**3.3. Performance evaluation**

Results highlighted in this section have been computed with the simulation model presented in Section 2.5 in the scenarios depicted in Fig. 2. The aim of this section is to demonstrate the regulative character and the interference compensation functionality of the PRC approach. Therefore, its performance under increasing offered system load is compared to the one obtained with the ALOHA MAC protocol without feedback. Note that the IEEE 802.15.4a proposes ALOHA with optional feedback as the standard MAC protocol for IR-UWB physical layers.

0.0e 5.0e4 1.0e5 1.5e5 2.0e5 2.5e5 3.0e5

character of the link's BER.

*Pe*,*<sup>i</sup>* ≤ *βCH*, is fulfilled for all links.

time-varying number of players9.

exhibits the best channel gain and the lowest BER.

a random variable controlled by the exponential traffic distribution.

PRF [Hz]

0 20 40 60 80 100 120

0e

0 10 20 30 40 50 60

BER constraint

time [s]

(b)

5e-4

BER

**Figure 3.** Existence of an equilibrium in Scenario 1 - Continuous transmission. Each link is represented with a different color: (a) Convergence to the unique equilibrium *prf* allocation; (b) Upper-bounded

the node topology is symmetric. Figure 3(b) confirms that the QoS constraint on the BER,

**Time-varying number of players** This setting considers the factory hall scenario described in Section 2.4. There are a total of 100 heterogeneous nodes, some of them are moving along a production line, others are fixed in known positions and the rest is moving around the CH following a random waypoint movement model. The information data rate has been chosen low enough to guarantee that nodes do not always have packets to send; this results in a

Figure 4 presents results for two values of the information data rate which are 10 kbit/s and 15 kbit/s, respectively. With 10 kbit/s information data rate, the CH is exposed to a low system pulse load measured in pulses per second (Pps). Figure 4(a) suggests the existence of a stable equilibrium in game ΓPRC, and that the PRC algorithm converges to it. The *prf* equilibrium allocation is again symmetric, with all links converging to the maximum possible *prf* of 1 MHz. Figure 4(b) confirms that the QoS constraint on the BER is satisfied for all links. Note that the BER curves of all links except one follow almost an identical course. The outlier curve corresponds to the sensor node which is located closest to the CH (at 3m) and therefore

By increasing the user data rate to 15 kbit/s, the system pulse load achieves a level which is not compliant with the problem's QoS constraint. The PRC algorithm should then reduce the links' pulse rates to a sustainable level, which is identified by the congestion cost factor *πCH*. Figure 4(c) illustrates the regulative effect of the PRC algorithm. In spite of the heterogeneous network character, the *prf* equilibrium corresponds to a *max-min* socially fair allocation induced by the choice of a common pricing factor for all links. Besides, it can be observed that in Figure 4(d) some of the links violate the QoS constraint. Still, if we consider the average BER over all links (see Figure 8(b)) it is below the upper bound *βCH*. This is

<sup>9</sup> Note that the number of players in the game coincide with the number of active links in each superframe, and this is

1e-3

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 67

Analytical NE

time [s]

(a)


**Table 1.** List of main simulation parameters.

### *3.3.1. Existence of an equilibrium*

First a simplified setting with a constant number of players, as usually assumed in game theory, is considered. Then, a more realistic setting where the number of players is a random variable controlled by the traffic distribution is explored. For both settings simulation results<sup>8</sup> confirm the existence of an NE regardless of the algorithm's initialisation parameters.

**Constant number of players** This setting considers the continuous transmission scenario described in Section 2.4. Recall that in this scenario up to 10 nodes are collocated along a circumference of 10m radius with the CH in the centre. It is assumed that all nodes have always packets to send; this is assured by configuring the source traffic generator to an information data rate of 1Mbps.

Figure 3 confirms the existence of a stable equilibrium in game ΓPRC, and that the PRC algorithm converges to it. Figure 3(a) depicts a symmetric *prf* allocation, which agrees with the analytically predicted NE; the equilibrium *prf* per link is approximately 220 kHz. In fact, a symmetric equilibrium was expected since the congestion cost is the same for each link and

<sup>8</sup> These results were obtained in [11].

