Performance Forecasting for Multi-Server Retrial Queue with Possibility of Processing Repetition and Server Reservation for Repeating Users

Abstract

This study focuses on forecasting and optimizing the performance of a real-world object modelled by a multi-server queueing system that processes two types of users: primary (new) users and repeating users. The repeating users are those who succeeded in entering processing upon arrival and then decided to repeat it. These users have privilege and can enter processing when they wish once at least one device is idle. The primary user is admitted to the system only if the number of occupied devices is less than some threshold value and the quantity of repeating users residing in the system does not exceed certain thresholds. Repeating users are impatient and non-persistent. Arrivals of primary users are described by the Markovian arrival process. Processing times of primary and repeating users have distinct phase-type distributions. Utilizing the concept of the generalized phase–time distributions, the dynamics of this queueing system are formally characterized by the multidimensional Markov chain, which is examined in this paper. The ergodicity condition is derived. The relation of the key performance characteristics of the system and the thresholds defining the policy of the primary user’s admission is numerically highlighted. Optimal threshold selection is demonstrated numerically.

1. Introduction

Queuing theory is a powerful tool for capacity planning and performance evaluation of a wide range of real-world systems based on available information about the pattern of the users’ arrival process and service discipline; see, for example, [1,2,3,4,5,6,7]. The results of analysis can be further used for managerial goals to optimize performance of the system and the revenue of the service provider.

In this paper, a novel multi-server queueing model is formulated and analyzed. Users processed in the system are divided into two categories. The primary users are those who just arrived to the system. The repeating users are those who have already processed in the system but have decided to receive it again (probably an unlimited number of times). The repeating users have priority over the primary users. These users have privilege and can enter processing when they wish once at least one device is idle. The intuitive motivation for giving priority to these users is clear. A primary user arrives at the system for the first time; it was not processed in the system earlier, and it does not have any formal or informal relations with the system. Such a user can be considered an ad hoc client. A repeating user has already been processed in the system earlier. This user is registered in the system’s database and has established a relationship with it by receiving benefits such as a discount card, service subscription, or season ticket. A repeating user can be considered a permanent client, and a priori, it deserves a higher level of service than the ad hoc client.

Realization of the privilege of the repeating user is implemented via the different admission discipline. The repeating user is always admitted for processing if there are available (non-occupied) devices. Only when all devices are occupied during the epoch of the repeating user’s arrival does this user not receive access to devices and moves to a certain virtual place called the orbit to repeat attempts to catch an idle device. The admission of the primary users is restricted in the following way: we assume that the system reserves several devices exclusively for the processing of repeating users. An arriving primary user is admitted to the system only if the quantity of occupied devices does not exceed the predetermined threshold and the quantity of repeating users is less than the corresponding threshold. A more exact description of the primary user’s admission is given in the formal description of the model below.

It is worth clarifying the terminological issues. The notion of repeating attempts to occupy the device (retrials) is well known in queueing theory. Surveys of the relevant literature can be found, e.g., in [8,9,10,11,12,13,14]. The notion of repeating users used in this paper is close to the traditional notion of users that make repeated attempts to enter the processing. In both cases, the users retry entering the processing during their stay in the orbit, and the inter-retrial times of individual users are the independent identically distributed random variables. For ease of mathematical analysis, the exponential distribution is practically always suggested. The main difference between classical retrying users and those in the model under study is as follows: In traditional retrial queues, a user makes the retrials before entering the device. In our model, a user becomes the retrying user only after processing for the first time.

Repeated processing is sometimes called user feedback. In real-world systems’ modelling, the user feedback can be interpreted in different ways. One way is to consider a phenomenon of possible repetition of processing as the result of registration of a primary user, upon arrival, as a permanent user that will repeatedly request processing in the system during some period of time. Another possible way is to consider the repetition of processing as the consequence of a user’s dissatisfaction with the quality of service. In this interpretation, the considered system falls into the category of so-called unreliable queueing systems. Providing a priority to the repeated users in these systems can be explained by the natural desire of a service provider to completely satisfy a user within a minimally possible period of time.

As works devoted to queues with feedback, the papers [15,16,17,18] can be mentioned. A review of some related research can be found in [19]. Multi-server queues with user retrials and feedback are considered, e.g., in [20,21]. The system analyzed in [20] assumes the exponential distribution of inter-arrival and processing times in the presence of one or two devices. A similar model but with an arbitrary finite quantity of devices is dealt with in [20,21]. The model considered in our paper assumes significantly more general arrival and processing processes and the possibility of control by the primary users’ admission. More information about the related research is also given below.

The existing literature considers several different scenarios of feedback implementation. The case of immediate feedback, where a user starts another processing immediately after completion with a certain probability, is less interesting from a theoretical perspective because analyzing a queuing system with this type of feedback can be simplified to analyzing the corresponding system without feedback by recalculating the processing time distribution. Feedback is often analyzed in tandem queues, where users return to the same and previous stations for repeated processing. Here, we assume that the user, who decides to repeat processing, goes to the orbit and makes the attempts after exponentially distributed intervals of time, in competition with other orbiting users. A similar model was considered in [22,23]. Our model is more general than the one analyzed in [22] in the following three aspects: (i) A primary user in [22] is always admitted when there is at least one idle device. We assume controlled admission of primary users by reserving part of the users only as retrying users and making the decision of whether to admit or reject the primary user depend on the quantity of retrying users. (ii) The exponential distribution of processing times is assumed in [22], while we assume a more general phase-type distribution, different for the primary and retrying users. (iii) The capacity of the orbit is assumed to be finite in [22], while we suggest that this capacity is infinite. The model considered in [23] assumes the existence of two devices, fast and slow ones, and no retrials. Here we consider the model with an arbitrary finite quantity of identical devices and focus on the possibility of feedback of retrying users. The model considered in [24] assumes the necessity of the availability of inventory units for processing.

A brief outline of this text is as follows: In Section 2, the formal description of the considered system and user admission discipline are given. The process describing the states of the system is introduced via the use of the so-called generalized $PH$ distribution in Section 3. The generator of this process, which is the multidimensional Markov chain with continuous time, is derived. The ergodicity of this chain is derived, and the way of solving the problem of computation of its steady-state distribution is advised in this section. Formulas for computing the values of various performance characteristics of the queueing system under study and the numerical results are presented. The relation of the system performance characteristics and the thresholds defining the strategy of the primary user’s admission is numerically illustrated here. The possibility of using the obtained results for optimization of this strategy is demonstrated. Section 4 concludes the paper.

2. Materials and Methods

Consider a queuing system of N identical devices with no input buffer. Figure 1 illustrates the system’s structure.

The arrival of primary users into the system is presumed to be characterized by the $MAP$, which is determined by the fundamental Markov chain ($MC$) ${\nu }_t,t\ge 0,$ with a finite state space $\{1,2,\dots ,W\}$.

