text:
\section*{Abstract}
For a sensor network, a tractable spatially-dependent node deployment model is presented with the property that the density is inversely proportional to the sink distance. A stochastic model is formulated to examine message advancements under greedy routing in such a sensor network. The aim of this work is to demonstrate that an inhomogeneous Poisson process can be used to model a sensor network with spatially-dependent node density. Elliptic integrals and asymptotic approximations are used to describe the random behaviour of hops. Types of dependence that affect hop advancements are examined. We observe that the dependence between successive jumps in a multihop path is captured by including only the previous forwarding node location. We include a simple uncoordinated sleep scheme, and observe that the complexity of the model is reduced when enough nodes are asleep. All expressions involving multidimensional integrals are derived and evaluated with quasi-Monte Carlo integration methods based on Halton sequences and recently-developed lattice rules. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The ensuing results agree extremely well with simulations.

\section{Introduction}
Advancements in networking technologies are leading to sensor networks being a feasible and common technology. However, a significant issue for sensor networks is developing routing methods that can handle the dynamic topologies of sensor networks. A common approach in sensor networks and ad hoc networks in general is geometric or position-based routing \cite{MAUVE:2001,STOJ:2002}. The operation of these algorithms is based on the assumption that each node knows its geographical location in relation to the main communication station, known as a \emph{sink}, and the location of neighbouring nodes within its transmission radius. Geometric routing is often considered attractive because of its localized nature and scalability. A natural geometric routing approach is for the source node to forward a data message to the node that is geographically the closest to the sink, and repeat this step until the message finally reaches the target sink. This approach serves as a \emph{greedy} routing method in itself or forms the basis for more intelligent routing methods in wireless ad hoc networks \cite{STOJ:1999,KUHN:2003,KARP:2000,ZORZI1:2003}.

Sensor nodes are often randomly scattered over the sensor field. To conserve power-consumption, a subset of sensor nodes may randomly fall into a low energy-consuming \emph{sleep} state, in which they cannot relay messages. The inherent randomness in sensor networks motivates the need for a suitable stochastic model to determine the ability of a routing scheme to successfully deliver data. In recent years, stochastic models and methods are being increasingly employed in analyzing communication networks. In particular, there is a special issue \cite{HABDF:2009} on stochastic geometry and related fields applied to communication networks, as well as a two-volume monograph by Baccelli and Blaszczyszyn \cite{BB1:2009,BB2:2009}.

The majority of this work, however, is under the assumption that sensor nodes are deployed according to a homogeneous Poisson process. Although such a tractable model might not capture the underlying node density variation, it can serve as a first approximation for studying network characteristics. We wish to extend the standard model by examining inhomogeneous or nonhomogeneous node deployment such that the node density is spatially-dependent.

There are various reasons why inhomogeneous deployment models may be necessary. The deployment of sensor nodes depends on the environment and application of the sensor network. Consequently, sensor nodes may need to be deployed more densely in important sensing regions. The obstacles in the network surroundings may prevent nodes being positioned in certain regions. Inconsistent system parameters (such as battery lifetime) and interference can reduce the effective node density in certain regions. Furthermore, the nature of the actual node deployment influences the node density. For example, an aerial dispersal of nodes could result in the nodes obeying some type of diffusion process. Also, there may be network protocols that require positioning the nodes in certain regions, which lead to performance advantages.

The design and deployment of sensor networks must address the problem of message collisions. The data messages in sensor networks converge towards the sink. Nodes closer to the sink need act as relay nodes more than nodes away from the sink. One possible solution to this problem is to deploy more nodes in these heavy traffic regions. Consequently, the node density would decay at some rate that is dependent on the distance to the sink.

It is in this last setting that we wish to examine greedy routing. We propose an approach using a tractable mathematical model similar to the one developed in previous work \cite{KEELER1,KEELER2}. Ideally, we want to offer a computationally quick and reliable way to obtain probabilistic descriptions of multihop paths in a sensor network with a simple stochastic sleep scheme. Moreover, we want to extend the model, analysis, and calculations methods from the homogeneous case to the inhomogeneous case, and demonstrate that the techniques are still valid.

To achieve these goals, we analyze greedy routing in randomly deployed networks under the multihop situation. We propose a tractable spatially-dependent node density function, and analyze the resulting stochastic characteristics. Furthermore, we examine the influence of a simple stochastic sleep scheme. More specifically, we examine the effects of having a certain proportion of nodes awake at any given time. We derive resulting probability integral expressions, and demonstrate their feasible evaluation via quasi-Monte Carlo integration methods based on Halton sequences and recently-developed lattice rules.

The work presented here is focused on the stochastic behaviour of multihop paths, the calculation methods, and mathematically representing the effects of a sleep scheme. Overall, we show the application of this mathematical formulation in describing the stochastic behaviour of message delivery in sensor networks with inhomogeneous node deployment. Additionally, we demonstrate an implementable calculation procedure based on quasi-Monte Carlo methods for evaluating multidimensional integrals.

\section{Background work}
The results presented here follow on from initial homogeneous Poisson model development and analysis \cite{KEELER1}, which was later extended by evaluating resulting integrals and analyzing a simple sleep scheme \cite{KEELER2}. Consequently, the majority of the formulation and techniques used here have been applied in the constant density setting.

More generally, Ishizuka and Aida \cite{IA:2004} performed some preliminary analysis of node placement approaches via simulations to gauge the fault tolerance of networks against random node breakdown and battery exhaustion. In particular, they assumed that individual nodes are scattered uniformly around the sink, and that the density decreased as the distance from the sink increased. Ishizuka and Aida examined two node density models: a Gaussian function and a simple power law (inversely proportional to the sink distance). They concluded that the simple power law placement resulted in good fault tolerance.

A concept closely related to connectivity is the sensing coverage of a sensor network, which is the ability of a sensor network to successfully sense or cover the entire sensor field. Solutions to coverage problems have been based on coverage processes such as the Boolean model; see Hall \cite{HALL:1988} and Stoyan, Kendall and Mecke \cite{SKM:1995} for more details. In the sensor network setting, a more recent example is that of Manohar et al. \cite{MRM:2009} who used coverage processes to examine the coverage of a sensor network with an exponentially decreasing node density.

The aforementioned work involving inhomogeneous node deployment cases did not examine the stochastic behaviour of any particular routing method. Furthermore, the work did not cover the effects that a sleep scheme has on stochastic dependencies in the routing model.

\section{Mathematical model}
We assume that nodes communicate data radially, and that a node's transmission radius is a constant that clearly cuts off at some distance, which implies that a node can relay data only to another node when it is within the forwarding node's transmission radius. For numerical calculations and simulations, we rescale all lengths with respect to the transmission radius, and hence the transmission radius is set to one. However, we denote the transmission radius $r$ in ensuing calculations and equations for clarity and future extensions.

We assume that nodes are scattered according to a two-dimensional Poisson process over a finite region, and that at any given time a random number of nodes are in sleep mode while the remaining are in their awake mode. Furthermore, we assume the awake node density, that is the average number of awake nodes per unit area, is a spatially-dependent function; that is, the nodes are scattered according to an inhomogeneous Poisson process \cite{SKM:1995}. We assume the nodes are scattered uniformly around the sink, but decreases in some way as the distance to the sink increases. Hence, the awake node density is a radial function of the form
\begin{equation}\label{inhomlambda0}
\lambda(u)=\lambda q(u)\qquad u\in(0,\infty),
\end{equation}
where $u$ is the distance to the sink, and $q(u)$ is a weighting function. The function $\lambda(u)$ can be interpreted as the mean number of awake nodes per infinitesimal area element. We refer to the constant $\lambda$ as the \emph{initial} node density.

