Retrial Queueing System for Analyzing Impact of Priority Ultra-Reliable Low-Latency Communication Transmission on Enhanced Mobile Broadband Quality of Service Degradation in 5G Networks

Abstract

Fifth generation (5G) networks support ultra-reliable low-latency communications (URLLC) and enhanced mobile broadband (eMBB). The coexistence of URLLC and eMBB is often organized by non-orthogonal multiple access (NOMA), giving priority to URLLC and resulting in eMBB quality of service (QoS) degradation. In this paper, we address the issue of joint URLLC and eMBB transmission, focusing on the problem from the perspective of delay-tolerant eMBB. Due to the priority given to URLLC, we assume that an eMBB session may be interrupted if there are no free resources available for URLLC or delayed when a new eMBB session arrives. To make the scheme more flexible, we propose that interrupted and delayed eMBB sessions periodically check for free resources, rather than continuously. To analyze this scenario, we propose a retrial queuing system with two retrial buffers (orbits) for interrupted and delayed eMBB sessions. The stationary probability distribution, provided in matrix form by recursive formulas, is presented. The paper concludes with a numerical example showing that the scheme with two buffers, compared to one buffer, practically doubles the average number of active eMBB sessions while keeping the interruption probability below 0.001. We provide an illustration of the configuration of eMBB retrial rates to meet its QoS requirements.

1. Introduction

Fifth generation (5G) networks offer significant advantages, including higher bandwidth, higher bit rates, and solutions to issues related to latency, power consumption, and reliability [1,2]. These developments have led to new usage scenarios that require very high data rates, numerous connected devices, and applications with ultra-low latency and high reliability [3,4]. This includes massive machine-type communications (mMTC), ultra-reliable low-latency communications (URLLC), and enhanced mobile broadband (eMBB) [5]. mMTC scenarios generally involve numerous connected low-cost devices transmitting a small amount of data. URLLC scenarios place stringent demands on bandwidth, latency, and availability, such as wireless control of industrial production, remote medical surgery, and transport security [6,7]. eMBB scenarios are people-centric and involve access to multimedia content, services, and data that require wide coverage, high user density, and high bit rates [8,9].

The coexistence of multiple scenarios makes it essential to jointly provide various services with different quality of service (QoS) requirements, including URLLC and eMBB [10,11]. This raises the question of how to multiplex the frequency channel between URLLC and eMBB sessions, given that eMBB traffic resources are dynamically allocated, but URLLC has stringent packet delay requirements [12]. URLLC traffic is highly sensitive to delay, regardless of the size of the transmitted data. In the context of the tactile Internet, where data packets can be quite large, network slicing can be a highly effective approach. However, in scenarios where URLLC has a small data volume, employing network slicing could negatively impact channel capacity. Therefore, non-orthogonal multiple access (NOMA) could be used. As for eMBB traffic, it should provide a high transmission rate, while other requirements may vary depending on the specific usage scenario. For virtual reality, for instance, data transmission delays are intolerable, but there are no specific requirements for data buffering.

This paper focuses on the joint transmission of URLLC and eMBB traffic using dynamic multiplexing of time-frequency resources based on NOMA [13]. We address the scenario of URLLC with a small data volume, the possibility of interrupting and resuming eMBB traffic transmission, as well as the delay in initiating eMBB traffic transmission.

The main contributions of our study are as follows:

• A mathematical framework based on queueing theory to analyze the impact of URLLC priority traffic on eMBB traffic data transmission.
• An algorithm for calculating metrics of interest based on using a matrix-geometric solution.
• Numerical analysis of the main metrics of interest: eMBB session blocking and interruption probabilities to assess the impact of URLLC traffic on eMBB traffic transmission.

The rest of the paper is organized as follows. Section 2 reviews the current state of the field. In Section 3, we formulate our system model. A mathematical model as a retrial queueing model is presented in Section 4. Section 5 is devoted to numerical analysis. Conclusions are drawn in the last section.

2. Related Work

In the context of coexisting URLLC and eMBB, there are various approaches toward resource allocation. One such approach is proposed in [14,15], where network slicing is suggested to allocate resources between different traffic types, ensuring their isolation and providing specific performance levels. Another approach is shared service models based on deep reinforcement learning (DRL), proposed in [16,17], which can learn optimal solutions without pre-modeling the problem. DRL approaches can solve non-deterministic problems and make real-time decisions, aiding resource allocation and decision making under uncertainty [18].

Multiplexing time–frequency resources is another challenge in URLLC and eMBB coexistence. Due to the significant difference in the URLLC transmission timescale, the channel time domain is divided into time slots, and further divided into mini-slots. The allocation of frequency resources between URLLC and eMBB requires multiplexing schemes to manage different traffic types [19,20]. Two such schemes are orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA). OMA assigns non-overlapping frequency sub-bands to each traffic type, while NOMA serves both URLLC and eMBB in the same frequency–time sub-bands [11]. Studies such as [12,14] have compared these technologies, showing that, under OMA, the channel dedicated to URLLC remains idle when there are few transmissions. On the other hand, research such as [21,22] explores the coexistence of URLLC and eMBB using NOMA technology, while [23] proposes a hybrid joint service scheme based on both OMA and NOMA.

A more detailed summary of the selected recent papers on the joint transmission of eMBB and URLLC is presented in

Table 1. Summary of selected works on joint eMBB and URLLC transmission.