The jump rates of this $MC$ inside its state space are delineated by the square matrices $D_0$ and $D_1$. The entries of the matrix $D_1$ denote the jump rates associated with user arrivals. The off-diagonal entries of the matrix $D_0$ represent the jump rates between states in the absence of user arrivals. The diagonal entries of the matrix $D_0$ are negative. Their modules ascertain the rate of departure from the respective state. The summation of the matrices $D_0+D_1$ constitutes the generator of the $MC$${\nu }_t,t\ge 0.$

The mean user arrival rate $\lambda$ is calculated using the equation $\lambda =\mathit{\theta }{D}_1\mathbf{e}$, where $\mathit{\theta }$ represents a row vector of steady-state probabilities of the $MC$${\nu }_t,t\ge 0.$ This vector constitutes the only possible solution to the system $\mathit{\theta }(D_0+D_1)=\mathbf{0},\mathit{\theta }\mathbf{e}=1.$ In this context, $\mathbf{0}$ denotes a row vector of suitable dimensions filled with zeros, while $\mathbf{e}$ represents a column vector of sufficient size composed entirely of ones.

For more information about the $MAP$, see, e.g., [25,26,27,28,29]. When the results of users’ arrival observations (sample or time-stamps) in a real-world system are available, various statistical methods are applicable to solve the problem of constructing the matrices $D_0$ and $D_1$ which characterize the $MAP$ that is a good model of the given real arrival process, see, e.g., references [30,31,32,33,34].

An arriving primary user is admitted for processing only under fulfillment of a certain condition (acceptance rule) that is described briefly below. If this condition is fulfilled, the primary user starts processing. If this condition does not hold true, the user is rejected and permanently abandons the system.

After processing, a primary user leaves the system with a probability of $p_1,0\le p_1\le 1,$ and the complementary probability indicates that the user becomes a repeating user. This means that the user visits a virtual place referred to as the orbit.

The orbit has a limitless capacity. Each repeating user tries to re-enter the processing independently of other users remaining in the orbit. The intervals between attempts possess a random duration. At any given time, if the quantity of users in the orbit is $i,i>0,$ then the overall rate of repeated attempts is $i\alpha ,\alpha >0.$ This means that the probability of a retrial attempt generation in any interval $(t,t+\mathrm{\Delta })$ of an infinitesimal length $\mathrm{\Delta }$ is equal to $i\alpha \mathrm{\Delta }+o\left(\mathrm{\Delta }\right).$ The attempt is considered successful if at least one device is idle. The user occupies a random idle device and initiates processing. In the case of an unsuccessful attempt, meaning all devices are occupied, the retrying user abandons the system with the probability $q,0\le q\le 1$, whereas with the complementary probability $1-q$ it remains in the orbit.

Upon exiting the processing facility, any repeating user either leaves the system permanently with the probability $p_2,0\le p_2\le 1,$ or joins the orbit again with the complementary probability.

The primary user acceptance rule mentioned above is as follows: the decision to admit or reject a primary user is dependent on the quantity of repeating users present in the orbit at the arrival moment, as well as on the quantity of occupied devices, and is determined by the parameters below.

Let K be an integer number such that $0\le K\le N.$ The parameter K determines the maximum quantity of devices that can be reserved exclusively for processing of repeating users. The value $K=0$ means that the system has no device reservation. The value $K=N$ corresponds to a system where the access of the primary users can be completely blocked.

Let some integer values (thresholds) $G_1,G_2,\dots ,G_K$ also be fixed such that

$$0=G_0<G_1\le G_2\cdots \le G_K<G_{K+1}=\infty .$$

If the quantity of repeating users in the orbit belongs to the interval $[G_k,G_{k+1})$, then $k,k=\overline{0,K}$ devices are reserved for processing of repeating users only. If, at the primary user arrival moment, the quantity of users in the orbit belongs to the interval $[G_k,G_{k+1})$ and there are less than $N-k$ idle devices, the arriving user is rejected. If not, it is admitted into the system and starts processing.

The processing time of a primary and retrying user by any device has a phase-type distribution with the irreducible presentations $({\mathbf{b}}_1,S_1)$ and $({\mathbf{b}}_2,S_2),$ respectively. This distribution is specified by the $MC$$m_t^{\left(1\right)},t\ge 0,$ for the primary users and by the $MC$$m_t^{\left(2\right)},t\ge 0,$ for the retrying users. The process $m_t^{\left(l\right)},t\ge 0,l=1,2,$ has the set $\{1,2,\dots ,M_l\}$ of transient states and an absorbing state $M_l+1.$ The mean processing duration for a primary user is denoted as $b_1^{\left(1\right)}={\mathbf{b}}_1{(-S_1)}^{-1}\mathbf{e}.$ The mean processing time for a retrying user is determined as $b_1^{\left(2\right)}={\mathbf{b}}_2{(-S_2)}^{-1}\mathbf{e}.$ For comprehensive information regarding the phase-type distribution, see [35,36,37,38]. The statistical methods for constructing the irreducible presentation of the phase-type distribution of service times are similar to the above-cited methods for fitting the real-world arrival processes to the $MAP.$

Users remaining in the orbit may abandon the orbit and the system due to impatience after a duration that follows an exponential distribution with the parameter $\gamma ,\gamma >0.$

We proceed to analyze the given mathematical model.

3. Results

To greatly facilitate the investigation, we employ the concept of the generalized phase-type distribution, as referenced in [39,40]. Rather than separately analyzing the processing processes for each user type (we refer to the primary user as a type-1 and the repeating user as a type-2 user), we posit that the processing time for an arbitrary user at a device follows a generalized $PH$ ($GPH$) distribution characterized by the parameters $({\mathit{\beta }}_1,{\mathit{\beta }}_2,S)$, where the vectors ${\mathit{\beta }}_1$ and ${\mathit{\beta }}_2$, along with the sub-generator S, are determined by specific formulas:

$${\mathit{\beta }}_1=({\mathbf{b}}_1,\underset{M_2}{\underbrace{0,0,\dots ,0}}),{\mathit{\beta }}_2=(\underset{M_1}{\underbrace{0,0,\dots ,0}},{\mathbf{b}}_2),S=\left(\begin{array}{cc}{S}_1& O\\ O& S_2\end{array}\right).$$

The processing duration is governed by the fundamental process $m_t$ with the state space $\{1,\dots ,M\}$, where $M=M_1+M_2$.