Ideally, the $q(u)$ function should be amenable to analytical and asymptotic methods while reflecting a realistic node distribution. We study a node density that is inversely proportional to the sink distance
\begin{equation}\label{inhomlambda}
q(u)=\frac{1}{u},
\end{equation}
which is the model proposed by Ishizuka and Aida \cite{IA:2004}. However, we  chose this function also because we believe that the positioning of nodes will better accommodate the convergence of messages near the sink.

We introduce a Poisson process \emph{mean} measure $\Lambda$, which for a bounded Borel set $B\subset\mathbb{R}^2$ with area $A$ is the density function integrated over the region $B$, thus in polar coordinates
\begin{align}
\Lambda(B)&=\lambda \iint_Bq(u)udud\theta,\\
&= \lambda Q(B),
\end{align}
where $Q$ is referred to as the \emph{rescaled} mean measure, and is used such that the notation is analogous to the results based on the constant density case when $Q$ reduces to the area $A$ of the region \cite{KEELER1,KEELER2}. We emphasize that all the $Q$-type expressions we consider in this work are derived in a similar manner to the corresponding area expressions under the homogeneous model by integrating the density over specific regions \cite{KEELER1,KEELER2}.

Under a sleep model, the initial node density parameter can be written as $\lambda=p\alpha$ where $p$ is the probability of a node being awake and $\alpha$ is the underlying (that is, sleep and awake) node density parameter. The awake nodes form a thinned Poisson process. As in the homogeneous case, both the number of points kept and the number of points removed form random variables that are independent of each other \cite{SKM:1995}. Hence, in our model the number of awake nodes is independent of the number of asleep nodes.

The number of awake nodes $N_B$ located within some region $B$ is a inhomogeneous Poisson random variable with a probability mass function
\begin{equation}\label{inhomPoisson1}
\mathbb{P}(N_B=n)=\frac{\left(\lambda Q(B)\right)^n}{n!}e^{-\lambda Q(B)}.
\end{equation}

\section{Single hop analysis}
We introduce the parameter $\gamma$ to represent the distance between a forwarding node and the sink (refer to Fig. \ref{overlapcircle5}). The sink distance parameter is set to $\gamma=\ell$ when the forwarding node is the source node. Often we shall present results such that the initial node density parameter $\lambda$ is a product of the source-sink distance $\ell$ and some positive constant.
\\
\begin{figure}[h]
\begin{center}
\begin{overpic}[scale=0.60]{overlapcircle.EPS}
\put(27,42){$\I_{\gamma}(u)$}
\put(52,31){$u$}
\put(13,30){$r$}
\put(52,5){$\gamma$}
\put(93,36){$2\psi_{\gamma}(u)$}
\end{overpic}
\caption{Integrate over the region $\I_{\gamma}(u)$ to derive $\Lambda(\I_{\gamma}(u))$ for the forwarding node.\label{overlapcircle5}}
\end{center}
\end{figure}

For a forwarding node with a sink distance $\gamma$, let $\I_{\gamma}(u)\subset\mathbb{R}^2$ be its partial feasible region as a function of $u$. This region is formed by the intersection of two circles of radii $r$ and $u$ centered at the source node and the sink respectively (as illustrated in Fig. \ref{overlapcircle5}). The area of $\I_{\gamma}(u)$ is denoted by $A_{\gamma}(u)$, and given by the integral
\begin{equation}\label{area5}
A_{\gamma}(u)=2\int_{\gamma-r}^u \int_0^{\psi_\gamma(w)}wd\theta dw,
\end{equation}
where the sink angle function $\psi_{\gamma}(u)$ is defined as
\begin{equation}\label{psi5}
  \psi_{\gamma}(u)=\arccos\left(\frac{u^2+\gamma^2-r^2}{2u\gamma}\right).
\end{equation}
Furthermore, the mean measure of the region $\I_{\gamma}(u)$ is given by the integral
\begin{equation}\label{Lambda0}
\Lambda(\I_{\gamma}(u))=2\int_{\gamma-r}^u \int_0^{\psi_\gamma(w)}\lambda(w)wd\theta dw,
\end{equation}
which for the functions (\ref{inhomlambda0}) and (\ref{inhomlambda}) reduces to
\begin{equation}\label{Lambda1}
\Lambda(\I_{\gamma}(u))=2\lambda\int_{\gamma-r}^u \int_0^{\psi_\gamma(w)}d\theta dw.
\end{equation}
Henceforth, we write
\[
\Lambda_{\gamma}(u)= \Lambda(\I_{\gamma}(u)), \qquad
Q_{\gamma}(u)= Q(\I_{\gamma}(u)),
\]
to refer to the mean measure and the rescaled mean measure respectively, the motivation of which will become apparent by the analogous results that follow.

Let the random variable $U$ be the sink distance of the forwarding node after a single message hop. The standard nearest neighbour \cite{KEELER1,SKM:1995} method leads to the probability distribution
\begin{equation}\label{inhomFu}
F_{\gamma}(u) = \left\{ \begin{array}{ll}
1-e^{-\lambda Q_{\gamma}(u)} & \gamma-r \leq u <\gamma\\
1 & u\geq \gamma\\
0& u<\gamma-r.
\end{array} \right.
\end{equation}
There is a jump discontinuity in the distribution at $u=\gamma$ owing to the positive probability that no nodes lie within the feasible region. On the support where the distribution (\ref{inhomFu}) is absolutely continuous the probability density is defined by
\begin{equation}\label{inhomfu}
f(u)=\lambda Q_{\gamma}'(u)e^{-\lambda Q_{\gamma}(u)},\qquad \gamma-r \leq u < \gamma,
\end{equation}
where the derivative of the rescaled mean measure
\[
Q_{\gamma}'(u)=2\psi_{\gamma}(u).
\]

For a node of sink distance $\gamma$, it follows from integral (\ref{Lambda1}) that the mean measure as a function of $u$ is calculated by the integral
\begin{equation}\label{Lambda}
\Lambda_{\gamma}(u)=2\lambda\int_{\gamma-r}^u \psi_\gamma(w)dw.
\end{equation}
The corresponding indefinite integral has the analytic solution
\begin{align}
\int \psi_\gamma(w)dw= & \phantom{i} w\psi_{\gamma}(w)\\
& + i E\left( \sin ^{-1}\left(\frac{w}{\gamma+r}  \right);\frac{(\gamma+r)}{(\gamma-r)}\right)(\gamma-r)  \\
& + iF\left( \sin ^{-1}\left(\frac{w}{\gamma+r} \right);\frac{(\gamma+r)}{(\gamma-r)}\right)(2 r),
\end{align}
where $i=\sqrt{-1}$, and $F$ and $E$ denote incomplete elliptic integrals of the first and second kind defined in Legendre form as
\begin{equation}
F(\phi;k)=\int_0^{\phi}\left(1-k^2\cos^2\theta\right)^{-1/2}d\theta,
\end{equation}
\begin{equation}
E(\phi;k)=\int_0^{\phi}\left(1-k^2\cos^2\theta\right)^{1/2}d\theta .
\end{equation}
A number of standard computer packages have pre-written routines for numerically evaluating elliptic integrals, however, care must be taken as their definitions vary depending on the package. The results presented here are obtained via our purposely-written elliptic integral functions based on Carlson symmetric elliptic integrals \cite{CARLSON:1995}. For more details see the papers by Carlson \cite{CARLSON:1979,CARLSON:1995}, which give algorithms that are readily implementable and can handle both real and complex values under specified variable and parameter regimes.

