Throughput and delay analysis of network coded ALOHA in wireless networks
© Lee et al.; licensee Springer. 2012
Received: 9 December 2011
Accepted: 9 August 2012
Published: 30 August 2012
The purpose of this article is to analyze the impact of network coding in wireless networks. We consider a network coded ALOHA that performs bi-directional network coding over the ALOHA MAC protocol in a star topology network. The transmission probabilities of each outer node and the center node, and the target signal-to-interference-plus-noise ratio (SINR) are jointly optimized to achieve the maximum throughput of coded ALOHA. We analyze and compare the optimal performance of slotted and coded ALOHA. Under the unsaturated traffic condition at the center node, we derive practical throughput and delay, considering the network coding opportunity and the maximum queue length of the center node. Under the saturated traffic condition, we obtain a throughput upper bound of coded ALOHA to judge the ideal gain of network coding. The impact of asymmetric topology is evaluated with simulations.
The demand for high-quality mobile services has stimulated the development of new wireless network technologies. In contrast to wired networks, in which the quality of service (QoS) can be improved by increasing infrastructure density (server, communication line, router, and switch, etc.), a wireless (access) network is not easy to improve by means of infrastructure deployment. One reason is that the radio frequency is restricted and is a finite resource, and the same frequency channel cannot be used at the same time and in the same space. Due to this scarcity of radio resources, many researchers have tried to improve the performance of wireless links.
Medium access control (MAC) has been studied in order to allocate radio resources by resolving the conflicts from multiple users’ access to the common medium and improve the channel access efficiency in wireless networks. ALOHA , a classic MAC, was created in 1970, and has been widely used and highlighted for its simplicity. Many variations to ALOHA have also been proposed. Currently, this protocol is used in RFID . The most importantly, ALOHA formed the basis for the random access MAC in wireless sensor networks and wireless LAN. However, despite its simplicity of operation, the original ALOHA has low throughput, while the slotted ALOHA, an improvement over the original, has more or less double the capacity.
The relatively low performance of ALOHA originates from its random access nature. The node having a packet accesses the medium through a Bernoulli trial with a given probability. High access probability makes for high collisions, while a low probability results in under-utilization of the radio medium. The resulting optimal access probability still shows low throughput, even in a slotted ALOHA. The variations of ALOHA mostly focus on improving time-synchronization [3, 4], channel reservation [5–7], and cross-layer optimization [8–10]. Physical layer technologies such as MIMO [11, 12] and cognitive radio [13, 14] are combined with ALOHA, and their performances are analyzed. Most of these efforts, with the exception of the adoption of new physical layer characteristics, aim to reduce idle or collision time in the overall time scale.
In the previous studies, the complex operation is required to increase the throughput of ALOHA. However, it may reduce the merit of ALOHA that is simple and easy to implement. In this article, we analyze a network coded ALOHA, which is a different type of remedy for the low throughput of ALOHA. Network coding  is a promising technique in improving network throughput in multi-hop networks. Intermediate nodes in the network combine code packets, and intermediate or destination receivers decode these coded packets using known packets. From the medium usage point of view, network coding gives better performance by using the medium less when compared to the traditional communication method, in which each packet uses the medium exclusively [16, 17].
To verify the effect of network coding, we investigate and analyze embedding the network coding into the ALOHA protocola in the star topology, where the center node communicates with source/destination nodes one hop away. The network throughput can be improved when the network coding opportunity in the center node increases. Outer nodes put more packets into the center node in order to increase the network coding opportunity. Therefore, the optimal access probability for each node should be calculated in the network coded situation. Besides, the impact of network coding on the delay performance should be considered. With the simulation, we also evaluate the impact of asymmetric distance in the star topology, and find the tendency of the optimal transmission probability.
There has been research into combination of ALOHA and network coding. A similar model was considered in previous studies [18, 19]. In , routing, MAC, and network coding are jointly optimized, but both [18, 19] did not consider the queuing effects for network coding at the intermediate node. Other than this, the most relevant article  analyzed the throughput and delay of network coded ALOHA. However, they considered a protocol (collision) channel model and restricted the topology to two-way relay networks. The protocol channel model is not sufficient to represent the wireless channel (e.g., transmission errors due to the channel gain and capture effect). These assumptions are relaxed in our article. Network coding gain has been analyzed  within the CSMA/CA MAC layer, and we determined the set of the target signal-to-interference ratio (SIR) value and the carrier-sensing range that jointly maximize the throughput. Our results in this article are summarized as follows: (1) The maximum throughput of coded ALOHA is greater than that of the conventional ALOHA (coding gain is about 1.3). (2) Under the same parameters (transmission probability and target SINR), coded ALOHA shows less delay. (3) With the optimal parameters that maximize the throughput, coded ALOHA has a tendency to increase delay. (4) A throughput upper bound on coded ALOHA is obtained to judge the gain of network coding.
