Framework for Analysis of Queueing Systems with Correlated Arrival Processes and Simultaneous Service of a Restricted Number of Customers in Scenarios with an Infinite Buffer and Retrials

Abstract

In this paper, we create a framework for the uniform algorithmic analysis of queueing systems with the Markov arrival process and the simultaneous service of a restricted number of customers, described by a multidimensional Markov chain. This chain behaves as the finite-state quasi-death process between successive service-beginning epochs, with jumps occurring at these epochs. Such a description of the service process generalizes many known mechanisms of restricted resource sharing and is well suited for describing various future mechanisms. Scenarios involving customers who cannot enter service upon arrival, access via waiting in an infinite buffer, and access via retrials are considered. We compare the generators of the multidimensional Markov chains describing the operation of queueing systems with a buffer and with retrials and show that the sufficient conditions for the ergodicity of these systems coincide. The computation of the stationary distributions of these chains is briefly discussed. The results can be used for performance evaluation and capacity planning of various queueing models with the Markov arrival process and a variety of different service mechanisms that provide simultaneous service to many customers.

1. Introduction

Single-server queueing systems, where customers receive service one by one, have received significant attention in the existing queueing literature. However, impressive progress in service provision technologies in various real-world systems, particularly telecommunication systems and networks, has made it important to investigate queueing systems that can provide service to many customers simultaneously. The simplest case of such systems is the standard multi-server queueing system, where the system resource is divided into a fixed number, say, N, of parts (servers), each of which can provide service to customers independently of the other servers. This type of system has also been well investigated in the literature since the pioneering work of A.K. Erlang in the early 1900s. A shortcoming in the analysis of multi-server queueing systems compared to single-server queueing systems is that single-server systems are well studied in the case of arbitrary distributions of service times and even mutually dependent service times, while for multi-server systems, except for the model without a buffer (Erlang loss model), exact analyses of multi-server queues are only implemented when the service time has an exponential distribution or, at most, an essential generalization such as the phase-type ($PH$) distribution. For definitions and properties, see, e.g., [1,2].

Further progress in technology has led to the possibility and the necessity of analyzing other, more effective mechanisms of resource sharing for the simultaneous service of several customers, like processor sharing ($PS$), see, e.g., [3,4]. This mechanism assumes, in the simplest settings, that all customers receive service simultaneously at a rate inversely proportional to the number of customers. The classical $PS$ mechanism is not suitable for many real-world systems, e.g., telecommunication systems, where service at a very low rate may not be acceptable to customers. Therefore, the number of simultaneously serviced customers must be restricted. Thus, the discipline of limited processor sharing ($LPS$) was introduced. This discipline assumes that no more than N customers can receive service at the same time. The number N is sometimes called the multi-programming, concurrency level, or multiplicity. For details, see, e.g., [5,6,7,8,9].

The mixed-service discipline (for details see e.g., [10,11]) assumes that a limited number of customers (corresponding to the so-called non-elastic traffic) receive service at a constant rate, as in the standard multi-server system, while another finite group of customers is processed using the $LPS$ discipline.

As another example of systems in which up to a limited number of customers may receive service simultaneously, the so-called semi-open queueing networks can be mentioned (see, e.g., [12,13,14,15]), in which only a finite number of customers can be processed in the inner network. Therefore, the range of queueing models with different mechanisms for the simultaneous service of customers is quite wide. The first contribution of this paper is the presentation of a framework for the uniform consideration of queueing systems with existing and future service disciplines.

Besides the service discipline, the performance characteristics of any queueing model essentially depend on the arrival process. Here, we assume a general Markov arrival process ($MAP$), which is a more general process than the stationary Poisson process and the renewal process with a $PH$ distribution of inter-arrival times. It is known that the $MAP$ is suitable for describing arrival processes with correlated inter-arrival times and times with large variance, which are typical of certain information flows in telecommunication systems and contact centers. For more information about the $MAP$, see, e.g., [16,17,18,19,20,21].

First, we precisely define the notion of simultaneous service of a restricted number of customers, which is described by a multidimensional Markov chain ($MC$), and present several examples of partial cases of such service. After that, we consider a system with the $MAP$, a general Markovian description of the simultaneous service process for a finite number of customers, and an infinite buffer. This buffer is designed for the temporary storage of customers who arrive when service capacity is exhausted, i.e., the number of customers receiving service is equal to $N.$

It is well known that in some real-world systems, e.g., in wireless communication networks, there is no opportunity to store customers. Instead, customers have the option to temporarily leave the service area and make repeated attempts to enter service after random time intervals. This type of queueing system is called a retrial queue. For references and some known results, see the books [22,23] and the papers [24,25,26,27,28].

Retrial queues are a more complicated research subject than queues with buffers. This statement is explained in detail in [29], where a comparison of standard and retrial queues was conducted. The second contribution of our paper is an analysis of the discipline with simultaneous service for both queueing systems with an infinite buffer and systems with customer retrials. We show how to easily obtain the generator of the multidimensional continuous-time $MC$ describing the system with a general Markov description of customer processing and customer retrials, based on the known generator of the $MC$ describing a similar system with an infinite buffer. As the byproduct, we illustrate why the analysis of a retrial queue is much more complicated from both theoretical and computational points of view. Also, we show that the sufficient conditions for the ergodicity of both MCs coincide if the rate of customer retrials tends to infinity as the number of retrying customers tends to infinity.

Researchers who analyze specific cases of simultaneous service for multiple customers, which can arise in the analysis of real-world systems, may find the presented results useful. The single task of a researcher is to write down three groups of matrices describing the simultaneous service process. Using the results of this paper, this researcher immediately obtains the explicit form of the generator of the Markov chain describing the behavior of the system, the sufficient condition for the egrodicity of this chain, and the manual for computation of its stationary distribution in scenarios with the existence of an infinite buffer or customers retrials (with an infinitely increasing or constant retrial rate).

The outline of the results presentation is as follows. In Section 2, the Markov mathematical model of simultaneous service of a finite number of customers is described, and the necessary notation is given. Several examples of a description are presented. In Section 3, the system with the $MAP$ arrival process, simultaneous service of a finite number of customers, and an infinite buffer is analyzed. The ergodicity condition for the multidimensional Markov process describing the behavior of this system is derived. The known algorithmic results relating to the computation of the stationary distribution of this system are presented. In Section 4, a similar system but with customer retrials is considered. A brief comparison of the generators and the multidimensional Markov chains describing the behavior of the systems with the infinite buffer and retrials is presented. The coincidence of sufficient conditions for the ergodicity of these two chains in the case of the infinitely increasing retrial rate is shown. The numerically stable algorithm for computation of the stationary distribution of the states of the system with retrials is briefly described. Required modifications in the generators of the chains analyzed in Section 3 and Section 4 in the case when waiting customers are impatient are discussed in Section 5. Section 6 concludes the paper.