The solution to the original integral (\ref{Lambda}) is
\begin{align}\label{meanU}
\nonumber\Lambda_{\gamma}(u)= &\phantom{o} 2\lambda u \psi_{\gamma}(u)\\\nonumber
& + 2i\lambda E\left( \sin ^{-1}\left(\frac{u}{\gamma+r}  \right);\frac{(\gamma+r)}{(\gamma-r)}\right)(\gamma-r)  \\\nonumber
&-2i\lambda E\left( \sin ^{-1}\left(\frac{\gamma-r}{\gamma+r}  \right);\frac{(\gamma+r)}{(\gamma-r)}\right)(\gamma-r)  \\\nonumber
& + 2i\lambda F\left( \sin ^{-1}\left(\frac{u}{\gamma+r} \right);\frac{(\gamma+r)}{(\gamma-r)}\right)(2 r)\\
&- 2i\lambda F\left( \sin ^{-1}\left(\frac{\gamma-r}{\gamma+r} \right);\frac{(\gamma+r)}{(\gamma-r)}\right)(2 r).
\end{align}
The final result gives a real value, although in practice an insignificant imaginary part may arise from rounding errors. We found that the solution is obtained quickly owing to the speedy convergence of the Carlson algorithms.

Let $C=\gamma-U$ be the distance advanced towards the sink when the originating node is at a distance $\gamma$. We let $\Fb$ denote the distribution of $C$, which leads to
\begin{equation}\label{inhomFc}
\Fb_{\gamma}(c) = \left\{ \begin{array}{ll}
e^{-\lambda Q_{\gamma}(\gamma-c)} & 0 < c \leq r\\
1 & c> r\\
0& c\leq 0.
\end{array} \right.
\end{equation}
 Given that $C$ is non-negative, the $m$-th moment expression
\begin{align*}
\mathbb{E}(C^m) & = m\int_0^r c^{m-1}\Prob(C>c)dc,\\
& =  r^m-m\int_{0}^r c^{m-1} e^{-\lambda Q_{\gamma}(\gamma-c)}dc,
\end{align*}
follows; with the first moment given by
\begin{equation}\label{inhomeC1}
\mathbb{E}(C)=r-\int_{0}^r e^{-\lambda Q_{\gamma}(\gamma-c)}dc,
\end{equation}
and the second moment given by
\begin{equation}\label{inhomeC2}
\mathbb{E}(C^2)=r^2-2\int_{0}^r ce^{-\lambda Q_{\gamma}(\gamma-c)}dc.
\end{equation}

\subsection{Asymptotic results}
We derive an asymptotic approximation to the rescaled mean measure $Q_{\gamma}(u)$, which gives a tractable and accurate expression. The angle function $\psi_{\gamma}$ expanded around $u=\gamma-r$ gives
\begin{equation}\label{approxpsi5}
\psi_{\gamma}(u)\approx b_0(u-\gamma+r)^{1/2}+b_1(u-\gamma+r)^{3/2}+
b_2(u-\gamma+r)^{5/2},
\end{equation}
where the expansion terms
\begin{align}
b_0 &= \left[\frac{2r}{\gamma (\gamma-r)}\right]^{1/2},\\
b_1 &= \left[\frac{2r}{\gamma (\gamma-r)}\right]^{1/2}\left[\frac{r^2-3 r\gamma -3 \gamma^2}{12 (\gamma^2 r- \gamma r^2)}\right],\\
b_2 &= \left[\frac{2r}{\gamma (\gamma-r)}\right]^{1/2}\left[\frac{3 r^4+25 r^2\gamma^2 -10r^3\gamma +30 \gamma^3 r-5 \gamma^4}{160 \gamma^2 (\gamma-r)^2 r^2}\right],
\end{align}
follow. The asymptotic result for the rescaled mean measure
\begin{equation}\label{approxQ}
Q_{\gamma}(u)\approx 4\left[\frac{b_0}{3}(u-\gamma+r)^{3/2}+\frac{b_1}{5}(u-\gamma+r)^{5/2}+
\frac{b_2}{7}(u-\gamma+r)^{7/2}\right],
\end{equation}
follows.

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{CompLambdaR1.eps}
\caption{Comparison of $\Lambda_{\ell}(u)$ asymptotic expressions ($\lambda=3\ell$, $r=1$ and $\ell=10$) .\label{CompLambdaR1}}
\end{center}
\end{figure}

The results of the second-order approximation are accurate for a unit radius (see Fig. \ref{CompLambdaR1}). Adding the third term only improves the results slightly. However, it may be needed for larger transmission radius models (as Fig. \ref{CompLambdaR5} suggests). Conversely, the two-term expansion gives accurate results (see Fig. \ref{AsymHopDist}) when substituted into the sink distribution (\ref{inhomFu}). In fact, it appears that the expansion (\ref{approxQ}) consisting of elementary functions can be used to give accurate results, which are clearly faster to evaluate than those based on elliptic integrals. However, we continue to use the exact solution of the integral (\ref{Lambda}), and later compare it and its approximation.

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{CompLambdaR5.eps}
\caption{Comparison of $\Lambda_{\ell}(u)$ asymptotic expressions ($\lambda=3\ell$, $r=5$ and $\ell=10$) .\label{CompLambdaR5}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{AsymHopDist.eps}
\caption{Comparison of $F_{\ell}(u)$ asymptotic expressions ($\lambda=3\ell$, $r=1$ and $\ell=10$) .\label{AsymHopDist}}
\end{center}
\end{figure}

We present asymptotic moment results for our spatially-dependent node density model.
\begin{thm}\label{AsymThm5}
For the mean measure (\ref{meanU}), provided $\gamma>r$, under greedy routing the first hop moment
\begin{equation}\label{aEc5}
\mathbb{E}(C)\sim r-\dfrac{\Gamma(5/3)}{\left(\lambda q_0\right)^{2/3}} ,
\end{equation}
and the second hop moment
\begin{equation}\label{aVc5}
\mathbb{E}(C^2)\sim r^2-2r\dfrac{\Gamma(5/3)}{\left(\lambda q_0\right)^{2/3}}+\dfrac{\Gamma(7/3)}{\left(\lambda q_0\right)^{4/3}} ,
\end{equation}
as the initial node density
\[
\lambda \rightarrow \infty,
\]
where $\Gamma(\cdot)$ is the gamma function, and
\[
q_0=\dfrac{4}{3}\left[\frac{2r}{\gamma (\gamma-r)}\right]^{1/2}.
\]
\end{thm}

\begin{proof}
Consider integrals of the form
\[
I(\lambda)=\int_a^b t^k e^{-\lambda Q(t)}dt,
\]
where $Q(t)>Q(a)$ for all $t$ $\in$ $(a,b)$, and asymptotically
\[
Q(t) \sim q_0 (t-a)^{\mu} \quad \textrm{as} \quad t\rightarrow a.
\]
The generalized Laplace's method \cite[page 58]{WONG:1989} applied to this integral gives the asymptotic result
\[
I(\lambda)\sim \dfrac{\Gamma(\tau)}{\mu(\lambda q_0)^{\tau}},\quad \lambda \rightarrow \infty.
\]
where
\[
\tau=\dfrac{2(k+1)}{3}.
\]
For Laplace's method, $Q(a)$ needs to be the minimum on the integral interval. Hence, the change of variable $t=u-\gamma+r=r-c$ with a slight abuse of notation leads to the area function expansion
\begin{equation}\label{asymareaT}
\begin{array}{c}
  Q_{\gamma}(t)\sim q_0t^{3/2}+q_1t^{5/2} +O(t^{7/2})
\quad\textrm{as }\quad t \rightarrow 0,
\end{array}
\end{equation}
where
\[
q_0=\dfrac{4b_0}{3},\qquad \mu=\dfrac{3}{2}.
\]
The first moment result (\ref{aEc5}) follows, and the change of variable applied to the second moment equation gives
\[
\mathbb{E}(C^2)=r^2-2\int_{0}^r(r-t)e^{-\lambda Q_{\gamma}(t)}dt,
\]
which leads to the second result (\ref{aVc5}).
\end{proof}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{AsymMomemt1.eps}
\caption{Numerical and asymptotic results for first moment ($r=1$ and  $\ell=10$).\label{AsymMomemt1}}
\end{center}
\end{figure}