The remainder of the article is organized as follows: In Section ‘System model’, we explain our system model. In Section ‘Network throughput analysis’, the throughput of ALOHA and coded ALOHA is analyzed under the finite and infinite buffers in the center node. In Section ‘Packet delay analysis’, we derive the packet delay. Our numerical results are described in Section ‘Numerical results’. Finally in Section ‘Conclusions’, we conclude the article.
We consider the capture effect so that, even if more than two nodes transmit at the same time, the data or information from source node can be decoded successfully at receiver when the signal-to-interference-plus-noise ratio (SINR) is greater than the target threshold Θ. The source nodes can acknowledge success or failure of the transmitted packet (ACK). We assume that such ACK feedback is through the out-of-band transmission.
Medium access control and bi-directional network coding
The time is divided into equal-sized slots, each of which corresponds to one packet. Every node is synchronized with these time slots. At the beginning of each time slot, every outer node transmits a packet with transmission probability p, and the center node transmits with p c . Each outer node always has packets to transmit to its destination on the opposite side. The center node relays received packets from outer nodes with the transmission probability p c to appropriate destination nodes. The center node has the finite queue length, M in the real situation. The infinite queue length case is investigated for the asymptotic analysis. If transmission fails, the node retransmits the same packet with the same transmission probability until success is achieved.
We use bi-directional network coding at the center node, where network coding is applied for two bi-directional packets—a header packet in the queue of the center node and an opposite packet. The opposite packet is sent by the opposite node of the header packet’s transmitter. For example, when node n1 and n5 are bi-directional pair, node n1 sends packet X1 to the center node in time slot 1, node n5 sends packet X5 to the center node in time slot 2, and the center node broadcasts coded packet (X1 XOR X5) in time slot 3. In this example, X1 is the header packet, and X5 is the opposite packet. The destination node performs the XOR operation on the coded packet with its own transmitted packet. The ACK packet from the destination node implicitly means that the coded packet is successfully delivered and decoded at the destination node.
However, when there is no opposite packet in the queue, the center node transmits the header packet as a native packet without network coding. This opportunistic scheduling is optimal in terms of the sum rate maximization for the bi-directional traffic .
Transmission success probability
In Figure 1, there are three types of transmissions. First, outer nodes inject packets to the center node. Second, the center node relays the received packets to one of the outer nodes without any network coding. Finally, the center node (network) codes two bi-directional packets and transmits the coded packet to the two outer nodes. Let us denote the probability that each transmission succeeds by Pin, Pout, and Pnc, respectively.
where n is the number of interfering nodes, d0 is the distance between a source and its destination node, d i is the distance from the interfering node i to the destination node, p i is the transmission probability of the interfering node i, P0 is the transmission power of all transmitting nodes, N0 is the noise power, and Θ is the target SINR threshold. In this article, we assume that every outer node has the same transmission probability p i = p and the center node has p c . This probability is conditioned that the source node transmits and the receiver node does not participate the transmission.
Network throughput analysis
In this article, the throughput is defined as the average amount of received data by the outer nodes in a unit time slot. It is the result of multiplying the average number of received packets at the outer nodes by the amount of information in a single packet. We can increase the amount of information in a single packet using a high target SINR.
However, at the same time, the high target SINR reduces the transmission success probability and the average number of received packets at the destination nodes. In this section, we obtain the maximum throughput under the both of saturated and unsaturated traffic conditions. The saturated traffic condition means that the center node always has enough packets for the network coding. This condition makes the maximum coding opportunity in the infinite queue length of the center node . From this, we can obtain a throughput upper bound. On the other hand, the unsaturated traffic condition reflects the actual dynamics of the center node’s queue, and gives us a precise throughput with the finite queue length and actual coding opportunity.
