blocks ${\tilde{\mathcal{Q}}}_{i,j}$, of the form:

$${\tilde{\mathcal{Q}}}_{i,i}={\mathcal{Q}}^0,{\tilde{\mathcal{Q}}}_{i,i+1}={\mathcal{Q}}^{+},{\tilde{\mathcal{Q}}}_{i,i-1}={\mathcal{Q}}^{-},i>i^0.$$

It is easy to see that the $MC$ ${\zeta }_t=\{i_t,n_t\}$ describing the dynamic of the system with impatient customers is the level-dependent $QBD$. The generator $\mathcal{Q}$ of this $MC$ is the block tri-diagonal matrix having the blocks ${\mathcal{Q}}_{i,j}$, which define transition rates between the states that belong to the level i and states that belong to the level $j,$ of the form:

$${\mathcal{Q}}_{i,i}={\mathcal{Q}}^0-{\alpha }_i{I}_{N+1},{\mathcal{Q}}_{i,i+1}={\mathcal{Q}}^{+},{\mathcal{Q}}_{i,i-1}={\mathcal{Q}}^{-}+{\alpha }_i\Psi ,i>i^0.$$

Lemma 4.

The level-dependent $QBD$ ${\zeta }_t$ belongs to the class of $AQTMC$s.

Proof. 

It is clear that the block tri-diagonal structure of the generator $\mathcal{Q}$ of the $MC$ ${\zeta }_t$ is the special case of the structure (1).

It is evident that the matrix ${\mathcal{U}}_i$ for the $MC$ ${\zeta }_t$ is defined by

$${\mathcal{U}}_i={\alpha }_i{I}_{N+1}-{\widehat{\mathcal{Q}}}^0,$$

where ${\widehat{\mathcal{Q}}}^0$ is the diagonal matrix with the diagonal entries defined by the diagonal entries of the matrix ${\mathcal{Q}}^0.$

Calculating the limiting matrices $Y_k$ in the definition of $AQTMC$, it is easy to see that these limits indeed exist and are defined by

$$Y_0=\Psi ,Y_1=O,Y_2=O.$$

Thus, the $MC$ ${\zeta }_t$ satisfies the definition of $AQTMC$s. The lemma is proven. □

Theorem 1.

The $MC$ ${\zeta }_t$ is ergodic for any choice of the system parameters.

Proof. 

To prove the theorem, we apply Lemma 2 or Lemma 3. If the matrix $\Psi$ is irreducible, then the ergodicity condition (4) turns to the inequality $0<1$, which is always true.

If the matrix $\Psi$ is reducible, then the ergodicity conditions (6) with an account of evident equalities $Y_1^{\{l,l\}}=Y_2^{\{l,l\}}=O,l=1,\dots ,m,$ also turn to inequality $0<1$, which is always true.

Thus, we have proven that the $MC$ ${\zeta }_t$ is ergodic for any choice of the system parameters. □

Now, let the $MC$ ${\tilde{\zeta }}_t^{\ast }=\{i_t,n_t\}$ not be the level-independent $QBD$ but the more general $M/G/1$ type $MC$ having the generator, which is a particular case of the generator of form (1).

Let us assume that for $i>i^0,$ the blocks of the generator $\tilde{\mathcal{Q}}$ of this chain are defined by the formulas

$${\tilde{\mathcal{Q}}}_{i,i}={\mathcal{Q}}^0,{\tilde{\mathcal{Q}}}_{i,i-1}={\mathcal{Q}}^{-},{\tilde{\mathcal{Q}}}_{i,i+k}={\mathcal{Q}}^{+k},k=1,\dots ,K,{\tilde{\mathcal{Q}}}_{i,i+k}=O,k>K.$$

Here, K is an integer number, $K\ge 1.$ When $K=1,$ the $MC$ ${\tilde{\zeta }}_t^{\ast }$ is the $QBD.$ The case of $QBD$ has been analyzed above. Now, let us assume that $K>1.$ Note that the requirement that the number K be finite is not essential.

It is easy to see that the blocks of the generator of the $MC$ ${\zeta }_t^{\ast }=\{i_t,n_t\}$ describing the corresponding system with customer impatience are defined by

$${\mathcal{Q}}_{i,i}={\mathcal{Q}}^0-{\alpha }_i{I}_{N+1},{\mathcal{Q}}_{i,i+k}={\mathcal{Q}}^{+k},k=1,\dots ,K,{\mathcal{Q}}_{i,i-1}={\mathcal{Q}}^{-}+{\alpha }_i\Psi ,i>i^0.$$

Lemma 5.