\subsection{Sink dependence}\label{sinkDep}
The node intensity function is clearly dependent on the forwarding node sink distance. Comparing the hop distributions (in Fig. \ref{CompHopDist5}) reveals that a message is relayed farther in a single hop if the forwarding node is closer to the sink. Intuitively, hops increase stochastically as the message approaches the sink as more potential forwarding nodes are available in the forwarding region. Geometrically, we observe that the integral kernel in the mean measure equation (\ref{Lambda}) is the angle function $\psi_{\gamma}$, which decreases as $\gamma$ increases; that is
\begin{equation}\label{psiIneq}
\psi_{\gamma_1}(u) \leq \psi_{\gamma_2}(u), \qquad \gamma_1\geq \gamma_2 \geq r.
\end{equation}
Conversely, $\psi_{\gamma}$ and, hence, the indefinite integral (\ref{meanU}) increases as the sink distance decreases. This dependence on the sink distance of the forwarding node is simply referred to as the \emph{sink} dependence.

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{CompHopDist.eps}
\caption{Comparison of distributions $C$ for different $\ell$ ($\lambda=30$ and $r=1$).\label{CompHopDist5}}
\end{center}
\end{figure}

The influence of the sink dependence can be observed by comparing the difference in two hop distributions with different sink distances $\gamma_1$ and $\gamma_2$. In previous work \cite{KEELER1}, the hop distribution dependence on the sink distance under the homogeneous model was examined by a Kullback-Leibler analysis. Subsequently, the Kullback-Leibler divergence \cite{KULLBACK:1959} applied to the mixed discrete-continuous hop distribution gives
\begin{equation}\label{KLInhom}
D(\gamma_1,\gamma_2)=\int_0^r \fb_{\gamma_2}(c)\log\left[\frac{\fb_{\gamma_2}(c)}{\fb_{\gamma_1}(c)} \right]dc+\Fb_{\gamma_2}(0^+)\log\left[\frac{\Fb_{\gamma_2}(0^+)}{\Fb_{\gamma_1}(0^+)} \right],
\end{equation}
where the routing void (that is, no nodes in the feasible region) probability
\begin{align*}
\Fb_{\gamma}(0^+)&=\lim_{\epsilon\rightarrow0}\Fb_{\gamma}(\epsilon),\\
&= e^{-\lambda Q_{\gamma}(\gamma)}.
\end{align*}
The Kullback-Leibler divergence is non-negative and is zero for identical distributions \cite{KULLBACK:1959}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{KullbackGamma10and5Lambda3.eps}
\caption{Kullback-Leibler analysis of two hop distributions for  ($\lambda=3\ell$, $r=1$ and  $\ell=10$).\label{kullbackInhom}}
\end{center}
\end{figure}

We calculated the integral in equation (\ref{KLInhom}) numerically to observe how the hop distribution is influenced when we set $\gamma_1=\ell$ and vary $\gamma_2=\gamma$. The comparison reveals that $D(\ell,\gamma)$ is high near the sink and decreases as $\gamma$ increases (refer to Fig \ref{kullbackInhom}). This is the same expected behaviour as observed under the homogeneous model \cite{KEELER1}, however, $D(\ell,\gamma)$ decays relatively slowly under the inhomogeneous model. This contrasts starkly with the homogeneous model where $D(\ell,\gamma)$ is mostly zero away from the sink, and only increases significantly near the sink \cite{KEELER1}. Furthermore, $D(\ell,\gamma)$ differs significantly for two different $\ell$; the forwarding node sink distance cannot differ greatly when calculating hop distributions. This distribution behaviour eliminates the possibility of using renewal processes to model message advancement over multihop routes \cite{ZORZI1:2003,KEELER1}.

\section{Multihop analysis}
For a multihop analysis, indexing for the random variables $U$ and $C$ is introduced. We set $U_0=\ell$ and $U_1=U$ such that the $i$-th hop advancement is $C_i=U_{i-1}-U_i$. Each $C_i$ depends on the forwarding node's sink distance $U_{i-1}$; thus, the sink distance of the source node clearly affects the first hop and subsequent hops. As noted in the previous section, comparing the hop distributions demonstrates that each $C_{i}$ is stochastically dominated by $C_{i+1}$. That is, for $i\geq0$, we have the stochastic ordering
\begin{equation}\label{algineq}
\mathbb{P}(C_{i+1}>c)\geq \mathbb{P}(C_{i}>c) , \qquad c\in(0,r).
\end{equation}
This inequality is the opposite to the equivalent result under the homogeneous model as noted by Zorzi and Rao \cite{ZORZI1:2003} and Keeler and Taylor \cite{KEELER1}.

\subsection{Path dependence}\label{pathDep}
Let the random variable $\Theta_i$ be the angle between the $i$-th node and the previous node in relation to the sink. We assign the point $X_i=(U_i,\Theta_i)$ to the $i$-th forwarding node. The source (or zeroth) node corresponds to the point $X_{0}=(\ell,0)$. A message travels $i$ hops along a path that corresponds to a sequence of random points $\vec{X}_i=(X_0,X_1,\dots,X_i)$.

Let $\I_i(u_{i+1})\subset\mathbb{R}^2$ be the feasible region of the $i$-th forwarding node as a function of $u_{i+1}$ under the independent model. After the first hop, the nature of greedy routing implies that another dependence arises in the distribution of $U_2$, which was observed under the homogeneous model \cite{KEELER1,KEELER2}. If a forwarding node is chosen, then there are no other nodes in the source feasible region closer to the sink. Hence, the intersection of the feasible regions of the source and the first node (that is, $\I_1\cap\I_{0}$ in Fig. \ref{stochdep}) has no awake nodes. This implies that $U_2$ is  dependent on both $U_1$ and $\Theta_1$, the angle between the first node and the source node in relation to the sink. We call this dependence in both the sink distance and the sink angle the \emph{path} dependence, and the hop model that includes both the path and the sink dependence simply the \emph{dependent} model. Conversely, the \emph{independent} model only includes the sink dependence.

\begin{figure}
\begin{center}
\begin{overpic}[scale=.8]{stochdep3.EPS}
\put(37,14){$\Theta_1$}
\put(47,18){$U_1$}
\put(7.5,14.5){$X_{0}$}
\put(13,21){$X_{1}$}
\put(34,32){$\I_1\setminus\I_{0}$}
\put(33.5,6.1){$\I_1\cap\I_{0}$}
\end{overpic}
\caption{No awake nodes in the intersection region $\I_1\setminus\I_{0}$ during the first message relay.\label{stochdep}}
\end{center}
\end{figure}

Under a sleep scheme, if nodes were asleep during the previous message relay, it is possible for recently awoken nodes to be present in this region during the current message relay. To perform the initial analysis we assume no sleep scheme exists (by setting $p=1$). After our analysis, we include a simple sleep scheme and examine how varying $p$ and $\alpha$ (while fixing $\lambda$) affects the path dependence.