The center node may have a queue state such that it cannot operate the network coding process and transmits the header packet as a native packet. The state of the center node should be described as a sequence of packets in the queue. Unfortunately, because the state dimension increases with the number of flows, it is difficult to derive the steady state probability. Thus, we approximate the queue state by considering only the number of packets in the queue (the accuracy of our approximation is verified by the numerical evaluation in Section ‘Numerical results’). Let us define X t as the number of packets in the queue at time t, where M is the maximum queue length (X t ≤M). Let Π m denotes the stationary probability of ALOHA and denotes that of coded ALOHA.
where denotes the amount of data successfully delivered through unit bandwidth.
In coded ALOHA, the state diagram of the center node is complicated. With the successful transmission of a coded packet, the queue state moves to the state with two fewer packets. The number of packets in the queue does not provide sufficient information to determine whether network coding is possible. There may be no opposite packet, even if there are many packets in the queue. To quantify this, we introduce q(m), the network coding probability when there are m packets in the queue.
When there is only one packet in the center node (m = 1), the transition probability simplifies to μ N (1) = p c (1 − p)Pout.
The detailed derivation of state probabilities and the throughput are included in Appendix 2.
where (21) is obtained by substituting (20) into (17).
By solving and , we find the maximal throughput and the optimal transmission probabilities p∗ and , respectively. Unfortunately, the differential equations do not have solutions as a closed-form. Instead, we provide an approximation for p∗ and as follows:
See Appendix 3.
Once we have an approximate value for p∗, we can derive the optimal value for (18) or (20). Our simulations in Section ‘Numerical results’ verify that the formula matches the numerical results, even in low target SINR Θ cases. □
Packet delay analysis
The packet delay of ALOHA, D, is identical to (22), after replacing by Π m . Both and Π m are derived in Appendix 2.
In the previous section, we found the optimal parameters that maximize the throughput of ALOHA and coded ALOHA. As shown in the numerical example section that follows, the optimal p∗ of coded ALOHA is higher than that of ALOHA while the optimal of coded ALOHA is lower than that of ALOHA. This can be also identified by Proposition 1 under the saturated traffic condition. With a higher p and lower p c , coded ALOHA has a higher packet injection rate from outer nodes to the center node than dose ALOHA. This rate results in more queued packets Enc[m] at the center node, a higher network coding opportunity, and increases the throughput of coded ALOHA accordingly. However, the packet delay is dominated by the queueing delay, and becomes larger than that of ALOHA. This is the trade-off between throughput and delay in coded ALOHA for the optimal throughput point. In the following section, we show that this trade-off property is not valid for non-optimal solution cases.
Under the same parameters of transmission probabilities p, p c , and the target SINR Θ, coded ALOHA has a less than or equal packet delay when compared to ALOHA, Dnc ≤ D.
ALOHA and coded ALOHA have the same access delay and arrival rate. However, due to the different service rates, ALOHA and coded ALOHA have a different number of queued packets in the center node (i.e., Enc[m] ≠ E[m]). From the throughput of ALOHA, (10), we can define the service rate of ALOHA as μ(9). Similarly, the service rate of coded ALOHA can be defined as μnc = 2μ C (m) + μ N (m), from the throughput of coded ALOHA, (14). When the coding opportunity q(m) = 0, the service rate of coded ALOHA is equal to that of ALOHA, μnc = μ. However, when the coding opportunity q(m) = 1, the service rate of coded ALOHA is μnc = 2μ. Because 0 ≤ q(m) ≤ 1, the service rate of coded ALOHA exists in the interval of μ ≤ μnc ≤ 2μ. The higher service rate of coded ALOHA can result in fewer queued packets in the center node, Enc[m] ≤ E[m], and less delay, Dnc ≤ D. □
As shown in Figure 4, there is a unique optimal transmission probability, p∗, which achieves the maximum throughput. With the optimal , the throughput of coded ALOHA (14) is maximized at p∗ = 0.18 and that of ALOHA (10) at p∗ = 0.15. Similarly, in Figure 5, there is a unique optimal point in coded ALOHA () while the throughput of ALOHA monotonically increases and converges to a certain value with the optimal p∗. The maximum throughput of ALOHA is achieved in the interval of 0.43 ≤ p c ≤ 1. These optimal parameters p∗and are obtained from numerical enumeration. In both figures, the maximum throughput values of ALOHA and coded ALOHA are and , respectively. Using the network coding, a higher throughput is achieved with a smaller transmission probability of the center node. It allows outer nodes to transmit for more packets to the center node, and this makes more coding opportunity in the center node. Although increasing p might cause more collision events with other outer nodes or the center node, the network coding gain is much higher than the cost of higher collision. However, for high p, both throughput curves converge to zero due to overmany collisions.