The $MC$ ${\zeta }_t^{\ast }$ belongs to the class of $AQTMC$s.

The proof of this lemma repeats the proof of Lemma 4. However, here, not just two matrices $Y_1$ and $Y_2$ in the definition of $AQTMC$ but all matrices $Y_k,k=1,2,\dots ,K+1,$ are equal to $O.$

Theorem 2.

The $MC$ ${\zeta }_t^{\ast }$ is ergodic for any choice of the system parameters.

The proof of this theorem repeats the proof of Theorem 1, taking into account that, here, all matrices $Y_k,k=1,\dots ,K+1,$ are equal to O, while the matrix $Y_0$ is stochastic.

3.3. The Case of the Impatience Rate Dependent on the Value of Both Components of the Chain

In the problem statement, we have assumed that if the current state of the queueing system belongs to level i, then, during an interval of very small length $\Delta ,$ with the probability ${\alpha }_i\Delta +o(\Delta )$, one customer permanently departs from the system, and the impatience rate ${\alpha }_i$ tends to infinity when i tends to infinity.

Analyzing the proof of the ergodicity of the considered $MC$ for all values of the system parameters, it is not difficult to see that the obtained result can be generalized as follows.

Let us assume that if the state of the $MC$ ${\zeta }_t^{\ast }$ is $(i,n),$ then the probability that one customer permanently departs from the system during an interval of a very small length $\Delta$ is equal to ${\alpha }_i^{\left(n\right)}\Delta +o(\Delta )$ and

$${\alpha }_i^{\left(n\right)}\to \infty \mathrm{for} \mathrm{all} \mathrm{values} \mathrm{of}n=0,1,\dots ,N.$$

As above, the component $n_t$ can make transitions defined by the stochastic matrix $\Psi$ at the moment of the customer reneging.

In other words, here, we consider a more general case than the one considered above by assuming the possibility of having different impatience rates under different states of the finite component of the $MC.$

It is easy to prove, by analogy with the previous statements, the following assertions.

Lemma 6.

The $MC$ ${\zeta }_t^{\ast }$ belongs to the class of $AQTMC$s.

Theorem 3.

If the impatience rates ${\alpha }_i^{\left(n\right)}$ tend to infinite when i approaches infinite for any value of $n,n=0,1,\dots ,N,$ then the $MC$ ${\zeta }_t^{\ast }$ and its particular case ${\zeta }_t$ are ergodic for any choice of the system parameters.

The proof repeats the proof of Theorem 1 because it is easy to check that the matrix $Y_0$ is equal to the stochastic matrix $\Psi$ while all other limiting matrices $Y_k,k\ge 1,$ are equal to zero.

3.4. The System with Customers Balking

Let us return to the queueing system described by the $MC$ ${\tilde{\zeta }}_t=\{i_t,n_t\}$ defined in Section 2.1. It is quite typical for many real-world systems that the queue length is visible for arriving customers, and they can join or abandon the queue with a probability depending on the current queue length. In this subsection, we briefly consider such a scenario when the customers can abandon (balk) the system upon arrival. Namely, let us assume that the customer (or a batch of customers) arriving to the system when the current number of customers in the queue is equal to i (i.e., the $MC$ ${\tilde{\zeta }}_t=\{i_t,n_t\}$ resides in the state belonging to the level i), joins the queue with the probability $q_i$ and abandons the system with the complimentary probability.

We omit consideration of the simpler case when $K=1$ and assume that $K\ge 1.$ Let us denote the $MC$ describing the queue with customers balking by ${\eta }_t^{\ast }.$ It is easy to verify that the blocks of the generator of the $MC$ ${\eta }_t^{\ast }=\{i_t,n_t\}$ are defined via the blocks of the $MC$ ${\tilde{\zeta }}_t$ as follows:

$${\mathcal{Q}}_{i,i}={\mathcal{Q}}^0+(1-q_i)\sum _{k=1}^K{\mathcal{Q}}^{+k},{\mathcal{Q}}_{i,i+k}=q_i{\mathcal{Q}}^{+k},k=1,\dots ,K,{\mathcal{Q}}_{i,i-1}={\mathcal{Q}}^{-},i>i^0.$$

Lemma 7.