The distribution of $U_{i+1}$ under the independent model is dependent only on the sink distance of the current forwarding node, hence we write
\begin{equation}\label{Fi}
F_{i}(u_{i+1})=\mathbb{P}_I(U_{i+1}\leq
u_{i+1}|U_i=u_i),
\end{equation}
while under the dependent model the distribution is dependent on the message path, and so we write
\begin{equation}\label{Gi}
G_{i}(u_{i+1})=\mathbb{P}_D(U_{i+1}\leq
u_{i+1}|\vec{X}_i=\vec{x}_i),
\end{equation}
where the subscripts $I$ and $D$ indicate probability measures under the two models (the subscript is dropped if a result applies to both models). We denote the rescaled mean measure of the feasible region under the independent and dependent models respectively as $Q_{i}(u_{i+1})$ and $\vec{Q}_{i}(u_{i+1})$.

The rescaled mean measure under the independent model is always given by the original equation (\ref{meanU}), and hence, the distribution and  probability density of $U_{i+1}$ are obtained by setting $\gamma=u_i$ in equations (\ref{inhomFu}) and (\ref{inhomfu}). Under the dependent model, if the rescaled mean measure is given after $i$ hops, the sink distribution
\begin{equation}\label{Gu}
G_{i}(u_{i+1}) = \left\{ \begin{array}{ll}
1-e^{-\lambda \vec{Q}_i(u_{i+1})} &\quad u_{i}-r \leq u <u_{i}\\
1 &\quad u\geq u_{i}\\
0&\quad u<u_{i}-r,
\end{array} \right.
\end{equation}
immediately follows, and where it is absolutely continuous its  probability density
\begin{equation}\label{gu}
g_{i}(u_{i+1}) =
  \lambda \vec{Q}'_i(u_{i+1})e^{-\lambda \vec{Q}_i(u_{i+1})},
\end{equation}
also follows.

The set representing the feasible region under the dependent model follows by excluding the intersections of previous feasible regions, namely
\[
\D_i(u_{i+1})=
\I_i(u_{i+1})\setminus\displaystyle\cup_{j=0}^{j=i-1}\I_j(u_{j+1}).
\]
Under the dependent model, it is possible to calculate the rescaled mean over the feasible region after one hop; see appendix for details. If $i\geq2$, we approximate the feasible region under the dependent model
\begin{equation}\label{DepSetApprox}
\D_i(u_{i+1})\approx
\I_i(u_{i+1})\setminus\displaystyle\I_{i-1}(u_{j+1}),
\end{equation}
where we refer to this approximation as the one-hop model. Results under the homogeneous model revealed that this approximation sufficiently captures the path dependence \cite{KEELER2}. Consequently, only the location of the previous node is needed to calculate the rescaled mean under the dependent model.

The path dependence depends on the precise locations of the previous nodes while the sink dependence which only depends on their distance from the sink. To describe the random behaviour of message hops we need the joint density of $U_i$ and $\Theta_{i}$. Under our inhomogeneous Poisson model, the joint probability density is
\begin{equation}
g_{i}(u_{i+1},\theta_{i+1})= \lambda_{\D_i} (u_{i+1},\theta_{i+1}) e^{-\lambda \vec{Q}_i(u_{i+1})},
\end{equation}
where the derivation is analogous to that of the homogeneous model \cite{KEELER2}; see appendix for details. The spatially-dependent initial density $\lambda_{\D_i}(u_{i+1},\theta_{i+1})=\lambda\mathbf{I}_{\D_i}(u_{i+1},\theta_{i+1})$ and the indicator function of the dependent feasible region
\begin{equation*}
\mathbf{I}_{\D_i}(u_{i+1},\theta_{i+1})= \left\{ \begin{array}{ll}
1, &\qquad  (u_{i+1},\theta_{0i})\in \D_i,\\
0,&\qquad \textrm{otherwise},
\end{array} \right.
\end{equation*}
where angular coordinate $\theta_{0i}$ is the angle between the source node and the $i$-th forwarding node in relation to the sink.

\section{Sleep model}
We outline a simple sleep scheme that has been analyzed under the homogeneous model \cite{KEELER1}. We assume that the probability that a node is awake on each hop is $p$ and the event that a node is awake during a transmission attempt is independent of the event that it is awake at other transmission attempts. Consequently, the intersection region ($\I_1\cap\I_{0}$ in Fig. \ref{stochdep}) has a thinned initial node density  $(1-p)\lambda$. The rest of the feasible region $\I_1\setminus\I_{0}$ has an initial density $\lambda$. It follows that the initial node density function after one message hop is given by
\begin{equation}\label{sleeplambda}
\lambda_{\D_1}(u_{2},\theta_2)= \lambda\left[ \Ind_{\I_1\setminus\I_{0}}(u_{2},\theta_2)
+(1-p)\Ind_{\I_1\cap\I_{0}}(u_{2},\theta_2)
\right],
\end{equation}
where the superscripts denote the indicator functions of the disjoint regions.

In the limit as $p$ approaches zero and $\alpha$ approaches infinity with $\lambda=p\alpha$ held constant, the locations of the nodes after each hop is re-sampled, thus, completely removing the path dependence and allowing each forwarding node to be treated like a source node. Its effects on the node density have been explored more thoroughly under the homogeneous model \cite{KEELER2}.

To calculate the Poisson mean measure over the region $\D_i$, the above node density function is integrated over the domain \cite{SKM:1995} leading to
\begin{equation}
\vec{\Lambda}_{\D_i}(u_{i+1})=\int_{u_{i}-r}^{u_{i+1}}\int_{-\psi_{u_i}(w_{i+1})}^{\psi_{u_i}(w_{i+1})} \lambda_{\D_i}(w_{i+1},\theta_{i+1})q (w_{i+1})w_{i+1}d\theta_{i+1} dw_{i+1},
\end{equation}
and we define
\begin{equation}
\vec{Q}_{\D_i}(u_{i+1})=\frac{\vec{\Lambda}_{\D_i}(u_{i+1})}{\lambda}.
\end{equation}
The joint probability density of $U_i$ and $\Theta_{i}$, that is
\begin{equation}\label{jointsleep5}
g_{i}(u_{i+1},\theta_{i+1})= \lambda_{\D_i}(u_{i+1},\theta_{i+1}) e^{-\lambda\vec{Q}_{\D_i}(u_{i+1})}.
\end{equation}

