Firefly Clock Synchronization in an 802.15.4 Wireless Network

3.1. Clock State Correction

The clock state synchronization is established by the RFA model and uses the definition of the smooth, monotonically increasing, and concave down state function of (1) to calculate the overall phase advance . Consider the dissipation factor and the pulse strength within , then the phase advance equals

(3)

The direct implementation of all these functions would result in a time-consuming calculation process. Therefore, we simplified the equation by inserting the inverse function in (3). Let and , then (3) can be transformed to

(4)

Assuming a strong dissipation factor and a small pulse strength s.t. , then we can replace by the first-order approximation of the Taylor expansion and thus is negligible. The phase advance then can be reduced to

(5)

As a result, we have a linear Phase Response Curve (PRC), where the coupling factor specifies the strength of coupling between the oscillators and depends on the product of the dissipation factor and the pulse strength . This result is similar to the simplified firing function described in [3].

In contrast to the original RFA algorithm, our approach achieves a better synchronization precision and a faster convergence time by indirectly performing a clustering of the received synchronization events. This is done by ignoring all events which are within the phase advance of the last event to which a node reacted. In fact, this corresponds to the introduction of a short refractory period. Additionally, we do not allow a node to react to firing events which would originally occur after the node reached the period end. This ensures that in the case of synchronized nodes, the fastest node then does not advance its phase anymore resulting in a better precision. The algorithm is formally analyzed in more detail and guarantees network synchronization as long as the bounds for several parameters are maintained. Algorithm 1 explains the behavior of this extended RFA (E-RFA) algorithm with the use of pseudocode. The refractory period is implemented by the condition in Line 9. The variable eventset contains the correct phase of all received firing messages and denotes the random amount for the preponed transmission with at most the maximum, respectively, minimum message staggering delay , respectively, .

Since the purpose of this work should demonstrate that such a synchronization approach works with an off-the-shelf communication stack without MAC-timestamping, we have to expect a delay jitter in the order of milliseconds due to the uncertainty in the application and MAC layer. It should be mentioned that Lundelius and Lynch have shown in [6] that in the presence of a maximum delay jitter , an assemble of clocks cannot be synchronized to a precision better than .

Lower Bound for the Coupling Factor

We assume that every processor consists of a hardware clock which generates the phase . This clock stays within a linear envelope of the real time. Note that whereas the hardware clock continuously increases, the phase is periodically reset to with respect to where denotes a dynamic offset value which changes due to the state correction algorithm. represents the granularity of the hardware clock which corresponds to the synchronization period . We therefore assume that there exists a positive constant (maximum drift rate) such that . Note that this definition of the bounded drift simplifies the calculation of the precision and may differ from literature. We further assume a fully connected network in which the message delay is always in the range , where denotes the constant part and the maximum delay jitter of the communication delay in real time. The lower bound for in an ensemble of nodes in a fully connected network then depends on the maximum drift rate , the message staggering delay , and the communication delay . Note that all parameters with a preceding are defined with respect to . However, for simplifaction we now always assume that is normalized to . Let , respectively, denote the maximum, respectively, minimum relative message staggering delay and . We now show that in the case of two clocks, the modified RFA provides a bounded precision . Therefore, for denotes the maximum time difference among all nodes in real time units between the time reached and the time reached .

Lemma 3.1.

Let and be the drift offset. In the case of two clocks and no message loss, if the coupling factor is lower bounded to and , then for and , Algorithm 1 keeps the network synchronized with a worst case precision bounded to

(6)

Proof.

Assume the clocks are initially synchronized to . W.l.o.g. let be the faster node. We further use as the reference for the precision , where denotes the real time when 's phase reached . We further assume that the next time reaches the threshold is at time . Let be the corresponding precision at . For we then have and . Let respectively, denote the relative message staggering delay the node , respectively, has calculated for the last transmission. If the last fire event of was at , then with respect to the communication delay received the phase at and consequently adds the offset leading to . Similarly, a fire event from with offset is received by at phase . Let be the minimum, respectively, maximum possible phases of the calculated firing events. If , then it is guaranteed that , respectively, . Since , we have as stated.

