On the Performance of Delay Aware Shared Access with Priorities

On the Performance of Delay Aware Shared Access with Priorities

Zheng Chen, Nikolaos Pappas, Marios Kountouris and Vangelis Angelakis Z. Chen is with the Laboratoire de Signaux et Systèmes (L2S, UMR8506), CentraleSupélec - CNRS - Université Paris-Sud, Gif-sur-Yvette, France.
Email: zheng.chen@centralesupelec.fr. N. Pappas and V. Angelakis are with Department of Science and Technology, Linköping University, Norrköping, Sweden.
Email: {nikolaos.pappas, vangelis.angelakis}@liu.se. M. Kountouris is with the Mathematical and Algorithmic Sciences Lab, France Research Center, Huawei Technologies Co. Ltd.
Email: marios.kountouris@huawei.com. This work was presented in part in [1] and [2].
Abstract

In this paper, we analyze a shared access network with a fixed primary node and randomly distributed secondary nodes whose distribution follows a Poisson point process (PPP). The secondaries use a random access protocol allowing them to access the channel with probabilities that depend on the queue size of the primary. Assuming a system with multipacket reception (MPR) receivers having bursty packet arrivals at the primary and saturation at the secondaries, our protocol can be tuned to alleviate congestion at the primary. We study the throughput of the secondary network and the primary average delay, as well as the impact of the secondary node access probability and transmit power. We formulate an optimization problem to maximize the throughput of the secondary network under delay constraints for the primary node, which in the case that no congestion control is performed has a closed form expression providing the optimal access probability. Our numerical results illustrate the impact of network operating parameters on the performance of the proposed priority-based shared access protocol.

Shared access, queueing analysis, throughput with delay constraints, stochastic geometry.

1 Introduction

Mobile device proliferation is creating tremendous pressure on the capacity of current wireless networks. Due to the scarcity of the radio spectrum, several flexible spectrum management approaches have emerged. Spectrum sharing, licensed-assisted access (LAA), licensed sharing access (LSA), and cognitive radio [3] are some novel paradigms providing efficient and flexible spectrum utilization. In a cognitive-inspired shared access network, the unlicensed (secondary) users opportunistically access the under-utilized spectrum of the licensed (primary) network and adjust their transmissions so as not to create harmful interference to the primary user. This network setting can also model underlay device-to-device (D2D) communication in cellular networks, which is seen as a key enabler for 5G mobile communication systems [4, 5, 6] and the Internet of Things [7].

The conventional access protocol for the secondary node is to vacate the spectrum when the primary node is active, in other words, the secondary node can only be active when the channel is idle in order to avoid collision with the primary transmission. However, due to the imperfect knowledge of the channel occupancy, collisions may be inevitable. Scheduling policies for the secondary user under partial channel state information are developed in [8, 9]. Random access protocols with multipacket reception (MPR) are proposed in [10], where secondary nodes make transmission attempts with a given probability. Compared to the traditional collision channel model, the MPR channel [11, 12, 13, 14] captures the interference at the physical layer in a more efficient way, because a transmission may succeed even in the presence of interference. Nevertheless, spectrum sharing between primary and secondary nodes in MPR channel unavoidably creates interference among concurrent transmissions [15, 16, 17]. Taking into account the interference caused by the secondary network and affecting the primary user, a judicious access protocol for the secondary node has to be carefully designed so that the quality-of-service (QoS) of the primary user is not degraded.

1.1 Related Work

In [1], we analyzed the throughput of the secondary network when MPR capability is enabled in a cognitive network with congestion control on the primary user. Using the collision channel scenario, throughput optimization with deadline constraints on a single secondary user accessing a multi-channel system is studied in [18]. The optimal stopping rule and power control strategy are provided in terms of closed-form expressions. In [19] the joint scheduling and power control is considered in order to minimize the sum average secondary delay subject to interference constraints at the primary user. However, prior work has not studied the random access protocol design which takes into account both the throughput of the secondary network and the delay of the primary one.

