Resource Allocation for Improved User Satisfaction with Applications to LTE

Abstract

Cellular networks have experienced a strong development in the past decades and the technology evolution to meet the continued steep increase of mobile traffic expected for the next years is an important challenge. In this context, cellular operators have as objective to increase the number of satisfied users in the system whereas users or subscribers aim at having fulfilled their expected quality of service. In order to increase the number of satisfied users in the system we identify radio resource allocation as a key functionality. Radio resource allocation is responsible for managing and distributing the available scarce resources of the radio interface to the active connections. In this chapter, we present radio resource allocation strategies with multiple antennas at the transmitter and/or receivers to increase the number of satisfied users in cellular networks based on two approaches: heuristic and utility-based strategies. While the heuristic design provides simple and quick solutions to the radio resource allocation problems, the utility-based approach is a flexible and general tool for radio resource allocation design. Simulation results show that the proposed algorithms following these design guidelines are able to achieve high number of satisfied users in modern networks.

Appendices

Appendix 1: Utility-Based Optimization Formulation for NRT Services

Let us consider a utility-based optimization problem in a scenario with NRT services formulated as:

$$\begin{aligned}&\underset{\fancyscript{K}_j}{\text {max}} \sum _{j=1}^{J} U\left( T_j\left[ n\right] \right) \end{aligned}$$

(2.35a)

$$\begin{aligned} \text {subject to} \quad&\bigcup _{j=1}^{J} \fancyscript{K}_j \subseteq \fancyscript{K}, \end{aligned}$$

(2.35b)

$$\begin{aligned}&\fancyscript{K}_i \bigcap \fancyscript{K}_j = \emptyset , \quad i \ne j, \quad \forall i,j \in \{1, 2, \ldots , J\}, \end{aligned}$$

(2.35c)

where $J$ is the total number of users in a cell, $K$ is the total number of resources in the system (sub-carriers, codes, or the like) to be assigned to the users, $\fancyscript{K}$ is the set of all resources in the system, $\fancyscript{K}_j$ is the subset of resources assigned to user $j$, and $U\left( T_{j}\left[ n\right] \right)$ is an increasing utility function based on the current throughput $T_{j}\left[ n\right]$ of the user $j$ in TTI $n$. Constraints (2.35b) and (2.35c) state that the union of all subsets of resources assigned to different users must be contained in the total set of resources available in the system, and that these subsets must be disjoint, i.e., the same resource cannot be shared by two or more users in the same TTI.

The throughput of user $j$ is calculated using an exponential smoothing filtering, as indicated below:

$$\begin{aligned} T_{j}\left[ n\right] = \left( 1 - f^{\mathrm {thru}}\right) \cdot T_{j}\left[ n-1\right] + f^{\mathrm {thru}} \cdot R_j\left[ n\right] , \end{aligned}$$

(2.36)

where $R_j\left[ n\right]$ is the instantaneous data rate of user $j$ and $f^{\mathrm {thru}}$ is a filtering constant.

Evaluating the objective function in (2.35a) and the throughput expression in (2.36), the derivative of $U\left( T_{j}\right)$ with respect to the transmission rate $R_j$ is given by:

$$\begin{aligned} \frac{\partial U}{\partial R_j} = \frac{\partial U}{\partial T_j} \cdot \frac{\partial T_j}{\partial R_j} = f^{\mathrm {thru}} \cdot \left. \dfrac{\partial U}{\partial T_j}\right| _{T_{j} = \left( 1 - f^{\mathrm {thru}}\right) \cdot T_{j}\left[ n-1\right] + f^{\mathrm {thru}} \cdot R_j\left[ n\right] }. \end{aligned}$$

In the case that $f^{\mathrm {thru}}$ is sufficiently small, the expression above can be simplified as follows [32]:

$$\begin{aligned} \frac{\partial U\left( T_{j}\left[ n\right] \right) }{\partial R_{j}\left[ n\right] } \approx f^{\mathrm {thru}} \cdot \left. \dfrac{\partial U}{\partial T_j}\right| _{T_j = T_{j}\left[ n-1\right] }, \end{aligned}$$

(2.37)

where the previous resource allocation totally determines the current values of the marginal utilities. Using the one-order Taylor formula for the utility function [25, 32] and considering (2.37), we have

$$\begin{aligned}&\sum _{j=1}^{J} U\left( T_{j}\left[ n\right] \right) \approx \sum _{j=1}^{J} U\left( T_{j}\left[ n-1\right] \right) +\nonumber \\&\sum _{j=1}^{J} \left. \dfrac{\partial U}{\partial T_j}\right| _{T_j = T_{j}\left[ n-1\right] } \cdot \left( f^{\mathrm {thru}} \cdot R_j\left[ n\right] - f^{\mathrm {thru}} \cdot T_j\left[ n-1\right] \right) \!. \end{aligned}$$

