The random access NUM with multiclass traffic
 Phuong L Vo^{1},
 Sungwon Lee^{1} and
 Choong Seon Hong^{1}Email author
https://doi.org/10.1186/168714992012242
© Vo et al.; licensee Springer. 2012
Received: 14 February 2012
Accepted: 18 July 2012
Published: 6 August 2012
Abstract
In this article, we consider the network utility maximization (NUM) problem for the random access network with multiclass traffic. The utilities associated with the users are not only concave, but also nonconcave functions. Consequently, the random access NUM problem becomes more difficult to solve. Based on the successive approximation method, we propose an algorithm that jointly controls the rate and the persistent probability of the users. The proposed algorithm converges to a suboptimal solution to the original problem which also satisfies the Karush–Kuhn–Tucker conditions. We also generalize the framework so that a broader choice of utility functions can be applied.
Keywords
Introduction
The early studies that deal with the inelastic traffic in the basic NUM problem for wired networks are[5, 6]. The authors utilize the standard dualbased algorithm to allocate the rate. Certainly, this algorithm does not result to an optimal solution because of the nonconvexity of the primal problem. The duality gap is not always zero and the result is suboptimal or even infeasible. Therefore, the authors of[5] offer a ‘selfregulate’ mechanism for the users to access the network without fluctuation. On the other hand, the authors of[6] find the conditions for which the dualbased algorithm converges to a global optimum. It turns out that the link capacity must be higher than a critical value. Then, they propose the link ‘capacity provisioning’ to satisfy those conditions. Another method to solve the basic NUM is using the sumofsquare method in[7]. The nonconvex NUM is relaxed and solved by semidefinite programming. However, this method requires a centralized and offline computation. Its framework is also difficult to integrate into the crosslayer optimization problem in which the dual decomposition approach has shown its efficiency[8]. Extending the work in[6] to the random access WLANs, the authors of[9] design a dualbased algorithm to jointly allocate the rate and the persistent probability of elastic and inelastic traffic. Consequently, their algorithm only converges just in the case where the link capacities are higher than critical values. Otherwise, only the lower bound and upper bound are specified.
In this article, we address the random access NUM for multiclass traffic using the successive approximation method. The solutions to the convex approximation problems converge to a suboptimal solution which also satisfies the Karush–Kuhn–Tucker (KKT) conditions of the original problem. The successive approximation method is first introduced in[10]. It is usually applied to geometric programming in the power control problems such as[11–13]. Similar to our previous work[14] which jointly controls the rate and power in a multihop wireless network with multiclass traffic, the nonconcave objective of the problem is approximated to a concave function. After solving a series of approximation problems, the algorithm converges. Moreover, we generalize our analysis framework and show that a broader choice of utilities can be obtained.
The rest of the article is organized as follows. Section ‘Design of the successive approximation algorithm’ introduces the network model and propose the successive approximation algorithm. Section ‘More general utility functions and analysis’ generalizes the framework analysis and finds the conditions on the utility. The numerical results and some discussions are presented in Section ‘Numerical results and discussions’. Finally, conclusions are given in Section ‘Conclusions’.
Notations: In this article, we use italic characters to denote variables and bold characters to denote vectors. For example,$\mathit{x}=[{x}_{1},\dots ,{x}_{\left\mathcal{N}\right}]$,$\mathit{p}=[{p}_{1},\dots ,{p}_{\left\mathcal{N}\right}]$, and$\mathit{c}=[{c}_{1},\dots ,{c}_{\left\mathcal{N}\right}]$ are$\left\mathcal{N}\right$dimensional vectors which elements are x_{ i }, p_{ i }, and c_{ i }, respectively. The words ‘user’ and ‘node’ are sometimes used interchangeably.
Design of the successive approximation algorithm
Network model
where U_{ i } is the utility function of user i. In this article, we assume that${\mathit{x}}^{min}$ is strictly greater than 0 to avoid dividing by zero in the mathematical analysis.
