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Studies of the IEEE 802.15.4 Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme have been received considerable attention recently, with most of these studies focusing on homogeneous or saturated traffic. Two novel transmission schemes—OSTS/BSTS (One Service a Time Scheme/Bulk Service a Time Scheme)—are proposed in this paper to improve the behaviors of time-critical buffered networks with heterogeneous unsaturated traffic. First, we propose a model which contains two modified semi-Markov chains and a macro-Markov chain combined with the theory of

In recent years, wireless sensor networks (WSNs) have revolutionized the world of distributed systems and enabled many new applications. WSNs play more and more decisive roles in various aspects such as wide-range environmental surveillance, short-range health monitoring, inventory tracking, military locating

In this work, we propose two novel access schemes named OSTS/BSTS to improve the heterogeneous performance of the time-critical network. First, we model these two slotted CSMA/CA schemes for a one-hop, beacon-enabled 802.15.4 star topology combining discrete time Markov chains and the theory of

The rest of this paper is structured as follows: Section 2 gives a summary of related works and analysis premise of our model. In Section 3, a brief overview of slotted CSMA/CA scheme of the IEEE 802.15.4 standard is described. OSTS/BSTS modeled by Markov chains and

Literature reviews presented here are three-fold: (1) references related to the performance analysis using Markov chain model; (2) references related to queuing performance analysis with buffered condition; (3) references related to performance analysis with heterogeneous traffic.

Among performance analyses of CSMA/CA backoff mechanisms using Markov chain models, a relatively early and comprehensive approach is presented in [

Queue-length distributions at arrival, departure and random epochs are proposed in detail in the serial schemes in [

Performance analyses of heterogeneous networks are mostly based on priorities, and the first performance analysis and modeling of 802.11 DCF with heterogeneous traffic based on fair contending chance presented in [

Comprehensive models adopting Markov chains and _{1}, _{2} nodes to sense the variables of temperature and humidity in our periodic monitor application and transmit them to a sink, respectively. Packets arrive at the nodes for transmission according to a Poisson process with arrival rate of _{1} and _{2} for _{1} and _{2}, respectively. System heterogeneity can be expressed as the node distributions, and the heterogeneity at the same node distribution can be denoted as the asymmetry which refers to the difference of packet arrival rates. Each node has a buffer with finite capacity

The main contributions in this paper are threefold. Firstly, two novel schemes—OSTS/BSTS—are proposed to improve the behaviors of time-critical heterogeneous buffered networks with non-priority unsaturated traffic. Secondly, comprehensive models combining Markov chains and

First, we briefly explain the slotted CSMA/CA mechanism of the IEEE 802.15.4 MAC [

The scheme to be implemented before accessing the channel is illustrated in _{min}) (step 1). Then, the MAC sublayer delays for a random number of periods uniformly distributed in the first backoff range [0, 2^{BE}^{min} − 1] (step 2). When the backoff counter is decreased to 0, the node performs the first CCA (step 3). If the channel is sensed idle after CCA1, _{i}_{i} = W_{0}2^{(}^{BE}^{+1)}). If _{m}_{m}

Before presenting system analytical models, several assumptions according to our actual applications are proposed.

ACK of MAC-level can be omitted for each packet for we consider two types of nodes transmitting packets to one sink (or coordinator) within one-hop star topology which is also presented in [

Empty probability denotes _{0} if there is no any packet in node buffer after a packet departure, which is not equal to the idle probability _{0} at a random period. The node can go to sleep with a probability of _{0} if its buffer is empty at any one of such three situations: end of successful transmission; reaching maximum backoff stage; reaching retry limits.

Packet arrival process in buffers can be modeled as a Poisson process. Only header packets can contend for the channel every time, which leads to the channel contending analysis partly simple regardless of the queue distributions.

We modify that all nodes contending to the channel should decrease their backoff counters to initial values once one of them transmits successfully or packets are dropped due to channel access unsuccessfully or collision, avoiding nodes with low contention windows always capture the channel once they catch the channel in the case of competing for the channel simultaneously [

In this section, two novel schemes with semi-Markov chain models describing slotted CSMA/CA scheme of IEEE 802.15.4 with retry limits and one macro-Markov chain model presenting macroscopic state transitions are proposed. The metrics of throughput, packet service time and energy consumption are partly determined by the network operating points _{n}_{1} and _{2}, respectively, for simplification.