The $MC$ ${\eta }_t^{\ast }$ belongs to the class of $AQTMC$s.

Proof. 

It is easy to see that, here, the matrix ${\mathcal{U}}_i$ is defined by

$${\mathcal{U}}_i=-I\circ ({\mathcal{Q}}^0+(1-q_i)\sum _{k=1}^K{\mathcal{Q}}^{+k}),i>0,$$

and the limiting matrices $Y_k$ exist and are defined by formulas

$$Y_0=-{(I\circ \mathcal{T})}^{-1}{\mathcal{Q}}^{-},Y_1=I-{(I\circ \mathcal{T})}^{-1}\mathcal{T},Y_k=O,k=2,\dots ,K,$$

where

$$\mathcal{T}={\mathcal{Q}}^0+\sum _{k=1}^K{\mathcal{Q}}^{+k}.$$

This implies that the $MC$ ${\eta }_t^{\ast }$ indeed belongs to the class of $AQTMC.$ □

Theorem 4.

If the probabilities $q_i$ tend to zero when i approaches infinity, then the $MC$ ${\eta }_t^{\ast }$ is ergodic for any choice of the system parameters.

Proof. 

We obtain that the stochastic matrix $Y\left(1\right)={\displaystyle \sum _{k=0}^K}{Y}_k$ is, here, the sum of only two sub-stochastic matrices, $Y_0$ and $Y_1$, and the matrix $Y_0$ is a non-zero matrix. Taking into account the explicit form of the matrices $Y_0$ and $Y_1,$ we obtain that the matrix $Y\left(1\right)$ is defined by the formula

$$Y\left(1\right)=I-{(I\circ \mathcal{T})}^{-1}({\mathcal{Q}}^{-}+\mathcal{T})=I-{(I\circ \mathcal{T})}^{-1}\mathbf{Q}.$$

It was supposed above that the matrix $\mathbf{Q}$ is irreducible. A multiplication of this matrix from the left by the diagonal matrix with all non-zero diagonal entries and summing up with the identity matrix cannot make the resulting matrix reducible. Thus, the matrix $Y\left(1\right)$ is irreducible.

Therefore, inequality (4) as a sufficient condition for the ergodicity of $AQTMC$s can be rewritten here in the form $\mathbf{y}{Y}_0\mathbf{e}>\mathbf{y}{Y}_2\mathbf{e}$ and always holds good because the matrix $Y_0$ is non-zero sub-stochastic, while the matrix $Y_2$ is a zero matrix. The theorem is proven. □

Remark 1.

The obtained result holds good also in a more general situation when the joining probabilities have the form $q_i^{\left(k\right)},$ i.e., they depend on both the queue length i at the arrival moment and the number k of customers in the arrived batch of customers. The $MC$ ${\zeta }_t^{\ast }$ and its particular case ${\zeta }_t$ are ergodic for any choice of the system parameters if the probabilities $q_i^{\left(k\right)}$ tend to zero for all values of $k,k=1,2,\dots ,K,$ when i approaches infinity.

4. Impact of Customers’ Impatience in the Multi-Server Retrial Queueing Systems

4.1. Problem Statement

Let us consider the N-server retrial queueing system with patient customers. A customer who arrives at the system when less than N servers are busy immediately starts service. A customer who arrives when all servers are busy moves to the special virtual place called orbit and retries to enter service after exponentially distributed times. Let the total retrial rate from the orbit be equal to ${\nu }_i$ when the number of customers staying in the orbit of an infinite capacity is equal to $i,i\ge 1.$ We assume that ${\nu }_0=0$ and that there exists a finite or infinite limit of values ${\nu }_i$ when i tends to infinity. The most popular in the literature dependencies of ${\nu }_i$ on i are the classical retrial policy, when ${\nu }_i=i\nu ,$ where $\nu$ is an individual retrial rate, and the constant retrial rate when ${\nu }_i$ is equal to the constant independent of $i.$

If some server is available at a retrial moment, the retrying customer occupies the server and departs from the system after service completion. If all servers are busy at the retrial moment, the retrying customer returns to the orbit.