16 Will-be-set-by-IN-TECH

In this sense, note that the utility function of game ΓPRC in equation 5, combines the network utility function of the original resource allocation problem (cf. to equation 2) with a linear price function. By exploiting the linear space properties of EPGs, the potential function in equation 14 preserves the properties (such as concavity and uniqueness of the global

linear price term aims at adjusting the unique NE of Γ˜ PRC so that the QoS constraint in the original problem is respected. Hence the NE is optimal from a network design perspective.

Results highlighted in this section have been computed with the simulation model presented in Section 2.5 in the scenarios depicted in Fig. 2. The aim of this section is to demonstrate the regulative character and the interference compensation functionality of the PRC approach. Therefore, its performance under increasing offered system load is compared to the one obtained with the ALOHA MAC protocol without feedback. Note that the IEEE 802.15.4a proposes ALOHA with optional feedback as the standard MAC protocol for IR-UWB physical

**Parameter Value Parameter Value**

*<sup>m</sup>*-PPM <sup>2</sup> *<sup>β</sup>CH* <sup>5</sup> · <sup>10</sup>−<sup>4</sup> *Lp* 400 bit *μ* 2 *<sup>δ</sup>* <sup>1</sup> · <sup>10</sup>−<sup>2</sup> *<sup>ω</sup>* 2.5 · <sup>10</sup>−<sup>3</sup>

First a simplified setting with a constant number of players, as usually assumed in game theory, is considered. Then, a more realistic setting where the number of players is a random variable controlled by the traffic distribution is explored. For both settings simulation results8

**Constant number of players** This setting considers the continuous transmission scenario described in Section 2.4. Recall that in this scenario up to 10 nodes are collocated along a circumference of 10m radius with the CH in the centre. It is assumed that all nodes have always packets to send; this is assured by configuring the source traffic generator to an

Figure 3 confirms the existence of a stable equilibrium in game ΓPRC, and that the PRC algorithm converges to it. Figure 3(a) depicts a symmetric *prf* allocation, which agrees with the analytically predicted NE; the equilibrium *prf* per link is approximately 220 kHz. In fact, a symmetric equilibrium was expected since the congestion cost is the same for each link and

confirm the existence of an NE regardless of the algorithm's initialisation parameters.

*<sup>i</sup>* 1 [MHz] *prf min*

*<sup>p</sup>* <sup>2</sup> · <sup>10</sup>−<sup>11</sup> [Ws] BW 1.5 [GHz] *<sup>T</sup> rcx* <sup>5</sup> · <sup>10</sup>−<sup>9</sup> [s] *prf* granularity 1 [kHz]

*<sup>i</sup>* 10 [kHz]

<sup>1</sup> *log*(*ri*(**prf**)). The addition of the

maximisers) of the original network objective function <sup>∑</sup>|*N*<sup>|</sup>

**3.3. Performance evaluation**

*Etx*

**Table 1.** List of main simulation parameters.

*3.3.1. Existence of an equilibrium*

information data rate of 1Mbps.

<sup>8</sup> These results were obtained in [11].

*prf max*

layers.

**Figure 3.** Existence of an equilibrium in Scenario 1 - Continuous transmission. Each link is represented with a different color: (a) Convergence to the unique equilibrium *prf* allocation; (b) Upper-bounded character of the link's BER.

the node topology is symmetric. Figure 3(b) confirms that the QoS constraint on the BER, *Pe*,*<sup>i</sup>* ≤ *βCH*, is fulfilled for all links.

**Time-varying number of players** This setting considers the factory hall scenario described in Section 2.4. There are a total of 100 heterogeneous nodes, some of them are moving along a production line, others are fixed in known positions and the rest is moving around the CH following a random waypoint movement model. The information data rate has been chosen low enough to guarantee that nodes do not always have packets to send; this results in a time-varying number of players9.

Figure 4 presents results for two values of the information data rate which are 10 kbit/s and 15 kbit/s, respectively. With 10 kbit/s information data rate, the CH is exposed to a low system pulse load measured in pulses per second (Pps). Figure 4(a) suggests the existence of a stable equilibrium in game ΓPRC, and that the PRC algorithm converges to it. The *prf* equilibrium allocation is again symmetric, with all links converging to the maximum possible *prf* of 1 MHz. Figure 4(b) confirms that the QoS constraint on the BER is satisfied for all links. Note that the BER curves of all links except one follow almost an identical course. The outlier curve corresponds to the sensor node which is located closest to the CH (at 3m) and therefore exhibits the best channel gain and the lowest BER.