2. Description of a Simultaneous Service Process

We consider a queueing system with the possibility of providing service simultaneously to up to N customers. Customers receive service and leave the system individually. The case of a possibility of group service is not considered here. Its consideration deserves attention and can be implemented in the future.

2.1. Description

The description of the service process is as follows. Let the number of customers in service at the moment t be $n_t,n_t=\overline{0,N}.$ This notation means that $n_t$ admits values in the set $\{0,1,\dots ,N\}.$ Let ${\eta }_t$ be the auxiliary finite-state process such that the two-dimensional process $\{n_t,{\eta }_t\}$ behaves as the quasi-death process between the successive service beginning epochs. The quasi-death process is the case of the well-known Quasi-Birth-and-Death ($QBD$) process, see [1].

Under the fixed state n of the process $n_t,$ $n=\overline{0,N},$ the process ${\eta }_t$ admits values from the set of cardinality $K_n,K_n<\infty .$ The quasi-death process $\{n_t,{\eta }_t\}$ under the fixed value n of the process $n_t$ is completely defined by two matrices, $H_n^0$ and $H_n^{-}.$ The non-diagonal entries of the square matrix $H_n^0,n=\overline{0,N},$ of size $K_n$ describe the transition rates of the process ${\eta }_t$ within its state space which do not lead to the end of customer service. The diagonal entries of the matrix $H_n^0$ are negative. The module of such an entry defines the rate of the process ${\eta }_t$ exit from the corresponding state. The matrix $H_n^{-},n=\overline{1,N},$ of size $K_n\times K_{n-1}$ describes transition rates of the process ${\eta }_t$ when one of n customers processed in the system departs.

Note that the matrix $H_n^0$ is the sub-generator and $H_n^0\mathbf{e}+H_n^{-}\mathbf{e}={\mathbf{0}}^T,n=\overline{1,N},$ where $\mathbf{e}$ and $\mathbf{0}$ are column and row vectors of 1’s and 0’s of a suitable size, respectively, and ^T denotes transposition of the vector.

Because when $n=0$ customers are present in the system, it does not make sense to speak about their deaths. Thus, below by default we assume that $K_0=1$, $H_0^0=0,$ $H_0^{-}=0.$

If the number of customers receiving service in the system is $n,n=\overline{0,N-1},$ and a new customer arrives, this customer is admitted to the system. The number of customers in service increases to $n+1$ and the process ${\eta }_t$ makes the transition with the probabilities that constitute the stochastic matrix $H_n^{+},n=\overline{0,N-1},$ of size $K_n\times K_{n+1},$ $H_n^{+}\mathbf{e}=\mathbf{e},n=\overline{0,N-1}.$

Let us briefly present the examples of the determination of the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}.$

Let customers’ service be provided by N independent servers. Service time by a server has a $PH$ distribution with the irreducible representation $(\mathit{\beta },S)$ where $\mathit{\beta }=({\beta }_1,{\beta }_2,\dots ,{\beta }_M)$ is the stochastic row vector of size M and S is a sub-generator of size $M.$ Denote ${\mathbf{S}}_0=-S\mathbf{e}.$ Such a service time can be interpreted as the time of reaching the single absorbing state $M+1$ by the underlying $MC$ ${\zeta }_t$ that admits the value $m,m=\overline{1,M},$ at the moment of service beginning with the probability ${\beta }_m$, transits in the set of transient states $\{1,2,\dots ,M\}$ with the rates defined by the entries of the matrix S and, finally, transits to the absorbing state with the rates defined by the entries of the column vector ${\mathbf{S}}_0.$ For more detailed information about $PH$ distribution, see, e.g., [1,17,20].

The most popular in the existing literature two ways to description of the service process of many customers, generate two examples of constructing the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ defining the process $\{n_t,{\eta }_t\}$ for this system with N identical independent servers and the $PH$ distribution of service times.

2.2. Example 1

The first way for the description called $TPFS$ (track phase for server), see [30], assumes permanent re-numeration of busy servers in the order of their occupation and tracking the phase of service in each busy server. Following this way, we obtain that when the number of customers receiving service is equal to $n,$ then the auxiliary process ${\eta }_t$ has the components $({\zeta }_t^{\left(1\right)},{\zeta }_t^{\left(2\right)},\dots ,{\zeta }_t^{\left(n\right)})$ where ${\zeta }_t^{\left(k\right)}$ is the current phase of service in the kth busy server, $k=\overline{1,n}.$ For the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+},$ we have the following expressions:

$$H_n^0=S^{\oplus n}\stackrel{def}{=}\underset{n}{\underbrace{S\oplus \dots \oplus S}},n=\overline{0,N},S^{\oplus 0}\stackrel{def}{=}0,$$

$$H_n^{-}={\mathbf{S}}_0^{\oplus n}\stackrel{def}{=}\sum _{m=0}^{n-1}{I}_{M^m}\otimes {\mathbf{S}}_0\otimes I_{M^{n-m-1}},n=\overline{1,N},$$

$$H_n^{+}=I_{M^n}\otimes \mathit{\beta },n=\overline{0,N-1},$$

where ⊗ is the sign of Kronecker’s product, and ⊕ is the sign of Kronecker’s sum of square matrices, see, e.g., [31,32], $I_K$ denotes the identity matrix of size K, $I_{M^0}=1.$

The retrial queue of $MAP/PH/N$ type with such a description of simultaneous service was analyzed in [33,34]. In [34], its application to the analysis of the organization of sharing the resource of hot spot in an airport is discussed.