Ref.	Resource Sharing	Methods	Impact on eMBB Traffic
[26]	Dynamic multiplexing, OFDMA	Machine learning, matching theory, advanced K-granularity coloring (AKGC)	Bandwidth reduction
[28]	Dynamic multiplexing, OFDMA	Puncturing method, slicing, mixed numerology	Transmission delay, reduction in sum of eMBB rates
[15]	Network slicing	Reconfigurable intelligent surface (RIS)	Transmission power reduction
[32]	OFDM, hybrid OMA and NOMA	Difference of convex (DC) programming	Performance degradation
[33]	Network slicing, NOMA	Difference of convex (DC) programming	Throughput reduction
[29]	RIS-aided multiplexing, OFDMA	RIS phase-shift matrix optimization, puncturing method	Transmission rate reduction
[30]	Novel latency-relaxation concept of multiplexing	Particle swarm optimization (PSO) algorithm, simulation analysis	Throughput reduction
[30]	Dynamic multiplexing	Adaptive modulation and coding (AMC)	Puncturing eMBB traffic
[34]	OFDM, optimal and sub-optimal approaches	Puncturing method	Puncturing eMBB traffic
[31]	OFDM	Multi-connectivity, modulation, and coding	Throughput reduction
[21]	NOMA	Puncturing technique, numerical simulations	Throughput reduction
[35]	Dynamic resource allocation and puncturing strategy (DRAPS)	Lyapunov drift-plus-penalty (DPP) method, block coordinate descent (BCD) algorithm	Transmission rate reduction
[36]	Risk-resistant model	M/G/1 queue model, simulation	Transmission rate reduction
[37]	Network slicing	Attention mechanism, DRL	Puncturing eMBB traffic
[38]	Intelligent resource management framework	Co-training DRL (CDRL), deep neural network (DNN), convolutional neural network (CNN)	Performance degradation
[25]	Network slicing	DRL, dueling deep Q network (Dueling DQN)	Bandwidth reduction
[39]	Mixed deep deterministic policy gradient (Mixed-DDPG)	DRL, preemptive puncturing	Puncturing eMBB traffic, transmission interruption
[27]	Dynamic multiplexing	DL, deep function approximator (DFA)	Throughput and performance reduction
[23]	Hybrid puncturing and superposition, NOMA	Simulation	Transmission rate reduction

. The table reflects the main approaches for resource sharing and the methods used to implement them, as well as how prioritized URLLC traffic affects eMBB traffic. Among these approaches, network slicing [24,25] is considered, along with other methods such as dynamic multiplexing [26,27] and reconfigurable intelligent surface (RIS)-aided multiplexing [15]. Due to the stringent delay requirements of URLLC traffic, it is typically prioritized, which can have various impacts on eMBB transmission. For example, in works such as [28,29,30], delay, transmission interruption, or a reduced bit rate of eMBB traffic can occur through puncturing or pre-emption methods. Alternatively, in [30,31], a reduction in throughput is observed. The type of impact can vary according to the usage scenario.

This paper examines the scenario of small data volumes of URLLC traffic and NOMA multiplexing. eMBB traffic may be interrupted and may have to wait to initiate data transmission. In contrast to many other papers, we concentrate on the effect of prioritized URLLC transmission on non-priority eMBB. This paper presents a continuation of our studies focused on various models for joint URLLC and eMBB transmission, analyzed as queuing systems. In previous works, such as [40], we presented models with interruption and bit rate degradation of eMBB sessions. Additionally, we proposed a model of radio admission control for URLLC and adaptive bit rate eMBB in [41]. In [42], we introduced a model as a resource queuing system, and in [43], we extended this model to consider eMBB bit rate degradation due to the impact of path loss, based on the methods used in [44,45].

3. System Model

In this section, we present the system model, starting with a description of resource sharing between URLLC and eMBB sessions, as well as the admission control and retrial policy.

3.1. General Assumptions

We consider a base station that has access to N physical resource blocks (PRBs), which are divided into b mini-slots in the time domain. URLLC and eMBB devices establish sessions. The maximum number of eMBB sessions is equal to the number of PRBs (N). Due to the stringent latency and reliability requirements of URLLC sessions, they are prioritized over eMBB sessions. This means that the transmission of an eMBB session may be delayed or interrupted due to a lack of resources and the arrival of a new URLLC session.

To handle these situations, the base station employs two buffers for eMBB sessions. The first buffer (1-buffer) is used to store delayed eMBB sessions that cannot be transmitted immediately upon arrival due to resource constraints. Both the 1-buffer and 2-buffer have limited capacities. The capacity of the 1-buffer, denoted as $C_1$, is greater than the number of PRBs N to increase the channel throughput and minimize the loss of eMBB sessions. Similarly, the capacity of the 2-buffer, denoted as $C_2$, is also greater than the number of PRBs N to prevent the loss of interrupted eMBB sessions.

Figure 1 illustrates a scheme representing the system under consideration. At the center of the figure is the base station. The upper portion of the figure displays a segment of the channel as a sequence of PRBs, highlighting the resources allocated for eMBB sessions in green and the resources allocated for URLLC sessions in red. Toward the lower section of the diagram, two base station buffers are shown, with the upper buffer designated for new eMBB sessions and the lower buffer designated for interrupted eMBB sessions.

eMBB and URLLC sessions arrive according to the Poisson process with rates ${\lambda }_{\mathrm{m}}$ and ${\lambda }_{\mathrm{u}}$, respectively. The duration of each session is exponentially distributed, with means ${\mu }_{\mathrm{m}}^{-1}$ for eMBB and ${\mu }_{\mathrm{u}}^{-1}$ for URLLC. Delayed and interrupted eMBB sessions retry the connection with retrial periods exponentially distributed with rates ${\gamma }_1$ and ${\gamma }_2$, respectively.

The parameters of the system mentioned above, as well as those for the continuous-time Markov chain (CTMC) described in Section 4.1, are summarized in

Table 2. Main notation.