The behavior of the system is described by the three-dimensional $MC$ ${\tilde{\xi }}_t=\{i_t,n_t,r_t\}$, where $i_t$ is the number of customers in the orbit, $i_t\ge 0,$ $n_t$ is the number of busy servers, $n_t=0,1,\dots ,N,$ $r_t$ is the state of some auxiliary components of the chain, $r_t=1,\dots ,R,$ at the moment $t.$ As in the previous section, we start analysis from the case when the generator $\tilde{\mathcal{Q}}$ of the $MC$ ${\tilde{\xi }}_t$ has the block tri-diagonal structure with the non-zero blocks ${\tilde{\mathcal{Q}}}_{i,j}$ defined by the formulas

$${\tilde{\mathcal{Q}}}_{i,i}=\mathcal{A}-{\nu }_i{\mathcal{I}}^0,i\ge 0,{\tilde{\mathcal{Q}}}_{i,i+1}=\mathcal{C},i\ge 0,{\tilde{\mathcal{Q}}}_{i,i-1}={\nu }_i\mathcal{B},i\ge 1.$$

Here, the subgenerator $\mathcal{A}$ defines the transition rates of the components $\{n_t,r_t\}$ of the $MC$ ${\tilde{\xi }}_t$ that do not lead to a change in the value of the component $i_t$ of this chain. The matrix $\mathcal{C}$ defines the rates of transitions of the components $\{n_t,r_t\}$ of the $MC$ ${\tilde{\xi }}_t$ that imply the increase in the value of the component $i_t$ by 1. The matrix $\mathcal{B}$ defines the transition probabilities of the components $\{n_t,r_t\}$ of the $MC$ ${\tilde{\xi }}_t$ that imply the decrease in the value of the component $i_t$ by 1. The matrix ${\mathcal{I}}^0$ is defined by the formula ${\mathcal{I}}^0=\tilde{I}\otimes I_R,$ where $\tilde{I}=\mathrm{diag}\{1,1,\dots ,1,0\},$$\mathrm{diag}\{\dots \}$ denotes the diagonal matrix with the diagonal entries listed in the brackets, and ⊗ is the symbol of Kronecker product of matrices, see [30].

In detail, the matrix $\mathcal{C}$ is defined by the formula $\mathcal{C}=(I-\tilde{I})\otimes C$ where the matrix C defines the transition rates of the component $r_t$ of the $MC$ ${\tilde{\xi }}_t$ at the moment of a customer’s arrival to the system in the presence of N busy servers. The matrix $\mathcal{B}$ is defined by the formula $\mathcal{B}={\mathrm{diag}}^{+}\{B_0,B_1,\dots ,B_{N-1}\}$ where the ${\mathrm{diag}}^{+}\{\dots \}$ denotes the matrix having all zero blocks except the blocks $B_n$ above the diagonal, $n=0,1,\dots ,N-1.$ The block $B_n$ defines the transition probabilities of the component $r_t$ of the $MC$ ${\tilde{\xi }}_t$ at the moment of a retrying customer service beginning in the presence of n busy servers.

The concrete form of the described blocks for the particular case of the considered general retrial queueing system, such as the $BMAP/PH/N$ type retrial queue, can be found in [21,31]. Here, $BMAP$ denotes the batch Markov arrival process; for details, see, e.g., [32,33,34]. The $PH$ denotes the phase-type distribution of service times. Details about the $PH$ distribution can be found, e.g., in [10,32,34,35].

In [21], the component $r_t$ of the $MC$ ${\tilde{\xi }}_t$ is the set of components such as the underlying process of the $BMAP$ flow of customers and the underlying processes of the $PH$ distribution of service time on all busy servers. In [31], the component $r_t$ of the $MC$ ${\tilde{\xi }}_t$ is the set of components such as the underlying process of the $BMAP$ flow of customers and the number of servers providing service at each phase. For more details about these two different ways for tracking the $PH$ distribution of service time in busy servers, see, e.g., [36]. For the purposes of this paper, the explicit form of the blocks $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ does not matter.

Let us now modify this queueing system by assuming that the customers staying in the orbit are impatient. The mechanism of orbiting customers reneging from the orbit is the same as the mechanism of waiting customers reneging from the buffer described in the previous section. The total rate of customers reneging from the orbit when the number of customers in the orbit is equal to $i,i\ge 1,$ is denoted by ${\alpha }_i.$ We suppose that $\underset{i\to \infty }{lim}{\alpha }_i=\infty .$ We denote the matrix of transition probabilities of the components $\{n_t,r_t\}$ at the moment of a customer reneging as $\Psi .$ Note that, in this section, in which we separate two finite components, the size of the square matrix $\Psi$ is $(N+1)R$. In the previous section, the corresponding matrix had a size $(N+1).$

The $MC$ describing the considered retrial queue in the presence of customer impatience is denoted by ${\xi }_t$.