First, we study the behaviors of one type of nodes using a three-dimensional Markov chain as in

When the backoff stage increases to _{i}_{1} for _{1}and _{2} for _{2}, respectively. Values

We denote actual state transitions by adopting solid ovals and solid arrows for the IEEE 802.15.4 CSMA/CA scheme using a Markov chain, such as _{2} in _{1} using the same Markov scheme paralleled to the actual one with dashed ovals and dashed arrows which do not exist in the actual state transitions seen from _{1} and _{2} after two successive idle backoff periods for _{1} and _{2}, respectively. We denote _{1} and _{2} as presenting the parallel transition procedure for all nodes must perform the common backoff process. Output variables involved in _{1}_{r}_{0} ∼ _{1}_{rm}_{2}_{r}_{0} ∼ _{2}_{rm}_{1} and _{2}, respectively. Variables _{10} ∼ _{1}_{r}_{20} ∼ _{2}_{r}_{1} and _{2}, respectively. Variables _{100} ∼ _{10}_{m}_{1}_{r}_{0} ∼ _{1}_{rm}_{200} ∼ _{20}_{m}_{2}_{r}_{0} ∼ _{2}_{rm}_{1} and _{2}, respectively. State transition probabilities for any one type of nodes associated with Markov chain of

Packet queues in the node buffers are modeled as _{1} and _{2}, respectively. The node which obtains the channel firstly can transmit the header packet in its queue, and it can again contend for the channel with other nodes to transmit its remaining packets after completing the current packet, denoted as one service a time scheme (OSTS). Macroscopic state transitions for OSTS are shown in

Macroscopic states involving backoff procedures of both types of nodes follow the same algorithm as _{0}_{n}

Expressions of independent parameters _{n}_{i,k,j}_{i}

Through normalized condition of Markov chains and steady-state probabilities according to each type of nodes, we obtain

Then, we can derive probability expressions of each block as follows. We assume there is no maximal delay exponent limitation for consideration of evaluation simplification:

_{0} is the probability that a node remains in a sleeping state without any packet arrival in a random slot time. From

Substituting _{01}, _{02} and _{0}. Variables
_{1} and _{2} for _{1} and _{2} respectively, which means that nodes in _{1} can start to access the channel at the boundary of the next slot with probability _{1} if there are no new packet arrivals of other type of nodes. The next transmission probability _{2} for _{2} is derived as the similar way. According to the Markov blocks of the macroscopic state transition in _{in}_{1} or _{2} in _{L0} which refers to idle state length in the state transition.

In such a way, we can derive all parameters in the system using a numerical method that solves the non-linear system equations given by _{trn}

We present probability expressions of _{1} and _{2} here for an early time, which are deduced elaborately in Section 4.2. Probability _{0} that there is no packet to send in a random slot, and probabilities _{01}, _{02} which means that the queue become empty after a departure of _{1} and _{2} respectively can be derived through the queuing theory analyzed in the next section.

The second scheme BSTS, denoting s bulk service a time scheme, means that a node is allowed to transmit all packets in its buffer with a burst mode once it successfully obtains the channel and reserves it. In this scheme, transmission packet length is simply considered as

Performance analysis of this scheme is similar to that of OSTS except for setting the parameter _{0}_{n}_{0}_{n}_{0}_{n}

We denote _{trn}_{jk}_{jk}_{0}_{k}_{k}_{jk}_{k}_{-}_{j}_{+1} (0 ≤ _{kn}

We also denote _{kn}_{kn}

And then, steady-state equations for state transitions are given as follows [

We find _{kn}

From above queue expressions, we can derive the probability _{0}_{n}

The probability _{0} that the queue is empty at arbitrary time can be derived unlike the way of the probability _{0}_{n}_{Kn}_{0} can be derived as follows:

We can derive the closed expressions for system depiction by substituting

System operating points are determined by parameters _{n}

We analyze the medium behavior based on every CPC for simplification. When the channel is sensed busy after CCA1 with probability _{1} for _{1} due to data transmission of other nodes, it means that at least one of (_{1} − 1) remaining nodes transmits in the same slot with the current transmitting node and none of _{2} transmit, or none of remaining _{1} − 1 transmits and at least one of _{2} transmits [_{2} for _{2} can be derived in a similar way. In this way, channel sensing probabilities are independent of types such as _{1} and _{2}, which refer to the channel sensed busy after CCA2 for _{1} and _{2} respectively, can be derived in the same way. It can be simplified to _{1} and _{2} transmits in current slot:

Network operating points determined by carrier sensing probability _{n}_{n}_{1}/_{2}, in which simulation setup and simulation parameters are presented later in Section 6.