Parameter	Description
System Parameters
N	Total number of slots (number of PRBs)
b	Number of mini-slots in one slot
$C=Nb$	Total number of mini-slots
$C_1>N$	Maximum number of delayed eMBB sessions (size of 1-buffer)
$C_2>N$	Maximum number of interrupted eMBB sessions (size of 2-buffer)
${\lambda }_{\mathrm{m}}$	Arrival rate of eMBB sessions, 1/s
${\lambda }_{\mathrm{u}}$	Arrival rate of URLLC sessions, 1/s
${\mu }_{\mathrm{m}}^{-1}$	Average duration of an eMBB session, s
${\mu }_{\mathrm{u}}^{-1}$	Average duration of an URLLC session, s
${\gamma }_1$	Retrial rate of delayed eMBB sessions (from 1-buffer), 1/s
${\gamma }_2$	Retrial rate of interrupted eMBB sessions (from 2-buffer), 1/s
Continuous-Time Markov Chain (CTMC)
$n_{\mathrm{m}}$	Number of active eMBB sessions
$n_{\mathrm{u}}$	Number of active URLLC sessions
$q_1$	Number of delayed eMBB sessions (in 1-buffer)
$q_2$	Number of interrupted eMBB sessions (in 2-buffer)
$\mathbf{x}=(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)$	State of the system
$\mathcal{X}$	Set of states
$\pi$	Stationary distribution

.

3.2. Resource Sharing

Resource sharing between eMBB and URLLC sessions, as well as the channel structure, is illustrated in Figure 2. As shown in Figure 2, the resource channel is represented by N PRBs, which are divided in time into OFDM (orthogonal frequency division multiplexing) symbols and in frequency into subcarriers. A resource unit (RU) is defined as one OFDM symbol x one subcarrier. The duration of one PRB is one slot (0.5 ms). Each slot is further divided into mini-slots, the number of which is equal to the number b of OFDM symbols in one PRB, where $b=7$, resulting in a total of $C=Nb$ mini-slots.

eMBB sessions are scheduled relative to slots, while URLLC sessions are scheduled relative to mini-slots. The joint scheduling for eMBB and URLLC sessions is carried out using NOMA technology, where the data transmission scheduling of different sessions occurs at different frequencies and times. We consider dynamic multiplexing, assuming that each eMBB session requires a slot and each URLLC session requires a mini-slot.

3.3. Admission Control

When a new session arrives, the scheduler analyzes the current state of the physical channel and buffers and schedules data transmission for the next slot accordingly. All possible scheduling cases for each type of session are illustrated in Figure 3. Thus, green lines represent new eMBB sessions without delay, orange lines represent new eMBB sessions with delay, blue lines represent interrupted eMBB sessions, and red lines represent URLLC sessions.

For a new eMBB session to be accepted, at least one free slot and two empty buffers must be available. If these conditions are met, the eMBB session will occupy one of the free slots. If there are no free slots or if there are sessions in one of the buffers and the 1-buffer is not full, the new eMBB session will be added to the 1-buffer and the session will be delayed. Otherwise, the session will be blocked. An example of this scenario is depicted in Figure 4, with the initial state denoted as point A.

When a new priority URLLC session arrives, data transmission should commence immediately, even during slots that are already scheduled. The session will be accepted without any impact on eMBB if at least one free mini-slot is available. If there are no free mini-slots, but there are eMBB sessions, one of them will be interrupted, freeing up b mini-slots. The interrupted eMBB session will then wait for the resuming transmission in the 2-buffer, retrying the session with the corresponding rate. An example of this scenario is shown in Figure 4, with the initial state indicated as point D. If there are no free mini-slots and no eMBB sessions present when the URLLC session arrives, it will be blocked.

3.4. Retrial Policy

Thereby, delayed eMBB sessions are added to the 1-buffer, and interrupted eMBB sessions are added to the 2-buffer. Let us describe in more detail how eMBB sessions retry to start and resume transmission.

A delayed eMBB session in the 1-buffer retries to start transmission with rate ${\gamma }_1$. The 2-buffer for interrupted eMBB sessions has priority over the 1-buffer for new delayed eMBB sessions since interrupted eMBB sessions should be resumed as soon as possible. Thus, the retrial will be successful if the 2-buffer is empty and there is at least one free slot. Otherwise, a delayed eMBB session continues to wait in the 1-buffer. An example of starting transmission is shown in Figure 4, where the initial state is shown as point C.

An interrupted eMBB session in the 2-buffer retries to resume transmission with rate ${\gamma }_2$. The retrial will be successful if there is at least one free slot. Otherwise, an interrupted eMBB session continues to wait in the 2-buffer. An example of resuming the transmission of an interrupted eMBB session, even though there is a new eMBB session in the 1-buffer, is shown in Figure 4, where the initial state is shown as point F.

4. Queueing Model

In this section, we present a retrial queuing system that describes joint eMBB and URLLC transmission. We provide a recursive algorithm to calculate its stationary distribution in matrix form and give formulas for blocking or interruption probabilities and for the average number of sessions in the system and in the buffers.

4.1. Continuous-Time Markov Chain

We model the system described above as a retrial queuing system with two orbits (buffers), as illustrated in Figure 5. Let $N_{\mathrm{m}}\left(t\right)$ denote the number of active eMBB sessions, $N_{\mathrm{u}}\left(t\right)$ denote the number of active URLLC sessions, $Q_1\left(t\right)$ denote the number of delayed eMBB sessions (in 1-buffer, 1-orbit), and $Q_2\left(t\right)$ denote the number of interrupted eMBB sessions (in 2-buffer, 2-orbit) at time t. Thus, the behavior of the system is described by a multidimensional continuous-time Markov chain (CTMC)

$$\mathbf{X}\left(t\right)=\left(N_{\mathrm{m}}\left(t\right),N_{\mathrm{u}}\left(t\right),Q_1\left(t\right),Q_2\left(t\right)\right),t\ge 0$$

(1)

with the state space $\mathcal{X}$ defined as follows:

$$\begin{array}{c}\hfill \mathcal{X}=\{(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2):n_{\mathrm{m}}\ge 0,n_{\mathrm{u}}\ge 0,0\le q_1\le C_1,0\le q_2\le C_2,\\ \hfill b{n}_{\mathrm{m}}+n_{\mathrm{u}}\le C,n_{\mathrm{m}}+q_2\le N\}.\end{array}$$

(2)

4.2. Transition Rates between CTMC States

Based on the system behavior, we summarize below all possible cases of model functioning. The corresponding transition rates between CTMC states $(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)\in \mathcal{X}$ with conditions are listed in

Table 3. Transition rates.