 1.The concave utilities for elastic traffic$U\left(x\right)=\left\{\begin{array}{c}\mathrm{ln}(x+1),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}\alpha =1,\\ \frac{{(x+1)}^{(1\alpha )}1}{1\alpha},\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}\alpha >0\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}\alpha \ne 1;\end{array}\right.$(1)
 2.The sigmoidal utilities for inelastic traffic$U\left(x\right)=\frac{{x}^{a}}{k+{x}^{a}},\forall a>1,k>0.$(2)
The sigmoidal function (2) has an inflection point at${x}^{\mathrm{in}}={\left(\frac{k(a1)}{a+1}\right)}^{1/a}$. It is convex in$({x}^{min},{x}^{\mathrm{in}})$ and concave in$({x}^{\mathrm{in}},{x}^{max})$. In the literature, sigmoidal function is usually used for the realtime utility because it is small when the rate is below x^{in} and increases quickly when the rate exceeds x^{in}. As a result, x^{in}is also considered the demand of a realtime connection (see Figure1).
Similar to the articles on utility optimality of multiclass traffic, e.g.,[5, 7, 9, 15], the concave utilities usually cannot take the conventional form of αfair utility which is ln(x) if α=1 and$\frac{{x}^{1\alpha}}{1\alpha}$ if α>0 and α≠1[16]. It is shifted 1 unit on the xaxis. With the present of sigmoidal utilities which are usually the same as (2) or$\frac{1}{1+{e}^{a(xb)}}$a b>0 in the literature, the utilities of the users are normalized or at least have close values at${x}^{max}$ in order to be comparable. Otherwise, the inelastic flows always take the advantage over the elastic flows because of the conventional αfair utility is negative as α>1. So the concave utilities usually have the form as (1) in these articles.
Approximation problem
where ν_{ i } is the multiplier associated with the constraint${x}_{i}\le {c}_{i}{p}_{i}\prod _{j\ne i}(1{p}_{j})$ for all$i\in \mathcal{N}$. We have the following result
Lemma 1
then$({\mathit{x}}^{\ast},{\mathit{p}}^{\ast},(\sum _{i\in \mathcal{N}}{U}_{i}\left({x}_{i}^{\ast}\right)\left){\mathit{\nu}}^{\ast}\right)$ is a KKT point of (P1).
Proof
when${\mathit{\mu}}^{\ast}=\left(\sum _{i\in \mathcal{N}}{U}_{i}\right({x}_{i}^{\ast}\left)\right){\mathit{\nu}}^{\ast}$, for all$i\in \mathcal{N}$. □
As we have mentioned earlier, there is a sequence of approximations. The superscript τ is used here to indicate that this is the τ th approximation problem, θ^{(τ)} is a fixed value in τ th approximation problem. It will be proved that, by updating θ and solving the approximation problem many times, the solution to the approximation problem converges. At the stationary point, the approximation becomes exact.
where${\u0168}_{i}({\stackrel{~}{x}}_{i};{\theta}_{i}^{\left(\tau \right)})\triangleq {\theta}_{i}^{\left(\tau \right)}\mathrm{ln}\left(\frac{{U}_{i}\left({e}^{{\stackrel{~}{x}}_{i}}\right)}{{\theta}_{i}^{\left(\tau \right)}}\right)$ is a function of${\stackrel{~}{x}}_{i}$ parameterized by θ_{ i }, and${\stackrel{~}{c}}_{i}\triangleq \mathrm{ln}\left({c}_{i}\right)$.
Lemma 2
The function${\u0168}_{i}({\stackrel{~}{x}}_{i};{\theta}_{i})$ is strictly concave for both concave and sigmoidal utilities (1) and (2).
Proof
See the Appendix for the proof. □
From Lemma 2, (P4^{ τ }) is a convex problem; therefore, it can be solved efficiently for an optimal solution. In the next section, we will solve (P4^{ τ }) using the dualbased decomposition approach.