The initial state of the process $m_t$ at the start of processing is dictated by the probability vector ${\mathit{\beta }}_l,l=1,2$, depending on whether processing is initiated by a type-l user. The jump rates into the absorbing state, signifying processing completion at one of the occupied devices, are denoted by the entries of the column vector

$${\mathbf{S}}_0=\left(\begin{array}{c}-S_1\mathbf{e}\\ -S_2\mathbf{e}\end{array}\right)=\left(\begin{array}{c}{\mathbf{S}}_0^{\left(1\right)}\\ {\mathbf{S}}_0^{\left(2\right)}\end{array}\right).$$

The usage of the $GPH$ allows us to substantially simplify the process of determining the dynamics of the system and, consequently, facilitate the study of the $MC$ that characterizes the system’s behavior. This continuous-time chain is represented as

$${\xi }_t=\{i_t,n_t,{\nu }_t,m_t^{\left(1\right)},\dots ,m_t^{\left(M\right)}\},t\ge 0.$$

where

$i_t$ is the quantity of users in the orbit, $i_t\ge 0$;

$n_t$ is the quantity of occupied devices, $n_t=\overline{0,N}$;

${\nu }_t$ is the state of the fundamental process of the $MAP,$${\nu }_t=\overline{1,W}$;

$m_t^{\left(l\right)}$ is the quantity of devices on the l-th phase of processing, $m_t^{\left(l\right)}=\overline{0,n_t}$, $l=\overline{1,M}$, ${\displaystyle \sum _{l=1}^M}{m}_t^{\left(l\right)}=n_t$

at time $t,t\ge 0.$

The $MC$${\xi }_t$ is regular and irreducible.

It is worth noting that, besides the idea to use the $GPH$ for a compact uniform description of simultaneous processing of users of two types, which was generated in [39,40], here we also use the idea of describing simultaneous processing of many homogeneous users not via tracking the processing phase in each occupied device but via monitoring the number of devices currently providing processing at all existing phases, which was proposed and used in [41,42]. The advantages of such monitoring are discussed in [43].

We list the states of the $MC$${\xi }_t,t\ge 0$ in direct lexicographic order of the components $\{i_t,n_t,{\nu }_t\}$ and in reverse lexicographic order of the components $\{m_t^{\left(1\right)},\dots ,m_t^{\left(M\right)}\}$. We call the set of states of the chain that possesses the value i of the component $i_t$ as leveli of the $MC$${\xi }_t$. The level i consists of sub-levels $(i,n),n=\overline{0,N},i\ge 0.$

We designate the infinitesimal generator of this chain as Q. The matrix Q encompasses the jump rates of the $MC$${\xi }_t$ during an infinitesimally small interval. This matrix includes the blocks $Q_{i,j},i,j\ge 0,|j-i|\le 1.$

Theorem 1.

The infinitesimal generator Q of the $MC$${\xi }_t,t\ge 0$ has a block tridiagonal structure

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

where its nonzero blocks are determined as follows:

• The diagonal blocks $Q_{i,i}$ have a block tridiagonal structure $Q_{i,i}=\left(Q_{i,i}^{(n,n^{\prime })}\right),|n-n^{\prime }|\le 1,n,n^{\prime }=\overline{0,N},$ with nonzero blocks:

  $$Q_{i,i}^{(n,n)}=D_0\oplus (A_n+{\mathrm{\Delta }}_n)+{\delta }_{n\ge N-k}{D}_1\otimes I_{T_n}-$$

  $$-i({\delta }_{n<N}\alpha +{\delta }_{n=N}q\alpha +\gamma )I_{W{T}_n},n=\overline{0,N},i\in [G_k,G_{k+1}),k=\overline{0,K},$$

  $$Q_{i,i}^{(n,n+1)}=D_1\otimes P_n\left({\mathit{\beta }}_1\right),n=\overline{0,N-k-1},i\in [G_k,G_{k+1}),k=\overline{0,K},$$

  $$Q_{i,i}^{(n,n-1)}=p_1{I}_W\otimes L_n^{\left(1\right)}+p_2{I}_W\otimes L_n^{\left(2\right)},n=\overline{1,N},i\ge 0;$$
• The updiagonal blocks $Q_{i,i+1}$ have only nonzero subdiagonal blocks $Q_{i,i+1}^{(n,n-1)}$, determined as

  $$Q_{i,i+1}^{(n,n-1)}=(1-p_1)I_W\otimes L_n^{\left(1\right)}+(1-p_2)I_W\otimes L_n^{\left(2\right)},n=\overline{1,N},i\ge 0;$$
• The subdiagonal blocks $Q_{i,i-1}$ have nonzero updiagonal blocks $Q_{i,i-1}^{(n,n+1)}$, given by

  $$Q_{i,i-1}^{(n,n+1)}=i\alpha I_W\otimes P_n\left({\mathit{\beta }}_2\right),n=\overline{0,N-1},i\ge 1,$$
  and the diagonal blocks

  $$Q_{i,i-1}^{(n,n)}=i(\gamma +{\delta }_{n=N}q\alpha )I_{W{T}_n},n=\overline{0,N},i\ge 1.$$

Here,

⊗ and ⊕ are the symbols of the Kronecker product and sum of matrices; see, for example, [44,45].

I denotes the identity matrix, while O represents the zero matrix, with its dimensions specified by a subscript when required.

$T_n=C_{n+M-1}^{M-1},n=\overline{1,N};$

${\delta }_a=\left\{\begin{array}{cc}1,\hfill & a\mathrm{is}\mathrm{true};\hfill \\ 0,\hfill & \mathrm{otherwise};\hfill \end{array}\right.$

${\mathbf{S}}_0^{\left(l\right)}=-S_l\mathbf{e},l=1,2.$

The matrices $L_n^{\left(l\right)}=L_n^{\left(l\right)}\left({\mathbf{S}}_0^{\left(l\right)}\right),$$n=\overline{1,N},$$l=1,2,$${\mathrm{\Delta }}_n,A_n,n=\overline{0,N},$ and $P_n\left({\mathit{\beta }}_l\right),l=1,2,n=\overline{0,N-1}$ are computed by means of the algorithms for matrices of the same designation presented in [46].

Proof. 

Theorem 1 is proven by taking into account the presented description of providing processing to the primary and the repeating users and the following probabilistic meaning of the matrices $L_n^{\left(l\right)},$${\mathrm{\Delta }}_n, A_n,$ and $P_n\left({\mathit{\beta }}_l\right).$

Let us denote the vector stochastic process ${\mathbf{m}}_t=\{m_t^{\left(1\right)},m_t^{\left(2\right)},\dots m_t^{\left(M\right)}\},t\ge 0.$ Given that n users receive processing in the system:

-

The matrix $L_n^{\left(l\right)}$ describes jump rates of the process ${\mathbf{m}}_t$ during the epoch of a primary user processing completion;

-

The matrix $L_n^{\left(2\right)}$ describes jump rates of the process ${\mathbf{m}}_t$ during the epoch of a repeating user processing completion;

-

The matrix $A_n$ describes jump rates of the process ${\mathbf{m}}_t$ without processing completion in any occupied device;

-

The matrix $P_n\left({\mathit{\beta }}_1\right)$ determines jump probabilities of the process ${\mathbf{m}}_t$ during the epoch of the start of a primary user processing;

-

The matrix $P_n\left({\mathit{\beta }}_2\right)$ determines jump probabilities of the process ${\mathbf{m}}_t$ during the epoch of a repeating user processing beginning;

-

The negative diagonal entries of ${\mathrm{\Delta }}_n$ have absolute values that determine the exit rates of the process ${\mathbf{m}}_t$ from its states.