Our aim is to prove that the $MC$ ${\xi }_t$ is affiliated to the class of $AQTMC$ and is ergodic for any value of the system parameters.

4.2. Problem Solution

As in the previous section, we start our analysis from the case when the generator $\mathcal{Q}$ of the $MC$ ${\xi }_t$ has the block tri-diagonal structure.

It can be verified that the non-zero blocks ${\mathcal{Q}}_{i,j}$ of this generator are defined by the formulas

$${\mathcal{Q}}_{i,i}=\mathcal{A}-{\nu }_i{\mathcal{I}}^0-{\alpha }_i{I}_{(N+1)R},i\ge 0,{\mathcal{Q}}_{i,i+1}=\mathcal{C},i\ge 0,{\mathcal{Q}}_{i,i-1}={\nu }_i\mathcal{B}+{\alpha }_i\Psi ,i\ge 1.$$

(8)

Lemma 8.

The $MC$ ${\xi }_t$ belongs to the class of $AQTMC$s.

Proof. 

Let us denote $\widehat{\mathcal{A}}$ as the diagonal matrix with the diagonal entries defined by the moduli of the diagonal entries of the matrix $\mathcal{A}$. Also denoted as ${\mathcal{U}}_i$ is the matrix ${\mathcal{U}}_i=\widehat{\mathcal{A}}+(({\nu }_i+{\alpha }_i)\tilde{I}+{\alpha }_i(I-\tilde{I}))\otimes I_R.$

It is not difficult to verify that the following limits exist:

$$Y_k=\underset{i\to \infty }{lim}{\mathcal{U}}_i^{-1}{\mathcal{Q}}_{i,i+k-1}+{\delta }_{k,1}I,k=0,1,2,$$

and are defined as

$$Y_1=Y_2=O,Y_0=\frac{\gamma }{\gamma +1}(\tilde{I}\otimes I_R)\mathcal{B}+(\frac{1}{\gamma +1}\tilde{I}\otimes I_R+(I-\tilde{I})\otimes I_R)\Psi .$$

(9)

Here,

$$\gamma =\underset{i\to \infty }{lim}\frac{{\nu }_i}{{\alpha }_i}.$$

If $\gamma =0,$ i.e., ${\alpha }_i$ tends to infinity more fast than ${\nu }_i,$ then, $Y_0=\Psi .$ If $\gamma =\infty ,$ i.e., ${\alpha }_i$ tends to infinity more slowly than ${\nu }_i,$ then, $Y_0=(\tilde{I}\otimes I_R)\mathcal{B}+((I-\tilde{I})\otimes I_R)\Psi .$ If $\gamma$ is a finite positive number, the matrix $Y_0$ is a stochastic matrix defined in Formula (9). □

Therefore, results from [20] can be used for the derivation of a sufficient condition for the ergodicity of the $MC$ ${\xi }_t.$ Note that when $K=1,$ inequalities (4) and (6) in the ergodicity conditions for $AQTMC$s take the form

$$\mathbf{y}{Y}_0\mathbf{e}>\mathbf{y}{Y}_2\mathbf{e},{\mathbf{y}}^{\left\{l\right\}}{Y}_0^{\{l,l\}}\mathbf{e}>{\mathbf{y}}^{\left\{l\right\}}{Y}_2^{\{l,l\}}\mathbf{e},l=1,\dots ,m.$$

Because, according to (9), the matrix $Y_0$ is a stochastic one while $Y_2=O,$ these inequalities are trivially fulfilled.

Therefore, the following statement is proven.

Theorem 5.

The $MC$ ${\xi }_t$ describing the retrial queueing model with impatient customers is ergodic for any choice of the system parameters.

Let ${\xi }_t^{\ast }$ be the $MC$ which is the generalization of the $MC$ ${\xi }_t$ to the case when the generator of the chain has not three but $K+1$ non-zero block diagonals, similar to the $MC$ ${\zeta }_t^{\ast }$ considered in the previous section as the generalization of the $MC$ ${\zeta }_t$.

