Analysis of the local delay with slotted-ALOHA based cognitive radio ad hoc networks
- Jing Gao^{1}Email author and
- Changchuan Yin^{2}
https://doi.org/10.1186/s13638-015-0267-8
© Gao and Yin; licensee Springer. 2015
Received: 22 October 2014
Accepted: 6 January 2015
Published: 14 February 2015
Abstract
We analyze the local delay of cognitive radio ad hoc networks in which secondary nodes are overlaid with a pair of primary nodes. Supposing slotted ALOHA multiple access is adopted by secondary nodes, we derive the closed-form expression of the local delay by modeling the channel occupied by primary nodes as continuous-time Markov on-off process. Furthermore, we obtain the asymptotic local delay for two special cases: large and small primary traffic. We theoretically prove that the local delay increases with the increasing primary packet arrival rate and decreases with the increasing primary packet departure rate. Numerical and simulation results show that the local delay could be approximated to be the result obtained with significant primary traffic in most cases, which is the steady of the channel idle state.
Keywords
1 Introduction
Delay is one of the important indicators to measure the quality of service (QoS) of wireless network. Throughput, reliability, and delay comprehensively measure the ability of the network to transfer information. The local delay is defined as the mean time (number of time slots) needed for a packet being received successfully from a transmitter to its nearest receiver. This also provides the base for researching end-to-end delay. In [1], Baccelli et al. first proposed the local delay in mobile ad hoc networks with ALOHA medium access control (MAC) protocol. Based on this framework, Martin [2] derived the local delay in both static and high-mobility networks, in which all nodes are assumed to be distributed as a Poisson point process (PPP) for each time slot, and he also proved that the local delay is always finite in highly mobile networks. Furthermore, Martin [3] obtained the closed-form expression of the local delay for four types of transmission strategies. In addition, the local delay [4] is regarded as the metric of an opportunistic routing protocol for multi-hop context in mobile ad hoc network, and then the opportunistic routing protocol is certified to be valuable through simulation.
All the above-mentioned studies are focused on homogeneous network models but not involved in heterogeneous networks. A practical network usually consists of interdependent, interactive, and hierarchical network components which lead to a heterogeneous network structure. Cognitive radio (CR) network is one type of heterogeneous networks which can efficiently solve the problem of spectrum shortage. Recently, many research results have been developed for the performance of CR network, for instance, 1) the scaling law of throughput and delay for the density of nodes in overlaid networks [5] and 2) the transmission capacity of spectrum sharing networks by employing stochastic geometry [6]. However, there is little research of the local delay in CR networks despite its importance for the analysis of end-to-end delay.
In this paper, we analyze the local delay of CR networks in which secondary nodes are overlaid with primary nodes and adopt slotted ALOHA MAC protocol to access the channel. The closed-form expression of the local delay is derived with furthest receiver routing protocol. The relationship is analyzed finally between the local delay and some important parameters of CR networks (such as the packet arrival (departure) rate of the primary network, the transmission probability and node density of the secondary network).
The paper is organized as follows: the ‘System model’ section gives the system model and the definition of some symbols. ‘The analysis of the local delay’ section analyzes the local delay of the secondary network. The ‘Numerical and simulation results’ section presents the numerical and simulation results with some observations of them. Finally, the conclusions are given in the ‘Conclusions’ section.
2 System model
Considering a CR network, a secondary network is overlaid with a primary network, i.e., the secondary nodes could be allowed to occupy the channel when it is not used by primary nodes. In order to analyze conveniently, we suppose a single wireless channel shared by both primary and secondary networks. And the results could be further extended to the scenario of multiple channels. In this section, the system model will be introduced first, and then the local delay of secondary nodes will be defined.
2.1 Primary network model
2.2 Secondary network model
2.2.1 The topology
Assume the secondary network is an ad hoc network and the slotted ALOHA MAC protocol is adopted (T is the time slot length). In each time slot, secondary nodes are modeled as a marked PPP \({\hat \Phi _{S}} = \left \{ {\left ({{x_{i}},{t_{{x_{i}}}}} \right)} \right \} \subset {\mathbb {R}^{2}} \times \left ({0,1} \right)\) [8], where Φ _{ S }={x _{ i }} is a homogeneous PPP with density λ _{ S }. Mark \({t_{{x_{i}}}}\) as independent Bernoulli distributions with parameter P(t=1)=p=1−q.x _{ i } is supposed to be a transmitting node when \({t_{{x_{i}}}} = 1\) and a receiving node when \({t_{{x_{i}}}} = 0\). According to the displacement theorem, all transmitting nodes in a time slot follow a PPP \({\Phi _{S}^{t}}\) with density λ _{ S } p and all receiving nodes in a time slot follow another PPP \({\Phi _{S}^{r}}\) with density λ _{ S }(1−p), correspondingly.
2.2.2 The receiving model
Secondary nodes could opportunistically access the channel only when the channel state is idle, which is determined by primary network. That is to say, secondary nodes are assumed to be able to acknowledge the primary activity at the beginning of each time slot.
where r is the link distance from node u to node v.
2.2.3 The selection strategy of receiving nodes
Based on (5), whether a typical node could successfully transmit its packets mainly depends on the signal-to-interference ratio of the corresponding receiving node. As in [9], the interference of all receiving nodes in each time slot could be characterized as the model of shot noise. On the other hand, the signal power catched by a receiving node is determined by both path-loss function and small-scale fading. So, the distance of the typical link, i.e., how to select a receiving node becomes the key factor impacting the receiving signal power. In the following, the distribution of the link distance will be analyzed.