The proof of the form of the blocks $Q_{i,j},i,j\ge 0$ of the generator Q and their sub-blocks $Q_{i,j}^{(n,n^{\prime })}$ is implemented via the careful analysis of all possible transitions of the $MC$${\xi }_t,t\ge 0$ during the infinitesimally small time interval and the use of the formula of total probability. The operations of the Kronecker product and sum of the matrices are very helpful for defining the simultaneous transitions of several independent one-dimensional continuous-time Markov processes.

Since during an infinitesimal time interval users may enter the orbit and depart it singly, the matrices $Q_{i,j},i,j\ge 0$ are zero matrices for all $i,j$ such that $|j-i|>1.$ The matrices $Q_{i,j},i,j\ge 0,$$|j-i|\le 1$ consist of the blocks $Q_{i,j}^{(n,n^{\prime })}$ of the jump rates of the $MC$${\xi }_t,t\ge 0,$ from the sub-level $(i,n)$ to the sub-level $(j,n^{\prime }),n,n^{\prime }=\overline{0,N}.$

The matrices $Q_{i,i},i\ge 0$ have a block tridiagonal structure $Q_{i,i}=\left(Q_{i,i}^{(n,n^{\prime })}\right),|n-n^{\prime }|\le 1,n,n^{\prime }=\overline{0,N},$ because the jump from the sub-level $(i,n)$ to the sub-level $(i,n^{\prime })$ for $n,n^{\prime },|n-n^{\prime }|>1$ is not possible because no more than one user can start or finish processing during an interval of infinitesimal length.

The diagonal entries of the diagonal blocks $Q_{i,i}^{(n,n)},n,n^{\prime }=\overline{0,N}$ are negative; their absolute values represent the exit rates of the $MC$${\xi }_t,t\ge 0$ from the corresponding states. The events leading to a state change are as follows:

(1)

The fundamental process ${\nu }_t,t\ge 0$ changes its states except in the case when this process transits from one state to the same state with a primary user arrival, and it is rejected. In this case, the exit from the state does not occur. Up to the sign, the rates of these events are determined by the diagonal entries of the matrix $(D_0+{\delta }_{n\ge N-k}{D}_1)\otimes I_{T_n},i\in [G_k,G_{k+1}),k=\overline{0,K}.$

(2)

A user departs the orbit in the case of a successful attempt (and it starts processing), an unsuccessful attempt when all devices are occupied (and it abandons the system permanently), or due to impatience. The corresponding jump rates, up to the sing, are represented by the diagonal entries of the matrices $i({\delta }_{n<N}\alpha +{\delta }_{n=N}q\alpha +\gamma )I_{W{T}_n}$, $n=\overline{0,N}$.

(3)

The processing process ${\mathbf{m}}_t$ departs from its present state (jump rates are specified by the matrix $I_W\otimes {\mathrm{\Delta }}_n,n=\overline{0,N}$).

The non-diagonal entries of the diagonal blocks $Q_{i,i}^{(n,n)},n=\overline{0,N}$ in the matrices $Q_{i,i}$ are non-negative. These entries represent Markov chain jump rates that keep the values of components i and n constant. These jumps can occur if the following occur:

(1)

The fundamental process ${\nu }_t,t\ge 0$ makes a jump without the arrival of a primary user or changes its state with the arrival of a primary user that is not admitted to the system and abandons it because the acceptance rule is not fulfilled. The rates of such events are given by the corresponding non-diagonal entries of the matrix $(D_0+{\delta }_{n\ge N-k}{D}_1)\otimes I_{T_n},i\in [G_k,G_{k+1}),k=\overline{0,K}.$

(2)

The processing process ${\mathbf{m}}_t$ makes a jump without processing completion in any occupied device. The rates of such events are specified by the entries of the matrix $A_n,n=\overline{1,N}$.

For the matrices $Q_{i,i}$, the blocks $Q_{i,i}^{(n,n+1)}$ determine the rates of the event that causes the quantity of occupied devices to rise by one, assuming that the quantity of users in the orbit remains unchanged. It occurs only if a primary user arrives, is admitted by the system (the acceptance rule is fulfilled), and starts processing. The corresponding rates are determined by the matrices $D_1\otimes P_n\left({\mathit{\beta }}_1\right),n=\overline{0,N-k-1},i\in [G_k,G_{k+1}),k=\overline{0,K}.$

The blocks $Q_{i,i}^{(n,n-1)},n=\overline{1,N}$ of the matrices $Q_{i,i}$ determine the rates of the event that leads to the reduction in the quantity of occupied devices by one, assuming that the quantity of users in the orbit remains unchanged. Such an event can happen if processing completes in one of the occupied devices and a processed user abandons the system forever after processing. The matrices $p_1{I}_W\otimes L_n^{\left(1\right)}$ and $p_2{I}_W\otimes L_n^{\left(2\right)}$ determine the corresponding rates if the processed user is primary and repeating, respectively.

Therefore, we obtain the form of the blocks $Q_{i,i},i\ge 0.$

The blocks $Q_{i,i+1},i\ge 0$ determine the rate of jumps that lead to a rise in the quantity of users in the orbit by one. They consist only of the nonzero subdiagonal blocks $Q_{i,i+1}^{(n,n-1)}$, $n=\overline{1,N},$ because this event is only possible in the case of a reduction in the quantity of occupied devices. The quantity of users in the orbit rises by one only when a user joins the orbit again after processing. The corresponding rates are determined by the matrices $(1-p_1)I_W\otimes L_n^{\left(1\right)}$ and $(1-p_2)I_W\otimes L_n^{\left(2\right)}$ in the case of a primary and repeating user, respectively.

The blocks $Q_{i,i-1},i\ge 1$ determine the rate of jumps that lead to a reduction in the quantity of users in the orbit by one. This event can lead to the following:

(1)

A rise in the quantity of occupied devices in the system (a user from the orbit makes a successful attempt and starts processing). So, the blocks $Q_{i,i-1}$ contain the subdiagonal blocks $Q_{i,i-1}^{(n,n+1)}$ which are given by the matrices $i\alpha I_W\otimes P_n\left({\mathit{\beta }}_2\right),n=\overline{0,N-1}.$

(2)

Not changing the quantity of occupied devices in the system (a user from the orbit makes an unsuccessful attempt when all devices are occupied and abandons the system permanently or abandons the system due to impatience). Thus, the blocks $Q_{i,i-1}$ include the diagonal blocks $Q_{i,i-1}^{(n,n)}$ which are determined by the matrices $i(\gamma +{\delta }_{n=N}q\alpha )I_{W{T}_n},n=\overline{0,N}.$

□

The important part of investigating any $MC$ with an infinite state space is the derivation of the condition of steady-state regime existence. Firstly, let us consider the case when repeating users are impatient or non-persistent. It means that at least one of the parameters, $\gamma$ or q, is greater than zero. In this scenario, the considered $MC$ is ergodic for any set of system parameters. The proof of this statement evidently follows from the results of the paper [47]. The intuition behind this statement is obvious. If the retrying customers are impatient ($\gamma >0$) or non-persistent ($q>0),$ then the rate of the flow of the retrying customers that terminate their stay in the system without entering service tends to infinity when the number of retrying customers infinitely increases. This implies the impossibility of the overload of the system.