The validity of the following corollaries is easily established.

Corollary 2.

The $MC$ ${\xi }_t^{\ast }$ is affiliated to the class of $AQTMC$s.

Corollary 3.

The $MC$ ${\xi }_t^{\ast }$ is ergodic for any choice of the system parameters.

Remark 2.

If arriving customer balking (refusal to go to orbit upon arrival if all servers are busy) would be incorporated into the considered retrial queueing system described by the $MC$ ${\tilde{\xi }}_t,$ and the probability of moving to orbit $q_i$ tends to zero, then, the modified retrial queueing model will be stable for all values of the system parameters.

5. Impact of Customers Non-Persistence in the Multi-Server Retrial Queueing Systems

5.1. Problem Statement

Let us consider the same three-dimensional $MC$ ${\tilde{\xi }}_t=\{i_t,n_t,r_t\}$ describing the behavior of the multi-server retrial queueing system as in the previous section. The generator $\tilde{\mathcal{Q}}$ of this chain is defined by the blocks ${\tilde{\mathcal{Q}}}_{i,j}$ given by Formula (8).

Let us now modify the queueing system by assuming that if a customer makes the retrial in the presence of i customers in the orbit, and all servers are busy, then, with the probability $p_i,i\ge 1$, this customer returns to the orbit, and with the complementary probability, it departs from the system permanently. We assume that there is a limit

$$p=\underset{i\to \infty }{lim}{p}_i$$

and $p<1.$

This assumption is fulfilled, in particular, when the customers staying in the orbit decide whether to return to the orbit in the case when all servers are busy at the retrial moment independently of each other with the probability $p,p<1.$

Notions of customer impatience and non-persistence are close. Both impatience and non-persistence cause the same effect, namely, that the customer permanently leaves the system. The difference is that the departure due to impatience can occur at an arbitrary moment during the customer staying in the orbit, while the departure due to non-persistence can occur at any retrial moment. In the context of retrial queues, it is more reasonable for the customer waiting in the orbit to check the status of the servers before departing, whether or not all servers are busy at this moment. However, sometimes, the customers can leave the system without making the “last” trial. For example, in modeling the cell of the mobile communication network, the retrying customer can decide to stop retrials after any unsuccessful retrial and can also leave the cell due to the loss of connection to the base station under the move to another cell.

Let us denote by ${\widehat{\xi }}_t$ the $MC$ describing the dynamics of the system with non-persistent customers. Additionally, let us assume that a customer departure from the system due to non-persistence does not cause any changes in the value of the components $\{n_t,r_t\}$ of the $MC$ ${\widehat{\xi }}_t.$ The contrary case can be considered analogously.

The aim of this section is to briefly show that the $MC$ ${\widehat{\xi }}_t$ is always ergodic.

5.2. Problem Solution

As in the previous sections, we start analysis from the case when the generator $\mathcal{Q}$ of the $MC$ ${\widehat{\xi }}_t$ has the block tri-diagonal structure.

It can be verified that the non-zero blocks ${\mathcal{Q}}_{i,j}$ of this generator are defined by formulas

$${\mathcal{Q}}_{i,i}=\mathcal{A}-{\nu }_i{\widehat{\mathcal{I}}}_i^0,i>0,{\mathcal{Q}}_{i,i+1}=\mathcal{C},i>0,{\mathcal{Q}}_{i,i-1}={\nu }_i{\widehat{\mathcal{B}}}_i,i>1,$$

where

$${\widehat{\mathcal{I}}}_i^0=(\tilde{I}+(1-p_i)(I-\tilde{I}))\otimes I_R,i\ge 1,$$

$${\widehat{\mathcal{B}}}_i=\mathcal{B}+(1-p_i)(I-\tilde{I})\otimes I_R,i>1.$$

Lemma 9.

The $MC$ ${\widehat{\xi }}_t$ belongs to the class of $AQTMC$s.

Proof. 