Since parameters _{n}

In the most heterogeneous condition which refers to the number of two type nodes are comparable to each other, we observe that channel accessing character is dominated by the difference of the two arrival rates. Channel is sensed busy with smaller probability for there are small total pending packets when traffic rate _{1} is much smaller than _{2}, such as ln_{1} and _{2}. Probability _{1} is equal to _{2} at different node distribution, and

Probabilities _{n}_{1} = _{2}. Decreasing the difference of arrival rates, _{1} and _{2} of BSTS are less slightly than those of OSTS under more asymmetry conditions.

From these figures, we observe that accessing behaviors are determined by the node distribution and system asymmetry. Analysis results are consistent with simulation results at more symmetry conditions for a great extent, while those of more asymmetry cases are inconsistent with analysis results slightly shown in

We denote

Successful transmission probability _{s}

Probabilities that nodes encounter the collisions in a random slot are not similar to successful transmission probabilities as follows:

Thus, throughput expression

_{n}_{trn} P_{sn} E_{n}_{tr}_{1}_{s}_{1} + _{tr}_{2}_{s}_{2} and collision transmission probability is _{c}_{1} + _{c}_{2}, respectively. _{s}_{c}_{tr}_{1}_{s}_{1} − _{tr}_{2}_{s}_{2} − _{c}_{1} − _{c}_{2})_{s}, T_{c}_{tr}_{1,2}, _{s}_{1,2} and _{c}_{1,2} depend on operating point parameters _{1,2} as shown in _{s}_{c}

Assume _{s}_{c}_{L}_{CCA}, t_{s}, t_{c}_{ex}

In low-rate wireless applications, packet service delay is also an important metric, and we pay more attention to improving the performance of delay in our time-critical applications. Generally, total delay in a communication network includes processing delay, queuing delay, access delay, and propagation delay. In this paper, we focus on average packet service delay which consists of the delay in queue waiting and delay for accessing the channel. Access delay is the time from the instant which the packet is at the head of its MAC queue and ready to be transmitted to the instant when coordinator receives packet, which is also elaborately discussed by many papers such as [_{tr}_{q}_{0} immediately after a departure and _{0} at a random period are of our consideration besides the queue distribution involving the tagged packet.

The PGF of access delay consists of three factors as shown in ^{2} in each part of _{c}_{c}_{ta}_{i}_{i}_{i}_{i}_{i}_{i}

We denote the discussed random arrival packet as a tagged packet in either type of nodes to account for packet queuing delay. Tagged packet has a distance of _{1} if the analyses base on only _{1}, shown in _{N1}, and the queue which consists of tagged packet is called tagged queue accordingly. Queuing delay of _{N1} consists of three parts: the time for transmitting (_{1} − 1) nodes and the time for transmitting _{2} nodes. Access delay for any non-tagged packet is the same as the tagged one analyzed as

According to the probability distribution of queue size at packet departure of _{m}_{t−}_{t+}

Now, we consider the tagged queue length distribution based on only _{1}, and the same analysis process can be applied to that of _{2}. According to the PGF of accessing time, packets arriving at tagged queue _{1} consist of two parts: the arrivals _{1} if the current service packet belongs to (_{1} − 1) and the arrivals _{2} if the current service packet belongs to _{2}. The joint probability distribution consists of two parts as following _{1} in mean packet access time
_{1} if the current service is one of _{1} − 1 for the first part, and arrival length _{2} in mean packet access time
_{2} if current service is one of _{2}:

The probability that _{1} arrive at queue _{1} during the service time of _{1} and the probability that _{2} arrive at queue _{1} during the service time of _{2} are derived as following

_{1,2} ≤ _{1} arriving at queue _{1} during the service time of _{1} and the second factor denotes _{2} arriving at queue _{1} during the service time of _{2}:

So, we can substitute

So, we can derive the LST (Laplace-Stieltjes transform) of service delay. Service time is the time from packet arriving at nodes to departure, and we assume that the probability distribution of the queue length at packet arriving epoch is the same as the probability distribution of the queue length at arbitrary epoch [_{1}(_{1} ≤ _{1} − 1) queues and the time for transmitting _{2}(_{2} ≤ _{2} queues, which shown in