Solution to the approximation problem and the algorithm
and the dual problem is$\underset{\mathit{\lambda}\succcurlyeq 0}{min}D(\mathit{\lambda};{\mathit{\theta}}^{\left(\tau \right)})$.
where γ(t) is the stepsize sequence,${\stackrel{~}{x}}_{i}^{\left(\tau \right)}\left(t\right)$ and${p}_{i}^{\left(\tau \right)}\left(t\right)$ are calculated according to (17) and (18), respectively, at time instant t.${\left[a\right]}^{+}=max(a,0)$. Once again, we use the superscript τ in (17)–(19) to indicate that they are the values in solving the τ th approximation problem. From the above analysis, we develop the successive approximation algorithm for the multiclass traffic in the onehop random access wireless network as described in Algorithm 1.
Algorithm 1 Successive approximation algorithm for multiclass traffic
 1.
Initialize from θ^{(0)}and any feasible point;
 2.
τ:=0;
 3.
loop
 4.
τ:=τ + 1;
 5.
t:=0;
 6.
repeat
 7.
t:=t + 1;
 8.
Set rate, persistent probability, and multipliers according to (17), (18), and (19) respectively;
 9.
until stationary;
 10.
${\theta}_{i}^{(\tau +1)}:=\frac{{U}_{i}\left({x}_{i}^{\left(\tau \right)\ast}\right)}{\sum _{k\in \mathcal{N}}{U}_{k}\left({x}_{k}^{\left(\tau \right)\ast}\right)}$;
 11.
${x}_{i}^{(\tau +1)}\left(0\right):={x}_{i}^{\left(\tau \right)\ast}$;
 12.
end loop
In Algorithm 1,${x}_{s}^{\left(\tau \right)\ast}$ is the stationary value of the τ th (outer)iteration. At step 10, the new value θ is calculated by the stationary rate of previous outeriterations. Moreover, the initial value of a new outeriteration is the stationary value of the previous outeriteration at step 11.
Theorem 1
If the step size satisfies γ(t)>0,$\underset{t\to \infty}{lim}\gamma \left(t\right)=0$, and$\sum _{t=1}^{\infty}\gamma \left(t\right)=\infty $, then Algorithm1 monotonically increases the aggregate utility in each outeriteration and converges to a stationary point satisfying the KKT conditions of (P1).
Proof
See the Appendix for the proof. □
We have some discussions on the distributed implementation and the message passing mechanism of the proposed algorithm. There are two kinds of updates in Algorithm 1, the innerupdates (17)–(19) and the outerupdates (14). In each inneriteration, a user uses the information$\sum _{j\in \mathcal{N}}{\lambda}_{j}\left(t\right)$ to update its persistent probability according to (18). The persistent probabilities of all the nodes are also needed to update the user’s multiplier according to (19). Hence, after each inneriteration, each user broadcasts its information (p_{ i } and λ_{ i }) to all the other users in the network. At the outeriteration, each user needs the information of total utility of all the users to update its θvalue according to (14). Therefore, each user also broadcasts its current utility value to all the other users in each outeriteration. Note that, the users update their θvalues as recognizing the stationary of the inneriterations. The following technique can be used for the users to recognize the stationary. The users broadcast their utility periodically after each T timeslots. So, each user can always keep track of the aggregate utility value of the system. It only updates its θvalue as recognizing the stationary of this value.
 1.
Each node piggybacks its information p_{ i }, λ_{ i }, and θ_{ i }by inserting them into their data packets. Since all nodes are onehop neighbors to each other, the other odes can overhear these information and update their values based on the received information.
 2.