\section{Multihop distribution}
\subsection{Hop advancements}
We initially formulate this problem with the sink distance variabe since greedy routing is naturally based on it. However, hop advancement is a more intuitive variable in describing message progress over a multihop path. We adopt similar notation used for the sink distance random variables, hence $\Fb$ and $\Gb$ denote the distributions under the two models. The complement of the sink distribution yields the hop distribution under both the independent and dependent models; the latter being
\begin{equation}\label{Gc}
\Gb_{i}(c_{i+1})=\left\{ \begin{array}{ll}
e^{-\lambda \vec{Q}_{i}(u_i-c_{i+1})} \quad& 0 < c_{i+1} \leq r\\
1 \quad& c_{i+1} > r\\
0 \quad& c_{i+1} \leq 0.
\end{array} \right.
\end{equation}
and its probability density defined on the absolutely continuous part of the support
\begin{equation}\label{gc}
\gb_{i}(c_{i+1}) =
  \lambda \vec{Q}'_i(u_i-c_{i+1})e^{-\lambda \vec{Q}_i(u_i-c_{i+1})},
\end{equation}
where a simple sum relates the hop and sink distance variables
\begin{equation}\label{usum}
u_i=\ell-\sum^{i}_{j=1}c_j.
\end{equation}

\subsection{Distribution of $Z_n$}
Let the random variable $Z_n$ represent the distance advanced by a message in $n$ hops
\begin{equation}\label{Zprimen5}
Z_n=\sum^n_{i=1}C_i.
\end{equation}
To calculate the distribution of $Z_n$ we use the joint probability density of the random variables $C_1$ to $C_{n}$ and $\Theta_1$ to $\Theta_{n}$, that is
\begin{equation}\label{jointg5}
\gb_{(n-1)}(c_1,.,c_{n},\theta_1,.,\theta_{n})=\prod_{i=1}^n \lambda_{\D_{i-1}}(u_{i-1}-c_{i},\theta_{i})e^{-\lambda \vec{Q}_{i-1}(u_{i-1}-c_{i})},
\end{equation}
which is defined for $c_i\in(0,r]$. The derivation of the joint probability density is similar to that under the homogeneous mode \cite{KEELER2}, which we have adapted and included in the appendix for completeness.

It follows that the distribution of message advancement after $n$ hops is expressed by
\begin{align}\label{ProbZn5}
\mathbb{P}_D(Z_{n}\leq z)=&\int_{0^+}^{\min(z,r)}dc_1\int_{-\psi_{0}(c_{1})}^{\psi_{0}(c_{1})}d\theta_1
\int_{0^+}^{\min(z-c_1,r)}dc_2\int_{-\psi_{1}(c_{2})}^{\psi_{1}(c_{2})}\dots\\
&\int_{0^+}^{\min(z-\sum_{i=1}^{n-1}c_i,r)}dc_n\int_{-\psi_{n-1}(c_{n})}^{\psi_{n-1}(c_{n})}
\gb_{(n-1)}(c_1,.,c_{n},\theta_1,.,\theta_{n}) d\theta_n\\
&+\Prob_D(C_1=0)+ \Prob_D((Z_1\leq z)\cap (C_2=0))+\dots\\ &+\Prob_D((Z_{n-1}\leq z)\cap (C_{n}=0)).
\end{align}
where
\begin{equation}\label{shortpsi5}
\psi_i(c_{i+1})=\psi_{u_i}(u_i-c_{i+1}),
\end{equation}
denotes the maximum angle for a sink distance given by the sink angle function (\ref{psi5}). The distribution of $Z_n$ under a sleep scheme is obtained by substituting the product of the joint probability densities (\ref{jointsleep5}).

The integral explicitly shown in the expression of $\mathbb{P}_D(Z_{n}\leq z)$ is the distribution of $Z_n$ conditioned on the event that all hops $C_i$ advance some positive distance. We will often refer to this integral simply as the \emph{conditional distribution} of $Z_n$, and denote it by
\begin{equation}\label{ProbZnPos}
\mathbb{P}(Z_{n}\leq z|+)=\mathbb{P}(Z_{n}\leq z|C_1>0,\dots,C_n>0).
\end{equation}

Under the independent model the joint probability density is not a function of any of the variables $\theta_1$ to $\theta_n$.  Hence, the equivalent integral can be analytically integrated over the sink angle domains \cite{KEELER1}, thus giving a simplified expression in the form of hop probability densities
\begin{align}\label{ProbZnInd5}
\mathbb{P}_I(Z_{n}\leq z)=&\int_{0^+}^{\min(z,r)}dc_1
\int_{0^+}^{\min(z-c_1,r)}dc_2\dots\\
&\int_{0^+}^{\min(z-\sum_{i=1}^{n-1}c_i,r)}\bar{f}_{0}(c_1)\dots \fb_{{n-1}}(c_n)dc_n\\
&+\Prob_I(C_1=0)+ \Prob_I((Z_1\leq z)\cap (C_2=0))+\dots\\ &+\Prob_I((Z_{n-1}\leq z)\cap (C_{n}=0)).
\end{align}

Under the dependent model, the probability of a message reaching a routing void after advancing $i$ hops
\begin{equation}
\mathbb{P}_D(C_{i+1}=0|\vec{X}_i=\vec{x}_i)=e^{-\lambda \vec{Q}_{i}(u_i)},
\end{equation}
follows. The routing void probability leads to the distribution of $Z_n$ conditioned on the event that the message meets a routing void on the last hop
\begin{align}\label{ProbZnNull5}
\nonumber\mathbb{P}_D((Z_{n}\leq z) \cap (C_{n+1}=0))&=\int_{0^+}^{\min(z,r)}dc_1\int_{-\psi_{0}(c_{1})}^{\psi_{0}(c_{1})}d\theta_1
\cdots\\
&\nonumber\int_{0^+}^{\min(z-\sum_{i=1}^{n-1}c_i,r)}dc_n\cdots\\
&\int_{-\psi_{n-1}(c_{n})}^{\psi_{n-1}(c_{n})}\gb_{(n-1)}(c_1,.,c_{n},\theta_1,.,\theta_{n})\mathbb{P}_D(C_{n+1}=0) d\theta_n.
\end{align}

Consider the event when a message does not advance, hence $X_i=X_{i+1}$. If there is a sleep scheme, a forwarding node can benefit by making multiple relay attempts. The number of different possible events results in the integral expressions needed to describe such a model soon growing to be intractable. We restrict the integrals by assuming that a message executes only one relay attempt. We point out that under the homogeneous model \cite{KEELER2}, stochastic `rules of thumb' have been proposed for how many re-attempts forwarding nodes should make before considering other options (such as message backtracking). Under the inhomogeneous model, the equivalent results can obtained by appropriately replacing the area terms with the corresponding integrals from the rescaled mean measures.

\subsection{Number of hops}
Let the random variable $N$ represent the total number of hops required for a message to reach the sink. The distribution of $N$ gives a different perspective of the performance of a routing method in a sensor network. The random variables $N$ and $Z_n$ are connected by a simple result \cite{KEELER1}, which has been used to calculate the distribution of $N$ from the distribution of $Z_n$ in the homogeneous setting \cite{KEELER2}. We use this relation to calculate the equivalent results under our spatially-dependent node density model.

\section{Integration methods}
To calculate the distributions of $Z_n$ under the dependent model, a $2n$-fold integral (\ref{ProbZn5}) needs to be evaluated. For low $n$, traditional numerical integration schemes can be used. Unfortunately, integration by these methods is too slow at higher hop numbers, which motivates us to employ quasi-Monte Carlo methods.

\subsection{Quasi-Monte Carlo}
The integration description that follows is similar to the more detailed account \cite[Chap. 4]{KEELERTHESIS} where quasi-Monte Carlo methods were used to evaluate similar integrals under the homogeneous model. In recent years, these integration methods have gained much interest owing to their speed and accuracy in evaluating high dimensional integrals.

