**3. MUI Mitigation by distributed pulse rate control**

Interference mitigation is a fundamental problem in wireless networks. In CDMA systems, a well-known technique for this is to control the nodes' transmit powers [22]. The work in [17] has shown that for wireless networks in the *linear regime*, and that allow fine-grained rate adaptation, the optimal power allocation is to let nodes either transmit at full power or do not transmit at all. IR-UWB conforms to both attributes, and thus, according to [17], the MAC layer should concentrate on alternative interference mitigation techniques such as scheduling and rate adaptation.

The term rate adaptation embraces all technical means in a system to adapt the transmission speed (rate) to the current quality of the radio link. In IR-UWB networks, rate control can be achieved by adapting the channel coding rate, the modulation order or the processing gain. In order to adapt these parameters, the link's transmitter must have an estimate of the level of interference at its intended receiver. In autonomous networks, most approaches make use of feedback information from the receiver to the transmitter, for example within ACK packets. This information can take various forms; conventionally, it is a function of the signal-to-noise-plus-interference-ratio (SNIR). However, measuring the SNIR is difficult in practice due to the very low transmit power of UWB signals. Therefore, recent approaches [9] rely on information provided by the channel decoder, namely on BER estimations. This work follows theses approaches and also considers BER instead of SNIR feedbacks.

#### 8 Will-be-set-by-IN-TECH 58 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks <sup>9</sup>

The processing gain of IR-UWB is twofold: the number of pulses per symbol (*Ns*) and the average frame duration (*Tf* ). The adaptation of IR-UWB processing gain impacts both rate and average emitted power. Adapting the processing gain in IR-UWB systems was first suggested by Lovelace et. al. in [10]. Lovelace proposed a technique for adjusting the number of pulses per symbol in a single-hop link affected by uncoordinated near pulse interferers. This technique requires a particular type of receiver, capable of selectively and passively blank large interfering pulses from symbol decisions. Also, the approach assumes a system with a large number of pulses per symbols, so that blanking a few of them has only a minimal impact on the resulting BER. The joint adaptation of both types of processing gain was first studied in the context of IEEE 802.15.3a WPANs to reduce the mutual interference among uncoordinated, collocated WPANs [23], and was lately extended to cluster based wireless sensor networks in [20]. The basic observation in [23] was that the larger the number of collocated WPANs the larger the average frame time, *Tf* , has to be in order to reduce the amount of impulsive interference.

**3.1. Distributed pulse rate control**

requirement.

*3.1.1. Problem formulation*

expressed by equation 2.

second; with fixed *Etx*

power.

Pulse rate control (PRC) can be realised in form of a link adaptation function whose goal is the improvement of the network throughput while satisfying a minimum per link BER

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 59

While adaptive modulation and channel coding are local decisions to a sender-receiver pair, PRC involves a cooperation among different links since the average<sup>7</sup> probability of pulse collision at the *i*-th receiver (*Pcoll*,*i*) does not depend on its own link's pulse rate, but on the pulse rates of transmitters in its vicinity [20]. Since the collision probability is an indirect measurement of the BER, it can be further assumed that the BER at the *i*-th receiver, *Pe*,*i*, does not directly depend on its link's pulse rate, but on the pulse rate of the neighbouring links. In the following we model, analyse and evaluate the distributed PRC approach. We first formulate PRC as a network logarithmic utility maximisation problem with quality of service (QoS) constraints. In order to solve the problem in a distributed manner, PRC is reformulated to a non-cooperative game with pricing. A distributed asynchronous algorithm is proposed

The objective of the PRC approach is to determine the maximum pulse rate allocation, such that the QoS demands - in terms of BER- of all network links are fulfilled. This goal can be

*<sup>i</sup>* , *prf max*

PRC assumes that each IR-UWB node can autonomously adapt its average pulse repetition

independent of the modulation scheme, and is chosen so that the IR-UWB node with the highest data rate completely exploits the FCC requirements in terms of EIRP and peak power. Controlling the source's *prf* is equivalent to controlling it's data rate (*r*) in terms of pulses per

QoS constraints in equation 2 limits the set of feasible pulse rate allocations. Since a higher pulse rate level from one transmitter increases the collision probability -and in turn the BER

<sup>7</sup> The average is taken over the links' asynchronism. In fact, the collision probability between two transmitters, for instance transmitter *j* and the reference transmitter *i*, depends on their pulse rates and on the relative delay time between the instants at which both transmitters start their transmissions. This relative delay is a random variable

*<sup>i</sup>* ], ∀ *i* ∈ **N**

*<sup>p</sup>* , it is also equivalent to controlling the source's average transmitted

(2)

*Tf*

*<sup>p</sup>* , is fixed and equal for all users,

). Aditionally,

*log* (*ri*(**prf**))

*Pe*,*i*(**prf**−*i*) <sup>≤</sup> *<sup>β</sup>i*, <sup>∀</sup> *<sup>i</sup>* <sup>∈</sup> **<sup>N</sup>**,

*prf <sup>i</sup>* <sup>∈</sup> **PRF***<sup>i</sup>* = [*prf min*

frequency, *prf* - that is the inverse of the average frame duration (*prf* = <sup>1</sup>

which converges to the globally optimal solution of the original problem.

max **prf**

it assumes that the energy per transmitted pulse, *Etx*

determined by the action of the users; it is represented by the time shift *τj*.


subject to:

The first, general, theoretical considerations about the performance tradeoff between the two types of IR-UWB processing gain were published by Fishler and Poor in [6]. Fishler and Poor examined this tradeoff as a function of the BER in a system with fixed processing gain over flat-fading and frequency-selective channels. The study concludes that in a coded system transmitting over a flat-fading channel, the BER is independent of the ratio between the two types of processing gain. In contrast, in an uncoded system over a flat-fading channel and in frequency-selective channels there is a trade-off. This trade-off favours systems with a low number of pulses per symbol, as the system BER considerably degrades as the number of pulses per symbol increases. Thus, regarding processing gain adaptation, and assuming that the energy per transmitted pulse is the same for all users, it is preferable to extend the signal's average frame time (reduce the signal's duty cycle) than to increase the number of pulses per symbol. Moreover, using large frame times help to reduce the system complexity since a lower sampling rate can be used.

With the exception of the work in [10], which requires a particular receiver technique, the distributed adaptation of IR-UWB processing gain in autonomous networks has not been addressed in the literature before. The remaining approaches referenced in this section rely on the presence of a coordinator node that implements the adaptation algorithm and instructs other nodes on how to scale their parameters. This work focuses on autonomous networks; although hierarchical structures are not ruled out here, they are not required and therefore adaptation schemes cannot rely on the presence of coordination entities, but must be distributed.

The author claims that in autonomous IR-UWB networks, due to its self-organising and asynchronous character, and due to the monotonically increasing throughput for increasing pulse rates, the system's local pulse load can become so high that bit errors may not be longer tractable with error coding schemes. Based on this assumption, and following previous theoretical considerations on processing gain adaptation, this work develops a novel mechanism -distributed Pulse Rate Control (PRC)- to coordinate the links' pulse rate levels to optimise the overall network performance, measured in terms of total network logarithmic utility (*proportional fairness*), while satisfying a minimum per link BER requirement.