Based on the current precision and the phase advance of and at time labeled by and , we are able to calculate the precision the next time reaches the threshold. That is, . However, we have to distinguish between three cases depending on . In detail, if , then and , or if , then also and , or finally if , then due to Line 4 of Algorithm 1 we have and . Note that the overlapping of and is volitional, because if , then both cases can occur and hence must be considered. Further note that the bound of ensures that the interception point of the phase of both nodes is within the last period. In order to keep the clocks within the precision, the inequality must be valid for all three cases. From the first case we get and . From the third case it follows and . Note that is always valid due to the definition of . From the second case, it can be derived that and . Again, ensures that is valid. The worst case precision with respect to these three cases then equals .

Note that the correctness of the proof requires that a node advances its phase at most once per period. However, if , then may initiate a firing event after already passed the threshold. Simply setting avoids this effect.

In order to get the worst case precision, we further have to incorporate the precision (I) and (II) for all three mentioned cases. In detail, for we additionally have to analyze for case if the equation holds and for case , if and are valid. Similarly for it must be ensured that . From these equations we can derive the following additional bounds: , and . Therefore, if we want bounded between , then must hold. Furthermore, in the case of , we have to adapt the worst case precision to which now equals the worst case upper bound, since all possible cases were considered.

Finally, it should be mentioned that the maximum relative message staggering delay must be smaller than . Otherwise, assume the case where both nodes are initially apart. Then both nodes will never perform a phase advance due to Line 4 of the algorithm.

Note that in the case of a fully connected network comprising more than two nodes, all nodes synchronize to the fastest one due to Line 4 and Line 9 of Algorithm 1. Especially the condition in Line 9 ensures if a node advances its phase due to some received firing event , then all events immediately following some short time after are ignored. This condition is necessary. Otherwise, assume nodes are perfectly synchronized. Consequently, a node would perform times a phase advance, which results in a mutual excitation in the case is very large.

Theorem 3.2.

Let and be the drift offset. In the case of clocks and no message loss, if the coupling factor is lower bounded to and , then for and , Algorithm 1 keeps the network synchronized with a worst case precision bounded to

(7)

Corollary 3.3.

If a fully connected network comprises only of perfect clocks ( ) and the communication network suffers from no delay jitter ( ), then the network keeps synchronized with a precision of , if .

Note that Corollary 3.3 states that it is sufficient that the network is connected.

Corollary 3.4.

If a fully connected network comprises only of perfect clocks ( ) and the communication network suffers only from delay jitter ( , ), then the network keeps synchronized with a precision of , if .

Corollary 3.5.

If a fully connected network comprises of clocks with a maximum drift rate of and the network suffers from no communication delay ( ) and , then the network keeps synchronized with a precision of , if .

Upper Bound for the Coupling Factor

One may ask why not setting such that a node immediately adjusts its phase to a neighboring clock every time receiving a firing message from this clock. However, the following lemmata shows that there exists a basic upper bound which holds for every network.

Definition 3.6.

A firing configuration of a fully connected network comprising nodes is defined to be the concatenation of the phase of node at the time when just reached the threshold for the th time and consequently applied the phase advance .

Lemma 3.7.

In a fully connected network comprising of perfect clocks, if the coupling factor , then the nodes may never become synchronized.

Proof.

The proof is based on the fact that if is too large, then the nodes will infinitely often enter the same firing configuration. Let and be the two participating processors where is the first node reaching the threshold. The initial firing configuration then is with . Next, reaches the threshold leading to with and . The next time reaches the threshold is at with and . Finally again reaches the threshold at with and .

If we assume that , then the phase advance can be reduced to . The same applies to and . Thus, if all three conditions are true, can be redefined to . In other words, the nodes will infinitely often enter the initial firing configuration. We now have to find the lowest where the inequation is valid. Equalizing all three conditions yields and . Thus we get .

Since the algorithm ignores all firing events immediately following some short time after a previous firing event due to Line 9, a node may realize a set of nodes as a single node and therefore Lemma 3.7 also applies to networks comprising more than two nodes. We now exploit the intuition behind Lemma 3.7 and extend this problem to a general network comprising nodes.