By increasing the user data rate to 15 kbit/s, the system pulse load achieves a level which is not compliant with the problem's QoS constraint. The PRC algorithm should then reduce the links' pulse rates to a sustainable level, which is identified by the congestion cost factor *πCH*. Figure 4(c) illustrates the regulative effect of the PRC algorithm. In spite of the heterogeneous network character, the *prf* equilibrium corresponds to a *max-min* socially fair allocation induced by the choice of a common pricing factor for all links. Besides, it can be observed that in Figure 4(d) some of the links violate the QoS constraint. Still, if we consider the average BER over all links (see Figure 8(b)) it is below the upper bound *βCH*. This is

<sup>9</sup> Note that the number of players in the game coincide with the number of active links in each superframe, and this is a random variable controlled by the exponential traffic distribution.

due to the global altruistic behaviour of the congestion cost factor, *πCH*, which works with an estimation of the average10 BER at each stage of the game.

In general, and due to the increasing MUI, the aggregated network throughput is expected to drop as the number of nodes in the network grows. Figure 5(a) illustrates the aggregated network throughput. It can be observed that with four to ten source nodes the aggregated network throughput remains almost constant. These results suggest an interference compensation effect of the PRC algorithm; Figure 6 confirms this effect. Additionally, Figure 5(b) shows that the cumulative BER scratches the QoS upper bound in all cases except in the case of having only two nodes. In this special case, even with nodes sending with the maximum *prf* the offered system load is low enough to guarantee an average BER far below

1e-4

2 4 6 8 10

Number of nodes (b)

3e-4

BER

**Figure 5.** System performance in Scenario 1 - Continuous transmission: (a) Aggregated network throughput measured at the CH's application layer; (b) Average BER per link measured at the CH.

factor, *πCH*, is. Finally, larger congestion factors lead to lower *prf* levels.

Figure 6 shows the mean11 *prf* per link; where each link has been represented with a different color. It can be observed that the PRC algorithm relaxes the mean *prf* per link as the number of nodes in the network grows, since consequently the pulse density in the system increases. Recall that a higher pulse density raises the probability of pulse collisions and this, in turn, the probability of bit errors. The larger the average BER at the CH, the larger the congestion

**Time-varying number of players** Since in Scenario 2 - Factory hall the number of sensor nodes is constant, in order to increase the offered system pulse load we progressively raise

In Figure 7 two different phases can be identified. As long as the information data rate is kept below 10 kbit/s, the mean *prf* per link does not drop, but remains close to the maximum allowed *prf* value (1 MHz). When the information data rate reaches 15 kbit/s, the mean *prf* per link drops down to approximately 20 kHz and remains there despite growing information data rate. These results are consistent with the PRC algorithm's behaviour illustrated in

In a similar way Figure 8 illustrates the regulative behaviour of the PRC algorithm. At information data rates up to 10 kbit/s the PRC algorithm converges to the maximum possible

5e-4

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 69

the QoS upper bound.

0.0e

2 4 6 8 10

Number of nodes

(a)

the information data rate per sensor node.

Figure 4(a) and Figure 4(c).

<sup>11</sup> Over the whole simulation time.

5.0e5

1.0e6

Throughput [bit/s]

1.5e6

**Figure 4.** Existence of an equilibrium in Scenario 2 - Factory hall. Each link is represented by a different color: Graphics (a) and (c) show the convergence to the unique equilibrium *prf* allocation when the information data rate per user is 10 kbit/s and 15 kbit/s, respectively; Graphics (b) and (d) show the temporal evolution of the BER per link when the information data rate per user is 10 kbit/s and 15 kbit/s, respectively.

### *3.3.2. Variation of the offered load*

The results in this clause show the effect of increasing offered system load on the aggregate network throughput and the average BER, both measured at the CH. Like in the previous clause, the game performance with a constant number of players is differentiated from the that with a time-varying number of players.

**Constant number of players** The information data rate is fixed and equal to 1 Mbps to ensure continuous packet transmission. In order to progressively increase the offered system pulse load, the number of sensor nodes collocated along the circumference of 10 m radius has been stepped up from 2 to 10.

<sup>10</sup> Over all links.