Let, as in the previous section, $\widehat{\mathcal{A}}$ be the diagonal matrix with the diagonal entries defined by the moduli of the diagonal entries of the matrix $\mathcal{A}.$

It is easy to see that the matrix ${\mathcal{U}}_i$ appearing in the definition of the $AQTMC$ is defined by the formula

$${\mathcal{U}}_i=\widehat{\mathcal{A}}+{\nu }_i{\widehat{\mathcal{I}}}_i^0.$$

It is not difficult to verify that the following limits

$$Y_k=\underset{i\to \infty }{lim}{\mathcal{U}}_i^{-1}{\mathcal{Q}}_{i,i+k-1}+{\delta }_{k,1}I,k=0,1,2,$$

indeed exist and are defined as

$$Y_1=Y_2=O,Y_0=\mathcal{B}+(I-\tilde{I})\otimes I_R.$$

This completes the proof of the lemma. □

Again, the matrix $Y_0$ is a stochastic one, and, irrespective of whether the matrix $Y_0$ is irreducible or reducible, the fact that $Y_1=Y_2=O$ implies the trivial fulfillment of inequalities (4) or (6) in a sufficient condition of ergodicity of the $MC.$

Therefore, the following statement is proven.

Theorem 6.

The $MC$ ${\widehat{\xi }}_t$ describing the retrial queueing model with non-persistent customers is ergodic for any choice of the system parameters.

Let ${\widehat{\xi }}_t^{\ast }$ be the $MC$ which is the generalization of the $MC$ ${\widehat{\xi }}_t$ to the case when the generator of the chain has not three but $K+1$ non-zero block diagonals, similar to the $MC$ ${\zeta }_t^{\ast }$ considered in Section 2 as the generalization of the $MC$ ${\zeta }_t$.

The validity of the following corollary is easily established.

Corollary 4.

The $MC$ ${\widehat{\xi }}_t^{\ast }$ is affiliated to the class of $AQTMC$s.

Corollary 5.

The $MC$ ${\widehat{\xi }}_t^{\ast }$ is ergodic for any choice of the system parameters.

Remark 3.

Theorem 5 states that the retrial queue is always stable (the corresponding $MC$ is ergodic) if the customers are impatient. Theorem 6 states that the retrial queue is always stable if the customers are non-persistent. It is easy to check that the retrial queue is always stable if the customers are both impatient and non-persistent.

For reader convenience, the random processes used and analyzed in the text are summarised in

Table A1. Processes denotation.

Process Denotation	System	Specifics	Section
${\tilde{\zeta }}_t$	QBD	NO	Section 2.2
${\zeta }_t$	QBD	IMP	Section 2.2
${\tilde{\zeta }}_t^{\ast }$	$M/G/1$	NO	Section 2.2
${\zeta }_t^{\ast }$	$M/G/1$	IMP	Section 2.2 and Section 2.3
${\eta }_t^{\ast }$	$M/G/1$	BALK	Section 2.4
${\tilde{\xi }}_t$	QBD retrial	NO	Section 3.1
${\xi }_t$	QBD retrial	IMP	Section 3.1
${\xi }_t^{\ast }$	$M/G/1$ retrial	IMP	Section 3.1
${\widehat{\xi }}_t$	QBD retrial	NONPER	Section 4.1
${\widehat{\xi }}_t^{\ast }$	$M/G/1$ retrial	NONPER	Section 4.1

in Appendix A.

6. Conclusions

In this paper, it is shown that for a great variety of queueing models, including a wide range of variants of the $BMAP/PH/N$-type queues, with an infinite buffer or orbit, the following is true. If the customers waiting in the buffer or the orbit are impatient with an infinitely increasing total impatience rate, then, the multidimensional $MC$ describing the behavior of the system is affiliated to the class of $AQTMC$s and is always ergodic. The same is valid for systems with customers balking upon arrival as well as for the retrial queue with non-persistent customers, with the limiting value of the probability to return to the orbit when all servers are busy at the retrial moment less than 1.

The account of customer impatience, balking or non-persistence usually leads, generally speaking, to an essential complication in the computation of the stationary distribution compared to the systems with patient customers due to the space-inhomogeneous behavior of the corresponding $MC.$ But, as it is shown in this paper, the obtained $MC$ belongs to the class of $AQTMC$s. This has the following two important implications: (i) the question about the existence of a stationary distribution (under non-restrictive assumptions) always has a positive answer; (ii) effective and numerically stable algorithms from [20,27,28,29] can be used to compute the stationary distribution of the considered queueing system instead of various truncation schemes popular in the existing literature.

The presented results are planned to be extended to the case when the impatience rate depends not only on the value of the denumerable component but also on the value of the finite component of the $MC$ and tends to infinity not mandatory for all but at least for some values of the finite component.