Most of the prior studies on cognitive radio and shared access networks either assume a single secondary node or multiple secondary nodes in a fixed network topology. To the best of our knowledge, the throughput and delay analysis of a large-scale shared access network with highly mobile secondary uses at random locations has not been reported in the literature. Using tools from stochastic geometry, the secondary node distribution can be modeled as a Poisson point process (PPP) [20, 21], which is a widely used spatial model for the node distribution in dense wireless networks. Existing results on the interference and outage distribution in PPP networks provide direct connection between the interference level and the node density, thus allowing us to characterize the spatially averaged throughput of the secondary network and interference as a function of the secondary node density. Therefore, the primary average delay can be well confined by adjusting the access probability of the secondary nodes in the random access protocol.

1.2 Contribution

This work extends and enhances our early works in [1, 2] in the following aspects.

  • We propose a delay-aware shared access network with congestion control in the primary network. A large-scale secondary network is considered in which the nodes are distributed according to a stochastic point process.

  • We derive the average queue size and delay of the primary user as function of the secondary node access probability and transmit power.

  • We introduce an optimization problem to maximize the throughput of the secondary network subject to the delay constraints on the primary user. We analyze the impact of different network parameters on the throughput and delay behavior of our studied network.

  • For the particular case with no congestion control, we provide closed-form expressions for the optimal access probability of the secondary nodes. The analytical results are shown through simulations to be very accurate, allowing us to optimize the performance of a large-scale shared access network with simple control schemes. For the case with congestion control, we evaluate with numerical methods the optimal solution for our shared access protocol design.

2 System Model

2.1 Network Topology

We consider a shared access network, in which one primary source-destination pair and many secondary communication pairs share the same spectrum, as shown in Fig.1. The network region we study is a circular disk with radius . The primary receiver is centered at the origin of . The primary transmitter is located at fixed location with distance to the primary receiver, which is common in infrastructure-based communication. We assume that the secondary transmitters are distributed in the two-dimensional Euclidean plane according to a homogeneous Poisson point process (PPP) with intensity , where denotes the location of the -th secondary transmitter. Their associated receivers are distributed at isotropic directions with fixed distance from their transmitters. For each realization of the PPP, the number of secondary transmitters in our network region is a Poisson random variable with mean value . The time is slotted and each packet transmission occupies one time slot. We assume that the receivers have multipacket reception (MPR) capabilities and that the secondary nodes can transmit simultaneously with the primary node [22].

The primary source has an infinite capacity queue for storing arriving packets of fixed length. The arrival process at the primary transmitter is modeled as a Bernoulli process with average rate packets per slot. The secondary node queue is assumed to be saturated, i.e., it always has a packet waiting to be transmitted.

Fig. 1: Shared access network topology: one primary receiver centered at the origin with PPP distributed secondary transmitters under a given density (here ).

2.2 Priority Based Protocol Model

We consider the following priority-based protocol, which is an extension of that proposed in [1]. The primary node transmits a packet whenever backlogged, while the secondary nodes access the channel with a probability that depends on the queue size of the primary node, such that will not deteriorate the performance of the primary user. Denote the queue size in the primary node, the activity of the primary and secondary transmitters in a time slot are controlled in the following cases:

  • Case 1: When , the primary transmitter does not have packet to transmit, thus remains silent. Secondary transmitters randomly access the channel with probability .

  • Case 2: When , the primary transmitter transmits one packet. Secondary transmitters randomly access the channel with probability .

  • Case 3: When , the primary transmitter transmits one packet. Secondary transmitters remain silent.

For brevity we use PT and PR to denote the primary transmitter and receiver respectively, and ST and SR for denoting the secondary transmitter and receiver.

The threshold plays the role of a congestion limit for the primary node, meaning that when the queue reaches this size, then the STs do not attempt to transmit any packet. When , the protocol model is simplified to the case without congestion control.

Note that we use two random access probabilities for the secondary nodes because the SRs experience different interference levels depending on whether the PT is active or not. Thus, the optimal access probabilities in these two cases need to be investigated separately.

3 Physical Model Successful Transmission Analysis

The MPR physical model is a generalized form of the packet erasure model. At the receiver side, a packet can be decoded correctly by the receiver if the received signal-to-interference-plus-noise ratio (SINR) exceeds a prescribed threshold . Given a set of nodes transmitting during the same time slot, the received SINR at the -th receiving node is given by

where denotes the power of the transmitting node ; denotes the small-scale channel fading from the transmitter to the receiver , which follows (Rayleigh fading); denotes the distance between the transmitter to the receiver . Here we assume a standard distance-dependent power law pathloss attenuation , where denotes the pathloss exponent. denotes the background noise power.

