Adaptive Flow Rate Control for Network Utility Maximization Subject to QoS Constraints in Wireless Multi-hop Networks

Abstract

In this paper, we study the problem of network utility maximization subject to QoS constraints in a wireless multi-hop network. Recently, virtual queues based cross-layer solution has been proposed to address this issue. Virtual queues can share the burden of the actual queues and also control the lengths of actual queues to ensure certain QoS constraints. In this paper, we introduce link reliability into the virtual queue models and optimization objective and accordingly present a fully distributed adaptive CSMA based flow rate control algorithm (AFCA) to achieve network utility maximization subject to QoS constraints. We analyze AFCA’s stability property and its near-optimality in network utility maximization while satisfying given QoS constraints. We reveal the relationship between various key parameters in AFCA and resulting network utility. Simulation results validate the effectiveness of our analytical results.

Appendix

1.1 Proof of Proposition 1

Proof: According to (3), expanding the first term in (4), and transforming the obtained expression to a form using common denominator, then we have the corresponding numerator becomes the following.

$$ \sum_{\boldsymbol{\upsigma}}\sum_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\left\{\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right){\lambda}_{\boldsymbol{\upsigma}}\right\}-\sum_{\boldsymbol{\upsigma}}\sum_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\left\{\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right){\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\right\} $$

(34)

where λ _σ  = ∑_{(n, m)∣σ} σ _(n, m) λ _(n, m), ${\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}={\sum}_{\left(n,m\right)\mid {\boldsymbol{\upsigma}}^{\mathbf{\prime}}}{\sigma}_{\left(n,m\right)}{\lambda}_{\left(n,m\right)}$, ${\lambda}_{\boldsymbol{\upsigma}}^{\ast }={\sum}_{\left(n,m\right)\mid \boldsymbol{\upsigma}}{\sigma}_{\left(n,m\right)}{\lambda}_{\left(n,m\right)}^{\ast }$, and ${\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast }={\sum}_{\left(n,m\right)\mid {\boldsymbol{\upsigma}}^{\mathbf{\prime}}}{\sigma}_{\left(n,m\right)}{\lambda}_{\left(n,m\right)}^{\ast }$.

Regarding the second term in (4), we use the same way of transformation as above, and obtain a corresponding numerator as follows.

$$ \sum_{\boldsymbol{\upsigma}}\sum_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\left\{\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right){\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right\}-\sum_{\boldsymbol{\upsigma}}\sum_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\left\{\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right){\lambda}_{\boldsymbol{\upsigma}}^{\ast}\right\} $$

(35)

Since both of the above transforms share the same denominator, we add the two numerators (34) and (35) together, and yield

$$ \sum_{\boldsymbol{\upsigma}}\sum_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\left\{\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right)\left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right)\right\}-\sum_{\boldsymbol{\upsigma}}\sum_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}\left\{\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right)\left({\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}+{\lambda}_{\boldsymbol{\upsigma}}^{\ast}\right)\right\} $$

(36)

Since ${\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast }$ is positively correlated to $\exp \left({\lambda}_{\boldsymbol{\upsigma}}+{\lambda}_{{\boldsymbol{\upsigma}}^{\mathbf{\prime}}}^{\ast}\right)$, we have the first term in (36) is larger than or equal to the second term. So (4) is proved.

1.2 Proof of Proposition 2

Proof: First, we prove that the queue length of source node s(c), ∀c∈C, has upper bound. Second, we prove that the queue length of s(c)‘s next hop node on the path for a session c∈C has upper bound.

First, we prove that, for a session c∈C, ${Q}_{s(c)}^c(t)\le {q}_M$, ∀t.

Since ${Q}_{s(c)}^c(0)=0$, we have ${Q}_{s(c)}^c(0)\le {q}_M$ when t = 0.

Suppose ${Q}_{s(c)}^c(t)\le {q}_M$ (t > 0), we need to prove that ${Q}_{s(c)}^c\left(t+1\right)\le {q}_M$.