(2.38)

Notice that maximizing (2.38) leads to the maximization of the original objective function (2.35a). Since $f^{\mathrm {thru}}$ is a constant and $T_j\left[ n-1\right]$ is known and fixed before the resource allocation at the current TTI $n$, the objective function of our simplified optimization problem becomes linear in terms of the instantaneous user’s data rate, and is given by

$$\begin{aligned} \underset{\fancyscript{K}_j}{\text {max}} \sum _{j=1}^{J} U^{'}\left( T_{j}\left[ n-1\right] \right) \cdot R_{j}\left[ n\right] , \end{aligned}$$

(2.39)

where $U^{'}\left( T_{j}\left[ n-1\right] \right) = \left. \dfrac{\partial U}{\partial T_j}\right| _{T_j = T_{j}\left[ n-1\right] }$ is the marginal utility (derivative of the utility function) of user $j$ with respect to its throughput in the previous TTI. The objective function (2.39) characterizes a weighted sum rate maximization problem [8], whose weights are adaptively controlled by the marginal utilities.

Notice that we started with an optimization formulation based on throughput given by (2.35a), made some logical assumptions and mathematical simplifications, and ended up with a linear optimization formulation based on instantaneous rates given by (2.39). According to these arguments, we claim that the instantaneous optimization maximizing (2.39) leads to a long-term optimization that maximizes (2.35a).

Appendix 2: Utility-Based Optimization Formulation for RT Services

Let us consider a utility-based optimization problem in a scenario with RT services formulated as:

$$\begin{aligned}&\underset{\fancyscript{K}_j}{\text {max}} \sum _{j=1}^{J} U\left( d_j^{\mathrm {hol}}\left[ n\right] \right) \end{aligned}$$

(2.40a)

$$\begin{aligned} \text {subject to} \quad&\bigcup _{j=1}^{J} \fancyscript{K}_j \subseteq \fancyscript{K}, \end{aligned}$$

(2.40b)

$$\begin{aligned}&\fancyscript{K}_i \bigcap \fancyscript{K}_j = \emptyset , \quad i \ne j, \quad \forall i,j \in \{1, 2, \ldots , J\}, \end{aligned}$$

(2.40c)

where $J$ is the total number of users in a cell, $K$ is the total number of resources in the system (sub-carriers, codes, or the like) to be assigned to the users, $\fancyscript{K}$ is the set of all resources in the system, $\fancyscript{K}_j$ is the subset of resources assigned to user $j$, and $U\left( d_j^{\mathrm {hol}}\left[ n\right] \right)$ is a decreasing utility function based on the current HOL delay $d_{j}^{\mathrm {hol}}\left[ n\right]$ of user $j$ at TTI $n$. Constraints (2.40b) and (2.40c) state that the union of all subsets of resources assigned to different users must be contained in the total set of resources available in the system, and that these subsets must be disjoint, i.e., the same resource cannot be shared by two or more users in the same TTI.

In order to understand the model used in this work for the calculation of the HOL delays, Fig. 2.12a is presented. This figure illustrates a packet queue for a given RT user. As it can be seen in the figure, the traffic model for RT services used in this work assumes a packet arrival rate of $L$ packets per second, i.e., a new packet of $b_j^{\mathrm {hol}}$ bits (fixed size) arrives in the buffer of user $j$ every $1/L$ s.

Fig. 2.12

Modeling of a RT user buffer a User buffer as a sequence of packets b User buffer as a sequence of time slices

Taking into account Fig. 2.12a and considering a generic user $j$, we propose in this work a recursive model for calculating an approximate value of the HOL delay. The recursive equation is

$$\begin{aligned} d_{j}^\mathrm {hol}\left[ n+1\right] = d_{j}^\mathrm {hol}\left[ n\right] + t^{\mathrm {tti}} - \frac{1}{L} \cdot \left( \frac{R_j\left[ n\right] \cdot t^{\mathrm {tti}}}{b_j^{\mathrm {hol}}}\right) , \end{aligned}$$

(2.41)

where $t^{\mathrm {tti}}$ is the duration of the TTI in seconds, $L$ is the packet arrival rate, $b_j^{\mathrm {hol}}$ is the packet size in bits, and $R_{j}\left[ n\right]$ is the instantaneous achievable transmission rate on TTI $n$. In this queue model, we assume that the packet size $b_j^{\mathrm {hol}}$ is sufficiently small, so that the queue can be represented ideally by a sequence of time slices with duration $1/L$ s each (see Fig. 2.12b). Notice that this assumption does not invalidate the mathematical and conceptual RRA framework, and makes the optimization model much more tractable.