	Transition Rate	Condition
1(a)	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}},n_{\mathrm{u}}+1,q_1,q_2)={\lambda }_{\mathrm{u}}$	$b{n}_{\mathrm{m}}+n_{\mathrm{u}}+1\le C$
1(b)	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}}-1,n_{\mathrm{u}}+1,q_1,q_2+1)={\lambda }_{\mathrm{u}}$	$b{n}_{\mathrm{m}}+n_{\mathrm{u}}+1>C,n_{\mathrm{m}}>0,q_2<C_2$
2(a)	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}}+1,n_{\mathrm{u}},q_1,q_2)={\lambda }_{\mathrm{m}}$	$b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1=0,q_2=0$
2(b)	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}},n_{\mathrm{u}},q_1+1,q_2)={\lambda }_{\mathrm{m}}$	$b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1<C_1,$
		$q_2\le C_2,$$(q_1>0\mathrm{or}{q}_2>0)\mathrm{or}$
		$\mathrm{or}b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}>C,q_1<C_1$
3	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}},n_{\mathrm{u}}-1,q_1,q_2)=n_{\mathrm{u}}{\mu }_{\mathrm{u}}$	$n_{\mathrm{u}}>0$
4	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}}-1,n_{\mathrm{u}},q_1,q_2)=n_{\mathrm{m}}{\mu }_{\mathrm{m}}$	$n_{\mathrm{m}}>0$
5(a)	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}}+1,n_{\mathrm{u}},q_1-1,q_2)={\gamma }_1$	$b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1>0,q_2=0$
6(a)	$a(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)(n_{\mathrm{m}}+1,n_{\mathrm{u}},q_1,q_2-1)={\gamma }_2$	$b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_2>0$

and presented by the transition rate diagram in Figure 6.

1.

When a new URLLC session arrives:

(a)

It will start if there is at least one mini-slot available.

(b)

It will start and one randomly selected active eMBB session will be interrupted and put into the 2-buffer if there are no free mini-slots and at least one eMBB session is active.

(c)

It will be blocked if there are no free mini-slots and no active eMBB sessions.

2.

When a new eMBB session arrives:

(a)

It will start if there are at least b mini-slots available and the 1- and 2-buffers are empty.

(b)

It will be delayed and put into the 1-buffer if the 1- and 2-buffers are not empty and there is at least one empty place in the 1-buffer.

(c)

It will be blocked in other cases.

3.

When an active URLLC session ends, it will release one mini-slot.

4.

When an active eMBB session ends, it will release b mini-slots.

5.

When a delayed eMBB session from the 1-buffer retries to start:

(a)

It will start if the 2-buffer is empty and there are at least b mini-slots available.

(b)

It will still wait in the 1-buffer if the 2-buffer is not empty or there are no b free mini-slots.

6.

When an interrupted eMBB session from the 2-buffer retries to resume:

(a)

It will resume if there are at least b mini-slots available.

(b)

It will still wait in the 2-buffer if there are no b free mini-slots.

For your information, Figure 7 presents the different views and components of the transition rate diagram. Each figure shows a fragment of the diagram in a general form under imposed conditions, as well as the possible transitions with specified intensities for a random state. The following are the details for each component:

A

New eMBB and URLLC session arrivals and endings for empty buffers (with fixed $q_1=0$ and $q_2=0$) when a new eMBB session starts. Accepting new eMBB sessions is possible since the buffers are empty, and the maximum number of eMBB sessions in the absence of URLLC sessions is N.

B

New eMBB and URLLC session arrivals and endings for non-empty buffers (with fixed $q_1\ne 0$ or $q_2\ne 0$) when a new eMBB session is delayed. New eMBB sessions are sent to the 1-buffer for delay since the buffers are not empty, so, in this case, only URLLC session arrivals are possible.

C

New eMBB sessions delaying into the 1-buffer (with fixed $n_{\mathrm{u}}$ and $n_{\mathrm{m}}$). The number of active sessions remains unchanged, and only the arrival of new eMBB sessions in the 1-buffer is considered, depending on the different states of the 2-buffer.

D

Delayed eMBB sessions retrials from the 1-buffer (with fixed $n_{\mathrm{u}}$ and $q_2$). New URLLC sessions do not arrive and cannot interrupt already active eMBB sessions, so if there are available resources, only delayed eMBB requests can be accepted.

E

Interrupted eMBB sessions retrials from the 2-buffer (with fixed $n_{\mathrm{u}}$ and $q_1$). New URLLC and eMBB sessions do not arrive, so if there are available resources, only interrupted eMBB sessions can be accepted.

F

New URLLC session arrivals due to eMBB session interruption (with fixed $q_1$). New eMBB sessions do not arrive, but new URLLC sessions interrupt an eMBB session and send it to the 2-buffer.