2.3. Example 2

The second way called $CSFP$ (count server for phase) in [30], assumes keeping track of the number of servers providing service at various phases. Following this way, we obtain that when the number of customers receiving service is equal to $n,$ then the auxiliary process ${\eta }_t$ is defined as ${\eta }_t=(x_t^{\left(1\right)},x_t^{\left(2\right)},\dots ,x_t^{\left(M\right)})$ where the component $x_t^{\left(m\right)}$ is the number of servers at the mth phase of service, $m=\overline{1,M}.$ Expressions for the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+},$ are derived in [35]. In formula (1) in [35], the $H_n^0$ corresponds to the matrix $A_n,n=\overline{0,N}$, $H_n^{-}$ corresponds to the matrix $L_{N-n},n=\overline{1,N}$, and $H_n^{+}$ corresponds to the matrix $U_{N-n},n=\overline{0,N-1}.$ The non-diagonal entries of the matrix $A_n$ define transition rates of the process ${\eta }_t$ which do not cause service completion on any of the n busy servers. The diagonal entries of the matrix $A_n$ define exit rates of the process ${\eta }_t$ from its states when n customers are processed. The matrix $L_{N-n}$ defines the rates of transitions leading to service completion in one of n busy servers. The matrix $U_{N-n}$ defines transition probabilities of the process ${\eta }_t$ when n servers are busy and a new service starts. Recursive formulas for computation of the matrices $A_n,L_{N-n},U_{N-n}$ given in [35] are simplified in [36]. The description of simultaneous service in the retrial queue of $MAP/PH/N$ type as given in example 2 was used in [37].

Expressions for matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ are more transparent and easily derived when the $TPFS$ way is used. However, when M is small, the size of these matrices is essentially smaller if the $CSFP$ way is used; for concrete formulas and numbers, see, e.g., [30]. Mention that if the service time has an exponential distribution, then the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ become scalars equal to $-n\mu ,$ $n\mu ,$ and 1, respectively.

2.4. Example 3

Let the service time of a customer, which receives service in the system alone, have the $PH$ distribution with an irreducible representation $(\mathit{\beta },S).$ Let the $LPS$ service discipline be used, and the maximal number of customers that can receive service simultaneously is equal to N. Recall that under this discipline, all admitted customers are processed at an equal rate that is inversely proportional to the number of serviced customers. Thus, if n customers receive service, then the transitions of the underlying process of service of each of these customers are described by the sub-generator ${\displaystyle \frac{1}{n}}S.$

Formulas for computation of the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ in the description of simultaneous service for such a system can be found in Lemma 1 in [6]. Namely,

$$H_n^0={\displaystyle \frac{A_n}{n}},n=\overline{1,N},H_n^{-}={\displaystyle \frac{L_{N-n}}{n}},n=\overline{1,N},H_n^{+}=U_{N-n},n=\overline{0,N-1},$$

where the meaning of matrices $A_n,L_{N-n},U_{N-n}$ is explained in the previous example and recursive algorithms for their computation are given in [35,36].

It is worth noting that besides the classical $LPS$ discipline, where all admitted customers equally share the system’s bandwidth (egalitarian $PS$), there exist modifications of this discipline, e.g., discriminatory $PS$ and the generalized $PS$, see [38,39]. These modifications can also be analyzed using the results of this paper.

2.5. Example 4

One more example of computation of the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ is as follows. Let the system’s bandwidth be divided into N equal parts (independent servers) as in the classical N-server queueing system. Let the service process be influenced by the random environment ($RE$) that is a continuous-time $MC$ with state space $\{1,2,\dots ,R\}$ and the generator $\Phi =\left|\right|{\phi }_{r,r^{\prime }}\left|\right|,r,r^{\prime }=\overline{1,R}.$ Under the fixed state, r, of the $RE$, the service time in each busy server has an exponential distribution with the rate ${\mu }_r,r=\overline{1,R}.$

It is easy to check that customer service in this system is defined by the set of the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ defined by

$$H_n^0=\Phi -n\mathbf{M},H_n^{-}=n\mathbf{M},H_n^{+}=I$$

where the matrix $\mathbf{M}$ is the diagonal one with the diagonal entries ${\mu }_r,r=\overline{1,R}.$ It is worth noting that some rates ${\mu }_r$ may be equal to zero. Therefore, this queueing model can be applied for the analysis of the systems that can work intermittently, i.e., where service can be unavailable during some random time, in particular, unreliable queueing systems.

It is easy to continue this list of examples. Thus, the problem of computation of the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ characterizing simultaneous service of customers can be solved for a variety of queueing models.

Below we show how they can be used for analyzing the performance of the corresponding systems with an infinite buffer and flow of customers defined by the $MAP.$

3. Analysis of the Model with the MAP, Simultaneous Service Process, and Infinite Buffer

Let us analyze the characteristics of the queueing system with the service process described in the previous section. We assume that the arrival process of customers is defined by the $MAP.$ Arrivals can occur only at the epochs of transitions of an underlying continuous-time $MC$ ${\nu }_t,t\ge 0.$ This $MC$ is irreducible and has the finite state space $\{1,2,\dots ,W\}$ and the generator $D\left(1\right)$, which is the sum of two matrices, $D_0$ and $D_1,$ the first of which is the sub-generator. The elements of the non-negative matrix $D_1$ define the rates of transitions of $MC$ ${\nu }_t$ that cause customer arrival. The invariant probability vector $\mathit{\theta }$ of $MC$ ${\nu }_t$ is calculated as a solution to the system $\mathit{\theta }D\left(1\right)=\mathbf{0},\mathit{\theta }\mathbf{e}=1.$ The fundamental rate $\lambda$ of the $MAP$ is given by the formula $\lambda =\mathit{\theta }{D}_1\mathbf{e}.$ Formulas for calculation of the higher moments of inter-arrival times and the coefficient of their correlation can be found, e.g., in [16,20].

The maximum number of customers that can be processed in the system simultaneously is $N.$ If the number of customers in the systems at the epoch of a customer arrival is less than $N,$ this customer joins the set of customers receiving service. If the number of customers in the systems at the epoch of a customer arrival is equal to $N,$ this customer joins the buffer of an unlimited size. The process of customer service is defined in Section 2.

Let us analyze the described queueing system. The behavior of the system is determined by the three-dimensional process

$${\xi }_t=\{i_t,{\nu }_t,{\eta }_t\},i_t\ge 0,{\nu }_t=\overline{1,W},{\eta }_t=\overline{1,K_{min\{i_t,N\}}},t\ge 0,$$

where $i_t$ is the number of customers in the system (in the buffer and in service), and ${\nu }_t$ and ${\eta }_t$ are the states of the underlying processes of arrivals and service at the moment $t,t\ge 0.$ It follows from definitions of arrival and service processes that the random process ${\xi }_t$ is the continuous-time $MC$.

Let us arrange the states of the $MC$ ${\xi }_t$ in the lexicographic order. The set of the states having value i of the component $i_t$ is called level $i,i\ge 0.$

Let the matrix ${\mathbf{Q}}_{i,j}$ contain transition rates from the states that belong to level i to the states that belong to level $j,i,j\ge 0,|i-j|\le 1.$

Lemma 1. 

