We now model the HM-based link adaptive relaying scheme described in Section 4.

### 5.1 Model

As indicated above, based on the channel conditions of users, different regions can be constructed by grouping different mean channel responses of users into a finite set of values *j*,*j* = 1,...,*J*; each region containing users with similar radio conditions and, thus, similar coding and modulation scheme (see, Figure 6) [12].

Let users arrive to the system following a Poisson process with mean intensity *λ*. Users in region *j* are granted *N*^{j}(**y**) subcarriers for a (finite) mean time duration *T*^{j}. **y** is a vector with entry *j* representing the number *y*^{j}of users present in region *j, j* = 1,...,*J*. The service duration *T*^{j}depends on the quantity of resources the users get, which in turn depends on the number and type of users that are simultaneously in progress in the system as well as how well they can take advantage of the resources they are granted, i.e., their radio conditions.

Recall that users are served using TDM. This ensures fairness in time among all users in the cell. In this case, the number of users in our system can be modeled as a Continuous Time Markov Chain (CTMC) with state vector **y**, again, denoting the number of users *y*^{j}in each region of the system, *j* = 1,..., *J*. This model is completely described by a Processor Sharing (PS) queue with S={\sum}_{j}{y}^{j} being the total number of users in the system.

We also apply an admission control scheme on the maximal number of users *s*_{max} that can be admitted to the system. This guarantees every admitted user some minimal number of subcarriers *n*_{min} equal to \frac{N}{{s}_{\text{max}}} which, in turn, guarantees some minimal throughput for each user in the system.

Let us denote by \stackrel{\u0304}{\rho} the total cell load and by {\stackrel{\u0304}{\rho}}^{j} the load corresponding to region *j*,*j* = 1,...,*J*. We have:

\stackrel{\u0304}{\rho}=\sum _{j=1}^{J}{\stackrel{\u0304}{\rho}}^{j}

(14)

where {\stackrel{\u0304}{\rho}}^{j}={\rho}^{j}/{c}^{j};{\rho}^{j}=\rho \pi \left({\left({r}^{j}\right)}^{2}-{\left({r}^{j-1}\right)}^{2}\right);\rho =\lambda E\left[F\right]. *r*^{j}is the radius of region *j, j* = 1,..., *J, E*[*F*] is the mean file size and *c*^{j}is the rate that a user *s* of type *j* can achieve when he is alone in the system. In a scheme without using relays and without HM, the maximum achieved throughput by user *s* in region *j*, is given by [13]:

{c}_{s}^{j}=\sum _{n=1}^{N}\frac{W}{N}{\text{log}}_{2}\left(1+\frac{{\mathsf{\text{SNR}}}_{s,n}^{j}}{\Gamma}\right)

(15)

where *W* denotes the total bandwidth, *N* denotes the total number of subcarriers and {\mathsf{\text{SNR}}}_{s,n}^{j} is the instantaneous SNR experienced by user *s* in region *j* on subcarrier *n*.

Let us consider, as in Section 4 and, again, without loss of generality, that we have two types of users in the system (*J* = 2): users of type 1 who enjoy good radio conditions and who are able to decode successfully their signals, and users of type 2, who present worse radio conditions and may not be able to decode successfully their signals through the direct link (BS-SS) only.

#### 5.1.1 Using classical relaying

Considering TDM scheduling, time slots are classified into three possibilities: one time slot to serve a user of type 1, one time slot to serve a user of type 2 if *P*^{R}= 0 and 2 time slots to serve the same user if *P*^{R}= 1. And so, given *P* and *P*^{R}, users of type 2 get a rate equal to ((1 - *P*^{R}) + *PP*^{R})*c*^{2} for \frac{{y}^{{-}_{2}}\left(\left(1-{P}^{R}\right)+P{P}^{R}\right)}{{y}^{{-}_{1}}+\left(1-{P}^{R}\right){y}^{{-}_{2}}+2{P}^{R}{y}^{{-}_{2}}} of the time where {\overline{y}}^{j},j=1,2, denotes the mean number of users of each type in the system, and is equal to \frac{{\stackrel{\u0304}{\rho}}^{j}}{1-\stackrel{\u0304}{\rho}}.

Without the need for relaying (successful decoding on the direct link), users of type 2 get a rate equal to *c*^{2} for \frac{{y}^{{-}_{2}}}{{y}^{{-}_{1}}+{y}^{{-}_{2}}} of the time.

Consequently, the capacity *c*^{2} of users of type 2 changes, and the capacity given by Equation (15) will now be multiplied by the following term:

\frac{\left({y}^{{-}_{1}}+{y}^{{-}_{2}}\right)\left(\left(1-{P}^{R}\right)+P{P}^{R}\right)}{{y}^{{-}_{1}}+\left(1-{P}^{R}\right){y}^{{-}_{2}}+2{P}^{R}{y}^{{-}_{2}}}

(16)

The change in *c*^{2} will, in turn, impact the performance of users of type 1 as they will now have one time slot less every time the transmission to users of type 2 uses a relay. Users of type 1 obtain a rate equal to *c*^{1} for \frac{{y}^{{-}_{1}}}{{y}^{{-}_{1}}+\left(1-{P}^{R}\right){y}^{{-}_{2}}+2{P}^{R}{y}^{{-}_{2}}} of the time.