Definition 3.8.

is called to be an infeasible firing configuration, if there exists a positive integer such that and the network is not synchronized.

Lemma 3.9.

The maximum phase advance a node can perform in a fully connected network comprising nodes equals .

Proof.

The maximum phase advance occurs if the firing events are at close quarters such that no event is ignored due to Line 9 of Algorithm 1. In detail, assume a node received the firing event at the phases . The first phase advance then equals , where . Due to Line 9 of Algorithm 1, the earliest next time the node performs a phase advance can only be at and equals . Generally, and for . Solving the recursion leads to and thus . Solving the equation for then yields . The overall phase advance thus equals . Since the maximum occurs when , we finally get .

A weak upper bound results from the fact that we do not want a node to perform a phase advance which is greater than and directly follows from Lemma 3.9.

Corollary 3.10.

In a fully connected network comprising of perfect clocks, if the coupling factor , then in every feasible execution a node will never perform a phase advance which is greater than .

Note that even if the weak bound is maintained, it can be shown that there exist infeasible firing configurations. However, due to imprecisions in calculations, the varying short-term drift, the delay jitter, and due to several other indeterministic environmental effects, this bound is generally applicable. A stronger bound results from empirical studies which have shown that infeasible firing configurations do not exist, if the maximum phase advance . The resulting bound for again can be deduced from Lemma 3.9.

Theorem 3.11.

In a fully connected network comprising of perfect clocks, if the coupling factor , then the nodes will never enter an infeasible firing configuration.

Rate of Synchronization

Theorem 3.15 analyzes the time to sync for the case of two oscillators. The authors of [4] have also analyzed the case of oscillators. However, considering a multihop topology requires a more sophisticated solution. For the following proofs, let and denote the initial phase difference between the clocks and with in network .

Lemma 3.12.

The infeasible firing configuration with and is a unique fixpoint and has a phase difference of .

Proof.

If we set , we get and and thus and .

Although this fixpoint is a repeller, the roundoff error in the calculation may cause a node to enter the fixpoint. This is especially a concern if the granularity of the hardware clock is very low. The rate of sync with respect to different initial phase differences is visualized in Figure 4. It is obvious that there exists a special initial configuration which causes the network to enter this fixpoint. To analyze this initial configuration, we first transform the recursion of the dynamic system into a closed term.

Lemma 3.13.

The phase difference of for equals

(8)

where , , , , , , and from Lemma 3.12.

Proof.

Let be the initial firing configuration with where and . The phase difference when reached the threshold for the th time is . From Lemma 3.7 we know that with and . If we substitute for and consider the phase difference of , we get and which yields for . The dissolving of the recursion is left to the reader and leads to the solution as stated.

Lemma 3.14.

There exists a unique initial phase difference where the network eventually enters the fixpoint of Lemma 3.12 and equals with from Lemma 3.13.

Proof.

If the network enters the fixpoint in at some , then we have a phase difference of for with from Lemma 3.12. Using (8) then yields . Since we get and thus . Using and from Lemma 3.13 results in . The initial phase difference then has to be as stated.

Theorem 3.15.

The number of iterations until synchrony is at most with , and from Lemma 3.13 and

(9)

Proof.

Note that either converges to or ) as visualized in Figure 4. Therefore, we simply equate (8) with if or with if . Since for and the multiplicative factor is smaller than , the term with respect to does not influence the rate of sync for larger and hence can be neglected. This leads to the equation as stated.

3.2. Clock Rate Calibration

The concept of clock rate calibration combats the problem of frequency deviations due to the high clock drift of the RC-oscillators usually used in low-cost devices. This approach should allow a longer resynchronization interval with the same synchronization precision. Note that the rate correction can be performed completely independent from the clock state correction scheme.