The generator $\mathbf{Q}={\left({\mathbf{Q}}_{i,j}\right)}_{i,j\ge 0,|i-j|\le 1}$ of the $MC$ ${\xi }_t$ contains the following non-zero blocks:

• ${\mathbf{Q}}_{i,i}=D_0\oplus H_i^0,i=\overline{0,N},$
• ${\mathbf{Q}}_{i,i}=D_0\oplus H_N^0:={\mathbf{Q}}_0,i\ge N+1,$
• ${\mathbf{Q}}_{i,i+1}=D_1\otimes H_i^{+},i=\overline{0,N-1},$
• ${\mathbf{Q}}_{i,i+1}=D_1\otimes I_{K_N}:={\mathbf{Q}}^{+},i\ge N,$
• ${\mathbf{Q}}_{i,i-1}=I_W\otimes H_i^{-},i=\overline{1,N},$
• ${\mathbf{Q}}_{i,i-1}=I_W\otimes H_N^{-}{H}_{N-1}^{+}:={\mathbf{Q}}^{-},i>N.$

The proof of the statement of Lemma 1 accounts for the probabilistic meaning of the matrices $D_0,D_1,$ characterizing the customer arrival process and of the matrices $H_n^0$, $H_n^{-},$ and $H_n^{+}$ defining the service process, as well as the definition and properties of Kronecker’s product and sum of matrices.

For future use, let us denote

$$\Omega =\mathrm{diag}\{{\mathbf{Q}}_{i,i},i=\overline{0,N}\}+{\mathrm{diag}}^{+}\{{\mathbf{Q}}_{i,i+1},i=\overline{0,N-1}\}+$$

$${\mathrm{diag}}^{-}\{{\mathbf{Q}}_{i,i-1},i=\overline{1,N}\}=$$

$$\mathrm{diag}\{D_0\oplus H_i^0,i=\overline{0,N}\}+{\mathrm{diag}}^{+}\{D_1\otimes H_i^{+},i=\overline{0,N-1}\}+$$

$${\mathrm{diag}}^{-}\{I_W\otimes H_i^{-},i=\overline{1,N}\}$$

where $\mathrm{diag}\{\dots \},{\mathrm{diag}}^{+}\{\dots \},$ and ${\mathrm{diag}}^{-}\{\dots \}$ denote the block-diagonal, up-block-diagonal, and sub-block-diagonal matrix with the block-diagonal, up-block-diagonal, and sub-block-diagonal blocks, respectively, given in the brackets.

The matrix $\Omega$ has size $W{\widehat{K}}_N$ where ${\widehat{K}}_N={\displaystyle \sum _{n=0}^N}{K}_n.$ The square blocks ${\mathbf{Q}}^0,{\mathbf{Q}}^{+},$ and ${\mathbf{Q}}^{-}$ have the size $W{K}_N.$

It follows from Lemma 1 that the $MC$ ${\xi }_t$ belongs to the class of level-independent $QBD$ processes, see [1]. Using the result of [1], it is possible to prove the following statement.

Theorem 1. 

The $MC$ ${\xi }_t$ is ergodic if and only if the following inequality holds good:

$$\lambda <\mathbf{u}{H}_N^{-}\mathbf{e}$$

(1)

where the row vector $\mathbf{u}$ is the unique solution to the system

$$\mathbf{u}(H_N^0+H_N^{-}{H}_{N-1}^{+})=\mathbf{0},\mathbf{u}\mathbf{e}=1.$$

(2)

Proof. 

According to [1], the criterion for ergodicity of the $MC$ ${\xi }_t$ is the fulfillment of the inequality

$$\mathbf{y}{\mathbf{Q}}^{-}\mathbf{e}>\mathbf{y}{\mathbf{Q}}^{+}\mathbf{e}$$

(3)

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

$$\mathbf{y}({\mathbf{Q}}^0+{\mathbf{Q}}^{-}+{\mathbf{Q}}^{+})=\mathbf{0},\mathbf{y}\mathbf{e}=1.$$

(4)

Using the so-called mixed product rule for Kronecker’s product of matrices, it can be verified that the vector $\mathbf{y}$ that is the solution of (4) can be represented in the form

$$\mathbf{y}=\mathit{\theta }\otimes \mathbf{u}$$

where $\mathit{\theta }$ is the invariant probability vector of the underlying $MC$ of the $MAP$ and the vector $\mathbf{u}$ is the solution of system (2). By substituting this expression for the vector $\mathbf{y}$, one can make clear that inequality (3) is transformed to the form (1). Theorem 1 is proven. □

The left-hand side of (1) defines the mean arrival rate. The right-hand side of (1) defines the mean rate of customers’ departure from the overloaded system when the number of simultaneously processed customers is permanently equal to $N.$

Let inequality (1) hold true. Then the stationary probabilities $\pi (i,\nu ,\eta )$ of the states of the $MC$ ${\xi }_t$ exist. Denote by ${\pi }_i$ the row vector of these stationary probabilities listed in the lexicographic order, $i\ge 0.$

Assertion 1. 

The vectors of the stationary probabilities ${\pi }_i,i\ge 0,$ of the $MC$ ${\xi }_t$ are calculated as follows:

the vectors ${\pi }_i,i\ge N+1,$ are computed as:

$${\pi }_i={\pi }_N{\mathbf{R}}^{i-N},i\ge N+1,$$

where the matrix $\mathbf{R}$ is the minimal non-negative solution of the matrix equation

$${\mathbf{R}}^2{\mathbf{Q}}^{-}+\mathbf{R}{\mathbf{Q}}^0+{\mathbf{Q}}^{+}=O$$

and the vector $({\pi }_0,{\pi }_1,\dots ,{\pi }_N)$ is the unique solution to the following system of linear algebraic equations

$$({\pi }_0,{\pi }_1,\dots ,{\pi }_N)\left(\Omega +\mathrm{diag}\{O,\dots ,O,\mathbf{R}{\mathbf{Q}}^{-}\}\right)=\mathbf{0},$$

$$({\pi }_0,{\pi }_1,\dots ,{\pi }_N)\mathbf{e}+{\pi }_N\mathbf{R}{(I-\mathbf{R})}^{-1}\mathbf{e}=1.$$

Proof of Assertion 1 immediately follows from [1]. The statement of this assertion can also be easily verified using the substitution of the presented expressions for the row vectors ${\pi }_i,i\ge 0,$ into the system of equilibrium (Chapman-Kolmogorov) equations:

$$({\pi }_0,{\pi }_1,\dots ,{\pi }_i,\dots )\mathbf{Q}=\mathbf{0},({\pi }_0,{\pi }_1,\dots ,{\pi }_i,\dots )\mathbf{e}=1.$$

The statements of Theorem 1 and Assertion 1 define an evident algorithm for verification of the existence of the stationary distribution of the system states under the fixed set of the system parameters and the algorithm for computation of this distribution if it exists.