### **3.1. Distributed pulse rate control**

8 Will-be-set-by-IN-TECH

The processing gain of IR-UWB is twofold: the number of pulses per symbol (*Ns*) and the average frame duration (*Tf* ). The adaptation of IR-UWB processing gain impacts both rate and average emitted power. Adapting the processing gain in IR-UWB systems was first suggested by Lovelace et. al. in [10]. Lovelace proposed a technique for adjusting the number of pulses per symbol in a single-hop link affected by uncoordinated near pulse interferers. This technique requires a particular type of receiver, capable of selectively and passively blank large interfering pulses from symbol decisions. Also, the approach assumes a system with a large number of pulses per symbols, so that blanking a few of them has only a minimal impact on the resulting BER. The joint adaptation of both types of processing gain was first studied in the context of IEEE 802.15.3a WPANs to reduce the mutual interference among uncoordinated, collocated WPANs [23], and was lately extended to cluster based wireless sensor networks in [20]. The basic observation in [23] was that the larger the number of collocated WPANs the larger the average frame time, *Tf* , has to be in order to reduce the amount of impulsive

The first, general, theoretical considerations about the performance tradeoff between the two types of IR-UWB processing gain were published by Fishler and Poor in [6]. Fishler and Poor examined this tradeoff as a function of the BER in a system with fixed processing gain over flat-fading and frequency-selective channels. The study concludes that in a coded system transmitting over a flat-fading channel, the BER is independent of the ratio between the two types of processing gain. In contrast, in an uncoded system over a flat-fading channel and in frequency-selective channels there is a trade-off. This trade-off favours systems with a low number of pulses per symbol, as the system BER considerably degrades as the number of pulses per symbol increases. Thus, regarding processing gain adaptation, and assuming that the energy per transmitted pulse is the same for all users, it is preferable to extend the signal's average frame time (reduce the signal's duty cycle) than to increase the number of pulses per symbol. Moreover, using large frame times help to reduce the system complexity since a lower

With the exception of the work in [10], which requires a particular receiver technique, the distributed adaptation of IR-UWB processing gain in autonomous networks has not been addressed in the literature before. The remaining approaches referenced in this section rely on the presence of a coordinator node that implements the adaptation algorithm and instructs other nodes on how to scale their parameters. This work focuses on autonomous networks; although hierarchical structures are not ruled out here, they are not required and therefore adaptation schemes cannot rely on the presence of coordination entities, but must

The author claims that in autonomous IR-UWB networks, due to its self-organising and asynchronous character, and due to the monotonically increasing throughput for increasing pulse rates, the system's local pulse load can become so high that bit errors may not be longer tractable with error coding schemes. Based on this assumption, and following previous theoretical considerations on processing gain adaptation, this work develops a novel mechanism -distributed Pulse Rate Control (PRC)- to coordinate the links' pulse rate levels to optimise the overall network performance, measured in terms of total network logarithmic

utility (*proportional fairness*), while satisfying a minimum per link BER requirement.

interference.

sampling rate can be used.

be distributed.

Pulse rate control (PRC) can be realised in form of a link adaptation function whose goal is the improvement of the network throughput while satisfying a minimum per link BER requirement.

While adaptive modulation and channel coding are local decisions to a sender-receiver pair, PRC involves a cooperation among different links since the average<sup>7</sup> probability of pulse collision at the *i*-th receiver (*Pcoll*,*i*) does not depend on its own link's pulse rate, but on the pulse rates of transmitters in its vicinity [20]. Since the collision probability is an indirect measurement of the BER, it can be further assumed that the BER at the *i*-th receiver, *Pe*,*i*, does not directly depend on its link's pulse rate, but on the pulse rate of the neighbouring links.

In the following we model, analyse and evaluate the distributed PRC approach. We first formulate PRC as a network logarithmic utility maximisation problem with quality of service (QoS) constraints. In order to solve the problem in a distributed manner, PRC is reformulated to a non-cooperative game with pricing. A distributed asynchronous algorithm is proposed which converges to the globally optimal solution of the original problem.

### *3.1.1. Problem formulation*

The objective of the PRC approach is to determine the maximum pulse rate allocation, such that the QoS demands - in terms of BER- of all network links are fulfilled. This goal can be expressed by equation 2.

$$\begin{aligned} \max\_{\mathbf{prf}} & \sum\_{i=1}^{|\mathbf{N}|} \log \left( r\_i(\mathbf{prf}) \right) \\ \text{subject to:} & \\ P\_{e,i}(\mathbf{prf}\_{-i}) & \le \beta\_{i\prime} \forall \, i \in \mathbf{N}\_{\prime} \end{aligned} \tag{2}$$
  $prf\_i \in \mathbf{PRF}\_i = [prf\_i^{\min}, prf\_i^{\max}], \forall \, i \in \mathbf{N}$ 

PRC assumes that each IR-UWB node can autonomously adapt its average pulse repetition frequency, *prf* - that is the inverse of the average frame duration (*prf* = <sup>1</sup> *Tf* ). Aditionally, it assumes that the energy per transmitted pulse, *Etx <sup>p</sup>* , is fixed and equal for all users, independent of the modulation scheme, and is chosen so that the IR-UWB node with the highest data rate completely exploits the FCC requirements in terms of EIRP and peak power. Controlling the source's *prf* is equivalent to controlling it's data rate (*r*) in terms of pulses per second; with fixed *Etx <sup>p</sup>* , it is also equivalent to controlling the source's average transmitted power.

QoS constraints in equation 2 limits the set of feasible pulse rate allocations. Since a higher pulse rate level from one transmitter increases the collision probability -and in turn the BER

<sup>7</sup> The average is taken over the links' asynchronism. In fact, the collision probability between two transmitters, for instance transmitter *j* and the reference transmitter *i*, depends on their pulse rates and on the relative delay time between the instants at which both transmitters start their transmissions. This relative delay is a random variable determined by the action of the users; it is represented by the time shift *τj*.

(*Pe*)- at neighbouring receivers, there may not be any feasible pulse rate allocation to satisfy the requirements of all users.

The logarithmic utility function in equation 2 captures the link's desire for higher data transmission rate. In an IR-UWB network, the raw data rate per link can be controlled by adapting the channel coding rate (*Ri*), the modulation order (*mi*) or the processing gain, that is the number of pulses per symbol (*N<sup>i</sup> s*) and/or the average frame duration (*T<sup>i</sup> <sup>f</sup>* ). Equation 3 depicts the dependency of the raw data rate on these parameters.

$$r\_i^{raw} = \frac{1}{N\_s^i \cdot T\_f^i} \cdot R\_i \cdot \log\_2(m\_i) \text{ [bit/s]} \tag{3}$$

The original logarithmic utility function reflects the level of a user's satisfaction from consuming the resource *prf* (directly related to the transmission data rate). The pricing factor, *πi*(**prf**), reflects the cost per unit of resource charged to user *i*. Hence, the new utility function can be interpreted as if each user *i* maximises the difference between its old net utility and a payment to other users in the network due to the interference (pulse collisions) it generates. With the new utility function *vi*(**prf**) a non-cooperative PRC game with pricing, denoted by ΓPRC = �**N**, **PRF**, **v**�, is developed. For ΓPRC the set of players **N** = {1, 2, ..., |*N*|} corresponds with the set of active links (users) in the network, so that the terms "player" and "user" are used as synonym. The vector of utilities corresponds to **<sup>v</sup>**(**prf**) = {*v*1(**prf**), *<sup>v</sup>*2(**prf**), ..., *<sup>v</sup>*|*N*|(**prf**)}, and the set of actions that players can choose,