The multiplier update (19) can be a local update as follows. We rewrite the update (19) by ${\lambda}_{i}^{\left(\tau \right)}(t+1)={\left[{\lambda}_{i}^{\left(\tau \right)}\left(t\right)\gamma \left(t\right)({\stackrel{~}{c}}_{i}+\mathrm{ln}({p}_{i}^{\mathrm{succ}}\left(t\right)){\stackrel{~}{x}}_{i}^{\left(\tau \right)}(t\left)\right)\right]}^{+}$, where ${p}_{i}^{\mathrm{succ}}\left(t\right)={p}_{i}^{\left(\tau \right)}\left(t\right)\prod _{j\ne i}(1{p}_{j}^{\left(\tau \right)}(t\left)\right)$ is the successful transmission probability of node i. The value p^{succ}can be estimated locally. For example, (1) ${p}_{i}^{\mathrm{succ}}\approx \frac{\text{number of successful transmissions of}i}{\text{number of transmissions of}i}$, or (2) we can estimate the probability that the channel is idle ${p}^{\mathrm{idle}}\approx \frac{\text{number of timeslots the channel is idle}}{\text{number of timeslots}}$ and the successful transmission probability will be ${p}_{i}^{\mathrm{succ}}\approx {p}^{\mathrm{idle}}\frac{{p}_{i}}{1{p}_{i}}$ due to ${p}^{\mathrm{idle}}=\prod _{i\in \mathcal{N}}(1{p}_{i})$. By estimating this parameter locally, the multipliers can be implicitly updated. Therefore, the amount of message passing in the network is reduced significantly.
More general utility functions and analysis

twice continuously differentiable and monotonically increasing function;

bounded function: U_{ i }(x_{ i })>0, ∀x_{ i }>0 and U_{ i }(x_{ i }) is bounded as x_{ i }is bounded.
where U_{ i }=U_{ i }(x_{ i }),${U}_{i}^{\u2033}=\frac{d{U}_{i}\left({x}_{i}\right)}{d{x}_{i}}$, and${U}_{i}^{\mathrm{\u2033\u2033}}=\frac{{d}^{2}{U}_{i}\left({x}_{i}\right)}{d{x}_{i}^{2}}$.
We note at the factor$\frac{{f}^{\mathrm{\u2033\u2033}}({U}_{i}/{\theta}_{i})}{{\theta}_{i}{f}^{\u2033}({U}_{i}/{\theta}_{i})}$ in (22). It is always positive because f is a monotonically increasing and concave function. The higher the factor, the quicker the slope of f changes, and the more relaxed the condition of utility.
 1.
α fair utility ${U}_{i}\left({x}_{i}\right)=\frac{{x}_{i}^{1\alpha}}{1\alpha}$with 0<α<1: although this function is a canonical αfair concave function, it cannot be applied to [1]. Lemma 1 therein requires a ‘sufficiently’ concave utility function, i.e., α>1 for the αfair family. However, with the transform function f(u)=−1/u(which corresponding to β=2) and the new approximation (21) instead of (13), our framework can be applied.
 2.
Linear/convex utility function ${U}_{i}\left({x}_{i}\right)={x}_{i}^{M}$: if β=1, ${\u0168}_{i}({\stackrel{~}{x}}_{i};{\theta}_{i})$ is a linear function, the analysis in Section ‘Design of the successive approximation algorithm’ cannot be applied. With the use of f(u)=−1/u which corresponds to β=2, ${\u0168}_{i}({\stackrel{~}{x}}_{i};{\theta}_{i})=\frac{{\theta}_{i}^{2}}{{e}^{M{\stackrel{~}{x}}_{i}}}$ is a strictly concave function. Note that this utility function certainly leads to the nonconvergence of the standard dualbased algorithm in [1, 9] because it is not a concave function.
 3.
Exponential utility ${U}_{i}\left({x}_{i}\right)={\mathrm{e}}^{{x}_{i}}$: it is clear that we cannot use the standard dualbased algorithm in [1] because of the same reason as the above examples. The inequality (22) becomes $\beta >1+\frac{1}{{x}_{i}}$. Therefore, if we choose β such that $\beta >1+\frac{1}{\underset{i}{min}{x}_{i}^{min}}$, then the exponential utility can still be applied.