4.3. Infinitesimal Generator

Let us choose the following lexicographic order on the set space $\mathcal{X}$:

$$\begin{array}{c}(n_{\mathrm{m}}^{\prime },n_{\mathrm{u}}^{\prime },q_1^{\prime },q_2^{\prime })>(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{q}_1^{\prime }>q_1\mathrm{or}\hfill \\ \hfill \mathrm{or}\left(q_1^{\prime }=q_1,\left(q_2^{\prime }>q_2\mathrm{or}\left(q_2^{\prime }=q_2,\left(n_{\mathrm{m}}^{\prime }>n_{\mathrm{m}}\mathrm{or}(n_{\mathrm{m}}^{\prime }=n_{\mathrm{m}},n_{\mathrm{u}}^{\prime }>n_{\mathrm{u}})\right)\right)\right)\right).\end{array}$$

(3)

Then, the infinitesimal generator of $\mathbf{X}\left(t\right)$ will be a block tridiagonal matrix

$$\mathbf{A}=\left(\begin{array}{ccccc}{\mathbf{A}}_{00}& {\mathbf{A}}_{11}& & & \mathbf{0}\\ {\mathbf{A}}_{21}& {\mathbf{A}}_{01}& {\mathbf{A}}_{12}& & \\ & {\mathbf{A}}_{22}& \ddots & \ddots & \\ & & \ddots & \ddots & {\mathbf{A}}_{1C_1}\\ \mathbf{0}& & & {\mathbf{A}}_{2C_1}& {\mathbf{A}}_{0C_1}\end{array}\right)$$

(4)

with sub-matrices ${\mathbf{A}}_{0k}$, ${\mathbf{A}}_{1k}$, and ${\mathbf{A}}_{2k}$ having the following dimensions:

$$\begin{array}{cc}\hfill {\mathbf{A}}_{0k}:& \dim \left({\mathcal{X}}_k\right)\times \dim \left({\mathcal{X}}_k\right),k=0,\dots ,C_1,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathbf{A}}_{1k}:& \dim \left({\mathcal{X}}_{k-1}\right)\times \dim \left({\mathcal{X}}_k\right),k=1,\dots ,C_1,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathbf{A}}_{2k}:& \dim \left({\mathcal{X}}_k\right)\times \dim \left({\mathcal{X}}_{k-1}\right),k=1,\dots ,C_1,\hfill \end{array}$$

(7)

where

$$\begin{array}{cc}\hfill & {\mathcal{X}}_k=\left\{(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)\in \mathcal{X}:q_1=k\right\},k=0,\dots ,C_1,\hfill \\ \hfill & \dim \left({\mathcal{X}}_k\right)=\sum _{q_2=0}^{C_2}\sum _{n_{\mathrm{m}}=0}^{N-q_2}(C-b{n}_{\mathrm{m}}+1),\hfill \end{array}$$

(8)

and nonzero elements defined by formulas:

$$\begin{array}{c}{\mathbf{A}}_{0k}[(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2),(n_{\mathrm{m}}^{\prime },n_{\mathrm{u}}^{\prime },q_1^{\prime },q_2^{\prime })]\hfill \\ =\left\{\begin{array}{cc}{\lambda }_{\mathrm{u}},\hfill & n_{\mathrm{m}}^{\prime }=n_m,n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}}+1,q_1^{\prime }=q_1,q_2^{\prime }=q_2\hfill \\ \hfill & \mathrm{or}{n}_{\mathrm{m}}^{\prime }=n_{\mathrm{m}}-1,n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}}+1,q_2^{\prime }=q_2+1;\hfill \\ {\lambda }_{\mathrm{m}},\hfill & n_{\mathrm{m}}^{\prime }=n_{\mathrm{m}}+1,n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}},q_1^{\prime }=q_1=0,q_2^{\prime }=q_2=0,\hfill \\ n_{\mathrm{u}}{\mu }_{\mathrm{u}},\hfill & n_{\mathrm{m}}^{\prime }=n_{\mathrm{m}},n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}}-1,q_1^{\prime }=q_1,q_2^{\prime }=q_2,\hfill \\ n_{\mathrm{m}}{\mu }_{\mathrm{m}},\hfill & n_{\mathrm{m}}^{\prime }=n_{\mathrm{m}}-1,n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}},q_1^{\prime }=q_1,q_2^{\prime }=q_2,\hfill \\ {\gamma }_2,\hfill & n_{\mathrm{m}}^{\prime }=n_{\mathrm{m}}+1,n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}},q_1^{\prime }=q_1,q_2^{\prime }=q_2-1,\hfill \\ *,\hfill & n_{\mathrm{m}}^{\prime }=n_{\mathrm{m}},n_{\mathrm{u}}^{\prime }=n_{\mathrm{u}},q_1^{\prime }=q_1,q_2^{\prime }=q_2,\hfill \end{array}\right.\hfill \end{array}$$

(9)

where

$$\begin{array}{cc}\hfill *=-& ({\lambda }_{\mathrm{u}}·\mathbf{1}(b{n}_{\mathrm{m}}+n_{\mathrm{u}}+1\le C)\hfill \\ \hfill & +{\lambda }_{\mathrm{u}}·\mathbf{1}(b{n}_{\mathrm{m}}+n_{\mathrm{u}}+1>C,n_{\mathrm{m}}>0,q_2<C_2)\hfill \\ \hfill & +{\lambda }_{\mathrm{m}}·\mathbf{1}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1=0,q_2=0)\hfill \\ \hfill & +{\lambda }_{\mathrm{m}}·\mathbf{1}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1<C_1,q_2\le C_2,q_1>0)\hfill \\ \hfill & +{\lambda }_{\mathrm{m}}·\mathbf{1}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1<C_1,q_2\le C_2,q_2>0)\hfill \\ \hfill & +{\lambda }_{\mathrm{m}}·\mathbf{1}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}>C,q_1<C_1)\hfill \\ \hfill & +n_{\mathrm{u}}{\mu }_{\mathrm{u}}+n_{\mathrm{m}}{\mu }_{\mathrm{m}}\hfill \\ \hfill & +{\gamma }_1·\mathbf{1}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1>0,q_2=0)\hfill \\ \hfill & +{\gamma }_2·\mathbf{1}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_2>0)),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\mathbf{A}}_{1k}[(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2),& (n_{\mathrm{m}}^{\prime },n_{\mathrm{u}}^{\prime },q_1^{\prime },q_2^{\prime })]={\lambda }_{\mathrm{m}},\hfill \\ \hfill & \hspace{1em}\hspace{1em}\hspace{1em}{q}_1^{\prime }=q_1+1,q_1>0\mathrm{or}{q}_2>0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathbf{A}}_{2k}[(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2),& (n_{\mathrm{m}}^{\prime },n_{\mathrm{u}}^{\prime },q_1^{\prime },q_2^{\prime })]={\gamma }_1,\hfill \\ \hfill & n_{\mathrm{m}}=n_{\mathrm{m}}+1,q_1^{\prime }=q_1-1,b{n}_{\mathrm{m}}+n_{\mathrm{u}}<C-b,q_2=0,q_1>0.\hfill \end{array}$$