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 61

As described in 2.4 this work considers a network model with a single centralised receiver (CH) and several uncoordinated sources. With a common pricing factor provided by the CH, *πi*(**prf**) = *πj*(**prf**) = *πCH*(**prf**), ∀ *i*, *j* ∈ **N**, each user in the network can be guided by the altruistic goal of maximising the cumulative network throughput at the CH, while keeping their average bit error rate, *Pe*,*i*, as close as possible to a target *βCH* = *βi*, ∀ *i* ∈ **N**. In this setting, the pricing term acts as a control parameter employed by the CH to discourage the overuse of the wireless resource *prf* and to keep the interference sustainable. It is expected that the choice of a common pricing factor for all links degenerates the *proportional-fairness* character of the original problem formulation into a *max-min fairness*

Generally, an average cumulative *Pe*,*CH* >> *βCH* suggests a congestion situation caused by an overload in the local pulse density. In contrast, *Pe*,*CH* << *βCH* suggests an underload situation in which the local pulse density is below the sustainable load for the given QoS criteria. Accordingly, the CH measures the cumulative bit error rate at each superframe and

*<sup>k</sup>*∈**AL***<sup>s</sup>*

where *s* represents the superframe index and **AL***<sup>s</sup>* is the set of active links during superframe

to the UWB nodes in the next beacon frame. The computation rule for the congestion cost is very simple. If there is congestion in the current superframe *s*, the congestion cost for the next superframe *s* +1 must be increased, in contrast, if there is underload in the current superframe *s*, the congestion cost for the next superframe *s* + 1 can be decreased. The congestion cost represents a common price factor for all players, *πCH* = *πi*, ∀ *i* ∈ **N**. The UWB nodes regulate their *prf*, and therewith their data rate, in response to the congestion cost feedback from the

*CHδ*, �*P<sup>s</sup>*

*<sup>μ</sup>* , �*P<sup>s</sup>*

*<sup>e</sup>*,*CH* > 0

*<sup>e</sup>*,*CH* < (*βCHω*|*AL*|

*<sup>e</sup>*,*CH*, the CH computes a congestion cost for superframe *s* + 1 and feedbacks it

*e*,*CH*

*s*)

, (7)

*Pe*,*<sup>k</sup>* − *βCH*, (6)

continuously tracks its deviation from the target value in the variable �*P<sup>s</sup>*

*<sup>e</sup>*,*CH* = ∑

*CH* <sup>+</sup> *μπ<sup>s</sup>*

*CH δ*

*CH* <sup>−</sup> *<sup>π</sup><sup>s</sup>*

�*P<sup>s</sup>*

Specifically, the following computation rule is proposed

(1.0 <sup>−</sup> *<sup>δ</sup>*)*π<sup>s</sup>*

(1.0 <sup>−</sup> *<sup>δ</sup>*)*π<sup>s</sup>*

⎧ ⎨ ⎩

*πs*+<sup>1</sup> *CH* =

**PRF** <sup>=</sup> *PRF*<sup>1</sup> <sup>×</sup> *PRF*<sup>2</sup> <sup>×</sup> ... <sup>×</sup> *PRF*|*N*|, is compact and convex.

solution.

*<sup>s</sup>*. Based on �*P<sup>s</sup>*

CH.

Following [6], this work focuses on systems with low number of pulses per symbol. For the sake of simplicity and without loss of generality we consider hereafter *N<sup>i</sup> <sup>s</sup>* = 1, ∀ *i* ∈ **N**. The useful (net) data rate per link is given in equation 4.

$$r\_i(\mathbf{prf}) = r\_i^{raw} \cdot \left(1 - P\_{e,i}(\mathbf{prf}\_{-i})\right) \text{ [bit/s]} \tag{4}$$

Since the bit error rate, *Pe*, is a nonlinear, and neither convex nor concave function of the links' pulse rates, the pulse rate optimisation problem is in general a nonlinear optimisation problem. The classical optimisation theory has no effective method for solving the general nonlinear optimisation problem, but several different approaches such as geometric programming (GP). Each of these approaches involves some compromise [1]; for instance, GP is limited to algorithms with a central single point of computation [2]. Game theory represents an alternative to GP and it is used in the next section to model the PRC problem (in equation 2) in a distributed manner.

### *3.1.2. Pulse rate control game*

From the author's point of view, distributed PRC can be interpreted as a resource allocation mechanism that regulates the link's number of transmitted pulses per second. Hence, the framework of non-cooperative game theory can be applied to model and analyse the problem that searches for the network's maximum pulse rate allocation that satisfies a certain set of per-link BER constraints. Next we show how a game theoretical formulation helps to provide the UWB devices with incentives to minimise impulsive emissions when the cumulative system pulse load excesses certain limits, which are determined by some QoS constraints.

In PRC an increase in a link's average *prf* directly and negatively affects the probability of pulse collision, and consequently the BER, of neighbouring links [20]. A game theorist would refer to this fact by saying that there are *negative externalities* in the system. In order to deal with QoS constraints in the presence of negative externalities cooperation among the autonomous users must be enforced. Pricing is one of the most commonly used incentives to regulate selfish user behaviour and establish cooperation. Keeping this in mind, the original logarithmic utility function *ui*(**prf**) = *log*(*ri*(**prf**)) in equation 2 is modified by adding a linear pricing function of the link's *prf*. The new utility function is given in equation 5.

$$v\_i(\mathbf{prf}) = \log(r\_i(\mathbf{prf})) - \pi\_i(\mathbf{prf}) \cdot \mathbf{prf}\_i \tag{5}$$

The original logarithmic utility function reflects the level of a user's satisfaction from consuming the resource *prf* (directly related to the transmission data rate). The pricing factor, *πi*(**prf**), reflects the cost per unit of resource charged to user *i*. Hence, the new utility function can be interpreted as if each user *i* maximises the difference between its old net utility and a payment to other users in the network due to the interference (pulse collisions) it generates.

10 Will-be-set-by-IN-TECH

(*Pe*)- at neighbouring receivers, there may not be any feasible pulse rate allocation to satisfy

The logarithmic utility function in equation 2 captures the link's desire for higher data transmission rate. In an IR-UWB network, the raw data rate per link can be controlled by adapting the channel coding rate (*Ri*), the modulation order (*mi*) or the processing gain, that

Following [6], this work focuses on systems with low number of pulses per symbol. For the

Since the bit error rate, *Pe*, is a nonlinear, and neither convex nor concave function of the links' pulse rates, the pulse rate optimisation problem is in general a nonlinear optimisation problem. The classical optimisation theory has no effective method for solving the general nonlinear optimisation problem, but several different approaches such as geometric programming (GP). Each of these approaches involves some compromise [1]; for instance, GP is limited to algorithms with a central single point of computation [2]. Game theory represents an alternative to GP and it is used in the next section to model the PRC problem (in equation 2)

From the author's point of view, distributed PRC can be interpreted as a resource allocation mechanism that regulates the link's number of transmitted pulses per second. Hence, the framework of non-cooperative game theory can be applied to model and analyse the problem that searches for the network's maximum pulse rate allocation that satisfies a certain set of per-link BER constraints. Next we show how a game theoretical formulation helps to provide the UWB devices with incentives to minimise impulsive emissions when the cumulative system pulse load excesses certain limits, which are determined by some QoS constraints.