In general, and due to the increasing MUI, the aggregated network throughput is expected to drop as the number of nodes in the network grows. Figure 5(a) illustrates the aggregated network throughput. It can be observed that with four to ten source nodes the aggregated network throughput remains almost constant. These results suggest an interference compensation effect of the PRC algorithm; Figure 6 confirms this effect. Additionally, Figure 5(b) shows that the cumulative BER scratches the QoS upper bound in all cases except in the case of having only two nodes. In this special case, even with nodes sending with the maximum *prf* the offered system load is low enough to guarantee an average BER far below the QoS upper bound.

**Figure 5.** System performance in Scenario 1 - Continuous transmission: (a) Aggregated network throughput measured at the CH's application layer; (b) Average BER per link measured at the CH.

Figure 6 shows the mean11 *prf* per link; where each link has been represented with a different color. It can be observed that the PRC algorithm relaxes the mean *prf* per link as the number of nodes in the network grows, since consequently the pulse density in the system increases. Recall that a higher pulse density raises the probability of pulse collisions and this, in turn, the probability of bit errors. The larger the average BER at the CH, the larger the congestion factor, *πCH*, is. Finally, larger congestion factors lead to lower *prf* levels.

**Time-varying number of players** Since in Scenario 2 - Factory hall the number of sensor nodes is constant, in order to increase the offered system pulse load we progressively raise the information data rate per sensor node.

In Figure 7 two different phases can be identified. As long as the information data rate is kept below 10 kbit/s, the mean *prf* per link does not drop, but remains close to the maximum allowed *prf* value (1 MHz). When the information data rate reaches 15 kbit/s, the mean *prf* per link drops down to approximately 20 kHz and remains there despite growing information data rate. These results are consistent with the PRC algorithm's behaviour illustrated in Figure 4(a) and Figure 4(c).

In a similar way Figure 8 illustrates the regulative behaviour of the PRC algorithm. At information data rates up to 10 kbit/s the PRC algorithm converges to the maximum possible

18 Will-be-set-by-IN-TECH

due to the global altruistic behaviour of the congestion cost factor, *πCH*, which works with an

5e-4

5e-4

BER

**Figure 4.** Existence of an equilibrium in Scenario 2 - Factory hall. Each link is represented by a different color: Graphics (a) and (c) show the convergence to the unique equilibrium *prf* allocation when the information data rate per user is 10 kbit/s and 15 kbit/s, respectively; Graphics (b) and (d) show the temporal evolution of the BER per link when the information data rate per user is 10 kbit/s and

The results in this clause show the effect of increasing offered system load on the aggregate network throughput and the average BER, both measured at the CH. Like in the previous clause, the game performance with a constant number of players is differentiated from the

**Constant number of players** The information data rate is fixed and equal to 1 Mbps to ensure continuous packet transmission. In order to progressively increase the offered system pulse load, the number of sensor nodes collocated along the circumference of 10 m radius has been

1e-3

BER

0 10 20 30 40 50 60

BER constraint

time [s]

0 10 20 30 40 50 60

BER constraint

time [s]

(d)

(b)

1e-3

estimation of the average10 BER at each stage of the game.

0 10 20 30 40 50 60

Analytical NE

time [s]

0 10 20 30 40 50 60

Analytical NE

time [s]

(c)

(a)

 0e 2e5 4e5 6e5 8e5 1e6

> 0e 2e4 4e4 6e4 8e4 1e5

15 kbit/s, respectively.

*3.3.2. Variation of the offered load*

stepped up from 2 to 10.

<sup>10</sup> Over all links.

that with a time-varying number of players.

PRF [Hz]

PRF [Hz]

<sup>11</sup> Over the whole simulation time.

20 Will-be-set-by-IN-TECH 70 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks <sup>21</sup>

0.0e 2.0e5 4.0e5 6.0e5 8.0e5 1.0e6 1.2e6 1.4e6

0.0e

0.0e

5.0e5

1.0e6

Throughput [bit/s]

1.5e6

2.0e6

5.0e5

1.0e6

Throughput [bit/s]

1.5e6

2.0e6

Throughput [bit/s]

0.5 1 5 10 15 30 50

0e 1e-4 2e-4 3e-4 4e-4 5e-4 6e-4

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 71

0.0e 4.0e-4 8.0e-4 1.2e-3 1.6e-3 2.0e-3

0.0e 4.0e-4 8.0e-4 1.2e-3 1.6e-3 2.0e-3

BER

BER

**Figure 9.** Performance comparison of ALOHA with and without PRC: (a) Aggregated network throughput in Scenario 1; (b) Average BER per link in Scenario 1; (c) Aggregated network throughput in

Figures 9(a) and 9(b) depict a performance comparison between ALOHA with and without PRC in the continuous transmission scenario with high offered pulse load and constant

BER

**Figure 8.** System performance in Scenario 2 - Factory hall: (a) Aggregated network throughput measured at the CH's application layer; (b) Average BER per link measured at the CH.