As we consider the tagged packet in _{2}, we can derive the service time distribution as that of tagged packet in _{1} by substituting the queue parameters of _{2} for that of _{1} in a similar way. For example, joint probability distribution consists of two parts as shown in the following _{1} arrives at tagged queue _{2} in mean packet access time
_{1} if the current service is one of _{1} for the first part, and _{2} arrives at tagged queue _{2} in mean packet access time
_{2} if current service is one of _{2} − 1. Other results can be derived as the similar way as that of _{1}, and we can omit this repetitive process for

Energy consumption is the most important metric in WSNs, and we also analyze it elaborately. We assume a node is sleeping in a backoff period while it is receiving in extra waiting period after a successful transmission or not. Thus, we assume a node does not consume any energy during backoff procedures. Moreover, energy consumption of turnaround process _{ta}_{RX}_{TX}

Now we present extensive simulations of slotted IEEE 802.15.4 to validate our scheme with heterogeneous and unsaturated traffic using the NS-2 simulator [

Randomly deployed in a circle area of radius 3 meters with one sink in the center receiving data, nodes are all in the range of each other transmitting packets to sink. The transmission range of the transceiver is about 7 m. Node model is initiated as related in [

Packet length is fixed to 7 units of backoff period including the PHY-header and MAC-header period. Backoff stage and retry number are fixed to 5 and 3, respectively. We study the asymmetry or heterogeneity and buffer characteristics of OSTS/BSTS schemes in this paper. We consider that the relative arrival rates _{1}, _{2} and relative numbers _{1}, _{2} which represent the system asymmetry and heterogeneity, and parameter

Performance is evaluated as the function of the aggregate offered load in different system size. Two different network sizes, _{1} = 13, _{2} = 12 and the least one _{1} = 23, _{2} = 2 of the network of _{1}, _{2} and numbers _{1}, _{2} for the same buffer capacity, and we cannot derive the variation tendency if these four parameters change at the same time without any datum mark. In this way, we can evaluate the performance as the functions of R = _{1}/_{2} in the datum mark of system aggregate offered load, which is fixed at each buffer length of system size _{0}(_{1}_{1} + _{2}_{2}), in which the parameter _{0} is an impact factor standing for adjustment of system size and transmission arrival rate. Preferable and comprehensive performance metrics as the functions of ln_{1}, _{2} may be intuitively large, such as ln_{1} = 0.135_{2}, but performance differentiation can be evened largely through node distribution of _{1} and _{2} in the same offered load. Performance curves are manifested to smoothness character in function of ln

According to

We firstly consider the asymmetry characters for the same heterogeneity conditions shown in _{1} is equal to _{2}. Node's buffer capacity also plays an important role on throughput illustrated in

Heterogeneity also plays a decisive role on system performance shown from the curves of _{1} = 13, _{2} = 12 is higher than that of _{1} = 23, _{2} = 2 at each ln_{1} = 13, _{2} = 12 is lower than that of _{1} = 23, _{2} = 2 at each ln_{1}/_{2}) = 0 means a network consisting of _{1} + _{2} identical nodes for _{1} equal to _{2} for a fixed network size, leading to each curve passing through the same point at _{1}/_{2} = 1 for the same load, which can be seen from the point of ln(_{1}/_{2}) = 0 in _{1}/_{2}) = 0 when _{1} = 13, _{2} = 12, which is the same value of _{1} = 23, _{2} = 2 for OSTS and BSTS, respectively, and throughput is 0.308 for OSTS scheme and 0.288 for BSTS scheme at ln(_{1}/_{2}) = 0 when _{1} = 23, _{2} = 2 in our system increases to a saturated value with _{1}/_{2} increasing, and then decreases with a marginal rate shown in _{1}/_{2}) < 0, while simulation results are higher than analysis results when ln(_{1}/_{2}) > 0, and the deflection is of 2.659% to 5.645%. These deflections are sustainable to our applications.

Delay is the most important character in our real-time monitoring system, and we always attempt to improve the behavior of delay in order to obtain the real-time monitoring. From _{1} + _{2} = 10, mean delay is much lower than that of large size such as _{1} + _{2} = 25 for nodes increasing results in more pending packets and more collisions.

Asymmetry and heterogeneity play decisive roles on the system delay performance from the curves in _{1} is equal to _{2} at different node distributions shown in

Heterogeneity also plays a decisive role on delay performance shown from these curves. Delay increases with the heterogeneity of the network increasing at ln_{1} = 13, _{2} = 12 is higher than that of _{1} = 23, _{2} = 2 at each ln_{1} = 13, _{2} = 12 is much lower than that of _{1} = 23, _{2} = 2 at each ln_{1} are much more than those of _{1} (such as _{1} = 23, _{2} = 2) and traffic rate of _{1} is much more than that of _{2}, which means system packets almost consisting with only _{1} and delay performance is almost determined by traffic rate _{1} of _{1}, which is presented from the comparison of _{1}/_{2}) = 0 for the network composed of _{1} + _{2} identical nodes related as above.