Let and be the transmit powers of the PT and the STs, respectively. In the following we refer to the primary node by node , while the secondary nodes are labeled with index . Denote the location of the PT and recall that the distribution of the STs is given by , then we have . Note that in this work when we refer to the set of locations of the transmitting nodes, it means the set of transmitting nodes at these locations.

Following the description of our access protocol presented in Section 2.2, to derive the success probability of the primary and secondary nodes we need to consider three cases.

3.1 Case 1

When , the PT is silent and the STs attempt packet transmission with probability . Denote the locations of active STs, as a result of independent thinning [21], follows a homogeneous PPP with intensity . Hence, we have the active transmitter set as .

Without loss of generality, we consider an arbitrary (typical) active secondary pair in our network region. Denote the success probability of the typical secondary pair when only the STs from are active, we have

(1)

Here, comes from and the probability generating functional (PGFL) of the PPP [23]. For a specific realization of the PPP, represents the percentage of active secondary pairs having successful transmission. It can also be seen as the probability of the typical active secondary pair to have successful transmission, averaging over different realizations of the PPP.

3.2 Case 2

When , both the PT and part of the STs are active. Similarly, with independent thinning probability , the locations of active STs follow another homogeneous PPP, denoted by , with intensity . In that case, the active transmitter set contains both the PT and the active STs, i.e., .

Denote and the success probabilities of the primary and secondary pairs when both types of nodes are active. With the help of existing results on the interference and outage distribution in PPP networks [21], we have the success probability of the primary transmission when the secondary network is active, given as

(2)

For the active secondary nodes, considering an arbitrary (typical) active secondary pair , we obtain the success probability in the following proposition.

Proposition 1.

The success probability of the typical secondary pair, when the active transmitters are , is given by

(3)

where .

Proof:

See Appendix A. ∎

3.3 Case 3

When , only the PT is active. Denote the success probability of the primary pair when all the STs are silent, we have

(4)

Note that and always hold.

4 Network Performance Metrics

In this section, we define several relevant metrics for the performance evaluation of the proposed priority-based protocol with congestion control.

4.1 Throughput of the Secondary Network

For the considered shared access network, we aim at evaluating the throughput of the secondary network, abbreviated as secondary throughput, which is the number of packets per slot that can be successfully transmitted by the active secondary nodes to their destinations. In order to be consistent with the PPP model where the secondary nodes are generated with a certain density , we define the secondary throughput as the throughput of the secondary network per unit area, given as

(5)

Recall that the active STs is with density when the primary queue is empty, i.e., . When the primary queue is , then the active STs have density . Hence, we have

(6)

where and are given in (1) and (3), respectively.

4.2 Primary Service Rate

The service rate of the primary given a certain SINR target can be defined as the percentage of successfully transmitted packets per time slot. Dividing the cases by the primary queue size greater or less than , when , we have the primary service rate given by

(7)

When , the service rate is

(8)

Combining the two cases, we have the average service rate of the primary, denoted by , given by

(9)

4.3 Primary Average Delay

The delay per packet at the primary node consists of the queueing delay and the transmission delay from the PT to the PR. From Little’s law, we obtain the queueing delay which is related to the average queue size per packet arrival. The transmission delay is inversely proportional to the average service rate [24].

Denote the primary average delay per packet, we have

(10)

where and are the average queue size and the average service rate of the primary, which will be analyzed with closed-form expressions in Section 5.

5 Analysis of the Primary Queue and Delay

From the definition of the metrics in Section 4, we see that the secondary throughput and the primary delay depends on the state of the primary queue size. Therefore, we need to derive first and .

We model the primary queue as a discrete time Markov Chain (DTMC), which describes the queue evolution and is presented in Fig. 2. Each state is denoted by an integer and represents the queue size. The packet arrival rate is always . The service rate is when , and is when . From our analysis in Section 3, we know that . All the metrics related to the rate are measured by the average number of packets per time slot.

Fig. 2: The Discrete Time Markov Chain which models the queue evolution at the primary node.

Denote the stationary distribution of the DTMC, where is the probability that the queue has packets in its steady state. We have the following lemma.