Numerical results and discussions
In this section, we use$\frac{{x}_{i}}{{x}_{i}+1}$ as elastic utility with α=2 and$\frac{{x}_{i}^{4}}{{x}_{i}^{4}+400}$ as inelastic utility with k=400 and a=4 (see Figure1). The rate unit for calculating utilities is Mbps. The inneriteration is considered stationary if$\left\frac{\mathit{x}\left(t\right)\mathit{x}(t1)}{\mathit{x}(t1)}\right\prec {10}^{4}$.${\mathit{x}}^{min}=0.01$ Mbps and${\mathit{x}}^{max}=\mathit{c}$ Mbps. The diminishing step size 0.001/t is used for Algorithm 1. λ^{0}(0) is 0.1.
Convergence of the algorithm
We can see that although two users are symmetric, i.e., the same utilities as well as link capacities, one of them accesses the channel most of the time whereas the other one is mostly abandoned. This result shows the major difference from the resource allocation of elastic flows in which all elastic flows are fairly allocated the resource. Therefore, by using the sigmoidal utilities, the admission control is implicitly integrated as we solve the NUM. This is an advantage of using the sigmoidal utility. Also we have a remark that we rarely have fairness among inelastic users. Intuitively, when there is not enough resource for both flows, it is better to drop one flow and keep the other one than to maintain both inelastic flows with bad quality. This unfairness is also similar to the realtime system with the explicit admission control scheme. Some realtime connections can be dropped to guarantee the system performance because of the lack of the resource.
To mitigate the unfairness among the users as well as to avoid the starvation of some users in the network, we can guarantee a minimum persistent probability for each user. The constraint$\mathbf{0}\preccurlyeq \mathit{p}\preccurlyeq \mathbf{1}$ is replaced by the new one${\mathit{p}}^{min}\preccurlyeq \mathit{p}\preccurlyeq \mathbf{1}$ where$\underset{i}{max}{p}_{i}^{min}\le 1/\left\mathcal{N}\right$ to avoid the infeasibility. As a result, the persistent probability update (18) for each user in the τ th outeriteration becomes${p}_{i}^{\left(\tau \right)}\left(t\right)=max\left(\underset{i}{\overset{min}{p}},\frac{{\lambda}_{i}^{\left(\tau \right)}\left(t\right)}{\sum _{j\in \mathcal{N}}{\lambda}_{j}^{\left(\tau \right)}\left(t\right)}\right)$ for all$i\in \mathcal{N}$. With the new lower bound${\mathit{p}}^{min}$, all the users have a minimum chance to access the channel.
A heuristic implementation
We implement a heuristic algorithm in this experiment by limiting the number of inneriterations in each τstep to a fixed value T. As we have seen, Algorithm 1 has two levels of convergence. The outeriterations update θ and the inneriterations solve the convex approximation problem. Theoretically, the number of inneriterations must be large enough for the convergence in every outerstep. In the heuristic algorithm, we limit the number of inneriterations to a fixed value T. Moreover, we also apply a constant step size to the subgradient update (19) since it usually has a faster convergence than the diminishing stepsize. It is known that with the dualbased subgradient algorithm using constant step size, the primal function sequence calculated from the running average primal values$\left\{{\widehat{\mathit{x}}}^{\left(\tau \right)}\right(t)=\frac{1}{t}\sum _{k=1}^{t}{\mathit{x}}^{\left(\tau \right)}(k),t=1,2,\dots \}$ converges to an optimal value (of P 3^{ τ }) within an error ([18], Sec. 1.2). The feasible violation of the running average primal sequence also converges to zero. So, in the heuristic algorithm, θ_{ i } corresponding to user i is updated according to${\theta}_{i}^{\left(\tau \right)}=\frac{{U}_{i}\left({\widehat{x}}_{i}^{(\tau 1)}\right(T\left)\right)}{\sum _{k\in \mathcal{N}}{U}_{k}\left({\widehat{x}}_{k}^{(\tau 1)}\right(T\left)\right)}$, the running average value of the previous outeriteration. The heuristic algorithm converges to the same solution as Algorithm 1 does in most of our experiments. However, we have a note that its convergence cannot be guaranteed theoretically. The reason is that with the dualbased subgradient update solving the approximation problem, the primal value${\widehat{x}}_{k}^{(\tau 1)}\left(T\right)$ can be infeasible. Therefore, the inequality (27) is no longer valid, i.e., we cannot guarantee a feasible improvement of the objective in every outeriterations.