Let us now consider the case $\gamma =0$ and $q=0,$ i.e., retrying customers are absolutely patient and persistent. In this case, the ergodicity of the considered $MC$ does not directly follow from [47]. Derivation of the ergodicity condition is based on the results for asymptotically quasi-Toeplitz $MC$s ($AQTMC$s) obtained in [48].

To use these results, we have to prove that the considered $MC$${\xi }_t,t\ge 0$ belongs to the class of $AQTMC$s. Thus, we have to prove the existence of the following matrices:

$$Y_0=\underset{i\to \infty }{lim}{R}_i^{-1}{Q}_{i,i-1},Y_1=\underset{i\to \infty }{lim}{R}_i^{-1}{Q}_{i,i}+I,Y_2=\underset{i\to \infty }{lim}{R}_i^{-1}{Q}_{i,i+1}$$

where $R_i$ is the diagonal matrix with the diagonal entries of the matrix $-Q_{i,i}.$

It can be verified that for $MC$${\xi }_t,t\ge 0,$ these matrices exist and have the following form:

$$Y_0=\left(\begin{array}{cccccc}O& I_W\otimes P_1\left({\mathit{\beta }}_2\right)& O& \dots & O& O\\ O& O& I_W\otimes P_2\left({\mathit{\beta }}_2\right)& \dots & O& O\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ O& O& O& \dots & O& I_W\otimes P_{N-1}\left({\mathit{\beta }}_2\right)\\ O& O& O& \dots & O& O\end{array}\right),$$

$$Y_1=\left(\begin{array}{ccccc}O& \dots & O& O& O\\ ⋮& \ddots & ⋮& ⋮& ⋮\\ O& \dots & O& O& O\\ O& \dots & O& \mathrm{\Psi }& \mathrm{\Phi }\end{array}\right),Y_2=\left(\begin{array}{ccccc}O& \dots & O& O& O\\ ⋮& \ddots & ⋮& ⋮& ⋮\\ O& \dots & O& O& O\\ O& \dots & O& \mathrm{\Gamma }& O\end{array}\right),$$

where

$$\mathrm{\Phi }=R^{-1}((D_0+D_1)\oplus (A_N+{\mathrm{\Delta }}_N))+I_{W{T}_N},$$

$$\mathrm{\Psi }=R^{-1}(p_1{I}_W\otimes L_N^{\left(1\right)}+p_2{I}_W\otimes L_N^{\left(2\right)}),$$

$$\mathrm{\Gamma }=R^{-1}((1-p_1)I_W\otimes L_N^{\left(1\right)}+(1-p_2)I_W\otimes L_N^{\left(2\right)}),$$

$$R=-\mathrm{\Lambda }\oplus {\mathrm{\Delta }}_N.$$

Here, the matrix $\mathrm{\Lambda }$ is the diagonal matrix with the diagonal entries of the matrix $D_0+D_1.$

Thus, the $MC$${\xi }_t,t\ge 0$ belongs to the class of $AQTMC$s.

The sufficient ergodicity condition for an $AQTMC$ with a block tridiagonal form of the generator has the form

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

where $\mathbf{y}$ is the only solution of the system $\mathbf{y}Y=\mathbf{y},\mathbf{y}\mathbf{e}=1,$ if the matrix $Y=Y_0+Y_1+Y_2$ is irreducible. In the case of the $MC$${\xi }_t,t\ge 0,$ the matrix $Y=Y_0+Y_1+Y_2$ is reducible. As it follows from Theorem 6 in [48], the ergodicity condition for the $MC$${\xi }_t$ takes the following form:

$$\mathbf{x}{X}_0\mathbf{e}>\mathbf{x}{X}_2\mathbf{e}$$

(1)

where $\mathbf{x}$ is the solution of the system

$$\mathbf{x}X=\mathbf{x},\mathbf{x}\mathbf{e}=1$$

(2)

where the matrix X is determined by $X=X_0+X_1+X_2$ and

$$X_0=\left(\begin{array}{cc}O& I_W\otimes P_{N-1}\left({\mathit{\beta }}_2\right)\\ O& O\end{array}\right),X_1=\left(\begin{array}{cc}O& O\\ \mathrm{\Psi }& \mathrm{\Phi }\end{array}\right),X_2=\left(\begin{array}{cc}O& O\\ \mathrm{\Gamma }& O\end{array}\right).$$

Let us represent the vector $\mathbf{x}$ in the form $\mathbf{x}=({\mathbf{x}}_0,{\mathbf{x}}_1).$ By substituting the vector $\mathbf{x}$ in this form into system (2) and using the mixed product rule, see [44], after performing some algebra we determine that

$${\mathbf{x}}_0={\mathbf{x}}_1(\mathrm{\Gamma }+\mathrm{\Psi }),$$

$${\mathbf{x}}_1{R}^{-1}(I_W\otimes (L_N^{\left(1\right)}+L_N^{\left(2\right)})P_{N-1}\left({\mathit{\beta }}_2\right)+(D_0+D_1)\oplus (A_N+{\mathrm{\Delta }}_N))=\mathbf{0}.$$

(3)

It can be verified by direct substitution that the solution of (3) is the vector ${\mathbf{x}}_1{R}^{-1}=c(\mathit{\theta }\otimes \mathit{\mu })$, where c is a positive constant, $\mathit{\theta }$ is the vector of steady-state probabilities of the $MAP$, and $\mathit{\mu }$ is the only solution of the following system:

$$\mathit{\mu }((L_N^{\left(1\right)}+L_N^{\left(2\right)})P_{N-1}\left({\mathit{\beta }}_2\right)+A_N+{\mathrm{\Delta }}_N)=\mathbf{0},\mathit{\mu }\mathbf{e}=1.$$

(4)

The vector $\mathit{\mu }$ gives the steady-state distribution of the quantity of devices at each phase of processing, conditioned on the fact that all N devices are permanently occupied and only type-2 users are admitted for processing. Since the processing of type-1 users is not performed under this condition, the vector $\mathit{\mu }$ has the form $\mathit{\mu }=(\mathbf{0},{\mathit{\mu }}^{\left(2\right)}),$ where ${\mathit{\mu }}^{\left(2\right)}$ is the steady-state distribution of the quantity of devices at the last $M_2$ phases of processing, and $\mathit{\mu }{L}_N^{\left(1\right)}=\mathbf{0}.$

Inequality (1) can be rewritten in the form

$${\mathbf{x}}_1{R}^{-1}(I_W\otimes (L_N^{\left(1\right)}+L_N^{\left(2\right)})P_{N-1}\left({\mathit{\beta }}_2\right))\mathbf{e}>{\mathbf{x}}_1{R}^{-1}{I}_W\otimes ((1-p_1)L_N^{\left(1\right)}+(1-p_2)L_N^{\left(2\right)})\mathbf{e}.$$

By substituting the vector ${\mathbf{x}}_1{R}^{-1}$ in the form ${\mathbf{x}}_{\mathbf{1}}{R}^{-1}=c(\mathit{\theta }\otimes \mathit{\mu })$ in this inequality, using the mixed product rule, and after performing some algebra, we determine that the ergodicity condition takes the form

$$p_2\mathit{\mu }{L}_N^{\left(2\right)}>0.$$

(5)

The vector $\mathit{\mu }$ is stochastic and the nonzero matrix $L_N^{\left(2\right)}$ has only non-negative entries, and at least one of its entries is positive. Therefore, equality (5) holds true if $p_2>0.$

Thus, we have proved the following assertion:

Theorem 2.