There are two cases regarding this.

• 1) If ${Q}_{s(c)}^c(t)\le {q}_M-T$ in scheduling cycle t, we can easily obtain that ${Q}_{s(c)}^c\left(t+1\right)\le {q}_M$ since ${s}_{\left(s,s\right)}^c(t)\le T$,

• 2) Else if ${q}_M-T<{Q}_{s(c)}^c(t)\le {q}_M$, then we have (30a) < 0, which means that ${s}_{\left(s,s\right)}^c(t)=0$. Accordingly, we still have ${Q}_{s(c)}^c\left(t+1\right)\le {q}_M$.

Thus, we have ${Q}_{s(c)}^c(t)\le {q}_M$, ∀t.

Second, we prove that the queue length of s(c)‘s next-hop on the route of session c, denoted by node m, has upper bound. Here, to ease the presentation, we let n = s(c).

Next, we proceed to prove that ${Q}_m^c(t)\le {q}_M/{\mathrm{Pr}}_{\left(n,m\right)}+T,\forall t$. The reason why such an upper bound holds at node m (i.e., source node’s next hop) will also be provided in the proof shown below. Let q _M ^′ = q _M /Pr_(n, m) + T.

It is obviously ${Q}_m^c(0)=0$ when t = 0. Thus, ${Q}_m^c(0)\le {q_M}^{\prime }$.

Suppose ${Q}_m^c(t)\le {q_M}^{\prime }$ (when t > 0) holds, we need to prove that ${Q}_m^c\left(t+1\right)\le {q_M}^{\prime }$.

If ${Q}_m^c(t)<{q}_M/{\mathrm{Pr}}_{\left(n,m\right)}$, we have (30b) > 0. That is, link (n,m) may have successful transmission scheduling (i.e., 0≤ ${s}_{\left(n,m\right)}^c(t)+{d}_{\left(n,m\right)}^c(t)\le T$). Then, in this case, we have ${Q}_m^c\left(t+1\right)={Q}_m^c(t)+{s}_{\left(n,m\right)}^c(t)<{q}_M/{\mathrm{Pr}}_{\left(n,m\right)}+T={q_M}^{\prime }$.

If ${q}_M/{\mathrm{Pr}}_{\left(n,m\right)}\le {Q}_m^c(t)\le {q}_M/{\mathrm{Pr}}_{\left(n,m\right)}+T$, we have (30b) < 0. Accordingly, the link (n, m) has no chance to be scheduled (i.e., ${s}_{\left(n,m\right)}^c(t)+{d}_{\left(n,m\right)}^c(t)=0$). Then, we still have ${Q}_m^c\left(t+1\right)={Q}_m^c(t)+{s}_{\left(n,m\right)}^c(t)\le {q}_M/{\mathrm{Pr}}_{\left(n,m\right)}+T={q_M}^{\prime }$.

Thus, we have we prove that ${Q}_m^c(t)\le {q}_M/{\mathrm{Pr}}_{\left(n,m\right)}+T,\forall t$.

In this way, we can recursively prove that the queue lengths of all downstream nodes on the path of a session is upper bounded. In addition, it is seen that the actual upper bounds of queue lengths at different nodes on a path for a session can be different.

Based on the above steps, (31) follows.

1.3 Proof of Proposition 3

Before analyzing the stability and near-optimality of the AFCA algorithm, we first give the following a lemma.

Lemma 2: By the AFCA algorithm, the transmission scheduling function (27) satisfies the following condition:

$$ {\varOmega}_{AFCA}(t)\ge \sum_c\left\{\frac{q_M-T}{q_M}{TU}^c\left(t-\tau \right)\left({r}_{\varepsilon}^{c,\ast }+\varepsilon \right)-\tau {T}^2\right\}+\varDelta (t) $$

(37)

where $\varDelta (t)=E\left\{\sum_{\left(n,m\right)}{w}_{\left(n,m\right)}^{\max, \ast }(t)\left({\overline{\sigma}}_{\left(n,m\right)}(t)-{\overline{\sigma}}_{\left(n,m\right)}^{\ast }(t)\right)\left|\mathbf{Y}(t)\right.\right\}$.