Varying the initial point
Compare to the standard dualbased algorithm
We compare the aggregate utility archived by Algorithm 1 to the lower and upper bounds calculated from the standard dualbased algorithm in[9] as the number of users in the network increases gradually. In[9], after logtransforming the rate variables of the original NUM, the standard dualbased algorithm (Algorithm 1 therein) can achieve the stationary value of the multipliers, i.e., λ^{∗}, due to the convexity of the dual problem. Therefore, the lower bound is calculated by$\sum _{i\in \mathcal{N}}{U}_{i}\left({x}_{i}^{\ast}\right)$, where${p}_{i}^{\ast}=\frac{{\lambda}_{i}^{\ast}}{\sum _{j\in \mathcal{N}}{\lambda}_{j}^{\ast}}$ and${x}_{i}^{\ast}={c}_{i}{p}_{i}^{\ast}\prod _{j\ne i}(1{p}_{j}^{\ast})$. The upper bound is the value of the dual function at the point λ^{∗}. Notice that this upper bound is absolutely not a feasible solution in case of nonzero duality gap.
Compare to binary exponential backoff MAC protocol
Setting parameters for subsection “Compare to binary exponential backoff MAC protocol”
Basic rate  1 Mbps 
Data rate (link capacity)  12 Mbps 
Slot time  9 μ s 
Propagation delay  1 μ s 
SIFS  16 μ s 
DIFS  34 μ s 
Packet size  512 bytes 
Physical header  48 bits 
MAC header  96 bits 
ACK  20 bytes 
Conclusions
Based on successive approximation method, we have proposed an algorithm that converges to a KKT solution to the nonconvex NUM problem of a random access WLAN serving multiclass traffic. The equivalent problem of the original one is approximated to a new convex problem, which is solved efficiently by the dualbased decomposition approach. The algorithm converges after a sequence of approximations. We specify the necessary condition on the utilities to be used in the framework and we also generalize the analysis framework. The simulations show that our algorithm can achieve the global optimum starting from many initial points.
Appendix
Proof of Lemma 2
We prove Lemma 2 by verifying the second derivative of${\u0168}_{i}({\stackrel{~}{x}}_{i};{\theta}_{i})$ in terms of${\stackrel{~}{x}}_{i}$. For clearly presentation, we transform back to the x space and omit the superscript τ.
 1.if α=1, then$\begin{array}{ll}\frac{{d}^{2}\stackrel{~}{{U}_{i}}({\stackrel{~}{x}}_{i};{\theta}_{i})}{{d{\stackrel{~}{x}}_{i}}^{2}}=& \frac{{d}^{2}}{{d{\stackrel{~}{x}}_{i}}^{2}}\left({\theta}_{i}\mathrm{ln}\left(\frac{\mathrm{ln}({e}^{{\stackrel{~}{x}}_{i}}+1)}{{\theta}_{i}}\right)\right)\phantom{\rule{2em}{0ex}}\\ =& \frac{{\theta}_{i}{x}_{i}}{{({x}_{i}+1)}^{2}{\mathrm{ln}}^{2}({x}_{i}+1)}\phantom{\rule{2em}{0ex}}\\ \times ({x}_{i}\mathrm{ln}({x}_{i}+1\left)\right)<0\phantom{\rule{2em}{0ex}}\end{array}$(23)
 2.If α>0 and α≠1, then$\begin{array}{ll}\phantom{\rule{12.0pt}{0ex}}\frac{{d}^{2}{\u0168}_{i}({\stackrel{~}{x}}_{i};{\theta}_{i})}{{d{\stackrel{~}{x}}_{i}}^{2}}=& \frac{{d}^{2}}{{d{\stackrel{~}{x}}_{i}}^{2}}\left({\theta}_{i}\mathrm{ln}\left(\frac{{({e}^{{\stackrel{~}{x}}_{i}}+1)}^{1\alpha}1}{{\theta}_{i}(1\alpha )}\right)\right)\phantom{\rule{2em}{0ex}}\\ =& (1\alpha )\left({({x}_{i}+1)}^{1\alpha}(1\alpha ){x}_{i}1\right)\phantom{\rule{2em}{0ex}}\\ \times \frac{{\theta}_{i}{x}_{i}{({x}_{i}+1)}^{1\alpha}}{{({({x}_{i}+1)}^{1\alpha}1)}^{2}}.\phantom{\rule{2em}{0ex}}\end{array}$(24)
From Bernoulli’s inequality,${({x}_{i}+1)}^{1\alpha}<1+(1\alpha ){x}_{i}$ if x_{ i }>0 and 0<α<1, and${({x}_{i}+1)}^{1\alpha}>1+(1\alpha ){x}_{i}$ if x_{ i }>0 and α>1, we have (24) is negative for all x_{ i }>0, α>0 and α≠1.