The $MC$${\xi }_t,t\ge 0$ is ergodic if the parameters $q,p_2$, and γ are not equal to zero at the same time.

It deserves mentioning that ergodicity or non-ergodicity of the system does not depend on the arrival, service, and retrial rate and is defined only by the values of the parameters $q,p_2$, and $\gamma .$ The system is non-ergodic only if the user cannot depart from it due to impatience, non-persistence, or after service completion. In such a situation, the retrying users never depart from the system, and it never becomes empty.

Further, we assume that the ergodicity condition holds true.

Let us now briefly outline the methods for computation of the steady-state distribution of the $MC$${\xi }_t.$ This distribution is determined by the steady-state probabilities determined as the limits

$$\mathit{\pi }(i,n,\nu ,m^{\left(1\right)},m^{\left(2\right)},\dots m^{\left(M\right)})$$

$$=\underset{t\to \infty }{lim}P\{i_t=i,n_t=n,{\nu }_t=\nu ,m_t^{\left(1\right)}=m^{\left(1\right)},m_t^{\left(2\right)}=m^{\left(2\right)},\dots m_t^{\left(M\right)}=m^{\left(M\right)}\},$$

$$i\ge 0,n=\overline{0,N},\nu =\overline{1,W},m^{\left(l\right)}=\overline{0,n},l=\overline{1,M},\sum _{l=1}^M{m}^{\left(l\right)}=n.$$

Let us denote by ${\mathit{\pi }}_i$ the row vector composed from the steady-state probabilities $\mathit{\pi }(i,n,\nu ,m^{\left(1\right)},m^{\left(2\right)},\dots m^{\left(M\right)})$ of the states that belong to the level i and by $\mathit{\pi }(i,n)$ the row vectors composed from the steady-state probabilities of the states that belong to the sub-level $(i,n).$ It is clear that ${\mathit{\pi }}_i=(\mathit{\pi }(i,0),\mathit{\pi }(i,1),\dots ,\mathit{\pi }(i,N)),i\ge 0.$

The vectors ${\mathit{\pi }}_i,i\ge 0$ satisfy the system of equations

$$({\mathit{\pi }}_0,{\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots )Q=\mathbf{0},({\mathit{\pi }}_0,{\mathit{\pi }}_1,{\mathit{\pi }}_2,\dots )\mathbf{e}=1.$$

(6)

Due to the infinite nature of this system and the absence of a block quasi-Toeplitz structure in the matrix Q, the $MC$${\xi }_t$ does not fall within the category of Quasi-Birth-and-Death processes, as detailed in [35], which have been extensively examined by M. Neuts. Consequently, resolving this system presents significant challenges. Frequently, in the literature, it is dealt with by some kind of system truncation.

Since the $MC$${\xi }_t$ falls into the category of asymptotically quasi-Toeplitz Markov chains, system (6) can be resolved utilizing the numerically stable techniques for such chains outlined in, for example, [28,49]. The high stability of the algorithms presented in [28,49] is guaranteed by the avoidance of the use of the subtraction operation during their implementation.

Once obtaining the vectors ${\mathit{\pi }}_i,i\ge 0,$ we can determine the values for various performance metrics of the considered queueing system.

The mean quantity of users residing in the orbit at an arbitrary instant is calculated as

$$L^{orb}=\sum _{i=1}^{\infty }i{\mathit{\pi }}_i\mathbf{e}.$$

The mean quantity of processing type-l users at an arbitrary instant is calculated as

$$L_l^{user}=\sum _{i=0}^{\infty }\sum _{n=1}^{N}n\mathit{\pi }(i,n)(I_W\otimes {\tilde{L}}_n^{\left(l\right)}\left({\tilde{\mathbf{S}}}_0^{\left(l\right)}\right))\mathbf{e}$$

where ${\tilde{\mathbf{S}}}_0^{\left(1\right)}={({\mathbf{e}}_{M_1}^T,{\mathbf{0}}_{M_2})}^T,$${\tilde{\mathbf{S}}}_0^{\left(2\right)}={({\mathbf{0}}_{M_1},{\mathbf{e}}_{M_2}^T)}^T,$ and the matrices ${\tilde{L}}_n^{\left(l\right)}\left({\tilde{\mathbf{S}}}_0^{\left(l\right)}\right)$ are calculated following the same method as the matrices $L_n^{\left(l\right)}\left({\mathbf{S}}_0^{\left(l\right)}\right),n=\overline{1,N},l=1,2.$

The mean quantity of occupied devices at an arbitrary instant is calculated as

$$L^{dev}=\sum _{i=0}^{\infty }\sum _{n=1}^{N}n\mathit{\pi }(i,n)\mathbf{e}=L_l^{user}+L_2^{user}.$$

The mean total quantity of users in the system (in the orbit and devices) is calculated as

$$L^{sys}=\sum _{i=0}^{\infty }\sum _{n=0}^N(i+n)\mathit{\pi }(i,n)\mathbf{e}=L^{orb}+L^{dev}.$$

The mean rate of the output flow of successfully processed primary users is calculated as

$${\mathit{\mu }}_1^{out}=\sum _{i=0}^{\infty }\sum _{n=1}^N\mathit{\pi }(i,n)(I_W\otimes L_n^{\left(1\right)})\mathbf{e}.$$

The mean rate of the output flow of successfully processed repeating users is calculated as

$${\mathit{\mu }}_2^{out}=\sum _{i=0}^{\infty }\sum _{n=1}^N\mathit{\pi }(i,n)(I_W\otimes L_n^{\left(2\right)})\mathbf{e}.$$

The mean rate of the output flow of successfully processed users is calculated as

$${\mathit{\mu }}^{out}=\sum _{i=0}^{\infty }\sum _{n=1}^N\mathit{\pi }(i,n)(I_W\otimes (L_n^{\left(l\right)}+L_n^{\left(2\right)}))\mathbf{e}={\mathit{\mu }}_1^{out}+{\mathit{\mu }}_2^{out}.$$