Proof: According to Proposition 1, the transmission scheduling by (29) and (30) is used to maximize Ω _AFCA (t), we have

$$ {\displaystyle \begin{array}{l}{\varOmega}_{AFCA}(t)\ge \sum \limits_{\left(n,m\right)}{w}_{\left(n,m\right)}^{\mathrm{max}}(t){\overline{\sigma}}_{\left(n,m\right)}^{\ast }(t)\\ {}\kern1.50em +\sum \limits_{\left(n,m\right)}{w}_{\left(n,m\right)}^{\max, \ast }(t)\left({\overline{\sigma}}_{\left(n,m\right)}(t)-{\overline{\sigma}}_{\left(n,m\right)}^{\ast }(t)\right)\end{array}} $$

Substituting the corresponding items in the above expression by (25c), we have,

$$ {\displaystyle \begin{array}{l}{\varOmega}_{AFCA}(t)\ge E\left\{\sum \limits_c\left(\frac{q_M-T-{Q}_s^c(t)}{q_M}\right){U}^c(t){s}_{s,s}^{c,\ast }(t)\right\}\\ {}\kern1.50em +\frac{1}{q_M}\sum \limits_c\sum \limits_{n\mid c}\left({Q}_n^c(t)-{Q}_m^c(t)\right){U}^c\left(t-\tau \right){s}_{\left(n,m\right)}^{c,\ast }(t)\\ {}\begin{array}{cccc}\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \left.+\frac{1}{q_M}\sum \limits_c\sum \limits_{n\mid c}{U}^c\left(t-\tau \right){Q}_n^c(t){d}_{\left(n,m\right)}^{c,\ast }(t)\left|\mathbf{Y}(t)\right.\right\}+\varDelta (t)\hfill \end{array}\end{array}} $$

where ${s}_{s,s}^{c,\ast }(t)$, ${s}_{\left(n,m\right)}^{c,\ast }(t)$ and ${d}_{\left(n,m\right)}^{c,\ast }(t)$, respectively, denote the values of their counterparts when the link scheduling vector picks a near-optimal vector ${\overline{\boldsymbol{\upsigma}}}^{\ast }$.

Rearranging the right side of the above expression, we have,

$$ {\displaystyle \begin{array}{l}E\left\{\sum \limits_c\left(\frac{q_M-T}{q_M}{U}^c(t){s}_{s,s}^{c,\ast }(t)-\tau {T}^2\right)+\varDelta (t)\right\}\\ {}\kern0.5em \left.+\sum \limits_c\frac{1}{q_M}{U}^c\left(t-\tau \right)\left[\sum \limits_{\left(n,m\right)\mid c}{Q}_n^c(t)\left({s}_{\left(u,n\right)}^{c,\ast }(t)-{s}_{\left(n,m\right)}^{c,\ast }(t)-{d}_{\left(n,m\right)}^{c,\ast }(t)\right)\right]\left|\mathbf{Y}(t)\right.\right\}\end{array}} $$

According to Lemma 1 in Section 4.3, we have,

$$ {\varOmega}_{AFCA}(t)\ge \sum_c\left(\frac{q_M-T}{q_M}{TU}^c(t){r}_{\varepsilon}^{c,\ast }-\tau {T}^2\right)+\varDelta (t) $$

According to the characteristics of near-optimal solution as we discussed in Section 4.3, (37) follows.