Lemma 1.

The stationary distribution of the DTMC described in Fig. 2 is given in the following cases:

  • For , we have

    (11)
  • For , we have

    (12)

where is the probability that the queue is empty, given by

(13)

The queue is stable if and only if .

Proof:

See Appendix B. ∎

Remark 1.

If without congestion control, the service rate is always . Obviously the condition to have stable queue is . The congestion control threshold increases the queue stability region to , implying that the maximum allowed arrival rate at the PT becomes higher. On the other hand, less opportunity will be given to the secondary nodes to be active, because no secondary transmission is allowed when the primary queue size exceeds .

In order to simplify the equations, we define . In the remainder of this work we will assume that , however the general expressions of our results hold also for , but one should replace the with the corresponding expression in this case.

Based on the results in Lemma 1, we have the following probabilities related to the primary queue size.

Lemma 2.

When the primary queue is stable and , the probability to have is

(14)

The probability to have is

(15)
Proof:

See Appendix C. ∎

We give the average queue size and the average delay of the primary in the following theorem.

Theorem 1.

The average queue size of the primary is given by

(16)

where

(17)

and

(18)

The primary average delay is given by

(19)
Proof:

See Appendix D. ∎

Remark 2.

For a certain packet arrival rate at the PT, is independent of . The primary queue size augments with because of the lower service rate , which leads to higher queueing delay. The transmission delay also increases with . As a result, is an increasing function of . Similarly, we know that also increases with .

6 Secondary Throughput Optimization with Primary Delay Constraints

In our considered shared access network, spectrum sharing between the primary and secondary users can be exploited in order to bring secondary throughput gains at the expense of increasing interference to the PR. In order to protect the QoS of the primary user, the secondary interference must be kept below a certain level, which corresponds to the thresholds on the ST access probability and transmit power .

In this section, we analyze the secondary throughput as a function of and with respect to the primary delay constraints.

6.1 General Case

From the definition of the secondary throughput in (6), with the help of the results in Lemma 1 and Lemma 2, we have

(20)

Considering the secondary throughput as a function of the access probability , it is obvious that there exists an optimal value , which is equivalent to , where is given in (1). From [25, 26] we have that the optimal access probability of the STs when the PT is silent is given by

(21)

which depends only on the ST density , secondary link distance and the pathloss exponent . Setting in (20), when the PT transmit power and the packet arrival rate are fixed, the secondary throughput depends only on the access probability and the transmit power .

As mentioned in Section 5, the primary average delay is an increasing function of and . When , i.e., the primary queue is stable, the delay constraints of the primary user can be translated to the feasible region of the two variables , defined as

(22)

where is the threshold of the primary average delay .

In order to achieve the maximum secondary throughput while keeping the primary average delay below the threshold, we formulate the following optimization problem:

(23)

subject to

where is the maximum available power for a ST.

Due to the complexity of the analytical results related to the primary queue, it is difficult to solve the above optimization problem in closed form. Hence, first we investigate the particular case without congestion control, i.e., . The solution to the optimization problem in the general case is evaluated numerically in Section 7.

6.2 Case with no Congestion Control ()

Without congestion control, the activity of the primary and secondary nodes is simplified into two cases:

  • When , the PT remains silent. STs randomly access the channel with probability .

  • When , the PT transmits one packet. STs randomly access the channel with probability .

Following the primary queue analysis in Lemma 2, we have the probability to have packets in the primary queue when it is in the steady state, given as

(24)

where

(25)

The primary queue is stable if and only if . Thus the feasible region of is defined by

(26)

The secondary throughput becomes

(27)

It is straightforward that the optimal value of is the same as in the case with congestion control, given in (21). Inserting in (27) and denoting the optimal per-node secondary throughput when , the secondary throughput can be written as a function of as follows

(28)

Our objective is to find the optimal access probability that maximizes the secondary throughput for fixed under the primary delay constraints. For that, the optimization problem is redefined as follows.

(29)

subject to

The following lemma provides the global optimal value of without considering the primary delay constraints.

Lemma 3.

When is verified, the global optimal value of that maximizes the secondary throughput in (28) is given by

(30)

where denotes the Lambert W function, . , , and are constant parameters related to the network setting.

Proof:

See Appendix E. ∎

Remark 3.