Having computed the vectors ${\pi }_i,i\ge 0,$ it is not difficult to compute the values of basic performance measures of the queueing system under any fixed mechanism of providing service to customers and parameters of this mechanism and arrival process.

4. Analysis of the Model with the MAP, Simultaneous Service Process and Retrials

Let us now assume that the described above system does not have a buffer. A customer, that arrives when the number of customers processing in the system is equal to $N,$ cannot be stored for waiting. Instead, he/she leaves the system (it is said that he/she moves to some virtual place called orbit) and tries to enter service after the random time interval. Duration of the intervals between retrials of customers from the orbit has an exponential distribution with the parameters ${\alpha }_i$ if the number of customers in the orbit is $i,i\ge 1.$ By default, ${\alpha }_0=0.$ If the system is full at the retrial epoch, the customer will make more attempts. The number of retrials is assumed to be unlimited. As it was mentioned above, the phenomenon of retrials is typical in many real-world systems, including wireless communication networks. This made the queues with retrials a very popular subject of research.

The behavior of the described system with retrials is described by the four-dimensional process

$${\chi }_t=\{j_t,n_t,{\nu }_t,{\eta }_t\},j_t\ge 0,n_t=\overline{0,N},{\nu }_t=\overline{1,W},{\eta }_t=\overline{1,K_{n_t}},t\ge 0,$$

where $j_t$ means the number of customers in the orbit, $n_t$ is the number of customers in service at moment $t,t\ge 0$, and the meaning of the components $\{{\nu }_t,{\eta }_t\}$ is the same as above.

It follows from definitions of arrival, service, and retrial processes that the random process ${\chi }_t$ is also the continuous-time $MC$.

Let us arrange the states of the $MC$ ${\chi }_t$ in the lexicographic order. The set of the states having value i of the component $j_t$ is called level i, and the set of the states having value $(i,n)$ of the components $(j_t,n_t)$ is called sub-level $(i,n).$ Because the important distinction of the retrial systems from the systems with a buffer is that the servers may stay idle while there are customers that would like to enter service, it is clear that all levels $i\ge 0$ consist of $(N+1)$ sub-levels $(i,n),n=\overline{0,N}.$

Lemma 2. 

The non-zero blocks of the generator $\tilde{\mathbf{Q}}={\left({\tilde{\mathbf{Q}}}_{i,j}\right)}_{i,j\ge 0}$ of the $MC$ ${\chi }_t$ are defined as follows:

• The diagonal blocks ${\tilde{\mathbf{Q}}}_{i,i}$ are given by the formula:

$${\tilde{\mathbf{Q}}}_{i,i}=\Omega -{\alpha }_i\widehat{I},i\ge 0,$$

where the block Ω was defined above and

$$\widehat{I}=I_{W{\widehat{K}}_N}-\mathrm{diag}\{O_{W{\widehat{K}}_{N-1}},I_{W{K}_N}\};$$

• The sub-diagonal blocks ${\tilde{\mathbf{Q}}}_{i,i-1}$ are given by the formula:

  $${\tilde{\mathbf{Q}}}_{i,i-1}={\alpha }_i{\mathrm{diag}}^{+}\{I_W\otimes H_n^{+},n=\overline{0,N-1}\},i\ge 1;$$
• The up-diagonal blocks ${\tilde{\mathbf{Q}}}_{i,i+1}$ are given by the formula:

  $${\tilde{\mathbf{Q}}}_{i,i+1}=\mathrm{diag}\{O_{W{\widehat{K}}_{N-1}},D_1\otimes I_{K_N}\},i\ge 0.$$

Proof. 

It is clear that transitions inside the level i occur in the retrial system in the same manner as in the corresponding system with the buffer when the buffer is empty. Only the exit from a given state can additionally occur due to a retrial from the orbit during the epoch when the number of customers receiving service is less than $N.$ If it is less than $N,$ the retrying customer is admitted for service. The customer departs from the orbit and a new service begins, i.e., the component $j_t$ of the $MC$ decreases by one, the component $n_t$ increases by one, and the state of the underlying process ${\eta }_t$ makes the corresponding transition. This explains the form of the block ${\tilde{\mathbf{Q}}}_{i,i-1}.$ The form of the block ${\tilde{\mathbf{Q}}}_{i,i+1}$ is evident because the increase in the number of customers in the orbit can happen only at a customer arrival moment when the number of customers in service is equal to $N.$

Lemma 2 is proven. □

Let us briefly discuss the relation of the generators $\mathbf{Q}$ and $\tilde{\mathbf{Q}}$ of the $MC$ ${\xi }_t$ describing the system with a buffer and the $MC$ ${\chi }_t$ describing the analogous system with retrials.

The generator $\mathbf{Q}$ of infinite size has the following non-zero blocks: the blocks ${\mathbf{Q}}_{i,i}$ are of size $(W{K}_i\times W{K}_i),i=\overline{0,N},$ the blocks ${\mathbf{Q}}_{i,i+1}$ are of size $(W{K}_i\times W{K}_{i+1}),i=\overline{0,N-1},$ the blocks ${\mathbf{Q}}_{i,i-1}$ are of size $(W{K}_i\times W{K}_{i-1}),i=\overline{1,N},$ the blocks ${\mathbf{Q}}^0,{\mathbf{Q}}^{+},{\mathbf{Q}}^{-}$ are square matrices of size $W{K}_N.$

Correspondingly, the procedure for computation of the stationary probability vectors given in Theorem 2 includes the iterative solution of the square matrix equation for the matrix $\mathbf{R}$ of size $W{K}_N,$ solution of the system of the linear algebraic equations for the vector $({\pi }_0,{\pi }_1,\dots ,{\pi }_N)$ of size $W{\widehat{K}}_N+1$ and the recursive computation of the vectors ${\pi }_i,i\ge N,$ via the formula ${\pi }_{i+1}={\pi }_i\mathbf{R},i\ge N.$

The generator $\tilde{\mathbf{Q}}$ has infinitely many large non-zero blocks ${\tilde{\mathbf{Q}}}_{i,j},i,j\ge 0,|i-j|\le 1,$ all of size $W{\widehat{K}}_N.$ Therefore, in general, the check of the existence of the stationary distribution of the $MC$ ${\chi }_t$ and computation of this distribution require more computer memory.