(12)

4.4. Stationary Distribution

Following the lexicographic order on the set space $\mathcal{X}$, the stationary probability distribution of $\mathbf{X}\left(t\right)$ can be written in the form:

$$\begin{array}{cc}\hfill \pi & =({\pi }_0,{\pi }_1,\dots ,{\pi }_k,\dots ,{\pi }_{C_1}),\hfill \\ \hfill {\pi }_k& =\left(\pi \right(0,0,k,0),\pi (0,1,k,0),\dots ,\pi (0,C,k,0),\pi (1,0,k,0),\dots \hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \dots ,\pi (N,C-bN,k,0),\pi (0,0,k,1),\dots ,\pi (0,C,k,C_2)),k=0,\dots ,C_1.\hfill \end{array}$$

(14)

This can be found by solving the following equations:

$$\begin{array}{cc}\hfill & \pi \mathbf{A}=\mathbf{0},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \pi {\mathbf{e}}^{\mathrm{T}}=1,\hfill \end{array}$$

(16)

where $\mathbf{e}$ represents a vector of suitable dimension with all values equal to 1.

Taking into account the block tridiagonal form of the infinitesimal generator $\mathbf{A}$, the components ${\pi }_k$ of the stationary distribution can be calculated as follows:

$${\pi }_k={\pi }_{C_1}\prod _{j=1}^{N-k}{\mathbf{B}}_{C_1-j},k=0,\dots ,C_1-1,$$

(17)

where matrices ${\mathbf{B}}_k$ follow the recursive formulas

$$\begin{array}{cc}\hfill {\mathbf{B}}_0& ={\mathbf{A}}_{11}·{\mathbf{A}}_{00}^{-1},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathbf{B}}_k& =-{\mathbf{A}}_{1,k+1}{\left({\mathbf{A}}_{0k}+{\mathbf{B}}_{k-1}·{\mathbf{A}}_{2k}\right)}^{-1},k=1,\dots ,C_1-1,\hfill \end{array}$$

(19)

and the probability ${\pi }_{C_1}$ is found by solving

$$\begin{array}{c}\hfill \sum _{k=0}^{C_1}\left({\pi }_{C_1}\prod _{j=1}^{C_1-k}{\mathbf{B}}_{C_1-j}\right){\mathbf{e}}^{\mathrm{T}}=1,\end{array}$$

(20)

$$\begin{array}{c}\hfill {\pi }_{C_1}\left({\mathbf{A}}_{0C_1}+{\mathbf{A}}_{2C_1}{\mathbf{B}}_{C_1-1}\right)=\mathbf{0}.\end{array}$$

(21)

4.5. Metrics of Interest

After calculating the stationary distribution $\pi$, we can compute several metrics of interest, including:

• The blocking probability of a new URLLC session:

  $$B_{\mathrm{u}}=\sum _{q_1=0}^{C_1}\sum _{q_2=0}^{C_2}\pi \left(0,C,q_1,q_2\right).$$

  (22)
• The probability that a new eMBB session will be delayed:

  $$D=\sum _{(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)\in {\mathcal{X}}_D}\pi (n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2),$$

  (23)

  where ${\mathcal{X}}_D$ represents the set of states where such a delay will occur.

  $$\begin{array}{cc}\hfill {\mathcal{X}}_D=\{& (n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2)\in \mathcal{X}:\hfill \\ \hfill & b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}\le C,q_1<C_1,q_2<C_2,\left(q_1\ne 0\mathrm{or}{q}_2\ne 0\right)\mathrm{or}\hfill \\ \hfill & \mathrm{or}(b(n_{\mathrm{m}}+1)+n_{\mathrm{u}}>C,q_1<C_1)\}.\hfill \end{array}$$

  (24)
• The interruption probability of an active eMBB session:

  $$\begin{array}{c}I=\sum _{n_{\mathrm{m}}=1}^N\sum _{q_1=0}^{C_1}\sum _{q_2=0}^{C_2}(\pi \left(n_{\mathrm{m}},C-b{n}_{\mathrm{m}},q_1,q_2\right)\hfill \\ \hfill \times \frac{{\lambda }_{\mathrm{u}}}{{\lambda }_{\mathrm{u}}+{\lambda }_{\mathrm{m}}+n_{\mathrm{u}}{\mu }_{\mathrm{u}}·\mathbf{1}\left({\mathcal{Y}}_1\right)+n_{\mathrm{m}}{\mu }_{\mathrm{m}}+{\gamma }_1·\mathbf{1}\left({\mathcal{Y}}_2\right)+{\gamma }_2·\mathbf{1}\left({\mathcal{Y}}_3\right)}·\frac{1}{n_{\mathrm{m}}}),\end{array}$$

  (25)

  where
  ${\mathcal{Y}}_1=\{n_{\mathrm{u}}>0\}$ indicates the presence of at least one active URLLC session;
  ${\mathcal{Y}}_2=\{b\left(n_{\mathrm{m}}+1\right)+n_{\mathrm{u}}\le C,q_1>0,q_2=0\}$ indicates the possibility of starting the delayed eMBB session;
  ${\mathcal{Y}}_3=\{b\left(n_{\mathrm{m}}+1\right)+n_{\mathrm{u}}\le C,q_2>0\}$ indicates the possibility of resuming the interrupted eMBB session.
• The blocking probability of a new eMBB session:

  $$B_{\mathrm{m}}=\sum _{n_{\mathrm{m}}=0}^N\sum _{n_{\mathrm{u}}=0}^C\sum _{q_2=0}^{C_2}\pi \left(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2\right).$$