The mean rate of users who join the orbit is calculated as

$${\lambda }_{orb}=(1-p_1){\mathit{\mu }}_1^{out}+(1-p_2){\mathit{\mu }}_2^{out}.$$

The probability that an arbitrary repeating attempt is successful is calculated as

$$P^{suc-ret}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{\infty }}i{\displaystyle \sum _{n=0}^{N-1}}\mathit{\pi }(i,n)\mathbf{e}}{{\displaystyle \sum _{i=1}^{\infty }}i{\displaystyle \sum _{n=0}^N}\mathit{\pi }(i,n)\mathbf{e}}}.$$

The mean quantity of reserved devices is calculated as

$$N^{res}=\sum _{k=1}^K\sum _{i=G_k}^{G_{k+1}-1}k{\mathit{\pi }}_i\mathbf{e}.$$

The loss probability of a primary user upon arrival is calculated as

$$P^{primary-loss}={\displaystyle \frac{1}{\lambda }}\sum _{k=0}^K\sum _{i=G_k}^{G_{k+1}-1}\sum _{n=N-k}^N\mathit{\pi }(i,n)(D_1\otimes I_{T_n})\mathbf{e}.$$

The loss probability of a repeating (secondary) user due to impatience is calculated as

$$P^{secondary-imp-loss}={\displaystyle \frac{1}{{\lambda }_{orb}}}\sum _{i=1}^{\infty }i\gamma {\mathit{\pi }}_i\mathbf{e}.$$

The loss probability of a repeating user due to non-persistence (departure due to the absence of idle devices) is calculated as

$$P^{secondary-non-pers-loss}={\displaystyle \frac{q\alpha {\displaystyle \sum _{i=1}^{\infty }}i\mathit{\pi }(i,N)\mathbf{e}}{{\lambda }_{orb}}}.$$

The loss probability of a repeating user is calculated as

$$P^{secondary-loss}=P^{secondary-imp-loss}+P^{secondary-non-pers-loss}.$$

The loss probability of an arbitrary user is calculated as

$$P^{loss}={\displaystyle \frac{1}{\lambda +{\lambda }_{orb}}}(\lambda P^{primary-loss}+{\lambda }_{orb}{P}^{secondary-loss})=1-{\displaystyle \frac{{\mathit{\mu }}^{out}}{\lambda +{\lambda }_{orb}}}.$$

The last formula presents two separate methods for calculating the probability $P^{loss}.$ This offers a mechanism for validating the precision of the computation of the steady-state distribution of system states and the values of the performance metrics of the system.

Now, we briefly present the results of two numerical experiments.

Experiment 1. In this numerical experiment, we aim to demonstrate the influence of the choice of the thresholds $G_k,k=\overline{1,K}$ that determine the device reservation policy on the main performance characteristics of the system and show that the optimal choice of these thresholds allows us to improve the system revenue.

We suggest that the arrival process of primary users is characterized by the $MAP$ delineated by the matrices

$$D_0=\left(\begin{array}{cc}-1.08208& 0\\ 0& -0.035128\end{array}\right),D_1=\left(\begin{array}{cc}1.07488& 0.0072\\ 0.019568& 0.01556\end{array}\right).$$

This choice of the matrices $D_0$ and $D_1$ is explained by the frequent appearance of similar matrices as a result of fitting the correlated arrival processes during implementation in the past in some applied projects not directly related to the system under study in this paper. Some characteristics of this process are as follows: the mean arrival rate $\lambda =0.800473$, the coefficient of correlation of successive inter-arrival times $c_{cor}=0.2$, and the coefficient of variation of inter-arrival times $c_{var}=12.34.$

The processing time of a primary user has a $PH$ distribution determined by the vector ${\mathit{\beta }}_1=\left(1\right)$ and the matrix $S_1=-0.5$. This distribution is exponential, with the mean processing time equal to 2.

The probability $p_1$ that a primary user leaves the system after the processing is equal to 0.4.

The retrial rate is $\alpha =0.2;$ the probability q that a repeating user abandons the system in the case of an unsuccessful attempt is equal to 0.7; and the impatience rate $\gamma$ of repeating users is equal to 0.002.

The processing time of a repeating user has a $PH$ distribution determined by the vector ${\mathit{\beta }}_2=\left(1\right)$ and the matrix $S_2=-1$. This distribution is also exponential, with the mean processing time equal to 1. The probability $p_2$ that a repeating user leaves the system after the processing is equal to 0.1.

We assume that the total quantity of devices is equal to $N=6$, and up to $K=2$ devices can be reserved for processing only repeating users. To achieve the formulated aim, let us vary the threshold $G_2$ over the interval $[2,30]$ with step 1, and the threshold $G_1$ over the interval $[1,G_2)$ also with step 1.

Figure 2 illustrates the relation of the mean rate ${\lambda }_{orb}$ of users who join the orbit and the thresholds $G_1$ and $G_2$. Figure 3 illustrates the relation of the probability $P^{suc-ret}$ that an arbitrary repeating attempt is successful and the thresholds $G_1$ and $G_2$. Figure 4 illustrates the relation of the mean quantity $N^{res}$ of reserved devices and the thresholds $G_1$ and $G_2$.

One can see from Figure 2 that the mean rate ${\lambda }_{orb}$ of users who join the orbit non-monotonically depends on the thresholds $G_1$ and $G_2$. The maximal value of ${\lambda }_{orb}$ is equal to 2.103581 and achieved when $G_1=13$ and $G_2=19.$ From Figure 3 and Figure 4, one can conclude that the probability of a successful retrial attempt and the mean quantity of occupied devices essentially reduce with a rise in the quantities $G_1$ and $G_2$. These dependencies are rather evident intuitively; our results allow us to estimate them quantitatively.

Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 illustrate the relation of the loss probability $P^{primary-loss}$ of a primary user, the total loss probability $P^{secondary-loss}$ of a repeating user, the loss probability $P^{secondary-imp-loss}$ of a repeating user due to impatience, the loss probability $P^{secondary-non-pers-loss}$ of a repeating user due to non-persistence, and the loss probability $P^{loss}$ of an arbitrary user and the thresholds $G_1$ and $G_2$.

It is seen from Figure 5 that the loss probability of primary users $P^{primary-loss}$ sharply reduces with the rise in the thresholds $G_1$ and $G_2$, since the growth in $G_1$ and $G_2$ leads to a reduction in the mean quantity of reserved devices; thus, more devices are available for primary users. At the same time, as seen from Figure 6, the rise in $G_1$ and $G_2$ leads to a rise in the loss probability of repeating users $P^{secondary-loss}$. From Figure 7 and Figure 8 we can conclude that the main reason for repeating user loss in this example is non-persistence, especially for large values of $G_1$ and $G_2$. The total loss probability $P^{loss}$ behaves as $P^{primary-loss}$, since in this numerical example the main reason for user loss is the rejection of primary users upon arrival.