In PRC an increase in a link's average *prf* directly and negatively affects the probability of pulse collision, and consequently the BER, of neighbouring links [20]. A game theorist would refer to this fact by saying that there are *negative externalities* in the system. In order to deal with QoS constraints in the presence of negative externalities cooperation among the autonomous users must be enforced. Pricing is one of the most commonly used incentives to regulate selfish user behaviour and establish cooperation. Keeping this in mind, the original logarithmic utility function *ui*(**prf**) = *log*(*ri*(**prf**)) in equation 2 is modified by adding a linear

*vi*(**prf**) = *log*(*ri*(**prf**)) − *πi*(**prf**)· *prf <sup>i</sup>* (5)

pricing function of the link's *prf*. The new utility function is given in equation 5.

<sup>1</sup> <sup>−</sup> *Pe*,*i*(**prf**−*i*)

*s*) and/or the average frame duration (*T<sup>i</sup>*

· *Ri*· *log*2(*mi*) [bit/s] (3)

*<sup>f</sup>* ). Equation 3

*<sup>s</sup>* = 1, ∀ *i* ∈ **N**. The

[bit/s] (4)

the requirements of all users.

in a distributed manner.

*3.1.2. Pulse rate control game*

is the number of pulses per symbol (*N<sup>i</sup>*

depicts the dependency of the raw data rate on these parameters.

*rraw <sup>i</sup>* <sup>=</sup> <sup>1</sup> *Ni <sup>s</sup>*· *<sup>T</sup><sup>i</sup> f*

useful (net) data rate per link is given in equation 4.

sake of simplicity and without loss of generality we consider hereafter *N<sup>i</sup>*

*i* · 

*ri*(**prf**) = *rraw*

With the new utility function *vi*(**prf**) a non-cooperative PRC game with pricing, denoted by ΓPRC = �**N**, **PRF**, **v**�, is developed. For ΓPRC the set of players **N** = {1, 2, ..., |*N*|} corresponds with the set of active links (users) in the network, so that the terms "player" and "user" are used as synonym. The vector of utilities corresponds to **<sup>v</sup>**(**prf**) = {*v*1(**prf**), *<sup>v</sup>*2(**prf**), ..., *<sup>v</sup>*|*N*|(**prf**)}, and the set of actions that players can choose, **PRF** <sup>=</sup> *PRF*<sup>1</sup> <sup>×</sup> *PRF*<sup>2</sup> <sup>×</sup> ... <sup>×</sup> *PRF*|*N*|, is compact and convex.

As described in 2.4 this work considers a network model with a single centralised receiver (CH) and several uncoordinated sources. With a common pricing factor provided by the CH, *πi*(**prf**) = *πj*(**prf**) = *πCH*(**prf**), ∀ *i*, *j* ∈ **N**, each user in the network can be guided by the altruistic goal of maximising the cumulative network throughput at the CH, while keeping their average bit error rate, *Pe*,*i*, as close as possible to a target *βCH* = *βi*, ∀ *i* ∈ **N**. In this setting, the pricing term acts as a control parameter employed by the CH to discourage the overuse of the wireless resource *prf* and to keep the interference sustainable. It is expected that the choice of a common pricing factor for all links degenerates the *proportional-fairness* character of the original problem formulation into a *max-min fairness* solution.

Generally, an average cumulative *Pe*,*CH* >> *βCH* suggests a congestion situation caused by an overload in the local pulse density. In contrast, *Pe*,*CH* << *βCH* suggests an underload situation in which the local pulse density is below the sustainable load for the given QoS criteria. Accordingly, the CH measures the cumulative bit error rate at each superframe and continuously tracks its deviation from the target value in the variable �*P<sup>s</sup> e*,*CH*

$$
\triangle P\_{\varepsilon, \mathsf{CH}}^{\mathrm{s}} = \sum\_{k \in \mathsf{AL}^{\mathrm{s}}} P\_{\varepsilon, k} - \beta\_{\mathsf{CH}^{\mathrm{s}}} \tag{6}
$$

where *s* represents the superframe index and **AL***<sup>s</sup>* is the set of active links during superframe *<sup>s</sup>*. Based on �*P<sup>s</sup> <sup>e</sup>*,*CH*, the CH computes a congestion cost for superframe *s* + 1 and feedbacks it to the UWB nodes in the next beacon frame. The computation rule for the congestion cost is very simple. If there is congestion in the current superframe *s*, the congestion cost for the next superframe *s* +1 must be increased, in contrast, if there is underload in the current superframe *s*, the congestion cost for the next superframe *s* + 1 can be decreased. The congestion cost represents a common price factor for all players, *πCH* = *πi*, ∀ *i* ∈ **N**. The UWB nodes regulate their *prf*, and therewith their data rate, in response to the congestion cost feedback from the CH.

Specifically, the following computation rule is proposed

$$
\pi\_{\rm CH}^{s+1} = \begin{cases}
(1.0-\delta)\pi\_{\rm CH}^s + \mu \pi\_{\rm CH}^s \delta, & \triangle P\_{e,\rm CH}^s > 0 \\
(1.0-\delta)\pi\_{\rm CH}^s - \frac{\pi\_{\rm CH}^s \delta}{\mu}, & \triangle P\_{e,\rm CH}^s < \left(\p\_{\rm CH}\omega |AL|^s\right)' \\
\end{cases}
\tag{7}
$$

12 Will-be-set-by-IN-TECH 62 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks <sup>13</sup>

where *μ* is a weight factor and *δ* is the smoothing factor of a weighted exponential-moving-average (EMA) algorithm [4]. In order to improve game convergence a tolerance region for the cluster congestion level has been defined in which no adaptation is done. The tolerance range is defined by |*AL*| *<sup>s</sup>*, the number of active links in superframe *s*, and a constant *ω* ≤ 0.

**3.2. Game analysis**

potential game (OPG).

*3.2.1. Existence of an equilibrium*

*to an exact potential game.*

*equilibria of* Γ˜*.*

least one NE.

is

1 and 2 and their respective proofs can be found in [14].

**Theorem 1.** *Identification of Ordinal Potential Games*

**Theorem 2.** *NE of Better-Response Equivalent Games*

∀ *<sup>i</sup>* ∈ **<sup>N</sup>***, a* ∈ **<sup>A</sup>***, vi*(*ai*, **<sup>a</sup>**−*i*) > *vi*(*bi*, **<sup>a</sup>**−*i*) ⇔ *<sup>v</sup>*˜*i*(*ai*, **<sup>a</sup>**−*i*) > *<sup>v</sup>*˜*i*(*bi*, **<sup>a</sup>**−*i*)*.*

With these results in mind, game <sup>Γ</sup>˜ *PRC* <sup>=</sup> �**N**, **<sup>A</sup>**, **˜v**� with utility function

*∂*<sup>2</sup>*ui*(**a**) *∂ai∂ak*

*v*˜*i*(**prf**) = *log*(*prf <sup>i</sup>*

**Definition 1.** *Better-Response Equivalence*

significantly facilitates the analysis.