The core concept of our rate calibration algorithm is that a processor implements a virtual clock which abstracts the hardware clock . The algorithm implemented on then only reads the time from the . We further denote the ticks from the by microticks and that from the by ticks. One tick of comprises several microticks which we denote by . By adjusting , the time duration of one tick can be increased or decreased. Let be the nominal threshold level and the absolute adjustment value s.t. . Note that the corresponding relative adjustment value for equals . In order to perform the rate calibration, every processor periodically broadcasts a synchronization messages . Let , respectively, denote the timestamps of when transmitted and , respectively, the timestamps of when received from . Let be the th message received from and the th message broadcasted. We further assume that is not received at some before is received for . The dependency between the virtual and the hardware clock with respect to some message is characterized as . Note that we assume that the hardware clock is a linear function of real time within a sufficient long period of time. In contrast, the virtual clock is periodically reset with respect to the resynchronization interval. This assumption is required in order to realize a pulse synchronization scheme.

The rate correction algorithm works as follows: based on the timestamp stored in , the receiving processor can calculate 's relative adjustment value in its own granularity, denoted by , with . Therein, the term denotes the latest received adjustment value from , that is, the relative adjustment value contained in the latest received message from . In order to reduce the impact of the delay jitter, we should choose the time interval between the two received messages as large as possible. Note that there still exists an upper limit due to the long-term stability of an oscillator, which is usually in the order of minutes. However, the optimal time interval also depends on the underlying oscillator type. In our case, we store the last received messages from each node and calculate the relative deviation with respect to the buffer size . To visualize the impact of the delay jitter, we replace by , where corresponds to the delay of message in 's clock granularity. From this it follows:

(10)

In our implementation we set reducing the impact of the jitter with respect to the resynchronization period to at about . The next step of the algorithm is to calculate the average relative phase adjustment value of all processors in the broadcast domain where , that is,

(11)

The main challenge, however, concerns the adjustments of to the new relative adjustment value of . Due to natural imprecisions and the delay jitter, a direct adjustment of is inappropriate resulting in a continuous increase or decrease of the real overall average relative adjustment value. This effect is also known as the common-mode drift. A better approach is to implement a parametrized adjustment by the use of a smoothing factor as shown in (12) which ensures that the virtual clocks smoothly converge to the overall average interval,

(12)

In our implementation we have chosen . However, it seems that the common-mode drift cannot be avoided but the effect can be minimized by carefully choosing with respect to the delay jitter. For this reason we have introduced bounds for the relative adjustment value, that is, . Furthermore, if the bound is exceeded, respectively, under-run, we decrease, respectively, increase by some small value before calculating .

3.3. Energy Awareness

The energy consumption is an important quality characteristic of each communication protocol used in sensor networks. Often more than 50 percent of energy is used for idle listening [7]. Therefore, it is necessary to reduce the major energy sources. Some Media Access Control (MAC) protocols have already incorporated such a concept (e.g., S-MAC, T-MAC, etc.). However, we assume that the underlying MAC layer is only responsible for the medium access control and not for energy improvements. For this reason, we assign the tasks for energy reduction to the upper layers.

A usual approach in reducing the consumed power is to periodically turn off the transceiver module if it is not required. A protocol using such a scheme is called a duty-cycle protocol. The duty-cycle is determined to be the ratio between the duration used for listening on synchronization messages to the medium and the duration of the complete period. As already mentioned, the bounded synchronization precision necessitates that the receiver module must be enabled some time prior, before any transmission takes place. This safety margin equals the synchronization window and should be greater than the upper bound of the synchronization precision. A node considers itself to be synchronized, if the maximum absolute deviation to all neighboring nodes is smaller than the specified synchronization window. With respect to the relative message staggering delays , respectively, and the period length , the duty-cycle equals . Note that after a number of periods, each node has to listen to the medium for a full period in order to avoid clique building.

4.1. Experiments and Results

The simulation results discussed in this chapter should give an overview of the achievable quality of our synchronization approach. For this reason, several network topologies have been developed and simulated. The results are compared with respect to different parameter choices, that is, the coupling factor and the number of nodes in the network.