Suppose that the quality of the system’s performance can be assessed based on the following cost criterion:

$$E=a_1{\mathit{\mu }}_1^{out}+a_2{\mathit{\mu }}_2^{out}-c_1\lambda P^{primary-loss}-c_2{\lambda }_{orb}{P}^{secondary-loss}$$

where $a_1$ and $a_2$ are the system’s revenue obtained via processing of each primary and repeating user, respectively, and $c_1$ and $c_2$ are the charges paid by the system for primary and repeating user loss, respectively.

Here, we suggest the following values of the cost coefficients: $a_1=2,a_2=3$, $c_1=3$, and $c_2=10$.

Figure 10 illustrates the relation of the cost criterion E and the thresholds $G_1$ and $G_2.$

As one can see from Figure 10, the optimal value of the cost criterion E is equal to 4.897. This value is achieved if the threshold $G_1$ is equal to 8 and the threshold $G_2=14.$ Thus, the system’s manager should reserve no devices if the quantity of users in the orbit is less than 8, should reserve one device if the quantity of users in the orbit is in the interval [8,14), and should reserve two devices otherwise. Note that the optimal device reservation can bring economic profit compared to processing provisioning without device reservation. For example, if we fix $G_1=29$ and $G_2=30,$ i.e., start reserving devices as late as possible (in the conditions of this example), the value $E(29,30)$ will be equal to 4.40968, which is less than the optimal value.

The computations are performed on a PC with an Intel Core i7-8700 CPU, 16 GB RAM, and Wolfram Mathematica 13.2. The computation time is 289 s for 435 pairs $(G_1,G_2),$ or about 0.66 s for one pair.

Note that running time under the use of the algorithms for the computation of the stationary probabilities of the states of the asymptotically quasi-Toeplitz Markov chains is difficult to predict in advance. It significantly depends on the load of the system, correlation, and variation in inter-arrival times. Higher load, positive coefficient of correlation, and a large value of the coefficient of variation lead to a larger number of non-negligible probabilities of the system states. In turn, this implies the increase in computation time.

Experiment 2. In the previous experiment, we assumed an exponential processing time distribution for primary and repeating users. Now, let us investigate how the dependence of the main performance characteristics changes if we consider, instead of the exponential processing time distribution, the $PH$ processing time distribution with the same mean processing rate but a high coefficient of variation.

To this end, let us assume that the processing time of primary users has a $PH$ distribution determined by the matrix $S_1=\left(\begin{array}{cc}-5& 0\\ 0& -0.05\end{array}\right)$ and the vector ${\mathit{\beta }}_1=(0.909091,0.0909091),$ and that the processing time of repeating users has a $PH$ distribution determined by the matrix $S_2=\left(\begin{array}{cc}-10& 0\\ 0& -0.1\end{array}\right)$ and the vector ${\mathit{\beta }}_2=(0.909091,0.0909091).$ These processing times have the same mean values, 0.5 and 1, correspondingly, as in Experiment 1, but a high coefficient of variation $c_{var}=17.2$ (not 1 as in Experiment 1).

Figure 11, Figure 12 and Figure 13 illustrate the relations of the loss probability $P^{primary-loss}$ of a primary user, the total loss probability $P^{secondary-loss}$ of a repeating user, and the loss probability $P^{loss}$ of an arbitrary user with the thresholds $G_1$ and $G_2$ in this case of high coefficients of variation of processing times.

Comparing Figure 11, Figure 12, Figure 13 and Figure 14 with Figure 5, Figure 6, Figure 9, and Figure 10, correspondingly, we can conclude that the high variation in the processing time has a positive impact on the main performance characteristics of the system. However, this impact is not very essential. The shape of dependencies of $P^{primary-loss},$$P^{secondary-loss}$, and $P^{loss}$ is the same as in Experiment 1; the slight difference is only in quantities.

Figure 14 illustrates the cost criterion E on the thresholds $G_1$ and $G_2$ in the case of high coefficients of variation of processing times.

As one can see from Figure 14, the optimal value of the cost criterion E is equal to 5.3608. This value is achieved when the threshold $G_1$ is equal to 7 and the threshold $G_2=16.$ It is worth noting that the optimal values of the thresholds $G_1$ and $G_2$ are different from the corresponding values, 8 and 14, respectively, in the above considered case of the exponential distribution of processing times. Therefore, approximation of the processing time distribution by the exponential one, generally speaking, leads to the bias in evaluation of the optimal strategy’s thresholds.

The computation time in the considered case of the $PH$ distribution is 6593 s for 435 pairs $(G_1,G_2),$ or about 15.15 s for one pair. Thus, due to the smaller size of the blocks of the generator, the simplifying assumption about the exponential processing time distributions allows us to significantly reduce computation time. This is especially significant when the quantity K of devices available for reservation is substantial and it is necessary to determine the optimal values of the thresholds $G_k,k=\overline{1,K}.$ However, if the precision of evaluation of the performance characteristics values is most important, the $PH$ distribution, which fits the value of the variance in processing times, should be used. It is worth noting that our numerical experience shows that basically, reservation of even a small (1–3) quantity of devices allows us to significantly raise the system’s revenue.

4. Conclusions

In this paper, a novel multi-server queueing system, appropriate for capacity planning and performance evaluation of a variety of real-world systems in which users who received earlier processing have easier access to devices than the just-arrived primary users, is analyzed under quite general assumptions. Namely, the flow of new users is described by the $MAP$ process that allows us to adequately account for the correlation and possible high variation in inter-arrival times. Processing times of primary and repeating users have a phase-type distribution that allows us to adequately account for the variance in and higher moments of these times. Distinct parameters of these distributions for primary and repeating users are supposed.

Repeating users may exhibit impatience, potentially leaving the system if their wait for processing is excessively long, and may also demonstrate a lack of persistence, ceasing further retries after each unsuccessful attempt to access the processing. Thus, to protect the interests of these users, it is suggested that the primary users arriving from outside can be rejected even when there are idle devices in the system if the quantity of repeating users is large. The threshold strategy for rejection of these users is determined. Under the specified parameters of this strategy, the system’s behavior is characterized by a multidimensional Markov chain. The choice of such a chain with an appropriately sized block for the generator is quite a challenging task. In this paper, it is made using the notion of the generalized $PH$ distribution. In combination with the approach from [41,42] for the description of phase-type processing in many devices, this choice allows us to drastically reduce the cardinality of the process describing processing compared to the choice of using a separate account of the quantities of the primary and repeating users receiving processing and their current phases of processing.

The generator of the constructed Markov chain is obtained. This enables us to ascertain that the Markov chain is classified as an asymptotically quasi-Toeplitz Markov chain, which permits the application of relevant results concerning the validation of its ergodicity and the calculation of its steady-state distribution. Formulas for the important performance metrics of the system are developed, and their relation to the parameters of the primary users’ admission strategy is numerically highlighted. The possibility of using the results to meet managerial needs is illustrated.

The results can be extended to the case of the bounded quantity of repeating users and the possibility of rejected primary user retrials, breakdowns of devices, existence of high-priority repeating users, etc.