  (26)
• The average number of active URLLC sessions:

  $${\overline{N}}_{\mathrm{u}}=\sum _{n_{\mathrm{u}}=1}^C{n}_{\mathrm{u}}\sum _{n_{\mathrm{m}}=0}^{⌊\frac{C-n_{\mathrm{u}}}{b}⌋}\sum _{q_1=0}^{C_1}\sum _{q_2=0}^{C_2}\pi \left(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2\right).$$

  (27)
• The average number of active eMBB sessions:

  $${\overline{N}}_{\mathrm{m}}=\sum _{n_{\mathrm{m}}=1}^N{n}_{\mathrm{m}}\sum _{n_{\mathrm{u}}=0}^{C-b{n}_{\mathrm{m}}}\sum _{q_1=0}^{C_1}\sum _{q_2=0}^{C_2}\pi \left(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2\right).$$

  (28)
• The average number of delayed eMBB sessions:

  $${\overline{Q}}_1=\sum _{q_1=1}^{C_1}{q}_1\sum _{n_{\mathrm{m}}=0}^N\sum _{n_{\mathrm{u}}=0}^{C-b{n}_{\mathrm{m}}}\sum _{q_2=0}^{C_2}\pi \left(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2\right).$$

  (29)
• The average number of interrupted eMBB sessions:

  $${\overline{Q}}_2=\sum _{q_2=1}^{C_2}{q}_2\sum _{n_{\mathrm{m}}=0}^N\sum _{n_{\mathrm{u}}=0}^{C-b{n}_{\mathrm{m}}}\sum _{q_1=0}^{C_1}\pi \left(n_{\mathrm{m}},n_{\mathrm{u}},q_1,q_2\right).$$

  (30)

5. Numerical Results

In this section, we provide a numerical example to evaluate the system and formulate its goals, while describing two scenarios. We assess the impact of URLLC transmission on eMBB QoS degradation using two key metrics: the blocking and interruption probabilities of eMBB sessions. These metrics are crucial in determining suitable arrival and retrial rate zones to ensure adequate QoS for eMBB sessions.

5.1. Considered Scenarios

The radio channel considered has a bandwidth of 1.4 MHz with $N=6$ PRBs. Each PRB consists of $b=7$ mini-slots, resulting in a total of $C=bN$ mini-slots. The sizes of the 1- and 2-buffers are both $C_1=C_2=9$. An eMBB session occupies either one slot or $b=7$ mini-slots, while a URLLC session occupies a single mini-slot. The eMBB and URLLC session durations are 1 min and 1 ms, respectively. URLLC sessions arrive k times more frequently than eMBB sessions, i.e., ${\lambda }_{\mathrm{u}}=k{\lambda }_{\mathrm{m}}$.

The numerical analysis has two main objectives. Firstly, it aims to evaluate the impact of eMBB and URLLC arrivals, as well as eMBB retrials, on eMBB transmission. Secondly, it aims to determine the appropriate arrival and retrial rates that ensure that the blocking probability of eMBB sessions falls below 4%, and the interruption probability of eMBB sessions remains below 0.04% ($4·{10}^{-4}$). We consider two cases. In the first scenario, the metrics were evaluated in relation to an increase in the arrival rate of URLLC sessions. In the second scenario, the metrics were evaluated in relation to an increase in the retrial rates of eMBB sessions. All input data are provided in

Table 4. Parameters for numerical example.

Parameter	Description	Value (Case 1)	Value (Case 2)
B	Bandwidth	1.4 MHz	1.4 MHz
N	Total number of slots (number of PRBs)	6	6
b	Number of mini-slots in one slot	7	7
$C=Nb$	Total number of mini-slots	42	42
$C_1$	Maximum number of delayed eMBB sessions (size of 1-buffer)	9	9
$C_2$	Maximum number of interrupted eMBB sessions (size of 2-buffer)	9	9
${\lambda }_{\mathrm{m}}$	Arrival rate of eMBB sessions	$0.06;0.08;0.1$ sess./s	$0.08$ sess./s
k	Multiplier for the arrival rate of URLLC sessions	$0\dots {10}^5$ sess./s	$5000;15,000;25,000$
${\lambda }_{\mathrm{u}}$	Arrival rate of URLLC sessions	$k·{\lambda }_{\mathrm{m}}$ sess./s	$k·{\lambda }_{\mathrm{m}}$ sess./s
${\mu }_{\mathrm{m}}^{-1}$	Average duration of an eMBB session	60 s	60 s
${\mu }_{\mathrm{u}}^{-1}$	Average duration of a URLLC session	1 ms	1 ms
${\gamma }_1$	Retrial rate of delayed eMBB sessions (from 1-buffer)	${10}^5$ sess./s	$0\dots {10}^5$ sess./s
${\gamma }_2$	Retrial rate of interrupted eMBB sessions (from 2-buffer)	${10}^5$ sess./s	$0\dots {10}^5$ sess./s

.

5.2. Impact of URLLC Arrivals

In the first scenario, we considered three cases with eMBB arrival rates ${\lambda }_{\mathrm{m}}=0.06$, $0.08$, and $0.1$ sess./s, and URLLC arrival rates ${\lambda }_{\mathrm{u}}=k{\lambda }_{\mathrm{m}}$ sess./s, where $k=0\dots {10}^5$. The retrial rates of eMBB sessions from the 1- and 2-buffers are fixed and equal to ${10}^5$ s${}^{-1}$. The eMBB retrial rates are also equal to ${10}^5$ s${}^{-1}$.