In order to compare the simulation results with the outcomes from [3], the evaluation metrics are similar. Therefore, the two important parameters are the amount of time until the system achieves synchronicity and the quality of precision. The time to sync defines the time until all nodes have entered the synchronization state and is determined by two parameters. These are the synchronization window and the number of required periods a node has to keep within this window. In the following we call the amount of required periods synchronization periods and is set to . A node only enters the sync-state, if the maximum absolute deviation with respect to the other nodes is within the synchronization window for out of the last firing iterations. The definition of the 50th and 90th Percentile Group Spread differs from that one defined in [3], because we only refer to one firing group. Therefore, the group spread in the simulation is defined to be the maximum absolute time difference between any two nodes in the network and thus cannot be greater than half the synchronization interval. We characterize the group spread distribution with the 50th and 90th percentile.

Incorrect results due to settling effects during the startup phase are avoided by postponing the start of the group spread measurement against the time to sync ( ) and the time the experiment ends ( ). On this account, the group spread measurement is performed only during the interval .

4.1.1. Parameter Settings

Several parameter settings are the same for all experiments and are adapted to simulate the behavior of our testbed environment. For instance, every virtual node is based on a virtual RC-oscillator. According to the datasheet, the real nodes have a nominal frequency of 8 MHz 10%. For this reason, every virtual node encounters a random initial clock drift between −100  milliseconds and +100  milliseconds per second. The general values of the other parameters are denoted in Table 2.

Parameter	Value
Oscillator technology	RC-oscillator
Initial clock drift [ppm]
Interval time [milliseconds]
Granularity (ticks/period) [ ]
Minimum message staggering delay [milliseconds]
Maximum message staggering delay [milliseconds]
Coupling factor
Constant transmission delay [milliseconds]
Maximum delay jitter [milliseconds]
Synchronization window [milliseconds]
Evaluation end [periods]

4.1.2. Simulation Results

The next paragraphs discuss the simulation results in dependence of several network topologies and parameter choices. Every configuration was simulated over periods.

The All-to-All Topology

The all-to-all communication topology is mainly used to measure the quality of the synchronization in dependence of the number of nodes and the coupling factor . Therein, every node is in the transmission range of each other.

The simulation results based on this topology give a good overview on the impact of different coupling factors. According to the diagrams in Figure 5, the time to sync decreases with an increasing coupling factor . If the factor is too big, then synchronicity will not be achieved. This effect is due to the upper bound of the coupling factor. Table 3 summarizes the upper bound for different values of with respect to Corollary 3.10. The stronger bound of Theorem 3.11 was neglected due to the fact that most network simulations using the weak upper bound with a random inital configuration have achieved synchronicity. The time to sync calculated in Table 3 are based on Theorem 3.15 for an initial maximum phase difference of . Note that a constant offset of was added so that the values can be compared with the simulation result, because the simulator declares a set of nodes synchronized only if they are within some precision for at least consecutive periods. The high time to sync bar in Figure 5(a) with and a number of 20 nodes comes from the fact that the coupling factor was too high.

Number of nodes	5	10	20	50	100
Upper bound for
Coupling factor	1.15	1.1	1.05	1.01	1.005
Estimated time to sync [s]

If we assume that the rate calibration scheme reduces the worst case drift to , then based on the parameters from Table 2 and Theorem 3.2, the worst case precision for equals . Note that without the rate calibration scheme, we would have . The group spread diagram complies with our theoretical result that the precision does not depend on if maintains the lower bound. Note that the simulations are based on a realistic radio model considering message collisions and transmission strength as well as the backoff scheme of the MAC layer. Therefore, the results show that if all network parameters (e.g., transmission delay, maximum drift rate, etc.) are correctly defined, then the worst case precision is maintained most of the time. However, there may exist outliers in the case of an omission failure of the fastest node, because this algorithm does not allow a node to adjust its clock backward.