On the other hand, without the need for relaying, they obtain a rate equal to *c*^{1} for \frac{{y}^{{-}_{1}}}{{y}^{{-}_{1}}+{y}^{{-}_{2}}} of the time. Consequently, the capacity for users of type 1, given by Equation (15) under TDM will now be multiplied by:

\frac{{y}^{{-}_{1}}+{y}^{{-}_{2}}}{{y}^{{-}_{1}}+\left(1-{P}^{R}\right){y}^{{-}_{2}}+2{P}^{R}{y}^{{-}_{2}}}

(17)

#### 5.1.2 Using HM-based link adaptive relaying

Considering the HM-based adaptive relaying scheme described previously, the TDM scheduling will also be altered due to the need for resources for relaying. As mentioned previously, the use of a resource allocation that is adapted to the radio conditions of the RS-SS link allows us to save some resources (in terms of time slots and power) compared to the classical relaying scheme and, in this case, during only one time slot, the RS forwards *β* different signals to the destination instead of only one signal. Recall that *β* is an integer equal to \frac{{c}^{R}}{{c}^{2}}, where *c*^{R}is the throughput achieved at the RS-SS link and *c*^{2} is the capacity of users of type 2. Recall that the ratio \frac{{c}^{R}}{{c}^{2}} is equal to the ratio \frac{{\text{log}}_{2}\left({M}^{\prime}\right)}{{\text{log}}_{2}\left(M\right)} (where *M* and *M*' stand for the constellation size without and with the use of link adaptation in the RS-SS link, respectively).

Here too, time slots can be divided into three possibilities: *β* time slots to serve a user of type 2 if *P*^{R}= 0, *β* + 1 time slots to serve a user of type 2 if *P*^{R}= 1 and *β* time slots to serve a user of type 1. And so, users of type 2 will get a rate equal to (*βPP*^{R}+ *β*(1 - *P*^{R}))*c*^{2} for \frac{{y}^{{-}_{2}}\left(\left(1-{P}^{R}\right)+P{P}^{R}\right)}{\beta {y}^{{-}_{1}}+\beta \left(1-{P}^{R}\right){y}^{{-}_{2}}+\left(\beta +1\right){R}^{R}{y}^{{-}_{2}}} of the time where {y}^{{-}_{j}},j=1,2, denotes, again, the mean number of users of each type in the system.

Hence, using HM-based link adaptive relaying, the capacity for users of type 2, *c*^{2}, given by Equation (15), will now be multiplied by:

\frac{\beta P{P}^{R}+\beta \left(1-{P}^{R}\right)}{\beta {y}^{{-}_{1}}+\beta \left(1-{P}^{R}\right){y}^{{-}_{2}}+\left(\beta +1\right){P}^{R}{y}^{{-}_{2}}}\left({y}^{{-}_{1}}+{y}^{{-}_{2}}\right)

(18)

As of users of type 1, their capacity *c*^{1}, given by Equation (15), will now be multiplied by a new term given by:

\frac{\beta \left({y}^{{-}_{1}}+{y}^{{-}_{2}}\right)}{\beta {y}^{{-}_{1}}+\beta \left(1-{P}^{R}\right){y}^{{-}_{2}}+\left(\beta +1\right){P}^{R}{y}^{{-}_{2}}}

(19)

### 5.2 Performance metrics

The steady-state probabilities are given by [18]:

\pi \left(x\right)=\frac{{\left(\stackrel{\u0304}{\rho}\right)}^{x}}{1+\stackrel{\u0304}{\rho}+\cdots +{\left(\stackrel{\u0304}{\rho}\right)}^{{s}_{\text{max}}}}

(20)

for 0 ≤ *x* ≤ *s*_{max}.

Using Equation (20), the mean transfer time *T*^{j}for a user in region *j, j* = 1,..., *J*, for a file of mean size *E*[*F*], is given by:

{T}^{j}=E\left[F\right]\frac{1-\left({s}_{\mathsf{\text{max}}}+1\right){\stackrel{\u0304}{\rho}}^{{s}_{\text{max}}}+{s}_{\text{max}}{\stackrel{\u0304}{\rho}}^{\left({s}_{\text{max}}+1\right)}}{{c}^{j}\left(1-\stackrel{\u0304}{\rho}\right)\left(1-{\stackrel{\u0304}{\rho}}^{{s}_{\text{max}}}\right)}

(21)

and the mean transfer time *T* in all the cell is given by

T=\frac{{\sum}_{j=1}^{J}{T}^{j}{\lambda}^{j}}{{\sum}_{j=1}^{J}{\lambda}^{j}}

(22)

where *λ*^{j}is the mean arrival intensity to region *j*.

The blocking probability *B* of a new flow, based on our admission control algorithm, is given by:

B=\frac{{\stackrel{\u0304}{\rho}}^{{s}_{\text{max}}}}{1+\stackrel{\u0304}{\rho}+\cdots +{\stackrel{\u0304}{\rho}}^{{s}_{\text{max}}}}

(23)