Until now, we assumed a general dependence of the total retrial rate ${\alpha }_i$ on the number i of customers in the orbit. A quite popular in the literature and easier for the study case is when ${\alpha }_i$ is equal to some constant $\gamma$ for all $i,i>0$ (the so-called constant retrial policy, see, e.g., [40,41,42]). This corresponds to the possible scenarios when all i customers staying in the orbit are allowed to make retrials independently of each other with the rate ${\displaystyle \frac{\gamma }{i}}$ or only one customer, e.g., the eldest one, is granted the possibility to retry with the rate $\gamma .$ It is evident that in this case, the $MC$ ${\chi }_t$ is, as the $MC$ ${\xi }_t,$ the level-independent $QBD$ and modifications of Theorem 1 and Assertion 1 are applicable for obtaining the ergodicity condition and computing the stationary distribution.

In particular, the following statement holds good.

Theorem 2. 

The criterion for ergodicity of the $MC$ ${\chi }_t$ is the fulfillment of the inequality

$$\overline{\mathbf{y}}{\overline{\mathbf{Q}}}^{-}\mathbf{e}>\overline{\mathbf{y}}{\overline{\mathbf{Q}}}^{+}\mathbf{e}$$

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

$$\overline{\mathbf{y}}({\overline{\mathbf{Q}}}^0+{\overline{\mathbf{Q}}}^{-}+{\overline{\mathbf{Q}}}^{+}),\overline{\mathbf{y}}\mathbf{e}=1$$

where the matrices ${\overline{\mathbf{Q}}}^0,{\overline{\mathbf{Q}}}^{-},{\overline{\mathbf{Q}}}^{+}$ have the form:

$${\overline{\mathbf{Q}}}^0=\Omega -\gamma \widehat{I},$$

$${\overline{\mathbf{Q}}}^{-}=\gamma {\mathrm{diag}}^{+}\{I_W\otimes H_n^{+},n=\overline{0,N-1}\},$$

$${\overline{\mathbf{Q}}}^{+}=\mathrm{diag}\{O_{W{\widehat{K}}_{N-1}},D_1\otimes I_{K_N}\}.$$

The vectors of the stationary probabilities are defined by the modification of Assertion 1, where all symbols $\mathbf{Q}$ are replaced with the symbols $\overline{\mathbf{Q}}.$

Therefore, the analysis of the retrial queue with the constant retrial policy is similar to the analysis of the system with an infinite buffer. Only computer implementation of the check of the ergodicity condition and computation of the stationary distribution becomes more complicated due to the more strict requirements for the computer memory and longer run time. In particular, the check of the ergodicity condition fulfillment for the system with an infinite buffer assumes the solution of system (2) for the vector $\mathbf{y}$ of size $W{K}_N.$ For the retrial queueing system, the size of the vector $\overline{\mathbf{y}}$ is $W{\widehat{K}}_N.$

Another popular in the literature and much more difficult for analysis dependencies of the total retrial rate ${\alpha }_i$ on the number i are of the forms ${\alpha }_i=i\alpha$ (the so-called classical retrial policy) and ${\alpha }_i=i\alpha +\gamma$ (the so-called linear retrial policy, see [43]) where $\alpha$ and $\gamma$ are the known constants. The use of these simple dependencies leads to the possibility of derivation of the differential equations for the vector-generating function of the stationary probabilities. However, these equations can be solved analytically only when $N=1$ and the cardinality of the state space of the components $\{{\nu }_t,{\eta }_t\}$ is equal to 1. This means that the arrival flow should be the stationary Poisson and the service time distribution be exponential.

Because we intend to consider a more general case, we would not obtain any benefit from assuming such simple dependence of ${\alpha }_i$ on $i.$ Thus, we suppose this dependence is arbitrary with the restriction that the limit $\underset{i\to \infty }{lim}{\alpha }_i$ exists and is equal to infinity.

It is evident that if ${\alpha }_i$ depends on i, then the $MC$ ${\chi }_t$ belongs to the class of level-dependent $QBD$ processes. Therefore, the famous results by M. Neuts, see [1], obtained for level-independent $QBD$ processes are not applicable here.

Instead, we will show that the constructed $MC$ ${\chi }_t$ belongs to the class of asymptotically quasi-Toeplitz MCs (AQTMCs), see [44], and exploit this fact.

According to the definition given in [44], the $MC$ with a generator $\tilde{\mathbf{Q}}$ having the blocks ${\tilde{\mathbf{Q}}}_{i,j}$ belongs to the class of AQTMCs if there exist the limits

$${\mathbf{Y}}_0=\underset{i\to \infty }{lim}{\mathcal{F}}_i^{-1}{\tilde{\mathbf{Q}}}_{i,i-1},{\mathbf{Y}}_1=\underset{i\to \infty }{lim}{\mathcal{F}}_i^{-1}{\tilde{\mathbf{Q}}}_{i,i}+I,{\mathbf{Y}}_2=\underset{i\to \infty }{lim}{\mathcal{F}}_i^{-1}{\tilde{\mathbf{Q}}}_{i,i+1},$$

and the matrix ${\displaystyle \sum _{k=0}^2}{\mathbf{Y}}_k$ is stochastic. Here, ${\mathcal{F}}_i$ is a diagonal matrix with positive diagonal elements defined as the moduli of the corresponding diagonal elements of matrix ${\tilde{\mathbf{Q}}}_{i,i}.$

Let us denote

$$\mathbf{U}=-I\circ (D_0\oplus H_N^0)$$

where ∘ denotes Hadamard’s product of matrices, see [45].