2.3 The local delay
where δ _{0}(n) indicates that (4) holds in time slot n.
3 The analysis of the local delay
3.1 The local delay of CR network
Combining the definition of the local delay in (8) with the CR network model in this paper, the local delay of CR network is given in the following.
where \({C_{\alpha }} = \frac {{2{\pi ^{2}}{\beta ^{2/\alpha }}}} {{\alpha \sin (2\pi /\alpha)}}.\)
Comparing (11) with (9), the average probability of successful transmission E(P _{1}(n)) at time slot n is a function related to n because of the existence of primary network (term P(∂ _{ n }=1)). D _{1} is derived by the law of total probability which is related to time slot n. Thus, we resort to the computer to show the relationship between D _{1} and network parameters.
3.2 The local delay for two special cases
In some cases, the local delay can be approximated to be a value unrelated to the time slot. In the following, we will further analyze the local delay under two specifical cases of primary traffic.
3.3 The optimization of CR local delay
According to the definition, the local delay is determined by both primary traffic and successful transmission probability of the secondary network. In the following, we will give the optimization of the local delay about the three network parameters.
Theorem 1.
The local delay D _{1} increases with increasing primary arrival rate λ and decreasing departure rate μ when the other parameters are fixed.
Proof.
Theorem 2.
where \(\phantom {\dot {i}\!}\lambda _{1} = {\lambda _{S}}\pi {R^{2}}, {\lambda _{2}} = {\lambda _{S}}{C_{\alpha } }{R^{2}}, {e_{1}} = {e^{- {\lambda _{S}}q\pi {R^{2}}}}, \text {and } {e_{2}} = {e^{- {\lambda _{S}}p{C_{\alpha } }{R^{2}}}}\).
Proof.
Since secondary nodes are overlaid with primary nodes, to prove the local delay is concave with respect to the transmission probability of secondary nodes p, all we have to do is to certify the second-order derivative of the average probability of successful transmission \({\bar p_{S}}\) in (13) which is less than zero, i.e., \({{{d^{2}}{{\bar p}_{S}}} \mathord {\left /{\vphantom {{{d^{2}}{{\bar p}_{S}}} {d{p^{2}} < 0}}}\right. \kern -\nulldelimiterspace } {d{p^{2}} < 0}}\). After calculating and classifying, we get \({{{d^{2}}{{\bar p}_{S}}} \mathord {\left / {\vphantom {{{d^{2}}{{\bar p}_{S}}} {d{p^{2}} < 0}}} \right. \kern -\nulldelimiterspace } {d{p^{2}} < 0}}\) when λ _{ S } meets the requirements of (17) if \(0 < p < \frac {\pi } {{\pi + {C_{\alpha } }}}\). Derivation is easy and omitted.
By solving the equation \({{d{{\bar p}_{S}}} \mathord {\left /{\vphantom {{d{{\bar p}_{S}}} {dp = 0}}} \right. \kern -\nulldelimiterspace } {dp = 0}}\), we derive the optimal probability of successful transmission to minimize the local delay as shown in (18).
The proof is completed.
Note that the probability p is always less than 1, and thus, the optimization of local delay makes sense.
Theorem 3.
Proof.
Since the method of proving is the same as that of Theorem 2, it is omitted.
4 Numerical and simulation results
Parameter settings
Parameters | Setting |
---|---|
Path loss factor α | 4 |
Transmission radius R | 20 m |
Time slot T | 125 μ s |
Decoding threshold β | 10 dB |
The density of secondary network λ _{ S } | 0.005 nodes/ m^{2} |
Transmission probability p | 0.02 |
4.1 Simulation scenario settings
The primary network is modeled as the Markov process described in the ‘System model’ section. The arrival rate λ and departure rate μ are given in the figures. The original state of primary network is idle. The secondary nodes are uniformly distributed on a finite plane with area [0, 2,000]m×[0, 2,000]m. In each time slot, the number of transmitting and receiving nodes follow a Poisson distribution with parameter λ _{ S } p nodes/ m^{2} and λ _{ S }(1−p) nodes/ m^{2}, respectively. Put a typical node on the center of the plane, the axis of which is [1,000, 1,000]. Consider the typical node begin to transmit packets with probability p once the channel state is detected to be idle in each slot. Regarding the number of time slots needed for a packet to be sent successfully as an example, the final simulation results are the mean value of 10,000 examples.
5 Conclusions
We conducted an analytical study of the local delay in cognitive radio ad hoc networks. We modeled the occupancy of the licensed channel by the primary network as a Markov process, and the secondary nodes opportunistically accessed the channel with the ALOHA protocol. The local delay is analyzed by employing the property of PPP in stochastic geometry.
We derived the analytical expression of the local delay and discussed the relationship between the local delay and some important network parameters which conclude the following: primary traffic (arrival and departure rates), transmission probability, and node density of the secondary network. Both numerical and simulation results are obtained for different primary traffic. We drew a conclusion that the local delay in most cases could be approximated to that for significant primary traffic which is important for further research of end-to-end delay.
Declarations
Acknowledgements
This work was supported in part by the National Research Foundation for the Doctoral Program of Higher Education of China under Grant 20120005110007, the NSFC under Grant 61379159, and Beijing Natural Science Foundation under Grant 4122034.
Authors’ Affiliations
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