Follower Utility Function

In the proposed Stackelberg game, multiple followers (D2D pairs) are considered, and the utility function of D2D pair i, uF,i, depends on its application or service,

where uF,i(x, αj,i, pdi) ∈ {uCa, uIa, uSa}, that is, uF,i(x, αj,i, pdi) maps directly to

either the casual class utility uCa,i, interactive class utility uIa,i, or the streaming

class utility uSa,i. These utility functions are arbitrarily different, and cannot be

scalarised and reduced into one single utility function. Initially, D2D pairs sort themselves into one application class that best describes the performance of the desired application or service. We assume that once D2D pairs have assigned themselves to a class, they cannot change classes during the finite stages of the proposed game. The utility function of follower i models the difference between D2D pair i’s performance for a particular application, and the leader’s charging price. The leader’s charging price is a function of BS satisfaction fee x and reuse fee αj,i. In order to reduce intra-cell interference in the network, D2D pairs aim

to find optimal transmit power. Hence, the individual finite action set for each follower i is transmit power, pdi, which is bounded by a minimum and maximum

transmit power. The D2D pairs in the proposed game have imperfect information, which means that all D2D pairs select their action simultaneously. Each D2D pair determines their utility with respect to all other D2D pairs sharing the same resource block.

The casual class utility is a power/SINR balancing function, as defined in [167], with an additional cost charged by the leader. The criterion set for the casual class (examples outlined in Table 3.2) requires low transmission power, and acceptable achievable spectral efficiency, reliability, and QoS. The applications or service as- sociated with this class are common/everyday functionalities of D2D users. The casual class utility uCa,i for D2D pairi is defined as:

uCa,i(x, αj,i, pdi) = γd γdi pdi−pdixδ(0.5 +αj,iGei,BS), (3.6)

56 Flexible D2D Application-Driven Resource Allocation

application class, as it provides large emphasis on minimising transmit power, while also guaranteeing QoS for D2D pairs.

The interactive class utility is a trade-off between maximising SINR and min- imising transmit power, as defined in [167], with an additional cost charged by the leader. The criterion set for the interactive class (examples outlined in Table 3.2) requires similar transmission power to the casual application class, however, it re- quires more reliability than the casual class, which is due to the transferring of data files between users, for example. Thus, this utility function will ensure good signal quality at the receiver without large latency periods. The interactive class utility uIa,i for D2D pair i is defined as:

uIa,i(x, αj,i, pdi) =−qpdi−k(γd−γdi)2−pdixδ(0.5 +αj,iGei,BS), (3.7)

where q and k are positive constants to ensure all parts of (3.7) have the same magnitude; and γ −γdi is the SINR error, which is the difference between target

SINR and received SINR of D2D pair i. If SINR error is less than 0, then the received SINR of D2D pair iis greater than target SINR, which means that QoS is guaranteed for D2D pairi. The utility function (3.7) was chosen for the interactive application class, as there is large emphasis on good signal quality in order to increase reliability, while also providing emphasis on reducing transmit power.

The streaming class utility is a trade-off between maximising achievable spectral efficiency and minimising transmit power, as defined in [78], with an additional cost charged by the leader. The criterion set for the streaming class (examples outlined in Table 3.2) allows much larger transmit power, and requires higher achievable spectral efficiency and reliability when compared to the other two classes, while also being able to adapt to changes in the network where QoS and reliability are guaranteed for each D2D pair. In particular, this utility function was chosen for the streaming application class, as it provides greater emphasis on achievable spectral efficiency, such that, content can be fully uploaded/downloaded/streamed in a reasonable timeframe, serious gamers won’t experience any latency concerns, and emergency services will have higher priority and reliability. The streaming class utility uSa,i for D2D pairi is defined as:

3.2 Flexible Resource Allocation Stackelberg Game 57

where rdi is the channel of D2D pair i; and v and w are positive scale factors to

ensure all parts of (3.8) have the same magnitude.

We observe that transmit power pdi of each D2D pair i belongs to a nonempty,

convex, and compact subset of Euclidean space R|D|_{. Each follower i finds the} best response (optimal) transmit power, by maximising its utility function uF ∈

{uCa, uIa, uSa} (that is, the utility function that best maps to the follower’s appli-

cation or service) with respect to transmit power, as follows:

maximise

pdi

uF,i(x, αj,i, pdi) ∀ i ∈ D

subject to pmin ≤pdi ≤pmax.

(3.9)

Conjecture 3.1 A non-scalarisation approach to multi-criteria optimisation – con-

sidering several, and arbitrarily different, utility functions for followers – can result in a best response outcome for all players when using a Stackelberg game.