In the flow rate control by AFCA algorithm, (28) maximizes the term ${\varPsi}_{AFCA}^c(t)$ over all feasible values of R ^c(t). Thus, we have the following inequality.

$$ {\displaystyle \begin{array}{l}{\varPsi}_{AFCA}^c(t)\ge V{\varphi}_c\left({h}^c{r}_{\varepsilon}^{c,\ast}\right)\\ {}\kern1.50em -{r}_{\varepsilon}^{c,\ast}\left[\frac{q_M-T}{q_M}{TU}^c(t)-{Z}^c(t){h}^c-{X}^c\left(t-\tau \right){\rho}^c\Big)\right]\end{array}} $$

(38)

Substituting the corresponding items in (24) by (26), (37) and (38), respectively, we have,

$$ {\displaystyle \begin{array}{l}\varDelta \varGamma \left(\mathbf{Y}(t)\right)- VE\ \left\{\sum \limits_c{\varphi}_c\left({h}^c{R}^c(t)\right)|\mathbf{Y}(t)\right\}\\ {}\le {B_M}^{\prime }+\varDelta (t)-\sum \limits_c\left(V{\varphi}_c\left({h}^c{r}_{\varepsilon}^{c,\ast}\right)\right)-\sum \limits_c{Z}^c(t)\left({r}_{\varepsilon}^{c,\ast }{h}^c-{m}^c\right)\\ {}\kern1.00em -\sum \limits_c{X}^c(t)\left[{r}_{\varepsilon}^{c,\ast }{\rho}^c-{L}_{\mathrm{max}}{q}^{\mathrm{max}}\right]\\ {}\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill -\sum \limits_c{U}^c(t)\left[\frac{q_M-T}{q_M} T\varepsilon -\frac{1}{q_M}{L}_{\mathrm{max}}{T}^2\right]\hfill \end{array}\end{array}} $$

where B _M ^′ = B _M  + ∑ _c τT ² + ∑ _c ρ ^c τL _max q ^max.

Given the two conditions in Proposition 3, we have

$$ {\displaystyle \begin{array}{l}\varDelta \varGamma \left(\mathbf{Y}(t)\right)- VE\ \left\{\sum \limits_c{\varphi}_c\left({h}^c{R}^c(t)\right)|\mathbf{Y}(t)\right\}\\ {}\le {B}_M+\varDelta (t)-V\sum \limits_cV{\varphi}_c\left({h}^c{r}_{\varepsilon}^{c,\ast}\right)-\theta \sum \limits_c\left\{{Z}^c(t)+{X}^c(t)+{U}^c(t)\right\}\end{array}} $$

Summing over all cycles in {0, 1, 2, …, M −1}, we have

$$ {\displaystyle \begin{array}{l}\varGamma \left(\mathbf{Y}\left(M-1\right)\right)-\varGamma \left(\mathbf{Y}(0)\right)-V\left\{\sum \limits_{t=0}^{M-1}E\left(\sum \limits_c{\varphi}_c\left({h}^c{R}^c(t)\right)\right)\right\}\\ {}\le {MB_M}^{\prime }+\sum \limits_{t=0}^{M-1}\varDelta (t)- MV\sum \limits_cV{\varphi}_c\left({h}^c{r}_{\varepsilon}^{c,\ast}\right)\\ {}\kern1.00em -\theta \sum \limits_{t=0}^{M-1}\sum \limits_c\left\{{Z}^c(t)+{X}^c\left(t-T\right)+{U}^c(t)\right\}\end{array}} $$

Taking the limit sup as M→+∞, (32) is proved.

The (32) means that the AFCA algorithm can achieve stable state, which means that ${\sum}_{t=0}^{M-1}\varDelta (t)$ will approach a finite value. Thus, for (33), since the utility function φ _c (⋅) is strictly concave, using the Jensen’s inequality, we have,

$$ \frac{1}{M}\sum_{t=0}^{M-1}E\left(\sum_c{\varphi}_c\left({h}^c{R}^c(t)\right)\right)\le E\left\{\sum_c{\varphi}_c\left(\frac{1}{M}{h}^c\sum_{t=0}^{M-1}{R}^c(t)\right)\right\} $$

According to the definition of r in (14) and the characteristics of network capacity region as we discussed in Section 4.2, (33) is proved.