We can observe that respective delay of _{1} and _{2} are not similar to the characters of system total mean delay. Asymmetry and heterogeneity also play important roles on the respective delay behaviors observed from the curves in _{1} decreases with increasing _{1}/_{2}, and its rate of decrease increases with the decreasing asymmetry. Delay of _{1} at _{1} for OSTS scheme is less than that of BSTS scheme when ln

In case of _{1} for OSTS scheme is more than that of BSTS scheme when ln_{1} for the OSTS scheme is less than that of BSTS scheme when ln_{1} increases with the increasing buffer capacity, and also increases with the network scale. Shown in _{1} is insensitive to the heterogeneity for ln_{1}/_{2}) = 0 for the same load.

Delay analysis of _{2} is similar to that of _{1}. We can observe that delay of _{2} increases with increasing _{1}/_{2}, and its rate of increase increases with the decreasing asymmetry as shown in _{2} at _{2} for the BSTS scheme is less than that of the OSTS scheme when ln_{2} for the BSTS scheme is more than that of the OSTS scheme when ln_{2} for the BSTS scheme is less than that of the OSTS scheme when ln_{2} for other node distributions can also be analyzed as shown in _{2} increases with the increasing buffer capacity, and also increases with the network scale. Heterogeneity plays a similar role on delay of _{2} as on delay of _{1}.

As shown in _{2} is insensitive to the heterogeneity for lnR > 0, while it increases with the increasing heterogeneity for lnR < 0. The respective curves pass through the same point at ln(λ_{1}/λ_{2}) = 0 for the same load.

Energy is a most important factor considered in WSNs that withdraw energy from batteries, and it is also analyzed elaborately in our time-critical system. We assume that nodes are sleeping in the backoff procedure for energy efficiency, without any energy consumption. Energy analysis is similar to the throughput analysis in a small system size shown in _{1} = 13, _{2} = 12 or the least heterogeneous one, which is shown in _{1} = 23, _{2} = 2, much more energy is consumed for more packets generated by _{1} with higher arrival rate _{1} when ln_{2} are relatively high, which results in energy consumption always increasing shown in

We also analyze system characteristics when the traffic rates are equal to each other, which means the homogeneous or symmetric condition mostly studied before [

In the same way, mean delay will increase slowly for small offered load and rapidly for large

According to the analysis and simulation results, we observe that the heterogeneity and asymmetry play decisive roles in system behavior, and buffer size also impacts largely on the characteristics of the schemes. Performance metrics are demonstrated to have different superiority when adopting different transmission modes, OSTS or BSTS. We can choose the appropriate scheme of OSTS and BSTS according to node distribution of the applications as shown in

Analysis and simulation results shown above are comprehensive for applications, and we can compare the performance metrics of our mechanism with those of other non-priority heterogeneous schemes. Our schemes are used for time-critical monitoring and detection application, in which minimized delay is the most important target. Different types of nodes contend for the the channel with a fair chance, and the fairness is also an improved requirement. Adopting the distinguished improvement of taking the global viewpoint into account, our schemes OSTS/BSTS excel in WSN networks with non-priority traffic. Through the comprehensive comparisons, we can derive that the delay and fairness performance metrics of our schemes are obviously improved over other schemes such as [

A performance analysis model of the IEEE 802.15.4 CSMA/CA scheme with heterogeneous traffic is presented in [

The most representative model of CSMA/CA scheme based on IEEE 802.11 with non-priority heterogeneous traffic is presented in [

We can present the access fairness comparisons based on the metrics of respective throughput and transmission probability according to the definition of [_{1} are the same as those of _{2} when _{1} = _{2} shown in

We can see from _{1} and _{2} is 0.1098 and 0.1338 at ln_{1} and _{2} is 0.1120 and 0.1343 at ln_{Sar}_{Ram}

Throughput comparisons are shown in

We find that delay metrics of OSTS and BSTS scheme are superior to those of Sarmiento's and Ramachandran's schemes from _{Sar}_{Ram}

Ramachandran's scheme consumes less energy than that of OSTS in the case of ln

We can find that the heterogeneous performance is improved greatly by adopting the novel schemes of OSTS/BSTS, in which the contending traffic, regardless of type, has no priority over each other. Delay and fairness of OSTS/BSTS are superior to those of other schemes, while throughput and energy efficiency are superior to others in more heterogeneous situations. In such a fair-required time-critical system, the schemes of OSTS/BSTS supply a satisfactory performance.

