Optimizing bandwidth allocation for heterogeneous traffic in IoT

Abstract

In Internet of Things (IoT), there generally exist two typical types of heterogeneous flows with different requirements: inelastic media flow (e.g., the monitoring flow) and elastic data flow (e.g., the environment info from measurement). These heterogeneous flows often coexist for sharing the limited bandwidth. Despite substantial works, a simple yet efficient approach to economically allocate bandwidth for these two types of flows is still not available. In this paper, we propose two methods to optimize the bandwidth allocation of these flows: the network-utility-maximization-based (NUM) method and the asymptotic analysis method. The NUM method provides a general solution to the optimization problem, but requires a certain computational complexity. The asymptotic analysis method delves into the inherent property of the network and explicitly expresses the solution in terms of protocol parameters and traffic requirements. Extensive simulations verify that the two methods are very accurate and can well achieve the desired objectives.

Appendix

In this appendix, we prove Theorem 1.

Proof of Theorem 1

Let $\varphi ^{o}\triangleq \beta _{2}{\sum }_{j=1}^{m}r_{j}$ represent the total attempt rate of the stations with elastic flows. We prove Theorem 1 in two steps below.

• Step 1: Express β _i+1 in terms of β ₂. From Eqs. 4, 7, and 8, since β _i ≪1, we have

  $$ r_{i}=\frac{{\Gamma}_{n+i}}{{\Gamma}_{n+1}}=\frac{L_{n+i}\beta_{n+i}(1-\beta _{n+1})}{L_{n+1}\beta_{n+1}(1-\beta_{n+i})}\approx\frac{L_{n+i}\beta_{n+i}} {L_{n+1}\beta_{n+1}}. $$

  (20)

  When n = 1, we can express β _i+1 as follows

  $$ \beta_{i+1}=\frac{r_{i}L_{2}\beta_{2}}{L_{i+1}}=\frac{r_{i}\varphi}{L_{i+1} {\textstyle\sum\nolimits_{j=1}^{m}} \frac{r_{j}}{L_{j+1}}},1\leq i\leq m. $$

  (21)

• Step 2: Express β _i in Eq. 16. We first express Γ₁ = a ₁ as Eq. 22. After substituting (21) into Eq. 22, then we can obtain an explicit relationship between β ₁ and β ₂, as shown in Eq. 23, where we apply the approximations: β _i ≪1 and (1−x)^y≈e ^−xy for x≪y. Combining (21) and (23), we obtain (16).

  $$ a_{1}=\frac{L_{i}\beta_{i}{\Pi}_{j\neq i}^{N}(1-\beta_{j})}{(\sigma-T_{c} ){\Pi}_{i=1}^{N}(1-\beta_{i})+\sum\nolimits_{i=1}^{N}({T_{s}^{i}}-T_{c})\beta_{i}{\Pi}_{j\neq i}^{N}(1-\beta_{j})+T_{c}}. $$

  (22)
  $$ \beta_{1}=\frac{\sigma-T_{c}+ {\textstyle\sum\nolimits_{i=1}^{m}} (T_{s}^{i+1}-T_{c})\beta_{i+1}+ \frac{T_{c}}{{\Pi}_{i=1}^{m}(1-\beta_{i+1})} }{\frac{L_{1}}{a_{1}}+\sigma-{T_{s}^{1}}} $$

  (23)
  $$ \approx\frac{\sigma-T_{c}+\frac{\varphi}{{\textstyle\sum\nolimits_{j=1} ^{m}}\frac{r_{j}}{L_{j+1}}}{\textstyle\sum\nolimits_{i=1}^{m}}(T_{s} ^{i+1}-T_{c})\frac{r_{i}}{L_{i+1}}+T_{c}e^{\varphi}}{\frac{L_{1}}{a_{1} }+\sigma-{T_{s}^{1}}}. $$

  (24)

• Step 3: Compute the optimal φ ^o, φ. First, we express ${\Sigma }_{i=1}^{N}{\Gamma }_{i}$ as in Eqs. 25 and 26 by applying (21). Second, setting the first-order derivative of ${\Sigma }_{i=1}^{N}{\Gamma }_{i}$ in terms of φ to 0, we obtain the equation of φ, B + T _c (e ^φ−φe ^φ)=0, and hence $(\varphi -1)e^{\varphi -1}=\frac {B}{T_{c}e}$. Then $\varphi -1=W_{0}(\frac {B}{T_{c}e})$ or $W_{-1}(\frac {B}{T_{c}e})$. We have $\varphi =W_{0}(\frac {B}{T_{c}e})+1$, because $W_{0}(\frac {B}{T_{c}e})>-1$ and $W_{-1}(\frac {B}{T_{c}e})<-1$ for $\frac {B}{T_{c}e}\in (\frac {-1}{e},0)$.

□

$$ {\Sigma}_{i=1}^{N}{\Gamma}_{i} =a_{1}+\frac{\beta_{2}}{\beta_{1}} a_{1}{\textstyle\sum\nolimits_{i=1}^{m}}r_{i} $$

(25)

$$ =a_{1}+\frac{(\frac{L_{1}}{a_{1}}+\sigma-{T_{s}^{1}})\frac{\varphi} {{\textstyle\sum\nolimits_{j=1}^{m}}\frac{r_{j}}{L_{j+1}}}a_{1} {\textstyle\sum\nolimits_{i=1}^{m}}r_{i}}{\sigma-T_{c}+\frac{\varphi} {{\textstyle\sum\nolimits_{j=1}^{m}}\frac{r_{j}}{L_{j+1}}}{\textstyle\sum \nolimits_{i=1}^{m}}(T_{s}^{i+1}-T_{c})\frac{r_{i}}{L_{i+1}}+T_{c}e^{\varphi}} $$

(26)