In this section, the utility function of ΓPRC is analysed in terms of existence and uniqueness of Nash equilibria. The fact that ΓPRC fits the framework of potential games ([13, 14])

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 63

A potential game is characterised by the existence of a function, denoted as the potential function, Φ : *A* → �, such that the change in the utility function of a player when it unilateraly deviates in its strategy (�*ui*) is reflected in a change in value of the potential function (�Φ). If for all unilateral deviations, �Φ = �*ui* the game is referred to as an *exact* potential game (EPG). If the relationship between the potential function and the utility functions is relaxed so that only sign changes are preserved, *sgn*(�Φ) = *sgn*(�*ui*), the game is called an *ordinal*

In order to prove the existence of an NE for game ΓPRC a definition and a powerful result concerning the identification of ordinal potential games are leveraged. Definition 1, theorem

*A game* <sup>Γ</sup> <sup>=</sup> �**N**, **<sup>A</sup>**, **<sup>v</sup>**� *is said to be better response equivalent to game* <sup>Γ</sup>˜ <sup>=</sup> �**N**, **<sup>A</sup>**, **˜v**�*, if*

*A game* Γ = �**N**, **A**, **v**� *is an ordinal potential game if and only if it is better response equivalent*

*If a game* Γ = �**N**, **A**, **v**� *is better response equivalent to game* <sup>Γ</sup>˜ <sup>=</sup> �**N**, **<sup>A</sup>**, **˜v**�*, then the Nash Equilibria of* <sup>Γ</sup>*, if any exist, coincide with the Nash*

is introduced. Since *log*(*xy*) = *log*(*x*) + *log*(*y*) and through definition 1, there is a trivial better response equivalence relationship between ΓPRC and Γ˜ *PRC*. Hence, from theorem 2 we know that by analysing the set of Nash equilibria of game Γ˜ *PRC* we are at the same time solving for the Nash equilibria of Γ*PRC*. Next, we prove that Γ˜ PRC is an EPG and, consequently thanks to theorem 1, ΓPRC is an OPG. Further, in [14] we find a powerful result concerning all potential games with a compact action space and a continuous potential function: The existence of at

From [14], a sufficient condition for the existence of a potential function in game Γ = �**N**, **A**, **u**�

<sup>=</sup> *<sup>∂</sup>*<sup>2</sup>*uk*(**a**) *∂ak∂ai*

) − *πCHprf <sup>i</sup>*

, (9)

, ∀ *i*, *k* ∈ **N**, ∀ **a** ∈ **A** (10)

### *3.1.3. Pulse rate control algorithm*

In a realistic distributed environment, at the start of a game it is not possible that a player *i* ∈ **N** has the complete price information (adjacent channel gains, link qualities) that is necessary to discover an NE inmediately. However, the player can make a guess, denoted by its selected action *prf <sup>i</sup>* , regarding its equilibrium average *prf* denoted by *prf* ∗ *<sup>i</sup>* . Then, assuming that the actions of the other players, **prf**−*<sup>i</sup>* , and its own price factor, *πi*, remain constant while it makes a decision, player *i* improves its guess by selecting a new action which maximises its utility function. This new guess results in a new approximation to *prf* ∗ *<sup>i</sup>* . When the deviations in all players' actions become negligibly small, the game can be assumed to have converged to an NE.

The PRC algorithm implements the adaptive behaviour described above and distributively solves the pulse rate allocation maximisation problem in equation 2.

### **PRC Algorithm**

• **Step 1: Initialisation**

For each user *i* ∈ **N** choose some *prf <sup>i</sup>* (0) ∈ *PRFi* and a price factor *πi*(0) ≥ 0.

• **Step 2: Price update**

At each iteration *t* ∈ *Ti*,*π*, user *i* updates its price according to equation 7.

• **Step 3: Pulse rate update**

At each iteration *t* ∈ *Ti*,*pr f* , user *i* updates *prf <sup>i</sup>* according to a best response decision rule:

$$\left[\text{prf}\_{i}^{\*}(t+1) = \text{BR}\_{i}\left(\text{prf}\_{-i}(t)\right) = \left[\frac{1}{\pi\_{i}(t)}\right]\_{\text{prf}^{\text{min}}}^{\text{prf}^{\text{max}}} \tag{8}$$

where the notation [*x*] *b <sup>a</sup>* means max{*a*, min{*b*, *x*}} and BR*<sup>i</sup>* is the set of best responses of player *<sup>i</sup>* to the strategy profile **prf**−*i*(*t*).

Notice that the price and *prf* adaptations do not need to happen at the same time. The price update instants are determined by the superframe beacon raster, while the *prf* update instants depend on the beacon raster as well as on the source traffic model and can therefore be asynchronous across users.

In the practical implementation of the algorithm, a discretisation of the action space (**PRF***i*) is unavoidable. The granularity of this discretisation process represents a trade-off for the convergence properties of the algorithm. An infinitely small granularity equals an infinite action space and guarantees that the NE action profile (if there is one) is considered in the search process; however, it increases the computation cost of the algorithm. With an increasing granularity the action space becomes finite, so that the search space for the algorithm shrinks and the computational cost is reduced. A brief discussion about the existence of Nash equilibria in discrete games can be found in [7]. In general it holds that, if the game with continuous action space has a stable equilibrium, the discrete pendant also has an equilibrium.

### **3.2. Game analysis**

12 Will-be-set-by-IN-TECH

where *μ* is a weight factor and *δ* is the smoothing factor of a weighted exponential-moving-average (EMA) algorithm [4]. In order to improve game convergence a tolerance region for the cluster congestion level has been defined in which no adaptation is

In a realistic distributed environment, at the start of a game it is not possible that a player *i* ∈ **N** has the complete price information (adjacent channel gains, link qualities) that is necessary to discover an NE inmediately. However, the player can make a guess, denoted by its selected

a decision, player *i* improves its guess by selecting a new action which maximises its utility

players' actions become negligibly small, the game can be assumed to have converged to an

The PRC algorithm implements the adaptive behaviour described above and distributively

At each iteration *t* ∈ *Ti*,*pr f* , user *i* updates *prf <sup>i</sup>* according to a best response decision rule:

Notice that the price and *prf* adaptations do not need to happen at the same time. The price update instants are determined by the superframe beacon raster, while the *prf* update instants depend on the beacon raster as well as on the source traffic model and can therefore be

In the practical implementation of the algorithm, a discretisation of the action space (**PRF***i*) is unavoidable. The granularity of this discretisation process represents a trade-off for the convergence properties of the algorithm. An infinitely small granularity equals an infinite action space and guarantees that the NE action profile (if there is one) is considered in the search process; however, it increases the computation cost of the algorithm. With an increasing granularity the action space becomes finite, so that the search space for the algorithm shrinks and the computational cost is reduced. A brief discussion about the existence of Nash equilibria in discrete games can be found in [7]. In general it holds that, if the game with continuous action space has a stable equilibrium, the discrete pendant also has an equilibrium.

**prf**−*i*(*t*)

 = 1 *πi*(*t*)

, regarding its equilibrium average *prf* denoted by *prf* ∗

At each iteration *t* ∈ *Ti*,*π*, user *i* updates its price according to equation 7.