0.5 1 5 10 15 30 50

User Data Rate [kbit/s] (b)

without PRC with PRC

2 4 6 8 10

Number of active links

0.5 1 5 10 15 30 50

User Data Rate [kbit/s]

(d)

(b)

without PRC with PRC

User Data Rate [kbit/s]

2 4 6 8 10

Number of active links

0.5 1 5 10 15 30 50

User Data Rate [kbit/s]

(c)

Scenario 2; (d) Average BER per link in Scenario 2.

(a)

without PRC with PRC

(a)

without PRC with PRC

**Figure 6.** Average *prf* per link in Scenario 1 - Continuous transmission.

*prf* per link, since the offered system pulse load is low enough to guarantee the QoS constraint on the BER (see Figure 8(b)). With higher information data rates, the PRC algorithm has to relax the effective system pulse load per user to avoid network congestion, and to be able to satisfy the QoS constraint.

**Figure 7.** Average *prf* per link in Scenario 2 - Factory hall.

### *3.3.3. Performance comparison with IEEE 802.14.5a MAC*

Finally, this clause is dedicated to compare the performance of the ALOHA MAC protocol, as described in IEEE 802.15.4a, with and without distributed PRC. For the simulations without PRC we have set the fixed *prf* per link to 1 MHz. IEEE 802.15.4a recommends ALOHA with optional feedback channel as the MAC protocol for low data rate sensor networks with IR-UWB physical layers. This work focuses on random access without feedback channel to keep the power consumption and the receiver complexity at the sensor nodes low. In exchange, the drawback has to be accepted that successful reception of data packets cannot be guaranteed since retransmission requests are not possible. Note that in the application field considered in this chapter, the relatively small throughput offered by the random access method without feedback channel is still satisfying.

20 Will-be-set-by-IN-TECH

2 4 6 8 10

Number of nodes

0.5 1 5 10 15 30 50

User Data Rate [kbit/s]

Finally, this clause is dedicated to compare the performance of the ALOHA MAC protocol, as described in IEEE 802.15.4a, with and without distributed PRC. For the simulations without PRC we have set the fixed *prf* per link to 1 MHz. IEEE 802.15.4a recommends ALOHA with optional feedback channel as the MAC protocol for low data rate sensor networks with IR-UWB physical layers. This work focuses on random access without feedback channel to keep the power consumption and the receiver complexity at the sensor nodes low. In exchange, the drawback has to be accepted that successful reception of data packets cannot be guaranteed since retransmission requests are not possible. Note that in the application field considered in this chapter, the relatively small throughput offered by the random access

*prf* per link, since the offered system pulse load is low enough to guarantee the QoS constraint on the BER (see Figure 8(b)). With higher information data rates, the PRC algorithm has to relax the effective system pulse load per user to avoid network congestion, and to be able to

0e

0e

**Figure 7.** Average *prf* per link in Scenario 2 - Factory hall.

*3.3.3. Performance comparison with IEEE 802.14.5a MAC*

method without feedback channel is still satisfying.

2e5

4e5

PRF [Hz]

6e5

8e5

1e6

**Figure 6.** Average *prf* per link in Scenario 1 - Continuous transmission.

2e5

4e5

6e5

PRF [Hz]

satisfy the QoS constraint.

8e5

1e6

**Figure 8.** System performance in Scenario 2 - Factory hall: (a) Aggregated network throughput measured at the CH's application layer; (b) Average BER per link measured at the CH.

**Figure 9.** Performance comparison of ALOHA with and without PRC: (a) Aggregated network throughput in Scenario 1; (b) Average BER per link in Scenario 1; (c) Aggregated network throughput in Scenario 2; (d) Average BER per link in Scenario 2.