In Figure 5, both throughput curves of ALOHA and coded ALOHA converge to 1.3, not to zero when p c goes to 1. It is because p c is a conditional transmission probability when the center node has packets in its queue. If there is no packet in the center node queue, the center node cannot participate the transmission even if p c = 1. Increasing p c makes higher service rate of the center node queue, which results in higher throughput. However, for high p c , throughput curve of ALOHA converges to 1.3 due to the limit of injection rate to the center node. For low p c , coded ALOHA can achieve higher throughput than ALOHA because coded ALOHA can send two packets at once using network coding. However, high p c reduces the number of packets in the center node and the coding opportunity. Thus, as p c increases, the coding gain converges to zero, and the throughput of coded ALOHA converges to that of ALOHA. The simulation results are averages of 100 independent samples over 10000 time slots. It can be seen that our analysis matches the simulation results well.
We can increase the throughput by selecting an optimal target SINR Θ of the PHY layer. The effect of selecting the target SINR Θ is shown in Figure 6. Throughput curves are concave and there is an optimal point in each curve. With a high target SINR, we can send the packet with high spectral efficiency, however, the transmission success probability of that packet is low. In contrast, with a low target SINR, we can send many packets that include little information. As a result of the numerical evaluation, the optimal target SINR that maximizes the throughput is 22.55 dB when transmit power to noise ratio P0/N0 = 30 dB, and is 14.77 dB when transmit power to noise ratio P0/N0 = 20 dB. When the noise level increases, the optimal target SINR should decrease to reduce the channel error.
In this article, we presented a coded ALOHA that performs the bi-directional network coding over ALOHA. We mathematically analyzed the performance of coded ALOHA in terms of throughput and delay. To maximize the throughput of coded ALOHA, the transmission probabilities (p and p c ) and the target SINR Θ were jointly optimized. The throughput of coded ALOHA was derived under the both unsaturated and saturated traffic conditions at the center node. Under the saturated condition, we obtained a throughput upper bound with the perfect network coding opportunity, and presented a closed form formula for the optimal transmission probability. The result of the unsaturated condition showed that the throughput of coded ALOHA converges to that of the saturated condition with a sufficiently large queue size. Additionally, as the topology becomes asymmetric, the optimal transmission probabilities of outer nodes try to balance the injection rates for higher coding opportunity. In slotted ALOHA, it can bring higher throughput by turning off the close node’s transmission.
With the same parameters, throughput of coded ALOHA is higher than that of ALOHA, and the delay of coded ALOHA is shorter than that of ALOHA. Moreover, with same throughput condition, coded ALOHA can have shorter delay than ALOHA. However, with their own optimal parameters, coded ALOHA has higher throughput and higher packet delay than those of ALOHA. For the optimal parameters, coded ALOHA has higher p and lower p c than those of ALOHA. This solution reflects the increased coding opportunity, since network coding has the benefit only when the system gets the opportunity to do it and this opportunity can be raised by stacking more candidate packets in the coding node repository. However, the packet delay is increasing by this solution since the average queueing delay of packets increases. The existence of significant queueing delay is the main reason of this throughput-delay tradeoff which looks weird at the first glance. In an engineering perspective, coded ALOHA is better to be used in the situation where the traffic is highly generated as of optimal p∗ and the time bound of application is relatively loose such as sensor network measuring a temperature or humidity of some region (delay tolerant network). Furthermore we compared four ways to consider the delay QoS, and found that increasing p c has the best performance in term of throughput loss and decreasing p can be a good solution in term of energy consumption.
aFor convenience, we shall refer to slotted ALOHA as ALOHA. bWe use the superscript “f” to denote the saturated case and “q” to denote the unsaturated case in (14).
Derivation of the transmission success probability
where Q i = g ij P j is the received power as an exponential random variable with mean , Aggregate interference is , and C i is a sequence of i.i.d. Bernoulli random variables with P(C i = 1) = p and P(C i = 0) = 1 − p. The coded packet is successfully delivered when both destinations have higher SINR than the target SINR Θ.
Queueing analysis of network coding
where z in (32) is a normalization factor. Combining (32)-(35) and the normalization equation, we can obtain the normalization factor z and the steady state probability .
Proof of proposition 1
This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency), (NIPA-2012-(H0301-12-1003)).
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