The Multihop Topology

This communication topology is the most important one, because in reality many sensor networks are based on a source-to-sink communication topology with a communication path consisting several hops. The simplest multihop scenario is a network comprising nodes, which are ordered in a chain and can only communicate with the immediate neighbors. We further call the chain size network diameter. Such a network with a big network diameter is often very problematic to synchronize, because every hop involves a communication delay which degrades the overall synchronization precision. Therefore, an estimation for the achievable precision with respect to Theorem 3.2 in a network with the network diameter is about . Our solution is based on grouped multihop networks. Therein, the nodes are replaced with groups comprising several nodes in all-to-all topology which all have a bidirectional communication link to all nodes in the immediate neighboring groups. Figure 6 is a snapshot of a running simulation regarding the grouped multihop topology with JProwler. Therein, a dot represents a node and an arrow visualizes that a node is currently transmitting data. The gray scale visualizes the deviation. Note that all grouped multihop topologies treated in our experiments have the same network diameter of 10 hops but vary in the group size.

The diagrams in Figure 7 shows the time to sync and the group spread in dependence of a different group size and coupling factor. The network diameter is always 10. These diagrams lead to the result that the precision increases with a bigger group size. This effect is caused by the better information about the interval drift due to the increased number of neighboring nodes. If a node has more neighboring nodes, then the node receives more information about the clock drift and can more precisely calibrate the interval duration, which also improves the synchronization precision. However, it is also important to have a preferably small coupling factor. On the one hand this increases the time to sync, but on the other hand this also increases the possibility that the network achieves synchronicity. To sum up, it is difficult to find the best parameter settings for a given multihop network, but it is definitely a good choice to have a group size of more than one node. This also increases the dependability and availability of the network.

5.1. Testbed Description

The testbed is based on Atmel's demonstration kit ATAVRRZ200 [9]. The kit features two component boards: The Display Board and the Remote Controller Board (RCB)s. The Display Board is based on an Atmega128 controller and features an LCD-module. This board also works as a docking station for programming the RCBs. The RCBs therefore are based on an Atmega1281 controller and contain an AT86RF230 (2450 MHz band) radio transceiver. The implementation of our approach is done with the AVR Z- 802.15.4/ZigBee nodes. The information about the synchronization precision is gathered via the established TDMA scheme. Therefore, beside energy savings, the time-triggered approach also serves as an evaluation protocol. For this, we used a modified version of the TTP/A protocol [10].

The synchronization algorithm was implemented analogously to the implementation in JProwler. A simple RC-oscillator-based 16-bit timer was used to generate the synchronization interval with a duration of one second. The only differences with respect to the parameter choices in Table 2 are a higher granularity of the virtual clock (31250 ticks/period) and a higher transmission delay of 896 microseconds.

We modified the initial settings of the MAC sublayer, that is, the minimum backoff exponent, to reduce the transmission delay. We further assumed that it is better to omit a message than to transmit postponed synchronization data since the underlying MAC layer does not support MAC timestamping. However, several measurements have shown that regardless of this configuration, some messages are still transmitted with a high delay. Therefore the tradeoff between the probability of an omission failure and a high delay jitter with respect to the precision degradation has to be chosen. In detail, whereas high delay jitter only degrades the deterministic worst case precision, omission failures may mostly provide a better precision, though it is more indeterministic and therefore results in a worse worst case precision. Note that omission failures are especially a problem in the case they occur at the fastest node and may degrade the worst case precision by at about , where denotes the maximum number of consecutive omission failures at the same node. In our implementation, every node is configured as a Full-Function Device (FFD) and no association process is required. This necessitates the use of individual predefined 16-bit short addresses for each node.

5.2. Experiments and Results

The evaluation metrics for the testbed experiments are similar to the one used for the simulation experiments. In order to observe the relative deviations over all nodes in a network, we decided that every node transmits its own maximum absolute deviation of the last period to a central evaluation node, which then calculates the maximum over all received deviations. These values over several minutes are then taken to compute the 50th and the 90th percentile group spread.

To be able to compare the testbed results with the simulator results, the parameter configuration has to be the same as used in the simulator experiments. Since in reality it is not possible to speedup the time, we reduced the experiment time to 720 periods.

5.2.1. Testbed Results

The All-to-All Topology

For the all-to-all topology experiment, we again measured the group spread in dependence of several coupling factors.