Looking at (2.41), first it can be seen that the HOL delay is always incremented by at least the duration of one TTI, no matter how many bits were transmitted in the current transmission interval. This represents the passing of time in the system, which means that all packets in the queue will be one TTI older. Second, the decrement of the HOL delay depends on the number of time slices (duration of $1/L$ seconds each) that is decremented due to the transmission in the current TTI. If user $j$ has not been served by any resource in TTI $n$, $R_{j}\left[ n\right]$ is equal to zero and no time slices are decremented. If the instantaneous transmission rate is such that the HOL packet is totally transmitted in the current TTI, it means that one time slice with duration of $1/L$ seconds should be decremented in the HOL delay. If the instantaneous transmission rate is sufficiently high so that many packets in the queue can be transmitted, the corresponding number of time slices should be decremented in the HOL delay.

Assessing the objective function in (2.40a) and the HOL delay expression in (2.41), we can see that the derivative of $U\left( d_{j}^\mathrm {hol}\right)$ with respect to the transmission rate $R_j$ can be expressed as

$$\begin{aligned} \frac{\partial U}{\partial R_j} = \frac{\partial U}{\partial d_j^\mathrm {hol}} \cdot \frac{\partial d_j^\mathrm {hol}}{\partial R_j} = \frac{\partial U}{\partial d_j^\mathrm {hol}} \cdot \left( -\frac{t^{\mathrm {tti}}}{L \cdot b_j^{\mathrm {hol}}} \right) . \end{aligned}$$

Using the result above and assuming that the TTI duration is sufficiently small, the Lagrange theorem of the mean can be used [15, 25], which says that

$$\begin{aligned}&\sum _{j=1}^{J} U\left( d_{j}^{\mathrm {hol}}\left[ n+1\right] \right) \nonumber \\&\approx \sum _{j=1}^{J} U\left( d_{j}^{\mathrm {hol}}\left[ n\right] \right) + \sum _{j=1}^{J} \left. \dfrac{\partial U}{\partial R_j}\right| _{R_j=R_j\left[ n-1\right] } \cdot \left( R_j\left[ n\right] - R_j\left[ n-1\right] \right) \nonumber \\&=\sum _{j=1}^{J} - \left. \dfrac{\partial U}{\partial d_{j}^\mathrm {hol}}\right| _{d_{j}^\mathrm {hol} = d_{j}^\mathrm {hol}\left[ n\right] } \cdot \frac{t^{\mathrm {tti}}}{L \cdot b_j^{\mathrm {hol}}} \cdot \left( R_j\left[ n\right] - R_j\left[ n-1\right] \right) \nonumber \\&=\sum _{j=1}^{J} \left. \left| \dfrac{\partial U}{\partial d_{j}^\mathrm {hol}} \right| \right| _{d_{j}^\mathrm {hol} = d_{j}^\mathrm {hol}\left[ n\right] } \cdot \frac{t^{\mathrm {tti}}}{L \cdot b_j^{\mathrm {hol}}} \cdot \left( R_j\left[ n\right] - R_j\left[ n-1\right] \right) \!. \end{aligned}$$

(2.42)

The absolute value operator was used in (2.42) because the utility function was assumed to be decreasing, which yields a negative marginal utility (derivative of the utility function) and cancels the negative sign in (2.42). Notice that the maximization of (2.42) leads to the maximization of (2.40a). Taking into account (2.42), we have that $t^{\mathrm {tti}}$, $L$, and $b_j^{\mathrm {hol}}$ are constants, and that $d_{j}^{\mathrm {hol}}\left[ n\right]$ and $R_j\left[ n-1\right]$ are known and fixed before the resource allocation at TTI $n$. Therefore, our simplified optimization objective function is given by

$$\begin{aligned} \underset{\fancyscript{K}_j}{\text {max}} \sum _{j=1}^{J} \left| U^{'}\left( d_{j}^\mathrm {hol}\left[ n\right] \right) \right| \cdot R_j\left[ n\right] \!, \end{aligned}$$

(2.43)

where $U^{'}\left( d_{j}^\mathrm {hol}\left[ n\right] \right) = \left. \dfrac{\partial U \left( d_{j}^\mathrm {hol} \right) }{\partial d_{j}^\mathrm {hol}}\right| _{d_{j}^\mathrm {hol} = d_{j}^\mathrm {hol}\left[ n\right] }$ is the marginal utility of user $j$ with respect to its current HOL delay. The objective function (2.43) is a weighted sum rate maximization [8], where the weights are given by the absolute value of the marginal utility with respect to the current HOL delay.

Notice that after some logical assumptions and mathematical simplifications made upon (2.40a), a linear optimization formulation based on instantaneous rates given by (2.43) was achieved. Therefore, we claim that the instantaneous optimization maximizing (2.43) leads to a long-term optimization that maximizes (2.40a).