*<sup>i</sup>* (*t* + 1) = BR*<sup>i</sup>*

function. This new guess results in a new approximation to *prf* ∗

solves the pulse rate allocation maximisation problem in equation 2.

*<sup>s</sup>*, the number of active links in superframe *s*,

, and its own price factor, *πi*, remain constant while it makes

(0) ∈ *PRFi* and a price factor *πi*(0) ≥ 0.

*<sup>a</sup>* means max{*a*, min{*b*, *x*}} and BR*<sup>i</sup>* is the set of best responses of

*prf max prf min*

*<sup>i</sup>* . Then, assuming that the

*<sup>i</sup>* . When the deviations in all

(8)

done. The tolerance range is defined by |*AL*|

and a constant *ω* ≤ 0.

action *prf <sup>i</sup>*

**PRC Algorithm**

• **Step 1: Initialisation**

• **Step 2: Price update**

• **Step 3: Pulse rate update**

where the notation [*x*]

asynchronous across users.

For each user *i* ∈ **N** choose some *prf <sup>i</sup>*

*prf* ∗

*b*

player *<sup>i</sup>* to the strategy profile **prf**−*i*(*t*).

NE.

*3.1.3. Pulse rate control algorithm*

actions of the other players, **prf**−*<sup>i</sup>*

In this section, the utility function of ΓPRC is analysed in terms of existence and uniqueness of Nash equilibria. The fact that ΓPRC fits the framework of potential games ([13, 14]) significantly facilitates the analysis.

A potential game is characterised by the existence of a function, denoted as the potential function, Φ : *A* → �, such that the change in the utility function of a player when it unilateraly deviates in its strategy (�*ui*) is reflected in a change in value of the potential function (�Φ). If for all unilateral deviations, �Φ = �*ui* the game is referred to as an *exact* potential game (EPG). If the relationship between the potential function and the utility functions is relaxed so that only sign changes are preserved, *sgn*(�Φ) = *sgn*(�*ui*), the game is called an *ordinal* potential game (OPG).

### *3.2.1. Existence of an equilibrium*

In order to prove the existence of an NE for game ΓPRC a definition and a powerful result concerning the identification of ordinal potential games are leveraged. Definition 1, theorem 1 and 2 and their respective proofs can be found in [14].

### **Definition 1.** *Better-Response Equivalence*

*A game* <sup>Γ</sup> <sup>=</sup> �**N**, **<sup>A</sup>**, **<sup>v</sup>**� *is said to be better response equivalent to game* <sup>Γ</sup>˜ <sup>=</sup> �**N**, **<sup>A</sup>**, **˜v**�*, if* ∀ *<sup>i</sup>* ∈ **<sup>N</sup>***, a* ∈ **<sup>A</sup>***, vi*(*ai*, **<sup>a</sup>**−*i*) > *vi*(*bi*, **<sup>a</sup>**−*i*) ⇔ *<sup>v</sup>*˜*i*(*ai*, **<sup>a</sup>**−*i*) > *<sup>v</sup>*˜*i*(*bi*, **<sup>a</sup>**−*i*)*.*

### **Theorem 1.** *Identification of Ordinal Potential Games*

*A game* Γ = �**N**, **A**, **v**� *is an ordinal potential game if and only if it is better response equivalent to an exact potential game.*

### **Theorem 2.** *NE of Better-Response Equivalent Games*

*If a game* Γ = �**N**, **A**, **v**� *is better response equivalent to game* <sup>Γ</sup>˜ <sup>=</sup> �**N**, **<sup>A</sup>**, **˜v**�*, then the Nash Equilibria of* <sup>Γ</sup>*, if any exist, coincide with the Nash equilibria of* Γ˜*.*

With these results in mind, game <sup>Γ</sup>˜ *PRC* <sup>=</sup> �**N**, **<sup>A</sup>**, **˜v**� with utility function

$$
\vec{v}\_{i}(\mathbf{prf}) = \log(p\mathbf{rf}\_{i}) - \pi\_{\mathbb{C}H} p\mathbf{rf}\_{i\prime} \tag{9}
$$

is introduced. Since *log*(*xy*) = *log*(*x*) + *log*(*y*) and through definition 1, there is a trivial better response equivalence relationship between ΓPRC and Γ˜ *PRC*. Hence, from theorem 2 we know that by analysing the set of Nash equilibria of game Γ˜ *PRC* we are at the same time solving for the Nash equilibria of Γ*PRC*. Next, we prove that Γ˜ PRC is an EPG and, consequently thanks to theorem 1, ΓPRC is an OPG. Further, in [14] we find a powerful result concerning all potential games with a compact action space and a continuous potential function: The existence of at least one NE.

From [14], a sufficient condition for the existence of a potential function in game Γ = �**N**, **A**, **u**� is

$$\frac{\partial^2 u\_i(\mathbf{a})}{\partial a\_l \partial a\_k} = \frac{\partial^2 u\_k(\mathbf{a})}{\partial a\_k \partial a\_l}, \forall i, k \in \mathbf{N}, \forall \mathbf{a} \in \mathbf{A} \tag{10}$$

#### 14 Will-be-set-by-IN-TECH 64 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks <sup>15</sup>

In Γ˜ PRC the utility functions are given by:

$$
\vec{v}\_{\text{i}}(p\eta f\_{\text{i}'} \mathbf{p} \mathbf{r} f\_{-\text{i}}) = \log(p\eta f\_{\text{i}}) - \pi\_{\text{CH}} p\eta f\_{\text{i}} \tag{11}
$$

Equation 14 is continuous since it is the sum of continuous functions, furthermore the action space of Γ˜ PRC is compact per definition. Hence, equation 16 verifies that (as any other EPG) Γ˜ PRC has at least one NE. Finally, applying theorem 2 it is proven that ΓPRC has at least one

From [14], it is known that an EPG following an asynchronous, myopic, best response decision rule converges to a pure strategy NE that maximises the potential function. Furthermore, if the potential function is strictly concave, it has a unique global maximum which is then the unique NE of the EPG. Based on these results the following proposition can be stated.

**Proposition 1.** *If* Φ˜ *PRC in equation 14 is strictly concave, the proposed PRC algorithm will always converge to the unique NE of game* Γ˜ *PRC, which in turn, is the unique global maximum*

*Proof.* The PRC algorithm can be interpreted as the players employing asynchronous myopic best response (MBR) updates. Thus, to demonstrate proposition 1 it suffices to prove the strict concavity of Φ˜ PRC in equation 14. As explained in [1, Section 3.1.4], this can be verified with

For the Hessian matrix in equation 18 the second derivatives of equation 14 are required.

. ... .

··· *<sup>∂</sup>*2Φ˜ PRC *∂prf* <sup>1</sup>*∂prf <sup>j</sup>*

··· *<sup>∂</sup>*2Φ˜ PRC *∂prf <sup>N</sup> ∂prf <sup>j</sup>*

The matrix is negative definite, *xTH*(Φ˜ )*x* < 0, since all diagonal elements are negative. Hence,

**Proposition 2.** *If* Φ˜ *PRC in equation 14 is strictly concave, the proposed PRC algorithm will always converge to the unique NE of game* Γ˜ *PRC, which, in turn, is the unique NE of game*

*Proof.* The proof of Proposition 2 results from combining the proof of Proposition 1 with

So far, there is no general result about the optimality of Nash equilibria in potential games or in any other more general class of games. However by designing the potential game in a way that its potential function complies with the network design function, a quite elegant way to demonstrate NE optimality is possible. In that case, any strategy that maximises the potential

.