Table 4 contains the simulation results and the testbed results with the same network configuration comprising 5 nodes in all-to-all topology. This demonstrates that the results are similar. For instance, the time to sync and the 50th percentile group spread of the testbed system are mostly better than the simulation results. Furthermore, the calculated worst case precision of is also maintained by the 90th percentile group spread, even though it is worse compared to the simulation results. The outliers come from the fact that the delay jitter of the communication delay due to the MAC stack was sometimes higher than expected. Note that we have already compensated the constant delays in the implementation. However, there may exist other delays which we have not considered. Simulation experiments have shown that a higher transmission delay or a higher delay jitter are the major reasons for the precision degradation due to the state correction algorithm and hardly affect the rate calibration scheme. Furthermore, the worst case precision may also result from an omission failure from a synchronization message of the fastest node. The results from an all-to-all topology comprising 9 nodes are comparable with those denoted in Table 4. Note that only a maximum number of 9 nodes were available for our experiments.

Parameter choice	Time to sync	50th percentile	90th percentile	Maximum deviation	Standard deviation
	(periods)	( )	( )	( )	( )
	105 (152)	672 (1000)	2005 (1300)	3456 (2200)	538 (257)
	79 (57)	704 (900)	1632 (1300)	2592 (2000)	410 (250)
	24 (35)	704 (900)	1973 (1300)	3040 (1900)	501 (262)
	33 (20)	672 (1000)	1723 (1400)	3104 (2000)	451 (267)
	14 (20)	732 (900)	1965 (1300)	3776 (1800)	565 (250)

The Multihop Topology

The results from the multihop experiments are important in order to get an overview about the limits of our synchronization approach. The first scenario was made up of 5 nodes ordered in a chain, where a node can only communicate with the immediate neighbors. The only difference between the simulation and the testbed environment is that the testbed environment does not have an omniscient observer, which is able to continuously measure the synchronization deviation among all nodes. For this reason, we decided to measure the time difference between the edge nodes with the aid of an oscilloscope, whereas each node periodically sets an output pin at the same phase state for a short time. Unfortunately, these measurements cannot be gathered automatically over several periods. Therefore, we manually made snapshots over several minutes and took those diagrams, which display the biggest time deviation. To simulate the multihop network, we have simply implemented a message filter.

The results show that the precision of a realistic multihop network with 4 hops is about 3  milliseconds and even maintains the calculated worst case precision of a fully connected network multiplied by . Interestingly, the simulation of the same network with a configured delay jitter of 1250 microseconds leads to the same result. To get an overview of the precision degradation with respect to the network diameter, another multihop experiment with 9 nodes was performed. The measurement setup is similar to the previous multihop network. The measurement results show a maximum deviation between the edge nodes of up to 14  milliseconds and is better than we have observed during the simulation where synchronicity was sometimes not achieved at all. Note that the measured worst case precision is again maintained by the calculated worst case precsion for a fully connected network multiplied by . However, such a high deviation was measured very seldom. In summary, the worst case precision of our synchronization algorithm degrades by at most the worst case precision for the corresponding fully connected network with each hop. Note that with respect to [3], the results can be compared with the simulation results regarding a regular grid topology containing nodes. Therein, they have measured a 90th percentile group spread of about 10  milliseconds. Compared to their results, our worst case deviation of 14  milliseconds is not so digressive.

We further measured the behavior of a grouped multihop network as shown in Figure 8 comprising 3 groups with a group size of 2 and additional two-edge nodes which have a communication link to the corresponding two edge groups. This was the only acceptable configuration with a number of 9 available nodes. The maximum deviation between the edge nodes measured over about 10 minutes was 7  milliseconds. In the simulator we had to configure a delay jitter of 7  milliseconds or in the case of no delay jitter an uncompensated additional transmission delay of 1.5  milliseconds to get the same result. Therefore, it is highly likely that beside the delay jitter, the testbed environment additionally suffers from a longer communication delay, which was not regarded in the current implementation. Unfortunately, due to the limited number of nodes, we were not able to make further experiments with a bigger group size. Thus we must rely on the simulation experiments which show us that a bigger group size usually results in a better synchronization precision.