The results of calculating metrics of interest are presented in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. Since our primary focus is on the blocking and interrupting probabilities of eMBB sessions, let us concentrate on these metrics. Figure 8, Figure 9, Figure 10 and Figure 11 show three areas with different behaviors for various metrics. For example, for the interruption probability of an eMBB session (Figure 9), the first area (${\lambda }_{\mathrm{u}}$ from 0 to 10,000) shows an increasing trend, the second area (10,000 to 30,000) shows a decreasing trend, and the third area (>30,000) shows an increasing trend again. Let us compare this with the average number of active eMBB sessions (Figure 11).

• For $k=0$… 10,000, the number of active eMBB sessions decreases, leading to an increase in the interruption probability of an eMBB session. This decrease in the number of eMBB active sessions is due to their interruption.
• For k = 10,000 … 30,000, the number of active eMBB sessions slowly decreases, indicating fewer interruptions.
• For $k>$ 30,000, the two plots behave similarly to the first area.

5.3. Impact of eMBB Retrials

In the second scenario, we selected three points from the first scenario, namely k = 5000, 15,000, and 25,000, and estimated the dependence of the blocking and interruption probabilities of eMBB sessions on the eMBB retrials. We assumed the arrival rate of eMBB sessions to be ${\lambda }_{\mathrm{m}}=0.08$ and the retrial rates ${\gamma }_1={\gamma }_2=0\dots {10}^5$. Figure 15 and Figure 16 show that the plots stabilize around ${\gamma }_1={\gamma }_2=$ 10,000.

The increase in the retrial rate of eMBB sessions leads to a decrease in the blocking probability of eMBB sessions (as shown in Figure 15). This indicates that higher retrial rates result in more eMBB sessions being accepted. However, as the retrial rates increase, the interruption probability of eMBB sessions also increases (as demonstrated in Figure 16), meaning that more eMBB sessions will be interrupted.

5.4. Discussion

The main goal of numerical analysis is to find the optimal configuration of retrial rates that meets the QoS requirements for eMBB sessions. QoS is measured by blocking and interruption probabilities, subject to certain constraints. Let us consider some examples of these constraints and the corresponding values for the retrial rates:

1.

When only the blocking probability of eMBB sessions is constrained, such as being less than 0.04 (as shown in Figure 15), ${\gamma }_1$ and ${\gamma }_2$ can range from 250 to infinity. However, the arrival of URLLC sessions should not exceed the arrival of eMBB sessions by more than 5000 times (i.e., $k\le 5000$).

2.

When there is only a constraint on the interruption probability of eMBB sessions—for example, less than $2·{10}^{-4}$, as shown in Figure 16c—suitable values can be: (i) ${\gamma }_1$ and ${\gamma }_2$ from 0 to 150 with $k\ge 5000$; (ii) ${\gamma }_1$ and ${\gamma }_2$ from 0 to 300 with $k\ge$ 15,000; (iii) ${\gamma }_1$ and ${\gamma }_2$ from 0 to 750 with $k\ge$ 25,000.

3.

When constraints are imposed on both blocking and interruption probabilities, the following cases arise:

• If the blocking probability should be less than 0.03 and the interruption probability less than $3·{10}^{-4}$, there are no values that satisfy these constraints.
• If the blocking probability should be less than 0.04 and the interruption probability less than $4·{10}^{-4}$, ${\gamma }_1$ and ${\gamma }_2$ should range from 250 to 750 when $k\le 5000$.

5.5. Comparing Different Schemes

To analyze the proposed model, we considered various service schemes, varying the size of buffers in our system relative to eMBB traffic, including possible interruptions and delays. We examined three cases: (i) there are no spaces in the buffers ($C_1=C_2=0$), meaning that new eMBB sessions will be blocked due to a lack of free resources and interrupted eMBB sessions will be lost; (ii) there are no spaces in the 1-buffer ($C_1=0,C_2=9$), resulting in new eMBB sessions being blocked due to a lack of free resources; (iii) there are available spaces in both buffers ($C_1$ = $C_2=9$), which corresponds to our proposed model.

In order to compare these strategies, we examined the probability of eMBB session interruption and the average number of eMBB sessions, the results of which are presented in Figure 17 and Figure 18. When considering the second service scheme ($C_1=0,C_2=9$), wherein there are available spaces in the 2-buffer for accommodating the eMBB sessions that have been interrupted, the probability of interruption is found to be the lowest; however, the average number of eMBB sessions also proves to be the smallest among all schemes. A similar situation is observed for the first scheme ($C_1=C_2=0$). For the third scheme, i.e., our proposed model, even though the interruption rate is the highest, it enables the servicing of a greater number of eMBB sessions while maintaining an interruption rate below $1·{10}^{-3}$. From this, our proposed model is the most advantageous strategy among others considered.

6. Conclusions

Based on the need to provide joint URLLC and eMBB transmission in 5G networks, we present a system model that incorporates delays, interruptions, and the resumption of eMBB sessions due to URLLC prioritization. This model utilizes dynamic resource multiplexing and technology NOMA. To analyze the system, we develop a mathematical model in the form of a retrial queueing system with two retrial orbits (buffers) for interrupted and delayed eMBB sessions. Additionally, we present recurrent formulas for calculating the stationary probability distribution using the matrix-geometric approach, along with formulas for calculating metrics of interest, such as blocking and interruption probabilities.

Our numerical results show that, with appropriate retrial rate settings, this model is able to provide a guarantee of successful eMBB session beginning and ending, as well as a guarantee of an uninterrupted service of an eMBB session. Additionally, a comparison of various model schemes with respect to the capacity of the buffers is presented. These results show that the scheme with possible interruptions and delays of eMBB sessions is the most effective among the others presented. It practically doubles the average number of active eMBB devices while keeping the interruption probability below ${10}^{-3}$.

Future research will focus on representing the joint eMBB and URLLC transmission model as a resource queueing system, as well as developing a resource allocation policy based on machine learning.