It is possible to directly check that the limits ${\mathbf{Y}}_k,k=0,1,2,$ indeed exist and are defined as follows:

$${\mathbf{Y}}_0={\mathrm{diag}}^{+}\{I_W\otimes H_n^{+},n=\overline{0,N-1}\},$$

$${\mathbf{Y}}_1=\mathrm{diag}\{O_{W{\widehat{K}}_{N-1}},I_{W{K}_N}+{\mathbf{U}}^{-1}(D_0\oplus H_N^0)\}+$$

$${\mathrm{diag}}^{-}\{O_{W{K}_0},O_{W{K}_1},\dots ,O_{W{K}_{N-1}},{\mathbf{U}}^{-1}(I_W\otimes H_N^{-})\},$$

$${\mathbf{Y}}_2=\mathrm{diag}\{O_{W{\widehat{K}}_{N-1}},{\mathbf{U}}^{-1}(D_1\otimes I_{K_N})\}.$$

It is easy to check that ${\displaystyle \sum _{k=0}^2}{\mathbf{Y}}_k\mathbf{e}=\mathbf{e},$ i.e., the matrix ${\displaystyle \sum _{k=0}^2}{\mathbf{Y}}_k$ is the stochastic one.

Therefore, the $MC$ ${\chi }_t$ indeed belongs to the class of $AQTMC$.

According to [44], the sufficient condition for ergodicity of the $MC$ ${\chi }_t$ is the fulfillment of the inequality

$$\mathbf{w}{\mathbf{Y}}_0\mathbf{e}>\mathbf{w}{\mathbf{Y}}_2\mathbf{e}$$

(5)

where $\mathbf{w}$ is the unique solution to the system of equations

$$\mathbf{w}=\mathbf{w}({\mathbf{Y}}_0+{\mathbf{Y}}_1+{\mathbf{Y}}_2),\mathbf{w}\mathbf{e}=1.$$

(6)

The condition

$$\mathbf{w}{\mathbf{Y}}_0\mathbf{e}<\mathbf{w}{\mathbf{Y}}_2\mathbf{e}$$

is sufficient for the $AQTMC$ to be non-ergodic.

Because each of the first $N-1$ block rows of the matrix ${\mathbf{Y}}_0+{\mathbf{Y}}_1+{\mathbf{Y}}_2$ has only one non-zero stochastic block, it is easy to see that the sub-vectors ${\mathbf{w}}_n$ in representation $\mathbf{w}$ in the form $({\mathbf{w}}_0,{\mathbf{w}}_1,\dots ,{\mathbf{w}}_N)$ are zero vectors for $n=\overline{0,N-2}.$

In turn, inequality (5) turns to the inequality

$${\mathbf{w}}_{N-1}(I_W\otimes H_{N-1}^{+})\mathbf{e}>{\mathbf{w}}_N{\mathbf{U}}^{-1}(D_1\otimes I_{K_N})\mathbf{e}.$$

(7)

Thus, to obtain the ergodicity condition, we need to calculate the non-zero components ${\mathbf{w}}_{N-1}$ and ${\mathbf{w}}_N$ of the vector $\mathbf{w}.$

From system (6), we obtain the following system of two equations for two unknown vectors ${\mathbf{w}}_{N-1}$ and ${\mathbf{w}}_N:$

$${\mathbf{w}}_{N-1}={\mathbf{w}}_N{\mathbf{U}}^{-1}(I_W\otimes H_N^{-}),$$

$${\mathbf{w}}_{N-1}(I_W\otimes H_{N-1}^{+})+{\mathbf{w}}_N{\mathbf{U}}^{-1}(D_0\oplus H_N+D_1\otimes I_{K_N})=\mathbf{0}.$$

Using these equations and introducing notation $\mathbf{z}={\mathbf{w}}_N{\mathbf{U}}^{-1},$ inequality (7) is reduced to the form

$$\mathbf{z}(I_W\otimes H_N^{-}){\mathbf{e}}_{W{K}_{N-1}}>\mathbf{z}(D_1\otimes I_{K_N}){\mathbf{e}}_{W{K}_N}$$

(8)

where the vector $\mathbf{z}$ is the stochastic solution to the equation

$$\mathbf{z}(D_0\oplus H_N^0+D_1\otimes I_{K_N}+I_W\otimes H_N^{-}{H}_{N-1}^{+})=\mathbf{0}.$$

By direct substitution, it is possible to show that the solution of this equation has the form

$$\mathbf{z}=\mathit{\theta }\otimes \mathbf{u}$$

where the vector $\mathbf{u}$ is the solution of system (2).

By substitution of this vector to inequality (8), we obtain inequality (1).

Thus, we proved the following statement.

Theorem 3. 

The sufficient condition for ergodicity of the $MC$ ${\chi }_t$ is the fulfillment of inequality (1), where the vector $\mathbf{u}$ is the solution of system (2). In other words, if $\underset{i\to \infty }{lim}{\alpha }_i=\infty$, then the sufficient conditions for ergodicity of the $MC$ ${\chi }_t$ describing the retrial queue coincide with the sufficient conditions for ergodicity of the $MC$ ${\xi }_t$ describing the corresponding queue with an infinite buffer.

Let the ergodicity condition be fulfilled. Then, similar to the previous section, the stationary probabilities $\pi (i,n,\nu ,\eta )$ of the states of $MC$ ${\chi }_t$ exist. Denote by $\pi (i,n)$ the row vector of these stationary probabilities that belong to sub-level $(i,n).$ Denote also

$${\pi }_i=(\pi (i,0),\pi (i,1),\dots ,\pi (i,N)),i\ge 0.$$

It is well-known that the row vectors ${\pi }_i,i\ge 0,$ satisfy the following system of equations:

$$({\pi }_0,{\pi }_1,\dots ,{\pi }_i,\dots )\tilde{\mathbf{Q}}=\mathbf{0},({\pi }_0,{\pi }_1,\dots ,{\pi }_i,\dots )\mathbf{e}=1.$$

(9)

The problem of solving infinite system (9) with tri-block-diagonal matrix $\tilde{\mathbf{Q}},$ which does not possess Toeplitz-like structure, is difficult. In [44,46,47,48], the numerically stable algorithms for solving system (9) are elaborated.

This algorithm is based on replacement of system (9) with the system having a two-block-diagonal matrix. This replacement exploits the construction of an infinite series of censored Markov chains (for definitions, see, e.g., [49,50]) for the $MC$ ${\chi }_t$ with different censoring levels. The short outline of one of the versions of this algorithm is as follows.