5.2.2. Energy Measurements

The energy consumption plays an important role for the device lifetime in battery-powered wireless networks, especially if no infrastructure is available. All RCBs are battery-powered with two 1.5 V AAA-batteries and thus have a voltage supply of 3 Volt. In order to get the device lifetime, the average power consumption is compared with the electrical energy of the batteries, which we assume to be about . The lifetime in hours is the ratio and can be reduced to the equivalent formula , where corresponds to the battery charge, denoted in mAh. If so, then the formula determines the device lifetime in hours and defines the average current consumption.

For further energy calculations, the current consumption during a complete period can be classified into four parts and were measured by the use of an oscilloscope and a current shunt resistor. These are the firing time, idle time, execution time, and transmission time.

The firing time results from the message staggering delay and corresponds to the interval where the transceiver is enabled and the nodes are allowed to transmit their synchronization messages. In our test application, this interval is also called part 1 of the firing time and equals the duration of 50  milliseconds. The consumed current during this time is about 20 mA. Note that there exists an interval between the end of the firing time and the period end which acts as a safety margin in the case a node starts a transmission exactly at the end of the firing time. If so, the transceiver must be enabled as long as the transmission continuous. This safety margin is further named part 2 of the firing time and consumes a current of 24 mA.

The idle time is the part, where the current drops to a minimum. The reason for the small current lies in the fact that the device is dormant, that is, the transceiver is disabled. With our ZigBee nodes, we measured a current of about 6.2 mA.

The execution time is the time where the RCB device executes some code. This is always the case at the end of each period, where the device has to execute the RFA. Other execution tasks must be configured in the RODL file. In the test application used for the energy measurement, the RODL file only contains one execution slot in each period. This task is responsible for data preparation. We measured a current of about 11 mA for a duration of 1  milliseconds. This energy part mainly depends on the amount of code of the executed tasks.

The transmission time corresponds to the time, where the device is transmitting data. Normally, this is always the case when the RCB wants to broadcast its synchronization message during the firing time. Other transmissions during the period must be registered in the RODL file. We further measure the energy consumption of a registered transmission slot, which is used to broadcast test data. We measured a current consumption of about over a time of . Note that this duration does not equal the real transmission time of about . This comes from the fact, that the transceiver requires some time for the startup phase and that the Carrier Sense Multiple Access with Colission Avoidance (CSMA/CA) scheme at the MAC layer may also cause some delay although the backoff scheme was disabled.

Table 5 sums up all different energy consumers with the corresponding battery discharge in mAs. In the case the period duration is exactly one second, the values defined in this table results in a battery discharge per cycle of about . Thus, the average current consumption also equals . Assuming that our batteries deliver a charge of about , then the resulting lifetime ( ) in hours can be calculated as follows:

(13)

The configured duty-cycle for this result is about 7 percent, but could be reduced by increasing the period time. If we assume that the time slices of the other consumers are for the most part constant, then a larger period time induces also a larger idle time. Note that a larger period time usually also entails a degradation in precision. The duty-cycle is defined to be the ratio between the sum of the two firing times ( ), the execution time ( ), and the transmission time ( ) and the complete period time ( ). Thus, the duty-cycle is hereinafter denoted by DC and corresponds to the equation

Energy consumer	[mA]	[s]	Battery discharge per cycle	Energy consumption per cycle
Firing time, part 1
Firing time, part 2
Execution time
Transmission time
Idle time

To follow up on our special energy example, we further want to calculate the improvement of the lifetime with respect to the lifetime as if no synchronization approach would be established, that is, the duty-cycle equals 100%. In that case, the average current consumption equals 23.752 mA. Consequently, a duty-cycle of 100% corresponds to a lifetime of about 50.5 hours. A comparison among the lifetime with a duty-cycle of 100% and the achieved lifetime with our configured duty-cycle of about 7% shows that the synchronization approach improves the lifetime by at least a factor of three.

To illustrate the dependence between the lifetime improvement and the period time, we introduce the improvement factor, denoted by . This factor is the ratio between the improved lifetime and the reference lifetime corresponding to a duty-cycle of 100% at the same period of time. The improvement factor equals and as visualized in Figure 9.