··· *<sup>∂</sup>*2Φ˜ PRC *∂prf* <sup>1</sup>*∂prf <sup>N</sup>*

Pulse Rate Control for Low Power and Low Data Rate Ultra Wideband Networks 65

··· *<sup>∂</sup>*2Φ˜ PRC *∂*2*prf <sup>N</sup>*

. . ⎞

⎟⎟⎟⎠

(17)

(18)

. ... .

if *i* = *j*

0 if *i* � *j*

NE - in fact the same as Γ˜ PRC).

*of* Φ˜ *PRC.*

the proof is complete.

*3.2.3. Optimality of the equilibrium*

Γ*PRC.*

Theorem 2.

*3.2.2. Uniqueness of the equilibrium*

the Hessian matrix and the second-order conditions.

*H*(Φ˜ ) =

⎛

*∂*2Φ˜ PRC *∂*2*prf* <sup>1</sup>

> . .

*∂*2Φ˜ PRC *∂prf <sup>N</sup> ∂prf* <sup>1</sup>

*∂*2Φ˜ PRC *∂prfi*

*∂prf <sup>j</sup>* = � <sup>−</sup> <sup>1</sup> *prf* <sup>2</sup> *i*

⎜⎜⎜⎝

Finally, proposition 1 can be reformulated to proposition 2.

function (any NE) maximises as well the network design function.

and

$$
\psi\_k(\text{prf}\_{k'} \text{prf}\_{-k}) = \log(\text{prf}\_k) - \pi\_{\text{CH}} \text{prf}\_k. \tag{12}
$$

In this network settings, users ignore any influence they may have on the price calculated at the CH. Hence, *∂πCH ∂prf <sup>i</sup>* = 0 ∀ *i* ∈ **N**, and it is easy to prove that

$$\frac{\partial^2 \vec{v}\_i (p\mathbf{r} f\_{i'} \mathbf{p} \mathbf{r} \mathbf{f}\_{-i})}{\partial p \mathbf{r} f\_i \partial p \mathbf{r} f\_k} = \frac{\partial^2 \vec{v}\_k (p\mathbf{r} f\_{k'} \mathbf{p} \mathbf{r} \mathbf{f}\_{-k})}{\partial p \mathbf{r} f\_k \partial p \mathbf{r} f\_i} = 0 \tag{13}$$

Characteristic for an EPG is the existence of a potential function Φ that exactly reflects any unilateral change in the utility of any player, that is �Φ(*a*) = �*ui*(*a*). Hence, starting from any arbitrary strategy vector **a** any unilaterally player's adaptation that increases its utility *ui*(*a*) identically translates in an increase of the potential function Φ(*a*).

The potential function of game Γ˜ PRC is given in equation 14.

$$\Phi\_{\text{PRC}}(\mathbf{prf}) = \sum\_{i \in N} \log(p\mathbf{rf}\_i) - \pi\_{\text{CH}} \sum\_{i \in N} p\mathbf{rf}\_i \tag{14}$$

Based on the potential game definition given in [13], the proof that Γ = �**N**, **A**, **u**� is an EPG requires that:

$$u\_i(a\_{i\prime}, a\_{-i}) - v\_i(b\_{i\prime}, \mathbf{prf}\_{-i}) = \Phi(a\_{i\prime}, a\_{-i}) - \Phi(b\_{i\prime}, a\_{-i}), \forall i \in \mathbf{N}, \forall \mathbf{a} \in \mathbf{A} \tag{15}$$

Equation 16 proves condition 15 for Γ˜ PRC. Note that all sum terms in the potential function that are independent of user *i*'s strategy are constant and can be grouped in an extra term denoted as *c*. By the substraction of the potential functions, the term *c* dissapears leaving the difference of the potential functions identical to the difference of the utility functions.

$$\begin{aligned} \tilde{v}\_{i}(\mathbf{pr}\_{i}^{\prime}, \mathbf{pr}\mathbf{f}\_{-i}) - \tilde{v}\_{i}(\mathbf{pr}\_{i}^{\prime}, \mathbf{pr}\mathbf{f}\_{-i}) &= \tilde{\mathbf{0}}\_{\text{PFC}}(\mathbf{pr}\_{f}^{\prime}, \mathbf{pr}\mathbf{f}\_{-i}) - \tilde{\mathbf{0}}\_{\text{PFC}}(\mathbf{pr}\_{f}^{\prime}, \mathbf{pr}\mathbf{f}\_{-i}) \\ &\quad \log(\mathbf{pr}\_{f}^{\prime}) - \pi\_{\text{CH}}\mathbf{pr}\_{f}^{\prime} - \log(\mathbf{pr}\_{f}^{\prime}) + \pi\_{\text{CH}}\mathbf{pr}\_{f}^{\prime} \\ &\quad + \underbrace{\sum\_{j \in \mathcal{N} \land j \neq i} \log(p\mathbf{r}\_{j}^{\prime}) - \pi\_{\text{CH}}\sum\_{j \in \mathcal{N} \land j \neq i} \mathbf{pr}\_{j}^{\prime}}\_{=:c} \\ &\quad - \underbrace{\log(p\mathbf{r}\_{f}^{\prime}) + \pi\_{\text{CH}}\mathbf{pr}\_{f}^{\prime}}\_{=:c} \\ &\quad - \underbrace{\sum\_{j \in \mathcal{N} \land j \neq i} \log(p\mathbf{r}\_{f}^{\prime}) + \pi\_{\text{CH}}\sum\_{j \in \mathcal{N} \land j \neq i} \mathbf{pr}\_{j}^{\prime}}\_{=:c} \\ &\quad \log(p\mathbf{r}\_{i}^{\prime}) - \pi\_{\text{CH}}p\mathbf{r}\_{i}^{\prime} - \log(p\mathbf{r}\_{f}^{\prime}) + \pi\_{\text{CH}}p\mathbf{r}\_{f}^{\prime} = \\ &\quad \log(p\mathbf{r}\_{f}^{\prime}) - \pi\_{\text{CH}}p\mathbf{r}\_{i}^{\prime} - \log(p\mathbf{r$$

Equation 14 is continuous since it is the sum of continuous functions, furthermore the action space of Γ˜ PRC is compact per definition. Hence, equation 16 verifies that (as any other EPG) Γ˜ PRC has at least one NE. Finally, applying theorem 2 it is proven that ΓPRC has at least one NE - in fact the same as Γ˜ PRC).

### *3.2.2. Uniqueness of the equilibrium*

14 Will-be-set-by-IN-TECH

In this network settings, users ignore any influence they may have on the price

Characteristic for an EPG is the existence of a potential function Φ that exactly reflects any unilateral change in the utility of any player, that is �Φ(*a*) = �*ui*(*a*). Hence, starting from any arbitrary strategy vector **a** any unilaterally player's adaptation that increases its utility

*log*(*prf <sup>i</sup>*

Based on the potential game definition given in [13], the proof that Γ = �**N**, **A**, **u**� is an EPG

Equation 16 proves condition 15 for Γ˜ PRC. Note that all sum terms in the potential function that are independent of user *i*'s strategy are constant and can be grouped in an extra term denoted as *c*. By the substraction of the potential functions, the term *c* dissapears leaving the

difference of the potential functions identical to the difference of the utility functions.

