Clock synchronization in symmetric stochastic networks

Abstract

We consider a stochastic model of clock synchronization in a wireless network of $N$ sensors interacting with one dedicated accurate time server. For large $N$ we find an estimate of the final time sychronization error for global and relative synchronization. The main results concern the behavior of the network on different timescales $t_{N}\rightarrow \infty$, $N\rightarrow \infty$. We discuss the existence of phase transitions and find the exact timescales for which an effective clock synchronization of the system takes place.

Appendix

Proof of Lemma 6

Let ${a_{1}}\not ={a_{2}}$. It is straightforward to check that

$$\begin{aligned} \sum _{n_{2}=0}^{n}{a_{1}}^{n_{2}}{a_{2}}^{n-n_{2}}=\left( {a_{1}}-{a_{2}}\right) ^{-1}\left( {a_{1}}^{n+1}-{a_{2}}^{n+1}\right) . \end{aligned}$$

Using (24) for $A={a_{1}}$ and $A={a_{2}}$ we find $U_{1}({a_{1}},{a_{2}})$.

Now let us calculate $U_{2}({a_{1}},{a_{2}})$. After some simple algebra we have identity

$$\begin{aligned} \sum _{\begin{array}{c} n_{1}\ge 0,\, n_{2}\ge 0\\ n_{1}+n_{2}\le n\end{array}}\,{a_{1}}^{n_{1}}{a_{2}}^{n_{2}}=\frac{{\displaystyle \frac{1}{1-{a_{1}}}}-{\displaystyle \frac{1}{1-{a_{2}}}}}{{a_{1}}-{a_{2}}}\,-\,\frac{{\displaystyle \frac{{a_{1}}^{n+2}}{1-{a_{1}}}}-{\displaystyle \frac{{a_{2}}^{n+2}}{1-{a_{2}}}}}{{a_{1}}-{a_{2}}}, \end{aligned}$$

Applying (25) for $A=1$, $A={a_{1}}$, and $A={a_{2}}$, we get that $U_{2}({a_{1}},{a_{2}})$ multiplied by $\delta _{N}^{2}$ is equal to

$$\begin{aligned}&\frac{{\displaystyle \frac{1}{1-{a_{1}}}}-{\displaystyle \frac{1}{1-{a_{2}}}}}{{a_{1}}-{a_{2}}}\,\left( 1-\frac{1+\delta _{N}t}{e^{\delta _{N}t}}\right) \\&-\,\frac{{\displaystyle \frac{\left( e^{-\delta _{N}t(1-{a_{1}})}-{\displaystyle \frac{1+\delta _{N}t{a_{1}}}{e^{\delta _{N}t}}}\right) }{1-{a_{1}}}}-{\displaystyle \frac{\left( e^{-\delta _{N}t(1-{a_{2}})}-{\displaystyle \frac{1+\delta _{N}t{a_{2}}}{e^{\delta _{N}t}}}\right) }{1-{a_{2}}}}}{{a_{1}}-{a_{2}}} \end{aligned}$$

By direct transformations and cancelation of terms this form can be reduced to the following one

$$\begin{aligned}&-\frac{{\displaystyle \frac{e^{-\delta _{N}t(1-{a_{1}})}-1}{1-{a_{1}}}}-{\displaystyle \frac{e^{-\delta _{N}t(1-{a_{2}})}-1}{1-{a_{2}}}}}{{a_{1}}-{a_{2}}}\\&\quad =\frac{1}{(1-{a_{1}})(1-{a_{2}})}-\frac{{\displaystyle \frac{e^{-\delta _{N}t(1-{a_{1}})}}{1-{a_{1}}}}-{\displaystyle \frac{e^{-\delta _{N}t(1-{a_{2}})}}{1-{a_{2}}}}}{{a_{1}}-{a_{2}}}. \end{aligned}$$

This proves the statement of Lemma 6 for ${a_{1}}\not ={a_{2}}$.

The case ${a_{1}}={a_{2}}$ is simpler than just considered case ${a_{1}}\not ={a_{2}}$. So we omit details here. Note that explicit expressions for $U_{i}({a},{a})$, $i=1,2$, correspond to formal limits of $U_{i}({a_{1}},{a_{2}})$ as ${a_{1}}\rightarrow {a}$, ${a_{2}}\rightarrow {a}$. $\square$