Figures 9(a) and 9(b) depict a performance comparison between ALOHA with and without PRC in the continuous transmission scenario with high offered pulse load and constant

#### 22 Will-be-set-by-IN-TECH 72 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks <sup>23</sup>

number of players. The network throughput obtained without PRC rapidly collapses<sup>12</sup> as the number of source nodes grows. This is comprehensible, since pulse collisions (and therewith bit errors, see Figure 9(b)) augment as the offered pulse load increases. In contrast, the interference compensation effect of the PRC approach under increasing offered load can be well observed. Notice how the PRC algorithm is able to limit the system pulse load to a level that ensures the QoS constraint.

**Author details**

**5. References**

*Germany*

Pérez Guirao María Dolores

*Technical report*.

*802.15.4-2006)* pp. 1–203.

*Conference on* pp. 195–199.

*Wiley & Sons Ltd* vol. 5: pp. 567–580.

[15] *O*MN*e*T*++, Discrete Event Simulation System.* [n.d.].

Hannover, Germany.

and State University.

URL: *www.omnetpp.org*

14: 124–143.

*Fundamentals*, Prentice Hall PTR.

*IEEE Transactions on Communications* 53(9): 1744–1753. [7] Fudenberg, D. & Tirole, J. [2001]. *Game Theory*, MIT Press.

*International Conference on*, pp. 396–405. TY - CONF.

*Institute of Communications Technology (IKT), Leibniz Universitaet Hannover (LUH), Hanover,*

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 73

[1] Boyd, S. & Vandenberghe, L. [2006]. *Convex Optimization*, Cambrigde University Press. [2] Chiang, M., Tan, C. W., Palomar, D., O'Neill, D. & Julian, D. [July 2007]. Power Control by Geometric Programming, *Wireless Communications, IEEE Transactions on* 6(7): 2640–2651. [3] di Benedetto, M. G. & Giancola, G. [2004]. *Understanding Ultra Wide Band Radio*

[4] Everette, S. & Gardner, J. [October-December 2006]. Exponential Smoothing: The State of the Art–Part II, *International Journal of Forecasting* 22, Issue 4: Pages 637–666. [5] Felegyhazi, M. & Hubaux, J.-P. [2006]. Game Theory in Wireless Networks: A Tutorial,

[6] Fishler, E. & Poor, V. [2005]. On the Tradeoff between two Types of Processing Gains,

[8] IEEE *Standard for Information Technology - Telecommunications and information exchange between systems - Local and metropolitan area networks - specific requirement Part 15.4: Wireless Medium Access Control (*MAC*) and Physical Layer (*PHY*) Specifications for Low-Rate Wireless Personal Area Networks (*WPAN*s)* [2007]. *IEEE Std 802.15.4a-2007 (Amendment to IEEE Std*

[9] Le Boudec, J.-Y., Merz, R., Radunovic, B. & Widmer, J. [2004]. DCC-MAC: a Decentralized MAC Protocol for 802.15.4a-like UWB Mobile Ad-Hoc Networks based on Dynamic Channel Coding, *Broadband Networks, 2004. BroadNets 2004. Proceedings. First*

[10] Lovelace, W. & Townsend, J. [16-19 Nov. 2003]. Adaptive Rate Control with Chip Discrimination in UWB Networks, *Ultra Wideband Systems and Technologies, 2003 IEEE*

[11] Luebben, R. [October 2007]. *Spieltheoretische Optimierung des Durchsatzes eines IR-UWB Sensornetzes*, Master's thesis, Institut für Kommunikationstechnik, Leibniz Universität

[12] Merz, R., Widmer, J., Le Boudec, J.-Y. & Radunovic, B. [2005]. A joint PHY-MAC Architecture for Low-Radiated Power TH-UWB Wireless Ad Hoc Networks, *Wireless Communications and Mobile Computing Journal, Special Issue on UWB Communications, John*

[13] Monderer, D. & Shapley, S. [1996]. Potential Games, *Games and Economics Behaviour*

[14] Neel, J. [2006]. *Analysis and Design of Cognitive Radio Networks and Distributed Radio Resource Management Algorithms*, PhD thesis, Faculty of the Virginia Polytechnic Institute

Figures 9(c) and 9(d) depict results in the factory hall scenario with low to moderate pulse load and time-varying number of players. Notice that the regulative effect of the PRC approach (see Figure 7) limits the maximum possible cumulative network throughput, while ALOHA with fixed *prf* cannot guarantee the design QoS constraint and breaks down as the information data rate per link increases to 30 kbit/s.