The vectors ${\mathit{\pi }}_i,i\ge 0,$ are calculated as:

$${\mathit{\pi }}_i={\mathit{\phi }}_i{\left(\sum _{k=0}^{\infty }{\mathit{\phi }}_k\mathbf{e}\right)}^{-1}$$

where the vectors ${\mathit{\phi }}_i$ are computed from the recursion

$${\mathit{\phi }}_i=-{\mathit{\phi }}_{i-1}{\tilde{\mathbf{Q}}}_{i-1,i}{({\tilde{\mathbf{Q}}}_{i,i}+{\tilde{\mathbf{Q}}}_{i,i+1}{\mathbf{G}}_i)}^{-1},i\ge 1,$$

with the initial vector ${\mathit{\phi }}_0$ given as the unique solution to the system

$${\mathit{\phi }}_0({\tilde{\mathbf{Q}}}_{0,0}+{\tilde{\mathbf{Q}}}_{0,1}{\mathbf{G}}_0)=\mathbf{0},{\mathit{\phi }}_0\mathbf{e}=1,$$

where the stochastic matrices ${\mathbf{G}}_i$ are computed from the backward recursion

$${\mathbf{G}}_i=-{({\tilde{\mathbf{Q}}}_{i+1,i+1}+{\tilde{\mathbf{Q}}}_{i+1,i+2}{\mathbf{G}}_{i+1})}^{-1}{\tilde{\mathbf{Q}}}_{i+1,i}$$

with the terminal condition

$${\mathbf{G}}_{\infty }=\mathbf{G}$$

where the matrix $\mathbf{G}$ is the stochastic solution to the matrix equation

$$\mathbf{G}={\mathbf{Y}}_0+{\mathbf{Y}}_1\mathbf{G}+{\mathbf{Y}}_2{\mathbf{G}}^2$$

solution of which can be found using different variants of the iterative method.

The numerical stability of this algorithm is provided via operation only with the non-negative matrices. More details relating to the realization of this algorithm can be found in [46,47,48].

5. Analysis of the Model with the MAP, Simultaneous Service Process, and Impatience of the Waiting Customers

Let us return to the model of Section 3 and modify it as follows. We assume that customers waiting in the buffer are impatient. More specifically, we assume that when i customers reside in the buffer, each of them, independently of others, may depart from the system without service. The importance of account of customer impatience phenomenon in real-world systems is explained, e.g., in [51,52,53,54,55,56,57,58,59,60].

The total departure rate due to impatience is assumed to be equal to ${\sigma }_i,i\ge 1,$ when i customers are waiting in the buffer. The most common in the literature case is when ${\sigma }_i=i\sigma ,i\ge 1,$ where $\sigma$ is the positive constant defining the individual impatience rate of a customer.

In this case, the behavior of the system is described by the $MC$ ${\xi }_t^{\left(imp\right)}$ that has the same state space as the $MC$ ${\xi }_t$ analyzed in Section 3. The generator of the $MC$ ${\xi }_t^{\left(imp\right)}$ is practically the same as the generator $\mathbf{Q}$ defined by Lemma 1 with the following two modifications: the blocks ${\mathbf{Q}}_{i,i}$ and ${\mathbf{Q}}_{i,i-1}$ now have the form

$${\mathbf{Q}}_{i,i}=D_0\oplus H_N^0-{\sigma }_{i-N}{I}_{W{K}_N},i>N,$$

$${\mathbf{Q}}_{i,i-1}=I_W\otimes H_N^{-}{H}_{N-1}^{+}+{\sigma }_{i-N}{I}_{W{K}_N},i>N.$$

These modifications imply that, in contrast to the $MC$ ${\xi }_t$ that is level-independent $QBD$, the $MC$ ${\xi }_t^{\left(imp\right)}$ is level-dependent and belongs to the class of $AQTMC.$

If the total impatience rate ${\sigma }_i$ tends to infinity when $i\to \infty ,$ it follows from [61] that the $MC$ ${\xi }_t^{\left(imp\right)}$ is ergodic for any values of the system parameters.

The stationary probabilities of the $MC$ ${\xi }_t^{\left(imp\right)}$ can be computed using the numerically stable algorithm described in the previous section.

It is worth noting that account of possible impatience of customers staying in the orbit can also be easily implemented. If we denote by $\widehat{\mathbf{Q}}$ the generator of the $MC$ describing the system with customers retrial (with retrial rate ${\alpha }_i$) and customers impatience (with impatience rate ${\sigma }_i$), it can be shown that the generator $\widehat{\mathbf{Q}}$ describing the system with retrials and impatience is practically the same as the generator $\tilde{\mathbf{Q}}$ for the system with retrials defined by Lemma 2 with the following two modifications: the blocks ${\widehat{\mathbf{Q}}}_{i,i}$ and ${\widehat{\mathbf{Q}}}_{i,i-1}$ now have the form

$${\widehat{\mathbf{Q}}}_{i,i}={\tilde{\mathbf{Q}}}_{i,i}-{\sigma }_i{I}_{W{K}_N},i\ge 1,$$

$${\widehat{\mathbf{Q}}}_{i,i-1}={\tilde{\mathbf{Q}}}_{i,i-1}+{\sigma }_i{I}_{W{K}_N},i\ge 1.$$

The $MC$ describing the retrial system with impatient customers with impatience rate ${\sigma }_i$ tending to infinity when $i\to \infty$ is always ergodic if the limit $\underset{i\to \infty }{lim}{\alpha }_i$ exists (and not necessary is finite). This follows from [61].

6. Conclusions

In this paper, we introduced the notion of simultaneous service of a finite number of customers and presented examples of such a service description. For the queueing system with such a service and the $MAP$ arrival process, we provided an algorithmic analysis. Two scenarios of customer admission when the number of serviced customers has the maximum available value are considered. One scenario assumes customers’ temporal store in an infinite buffer. Another one assumes customers retry after random intervals of time. For both scenarios, the generator of the multidimensional $MC$ describing the system’s behavior is obtained, the condition of ergodicity of $MC$ is derived, and the algorithm for computation of its stationary distribution is presented. In the case of an arbitrary retrial policy with an infinitely increasing total retrial rate, it is shown that the sufficient condition for the ergodicity of the system with retrials coincides with the correspondent condition for the system with an infinite buffer.

The obtained results create the background for the uniform analysis of a bunch of various queueing models with arbitrary Markov descriptions of simultaneous service of many customers. Systems with infinite buffers or retrials (with classical, linear, and constant retrial policies) can be analyzed. Results known in the literature for different particular cases of the discipline with simultaneous service of a finite number of customers defined in this paper, which are obtained for the systems with the stationary Poisson arrival process, can be easily generalized for the case when the arrivals are described by the essentially more general $MAP.$

The possible directions of further research include: generalization of the framework to the cases of batch arrivals, heterogeneous customers, the possibility of group service, the arrival of disasters; application of the created framework to the systems with unreliable servers, negative arrivals, the influence of a random environment, etc.