Quasi-Monte Carlo methods are based on purely deterministic sequences of quasi-random numbers such as Halton \cite{HALTON:1960} and Sobol \cite{SOBOL:1967} sequences. Mathematically, quasi-random sequences have low-discrepancy \cite{NIEDERREITER:1992,SM:1994}. Informally, such sequences appear `less random' than sequences produced by regular pseudo-random number generators as they occur more evenly spaced apart.

Previous numerical work \cite{KEELER2} has led us to use leaped Halton sequences to calculate integrals mostly. Often integrals were also calculated via regular Monte Carlo methods to check the quasi-random approach. We also give some results based on so-called \emph{lattice rules}, which lead to specific cases of quasi-random sequences. These rules can produce well-behaving quasi-random sequences and have been a research focus in recent years owing to their ability to counter the curse dimensionality \cite{KS:2005}. The points arising from lattice rules are used in a similar manner to the quasi-Monte Carlo approach. There are many suggestions for lattice rules, but we lightly examine only one known as \emph{rank-1 lattice rule}, which over the unit hyper-cube gives the integral estimate
\begin{equation}\label{LR}
\hat{I}_nf=\frac{1}{n}\sum_{k=1}^n f(\left\{k\frac{\mathbf{z}}{n}\right\}),
\end{equation}
where the \emph{generating vector} $\mathbf{z}\in \mathbb{Z}^s$, $n$ is the number of function samples, and the braces give the fractional part in $[0,1)$.

Given a generating vector, quasi-random sequences based on such a lattice rule can be clearly produced in an exceedingly fast manner. The way to quickly evaluate an integral is by choosing a suitable generating vector. However, the drawback is that lattice rules are based on input parameters known as weights, which depend on the nature of the function. In particular, the weights depend on how the function varies with respect to all its variables. Furthermore, the choice of some lattice rules require the total number of function samples before the integral calculation starts. This differs from regular quasi-Monte Carlo methods in which the integral estimate can be calculated continually until a sufficient number of function samples has been taken.

Under lattice rules, the number of function samples also influences the choice of the generating vector. Consequently, using the most suitable generating vector may not be a simple task as it involves analyzing the function of interest. However, a thorough analysis of which weights and quasi-random sequences are the most suitable in this setting is beyond the scope of this work. We simply give some complementary results based on the rank-1 lattice rule, and leave the analysis as a future task.

The research field of lattice rules has a relatively short history, and new lattice rules are being developed continually. For more information, we refer the reader to the introductory piece by Kuo and Soan \cite{KS:2005}, and an example of lattice rules used in a financial setting \cite{GKSW:2008}. Suitable lattice rules may offer a substantially faster way of evaluating the high dimensional integrals presented here and in previous work \cite{KEELER2}. However, we focus on reducing error regardless of the chosen sequences by employing importance sampling, which has been done under the homogeneous model \cite{KEELER2}.

\subsection{Importance sampling}
To reduce the variance of the integral estimate we employ importance sampling; that is, suitably generate $C_i$ values to sample the function in key regions. Based on previous work \cite{KEELER2}, we derive an importance sampling function which is similar in form to that of the homogeneous case. We recall the expansion of the $Q$ function (\ref{approxQ}) in which we use only the first term, hence
\begin{equation}
\begin{array}{c}
  Q_{\gamma}(u)\sim q_0(u-\gamma+r)^{3/2}+O(u)^{5/2}
\\
\quad\textrm{as }\quad u \rightarrow \gamma-r,
\end{array}
\end{equation}
where
\[
q_0 = \dfrac{4}{3}\left[\dfrac{2r}{\gamma (\gamma-r)}\right]^{1/2}.
\]
A change of variable $c=\gamma-u$ leads to the function
\[
\tilde{Q}_{\gamma}(c)=q_0(r-c)^{3/2},
\]
which leads to an approximate solution for the hop distribution
\[
\tilde{F}_{\gamma}(c)= e^{-\lambda \tilde{Q}_{\gamma}(c)}.
\]
The quantity $\tilde{F}_{\gamma}(0)$ is subtracted from the above expression and the result is divided by $\Delta\tilde{F}_{\gamma}=\tilde{F}_{\gamma}(c_{\max})-\tilde{F}_{\gamma}(0)$, to obtain an importance sampling function
\begin{equation}\label{ImpSampFc5}
\widehat{F}_{\gamma}(c)= \frac{1}{\Delta\tilde{F}_{\gamma}}\left[e^{-\lambda \tilde{Q}_{\gamma}(c)}-\tilde{F}_{\gamma}(0)\right],
\qquad 0\leq c \leq c_{\max}.
\end{equation}
where $c_{\max}$ is the largest hop value. The subtracting of the routing void term  has negligible effect for sufficiently large $\lambda$. The corresponding derivative needs to integrate to one, hence the rescaling step. The derivative
\begin{equation}\label{ImpSampFc5b}
\widehat{f}_{\gamma}(c)=\frac{3\lambda }{2\Delta\tilde{F}_{\gamma}}(r-c)^{1/2} e^{-\lambda \tilde{Q}_{\gamma}(c)},
\end{equation}
exists on the same domain as the sampling function. The inverse of the sampling function is given by
\begin{equation}\label{ImpSampFc5c}
\widehat{F}_{\gamma}^{-1}(t)=r-\left( \frac{-1}{\lambda a_1} \ln\left[ t\Delta\tilde{F}_{\gamma}+\tilde{F}_{\gamma}(0)\right] \right)^{3/2},
\qquad 0\leq t \leq r.
\end{equation}
Substituting a random variable from a uniform distribution, say $T\sim U(0,r)$, into the inverse of the importance sampling function
gives a random variable $C$ adhering to the importance sampling distribution (\ref{ImpSampFc5}).

After each hop is generated, the functions (\ref{ImpSampFc5b}) and (\ref{ImpSampFc5c}) should be updated by setting $\gamma$ to the current sink distance for each sample. This step was not necessary under the homogeneous model. However, under this inhomogeneous model each hop distribution varies more with respect to the sink distance. Furthermore, for high $\lambda$, the importance sampling step should improve as it it based on the independent model, and at high node density the intersection regions grow stochastically smaller \cite{KEELER1,KEELER2}.

\section{Simulation}
We used routing simulations to verify our stochastic model and calculations. Given a source node sink distance $\ell$, message relaying is simulated in a circular sensor field $\mathcal{C}_{\ell}\subset \mathbb{R}^2$ of radius $\ell$ with the sink located at the origin. The fact that messages only advance towards the sink under greedy routings implies that sensor field edges do not influence the message routing. The total number of nodes per simulation is a Poisson random variable with the parameter
\begin{align*}
\Lambda(\mathcal{C}_{\ell})=\,&\lambda\int_0^{\ell}\int_{0}^{2\pi}q(u)udud\theta\\ =\,& 2\lambda\pi\ell.
\end{align*}
In simulation, a node is assigned two independent random variables $\Theta_{\textrm{S}}$ and $R_{\textrm{S}}$ corresponding to their polar coordinates in relation to the sink. To simulate node deployment such that the nodes adhere to the spatially-dependent node density (\ref{inhomlambda}), both random variables are uniformly distributed
\begin{align*}
\Prob(\Theta_{\textrm{S}}\leq \theta)=\,&\dfrac{\theta}{\pi}, \qquad \theta \in[0,\pi],\\
\Prob(R_{\textrm{S}}\leq r)=\,&\dfrac{r}{\ell}, \qquad r\in[0,\ell].
\end{align*}

\section{Results}

All the numerical integration and routing simulations were performed in Matlab on a standard machine. The built-in Halton sequence generator was employed for the quasi-Monte Carlo integration. For a given value of $z$, it took between $10^2$ to $10^4$ points to obtain a quasi-Monte Carlo estimate of the conditional distribution $\Prob(Z_n\leq z|+)$. The importance sampling step improved the rate of convergence, particularly for high $\lambda$. The actual number of points depends on the number of hops; more hops require more points. All calculations took no longer than an hour to complete, and usually considerably less.