for all k,θ_{ i },a,x_{ i }>0.
Proof of Theorem 1
Define x^{(τ)}(0) to be the initial point of step τ, and x(τ)^{∗} to be the stationary point of step τ. First of all, we show that x(τ)^{∗} is obtainable in each outeriteration. Give θ, it is known that problem (P3^{ τ }) has a unique optimal solution because it is a strictly convex problem with a strictly concave objective. With the assumptions on the step size γ(t)>0,$\underset{t\to \infty}{lim}\gamma \left(t\right)=0$, and$\sum _{t=1}^{\infty}\gamma \left(t\right)=\infty $, the dualbased subgradient algorithm converges to the optimal point given θ^{(τ)} in each τstep according to ([17], Prop.8.2.5).
Equation (26) is obtained via the replacement of${\theta}_{i}^{\left(\tau \right)}=\frac{{U}_{i}\left({x}_{i}^{(\tau 1)\ast}\right)}{\sum _{j\in \mathcal{N}}{U}_{j}\left({x}_{j}^{(\tau 1)\ast}\right)}$ and${\stackrel{~}{\mathit{x}}}^{\left(\tau \right)}\left(0\right)={\stackrel{~}{\mathit{x}}}^{(\tau 1)\ast}$ into the righthand size. The inequality (27) is satisfied because x(τ)^{∗}is an optimal point of (P4^{ τ }) as well as (P3^{ τ }) given θ^{(τ)}. The inequality (28) is from (13). On the other hand, G(x) is a continuous function, so, G(x) is bounded as x is bounded. Moreover, the sequence {G(x(τ)^{∗}),τ=1,2,…} monotonically increases, therefore, it converges ([17], Prop.A.3). Hence, the sequence$\left\{\sum _{i\in \mathcal{N}}{U}_{i}\right({x}_{i}^{\left(\tau \right)\ast}),\tau =1,2,\dots \}$ also converges.
We can easily verify that the point$({\stackrel{~}{\mathit{x}}}^{\ast},{\mathit{p}}^{\ast},{\mathit{\xi}}^{\ast})$ also satisfies (4)–(7) which are the KKT conditions of (P2) if we replace${\theta}_{i}^{\ast}=\frac{{U}_{i}\left({x}_{i}^{\ast}\right)}{\sum _{k\in \mathcal{N}}{U}_{k}\left({x}_{k}^{\ast}\right)}$ and ξ^{∗}=ν^{∗}. Hence, the theorem is proved.
Declarations
Acknowledgements
This research was supported by the KCC (Korea Communications Commission), Korea, under the R&D program supervised by the KCA (Korea Communications Agency) (KCA20120891105002).