In this paper, two transmission schemes—OSTS and BSTS—are proposed to improve the performance of heterogeneous unsaturated networks. At first, accurate and comprehensive analyses for these two slotted CSMA/CA scheme using two semi-Markov and one macro-Markov models are made, along with a queuing model. These models contain a finite number of terminals and ideal channel, in which each node has a finite buffer capacity of

Moreover, we should gain deep insights into several problems in our future works. We know that MAC sublayer needs a finite amount of time to process data received by the PHY. To allow for this, two successive frames transmitted from a device shall be separated by at least an InterFrame Spaces (IFS) period. If the first transmission requires an ACK, the separation between the ACK frame and the second transmission shall be at least an IFS period. Two frames are seldom transmitted successively from a device in these schemes and no ACK is contained in them, therefore, the IFS between two frames can be ignored in our OSTS/BSTS schemes. The IFS should be taken into account for the appropriate successive transmissions/receptions or ACK transmissions in the future studies. Then, it is worth noting that the CFP is considered as the solution to delay-sensitive applications such as video services, and this time-critical mean can be used for our real-time applications in the future. Also, the distances among nodes are relatively close, and the propagation signal effect can be omitted in our current representations. However, the nodes which are used to sense the quantities to be measured can be away from each other for larger distances, and several situations should be taken into considerations. Firstly, the propagation model should be included into the simulation results, which is a major deviation of the analysis and simulation results. Secondly, the distances among nodes go to such an extent as to transmit the packet in two or more hops, which brings out hidden terminals or more complicated pending problems. Furthermore, heterogeneous queues should be resolved by the relay nodes and the coordinator, respectively. Currently, our research focuses on such intractable multi-hop access problems, and we shall devote ourselves to study and then improve the behaviors of these multi-hop wireless sensor networks with buffered heterogeneous traffic in our forthcoming research.

This work was supportted by the National Key Technology Research and Development Program of China (2006BAK03A17). We also wish to thank the anonymous reviewers for their valuable comments on our work.

^{a,Y}

The slotted CSMA/CA mechanism of 802.15.4.

Markov chain model for slotted IEEE 802.15.4 CSMA/CA scheme. If a tagged node has packets to transmit at the next backoff slot with the probability _{in}_{in}_{1} or _{2}. After random delay in range [0, _{0} − 1], the node can perform CCA1 with a probability of _{n}_{1,2} located in the end of backoff is to demonstrate the paralleled access behavior of the different types of nodes since all nodes regardless of types must perform the backoff process.

Macroscopic state transitions for OSTS scheme. Outputs within these blocks are those one-to-one corresponding outputs in _{0}_{n}_{0}_{n}_{0} if there is no any packet in any node. Nodes have packets to transmit at the next backoff slot with probabilities _{1} and _{2} for _{1} and _{2}, respectively.

Macroscopic state transitions for BSTS scheme. Outputs within these blocks are those one-to-one corresponding outputs in _{0} if there is no any packet in any node. Nodes have packets to transmit at the next backoff slot with probabilities _{1} and _{2} for _{1} and _{2}, respectively.

The behavior of parameters _{n}_{1} with ln_{2} with ln

The relation between the tagged packet and the header packet in one tagged buffer. After a tagged packet arrives at the tagged queue, it has a distance

Throughput as a function of _{1}/_{2} for fixed loads; (

Delay as a function of _{1}/_{2} for fixed loads. (

Delay for each system size of _{1} as a function of _{1}/_{2} for fixed loads. (

Delay for each system size of _{2} as a function of _{1}/_{2} for fixed loads. (

Energy consumption as a function of _{1}/_{2} for fixed loads. (

Performance when _{1}/_{2}. (

(

(

The parameters of our simulations.

_{data} |
2,240 μs | ||

_{CCA} |
640 μs | ||

_{slot} |
320 μs | ||

_{ta} |
12 symbols | ||

_{TX} |
17.4 mA | _{RX} |
19.7 mA |

_{CCAs}_{(inte}_{rval}_{)} |
18.5 mA | _{ta} |
18.5 mA |