, **prf**−*i*) = <sup>Φ</sup>˜ PRC(*prf <sup>i</sup>*

) <sup>−</sup> *<sup>π</sup>CHprf <sup>i</sup>* <sup>−</sup> *log*(*prf* �

*log*(*prf <sup>i</sup>*

<sup>−</sup>*log*(*prf* � *i*

*log*(*prf <sup>j</sup>*

) <sup>−</sup> *<sup>π</sup>CHprf <sup>i</sup>* <sup>−</sup> *log*(*prf* �

) <sup>−</sup> *<sup>π</sup>CHprf <sup>i</sup>* <sup>−</sup> *log*(*prf* �

∀ *i* ∈ **N**, ∀ *pr f* ∈ **PRF**

*log*(*prf <sup>j</sup>*

) − *πCHprf <sup>i</sup>* (11)

= 0 (13)

*prf <sup>i</sup>* (14)

*<sup>v</sup>*˜*k*(*prf <sup>k</sup>*, **prf**−*k*) = *log*(*prf <sup>k</sup>*) <sup>−</sup> *<sup>π</sup>CHprf <sup>k</sup>*. (12)

= 0 ∀ *i* ∈ **N**, and it is easy to prove that

<sup>=</sup> *<sup>∂</sup>*2*v*˜*k*(*prf <sup>k</sup>*, **prf**−*k*) *∂prf <sup>k</sup>∂prfi*

) − *<sup>π</sup>CH* ∑

*ui*(*ai*, *<sup>a</sup>*−*i*) <sup>−</sup> *vi*(*bi*, **prf**−*i*) = <sup>Φ</sup>(*ai*, *<sup>a</sup>*−*i*) <sup>−</sup> <sup>Φ</sup>(*bi*, *<sup>a</sup>*−*i*),<sup>∀</sup> *<sup>i</sup>* <sup>∈</sup> **<sup>N</sup>**, <sup>∀</sup> **<sup>a</sup>** <sup>∈</sup> **<sup>A</sup>** (15)

*i*

) − *<sup>π</sup>CH* ∑

) + *πCHprf* �

) + *πCH* ∑

*i*

*i*

) − *πCHprf <sup>i</sup>*

<sup>=</sup>*<sup>c</sup>*

<sup>=</sup>−*<sup>c</sup>*

*i*∈*N*

, **prf**−*i*) <sup>−</sup> <sup>Φ</sup>˜ PRC(*prf* �

*prf <sup>j</sup>*

*prf <sup>j</sup>*

*<sup>i</sup>* =

*i*

*<sup>i</sup>* =

) + *πCHprf* �

*j*∈*N* ∧ *j*�=*i*

*i*

*j*∈*N* ∧ *j*�=*i*

) + *πCHprf* �

) + *πCHprf* �

*i* , **prf**−*i*)

(16)

, **prf**−*i*) = *log*(*prf <sup>i</sup>*

In Γ˜ PRC the utility functions are given by:

calculated at the CH. Hence, *∂πCH*

and

requires that:

*v*˜*i*(*prf <sup>i</sup>*

, **prf**−*i*) <sup>−</sup> *<sup>v</sup>*˜*i*(*prf* �

*log*(*prf <sup>i</sup>*

+ ∑ *j*∈*N* ∧ *j*�=*i*

− ∑ *j*∈*N* ∧ *j*�=*i*

*log*(*prf <sup>i</sup>*

*log*(*prf <sup>i</sup>*

*v*˜*i*(*prf <sup>i</sup>*

*∂prf <sup>i</sup>*

*∂prfi*

, **prf**−*i*)

*∂prf <sup>k</sup>*

*ui*(*a*) identically translates in an increase of the potential function Φ(*a*).

<sup>Φ</sup>˜ PRC(**prf**) = ∑

*i*

*i*∈*N*

*<sup>∂</sup>*2*v*˜*i*(*prf <sup>i</sup>*

The potential function of game Γ˜ PRC is given in equation 14.

From [14], it is known that an EPG following an asynchronous, myopic, best response decision rule converges to a pure strategy NE that maximises the potential function. Furthermore, if the potential function is strictly concave, it has a unique global maximum which is then the unique NE of the EPG. Based on these results the following proposition can be stated.

**Proposition 1.** *If* Φ˜ *PRC in equation 14 is strictly concave, the proposed PRC algorithm will always converge to the unique NE of game* Γ˜ *PRC, which in turn, is the unique global maximum of* Φ˜ *PRC.*

*Proof.* The PRC algorithm can be interpreted as the players employing asynchronous myopic best response (MBR) updates. Thus, to demonstrate proposition 1 it suffices to prove the strict concavity of Φ˜ PRC in equation 14. As explained in [1, Section 3.1.4], this can be verified with the Hessian matrix and the second-order conditions.

For the Hessian matrix in equation 18 the second derivatives of equation 14 are required.

$$H(\bar{\Phi}) = \begin{pmatrix} \frac{\partial^2 \Phi\_{\text{PEC}}}{\partial^2 p r f\_1} & \cdots & \frac{\partial^2 \Phi\_{\text{PEC}}}{\partial p r f\_1 \partial p r f\_f} & \cdots & \frac{\partial^2 \Phi\_{\text{PEC}}}{\partial p r f\_1 \partial p r f\_N} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ \frac{\partial^2 \Phi\_{\text{PEC}}}{\partial p r f\_N \partial p r f\_1} & \cdots & \frac{\partial^2 \Phi\_{\text{PEC}}}{\partial p r f\_N \partial p r f\_f} & \cdots & \frac{\partial^2 \Phi\_{\text{PEC}}}{\partial^2 p r f\_N} \end{pmatrix} \tag{17}$$
 
$$\frac{\partial^2 \Phi\_{\text{PEC}}}{\partial p r f\_i \partial p r f\_j} = \begin{cases} -\frac{1}{p r f\_i^2} & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases} \tag{18}$$

The matrix is negative definite, *xTH*(Φ˜ )*x* < 0, since all diagonal elements are negative. Hence, the proof is complete.

Finally, proposition 1 can be reformulated to proposition 2.

**Proposition 2.** *If* Φ˜ *PRC in equation 14 is strictly concave, the proposed PRC algorithm will always converge to the unique NE of game* Γ˜ *PRC, which, in turn, is the unique NE of game* Γ*PRC.*

*Proof.* The proof of Proposition 2 results from combining the proof of Proposition 1 with Theorem 2.

### *3.2.3. Optimality of the equilibrium*

So far, there is no general result about the optimality of Nash equilibria in potential games or in any other more general class of games. However by designing the potential game in a way that its potential function complies with the network design function, a quite elegant way to demonstrate NE optimality is possible. In that case, any strategy that maximises the potential function (any NE) maximises as well the network design function.

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 maximisers) of the original network objective function <sup>∑</sup>|*N*<sup>|</sup> <sup>1</sup> *log*(*ri*(**prf**)). The addition of the 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.