Authors’ Affiliations
References
 Lee JW, Chiang M, Calderbank A: Utilityoptimal randomaccess control. IEEE Trans. Wirel. Commun 2007, 6(7):27412751.View ArticleGoogle Scholar
 MohsenianRad A, Huang J, Chiang M, Wong V: Utilityoptimal random access without message passing. IEEE Trans. Wirel. Commun 2009, 8(3):10731079.View ArticleGoogle Scholar
 MohsenianRad A, Huang J, Chiang M, Wong V: Utilityoptimal random access: reduced complexity, fast convergence, and robust performance. IEEE Trans. Wirel. Commun 2009, 8(2):898911.View ArticleGoogle Scholar
 Shenker S: Fundamental design issues for the future internet. IEEE J. Sel. Areas Commun 1995, 13(7):11761188.View ArticleGoogle Scholar
 Lee JW, Mazumdar R, Shroff N: Nonconvex optimization and rate control for multiclass services in the Internet. IEEE/ACM Trans. Netw 2005, 13(4):827840.View ArticleGoogle Scholar
 Hande P, Shengyu Z, Chiang M: Distributed rate allocation for inelastic flows. IEEE/ACM Trans. Netw 2007, 15(6):12401253.View ArticleGoogle Scholar
 Fazel M, Chiang M: Network utility maximization with nonconcave utilities using sumofsquares method. IEEE Conference on Decision and Control 2005, 18671874.Google Scholar
 Chiang M, Low S, Calderbank A, Doyle J: Layering as optimization decomposition: a mathematical theory of network architectures. Proc. IEEE 2007, 95(1):255312.View ArticleGoogle Scholar
 Cheung MH, MohsenianRad A, Wong V, Schober R: Random access for elastic and inelastic traffic in wlans. IEEE Trans. Wirel. Commun 2010, 9(6):18611866.View ArticleGoogle Scholar
 Marks BR, Wright GP: A general inner approximation algorithm for nonconvex mathematical programs. Oper. Res 1978, 26(4):681683. 10.1287/opre.26.4.681MathSciNetView ArticleMATHGoogle Scholar
 Chiang M, Tan CW, Palomar D, O’Neill D, Julian D: Power control by geometric programming. IEEE Trans. Wirel. Commun 2007, 6(7):26402651.View ArticleGoogle Scholar
 Papandriopoulos J, Dey S, Evans J: Optimal and distributed protocols for crosslayer design of physical and transport layers in manets. IEEE/ACM Trans. Netw 2008, 16(6):13921405.View ArticleGoogle Scholar
 Tran N, Hong CS: Joint rate and power control in wireless network: a novel successive approximations method. IEEE Commun. Lett 2010, 14(9):872874.View ArticleGoogle Scholar
 Vo PL, Tran NH, Hong CS: Joint rate and power control for elastic and inelastic traffic in multihop wireless networks. IEEE Globecom 2011, 15.Google Scholar
 Wang WH, Palaniswami M, Low SH: Applicationoriented flow control: fundamentals, algorithms and fairness. IEEE/ACM Trans. Netw 2006, 14(6):12821291.View ArticleGoogle Scholar
 Mo J, Walrand J: Fair endtoend windowbased congestion control. IEEE/ACM Trans. Netw 2000, 8(5):556567. 10.1109/90.879343View ArticleGoogle Scholar
 Bertsekas DP, Nedić A, Ozdaglar AE: Convex Analysis and Optimization. (Athena Scientific, Belmont, MA , 2003)Google Scholar
 Nedić A, Ozdaglar A: Cooperative distributed multiagent optimization. Convex Optimization in Signal Processing and Communications 2010. (Cambridge University Press, Cambridge, MA)Google Scholar
 Lee JW, Ao T, Jianwei H, Chiang M, Robert A: Reverseengineering mac: a noncooperative game model. IEEE J. Sel. Areas Commun 2007, 25(6):11351147.View ArticleGoogle Scholar
 Bianchi G: Performance analysis of the ieee 802.11 distributed coordination function. IEEE J. Sel. Areas Commun 2000, 18(3):535547.View ArticleGoogle Scholar
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.