The value of ST power has a significant impact to the global optimal value of . When is very high, the primary transmission can be severely harmed by excess interference. We assume here the practically relevant constraint that satisfies . This choice not only simplifies our analysis on the throughput optimization, but also reflects the evolution of wireless networks in deployments where D2D/M2M communication with very low power nodes could coexist with the traditional high-rate mobile users [27].

The average queue size of the primary in this case is

(31)

The primary average delay is thus given by

(32)

Then we obtain the feasible region of in the following lemma.

Lemma 4.

With respect to the maximum average delay of the primary and the queue stability condition, the feasible region of is given by

(33)

where , is defined in Lemma 3.

Proof:

See Appendix F. ∎

Theorem 2 provides the optimal which maximizes within the feasible region , as the solution to the optimization problem defined in (29).

Theorem 2.

The optimal access probability that maximizes the secondary throughput under primary delay constraints is given by

(34)

where is given in (30), is defined in Lemma 4.

Proof:

See Appendix G. ∎

7 Numerical Results

Parameters Values
ST density ()
Secondary link distance () m
Primary link distance () m
Cell size () m
Pathloss exponent ()
PT power () mW
Maximum ST power () mW
Noise power () dBm
SINR target () dB
Average delay threshold () time slots/packet
TABLE I: System Parameters

In this section we evaluate the secondary throughput as a function of the two variables within their feasible region that satisfies the delay constraints of the primary user. The primary delay and the feasible region boundary are also presented, showing the impact of the priority-based protocol design on the network performance. The values of the parameters are given in Table I.

Fig. 3: Success probabilities of the primary and secondary transmissions vs. ST access probability. is constant, is a function of , and are functions of . The ST power is set to mW.

In Fig. 3, we plot the success probabilities , , and as a function of the ST access probability or for ST power set to mW. The numerical values are obtained from (4), (2), (1) and (3), respectively. Recall that is a constant value, and depend only on , and depends only on . As expected, when the secondary network is active, the success probabilities decrease rapidly with and increasing, as a result of the increased interference level.

7.1 General Case

(a) ,
(b) ,
Fig. 4: Secondary throughput vs. under primary delay constraints. .
(a) ,
(b) ,
Fig. 5: Secondary throughput vs. under primary delay constraints. .
Fig. 6: The boundary of the feasible region of with and . Below each curve is the feasible region with respective values of and .
TABLE II: Optimal SU Setting with Throughput Maximization

In Fig. 4 and Fig. 5, we plot the secondary throughput under the primary delay constraints. The values of are obtained from (20) within the feasible region of defined in (22). The results are presented with congestion threshold and the packet arrival rate . Knowing that , we choose in order to satisfy the queue stability condition.

Our first remark is that the secondary throughput is not a monotonic function of and . There exists an optimal point that gives the maximum among the feasible choices of . We also observe a ceiling effect, i.e. once reaches a certain level, e.g., in Fig. 4, has very small variation with respect to variations of . This result implies that in order to have throughput gains, the necessary power for the secondary transmission should actually be quite low. Thus, the condition we used in Lemma 3 is validated.

Comparing the subfigures we observe that larger provides higher potential improvement for the secondary throughput, as the secondary links are more likely to be active. In order to validate our conclusion, in Table. II we give the numerical values of the optimal solution as well as the maximum SU throughput achieved with different and . We can see that for the same , larger increases the maximum achievable secondary throughput, and also the optimal values of and are higher.

Furthermore, in Fig. 6 we draw the boundary of the feasible region for the four cases with and respectively. The possible values of that satisfy the primary delay constraints are situated below each plot. We observe that, larger leads to more restricted feasible region, because in this case the congestion control is weaker, thus causes higher primary delay. Interestingly, we remark that the feasible region with and is larger than that with and . This means that for the same values of , the primary average delay obtained with is actually smaller than the case with . This is mainly due to the benefits of the congestion control in protecting the primary node transmission when the queue size is large. With high packet arrival rate, i.e., , the probability of having is very high, thus the STs will remain silent with high probability. In that case both the queueing delay and the transmission delay of the primary user will be reduced.

Fig. 7: Primary average delay vs. ST access probability . mW. .

In order to further understand the influence of and on the primary delay, in Fig. 7 we plot the primary average delay as a function of the ST access probability for different values of and . The ST power is set to mW. Note that all the results are obtained with in order to satisfy the queue stability condition. We have the following observations:

  1. With increasing, which corresponds to the case of the PT service rate decreasing, the primary delay increases rapidly at first, then saturating. The higher the arrival rate is, the lower saturated delay it gives.

  2. When is relatively small, which means relatively high service rate , the primary delay is higher in the case with higher arrival rate . However, when is relatively high, depending on the value of , this trend can be contrasting, e.g., in the case with , when , the primary average delay is lower than in the case with higher .

  3. Larger results in higher primary average delay, thus requires higher service rate (smaller ) in order to satisfy the delay constraints.

The main takeaway messages we have from these results are:

  1. With larger , the maximum secondary throughput is higher. However, larger put tighter constraints on the feasible values of .

  2. With higher arrival rate at the PT, both the ST access probability and transmit power should be set to be lower. By doing so, the primary user achieves higher service rate, thus the queue size decreases faster, which in turn gives higher chance to the STs to transmit during the next time slot.

  3. When the primary user is very sensitive to the delay, smaller is more beneficial in order to increase the primary transmission rate.

7.2 Case with no Congestion Control ()

(a)
(b)
Fig. 8: Secondary throughput vs. under primary delay constraints, in the simplified case without congestion control.
Fig. 9: The boundary of the feasible region of with . The real values are obtained with exhaustive search.
Fig. 10: Optimal access probability vs. . The real values are obtained with exhaustive search.

In Fig. 8, we plot the secondary throughput under the primary delay constraints in the case without congestion control. The results are presented for . The evolution of the secondary throughput follows the same trend as observed in the general case presented in Fig. 4 and Fig. 5.

Fig. 9 shows the theoretical boundary of the feasible region derived in Lemma 4 in comparison to the real boundary obtained by exhaustive search of the feasible values of under primary delay constraints. The results confirm the accuracy of our theoretical analysis on the feasible region of .

Fig. 10 shows the optimal access probability obtained with Theorem 2 in comparison to the the real optimal values obtained by exhaustive search of that maximizes the secondary throughput with respect to the primary delay constraints. This illustrate the accuracy of our analytical results in Theorem 2. Another observation is that with , has values close to . When is higher, declines rapidly with after reaches a certain value. This result is expected because above a certain value of , the optimal is equal to the maximum feasible value of which is at the boundary of .

8 Conclusions

This paper investigated a delay-constrained shared access network following a stochastic geometry approach. We proposed a priority-based protocol with congestion control and studied the throughput of the secondary network and the primary average delay, as well as the impact of the protocol design parameters on the throughput and delay performance. For the case without congestion control, we derived in closed-form the optimal access probability of the secondary node in terms of maximizing the throughput of the secondary network under primary delay constraints. The main contribution of this work was to analyze the performance of a shared access network with priorities using tools from stochastic geometry, as well as to extend prior work on throughput optimization in shared access networks to the case with primary delay constraints.

Appendix A Proof of Proposition 1

According to the definition of the success probability, for the typical active secondary pair , we have

(35)

Here, follows from . follows from , and the expectation is over . is the Laplace transform of interference coming from active STs with normalized transmit power.

With the help of the approximation in [26], the second term in (35) becomes

(36)

Depending on the distance from the PT to the active SRs, different SRs experience different interference levels caused by the primary transmission. The expectation of is over all the possible locations of the typical SR inside the network region .

The distribution of the active SRs depends on the locations of their associated STs, which follows a homogeneous PPP with intensity . For an arbitrary active SR, it can be approximately seen as uniformly distributed on the disk with radius . Hence, the pdf of the distance from the typical SR to the origin of , denoted by , is given by

(37)
Fig. 11: Geographical locations of the PT and the typical SR on the network region with radius .

The distance from the PT to the origin is . As shown in Fig. 11, using the law of cosine, we have the distance between the PT and the typical SR given by

(38)

where is a random variable uniformly distributed in . Averaging over and , we have the expectation of the distance given by

(39)

The third term in (35) is the Laplace transform of interference coming from nodes in with intensity . From existing results on the interference distribution in Poisson networks [23], we have

(40)

Substituting (36) and (40) in (35), together with (39), we have