We compared the dependent model to routing simulations of various ensemble sizes with and without a blinking sleep scheme. Generally, between $10^3$ to $10^5$ routing simulations were required. Similar to the integration process, higher hop numbers required more simulations to give converged results. The routing simulations are based on the same assumptions made in the mathematical model.

We observed under the inhomogeneous model that more function samples and routing simulations are needed to give similarly converged results compared to those obtained under the homogeneous model. An extensive empirical investigation is needed to see how fast quasi-Monte Carlo methods are compared to routing simulations. Also, more empirical and theoretical evidence is needed to elucidate the advantages and disadvantages of calculating probabilistic behaviour of greedy routing via our model.

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{CompOneHopExactVsEmp.eps}
\caption{Analytic and empirical distributions of the sink distance after one hop ($p=1$,$\lambda=3\ell$, $r=1$ and  $\ell=10$).\label{CompOneHopExactVsEmp}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{Comp2HopExactVsEmpZ.eps}
\caption{Results of $\Prob(Z_2\leq z|+)$ via quasi-Monte Carlo integration and simulations ($p=1$,$\lambda=3\ell$, $r=1$ and  $\ell=10$).\label{Comp2HopExactVsEmpZ}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{Comp2HopLRExactAsymVsEmpZ.eps}
\caption{Results of $\Prob(Z_2\leq z|+)$ via lattice rules integration (based on analytic and asymptotic expressions) and simulations ($p=1$,$\lambda=3\ell$, $r=1$ and  $\ell=10$).\label{Comp2HopLRExactAsymVsEmpZ}}
\end{center}
\end{figure}

For a single hop, the mathematical model results are practically identical to the simulation results (see Fig. \ref{CompOneHopExactVsEmp}). This implies that the model adequately captures the spatially-dependent node density function, and also vindicates the use of the elliptic integral functions.

The lattice rule approach was only used to calculate one set of results (see Fig. \ref{Comp2HopLRExactAsymVsEmpZ}). We used generator vectors based on fixed lattice rules for $2^{10}$ function sample points and equal weights \cite{KuoWeb}. We applied ten random shifts to the lattice rules; see Giles et al. \cite{GKSW:2008} for an example. This approach generally performed well, however, under importance sampling sometimes erratic results arose. This is possibly due to an over bias in function sampling or an unknown numerical artefact. A more thorough examination is needed, but we believe that the preliminary results using lattice rules are promising.

Despite the accuracy of elliptic integral functions, we found that the results based on them could be replaced with results that used the three-term approximations (\ref{approxpsi5}) and (\ref{approxQ}) with no discernible loss of accuracy (see the plots in Fig. \ref{Comp2HopLRExactAsymVsEmpZ}). Evaluating the approximations is considerably faster as the expressions only involve elementary functions. Consequently, the remaining results are based on these approximations (Fig. \ref{Pn2a} to Fig. \ref{Pn3a}).

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{NHopDistQMCCompGamma10Lambda2Asymn1e4P100.eps}
\caption{Independent and dependent model results of $\Prob(N\leq n)$ compared to simulations ($p=1$, $\lambda=2\ell$, $r=1$ and  $\ell=10$).\label{Pn2a}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{NHopDistQMCCompGamma10Lambda2Asymn1e4P10.eps}
\caption{Independent and dependent model results of $\Prob(N\leq n)$ compared to simulations ($p=0.1$, $\lambda=2\ell$, $r=1$ and  $\ell=10$).\label{Pn2b}}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{NHopDistQMCCompGamma10Lambda3Asymn1e4P100.eps}
\caption{Independent and dependent model results of $\Prob(N\leq n)$ compared to simulations ($p=1$, $\lambda=3\ell$, $r=1$ and  $\ell=10$).\label{Pn3a}}
\end{center}
\end{figure}

We calculated the distribution of $N$, and included a simple sleep scheme to see its influence on the path dependence. We compared the independent model to the dependent model by varying $p$ (and accordingly $\alpha$, hence holding $\lambda$ constant). The difference between the two models has an accumulative effect on $N$, hence it serves as a good indicator of path dependence.

Under the independent model, the distribution $\Prob(N\leq n)$ gives greater values for each $n$ than the equivalent result under the dependent model. Also, the path dependence clearly lessens as $p$ approaches zero (compare Fig. \ref{Pn2a} and Fig. \ref{Pn2b}). These results are analogous to those under the homogeneous density model \cite{KEELER2}.

For large $\lambda$, the difference between the two models is less. As is the case for the homogeneous, a larger density results in the next forwarding node being closer to the sink, thus reducing the intersection of the feasible regions and lessening the path dependence. Moreover, under the inhomogeneous model hops grows stochastically larger due to the increasing node density. Thus, we believe that the path dependence continues to decrease stochastically as the message approaches the sink.

In conclusion, the results reveal that the dependent model clearly captures the path dependence. The resulting expressions, both involving the elliptic integrals and the asymptotic expansions, give results that closely agree with simulations. We believe more numerical investigation is needed to choose the most appropriate quasi-random sequences in this setting.

\section{Future work}
In realistic settings, the constant node density assumption may often not be appropriate. Under the spatially-dependent model, the density function was chosen such that it was simple enough for analytic and asymptotic means, while still being a plausible node placement scenario. Other suggestions exist such as the node density decaying exponentially or according to some power of the sink distance.

Furthermore, under our model the sink was located at the maximum of the node density. Placing the sink at an arbitrary point in the sensor field results in the node density being dependent on the sink angle. This is an additional increase in the complexity of the density function. Moreover, the angle of an individual node would not be a uniformly distributed random variable, thus importance sampling might be needed when integrating over the angle domains. Consequently, these suggestions may result in analytic and asymptotic mean measures that can be used to model more realistic node deployment models.

Further investigation of the asymptotic approximations are needed. The approximation may break down when the radius is large compared to the sink distance. For a constant radius model, the lengths can always be rescaled with respect to the transmission radius. However, this may not be possible for a randomly varying radius model. We stress that including random transmission radii into our model would be an interesting and realistic model extension in itself.

We presented some integration results based on lattice rules. Despite these methods giving mostly agreeable results, further investigation is needed to gauge which lattice rules and quasi-random sequences are the most suitable for the integrals that arise from our model. This investigation would be both analytic and numerical in nature. This work may lead to our mathematical model considerably outperforming regular routing simulations.

Finally, an attractive feature of regular Monte Carlo methods is that the error is obtained by estimating the variance of the integral. Conversely, quasi-Monte Carlo methods lack a practical way of estimating the error, despite them generally have a faster convergence rate. The idea of `randomized' quasi-Monte Carlo methods seeks to combine the advantages of both approaches. Consequently, a future task lies in investigating these hybrid methods in evaluating hop integrals.

\section{Conclusion}
We presented a tractable inhomogeneous spatially-dependent density model. Inspired by previous work, we developed and examined a greedy routing model that incorporates both sink and path dependence. Moreover, the spatially-dependent density model verified that the formulation of the homogeneous model can be extended to an inhomogeneous case. This model is an alternative means of ascertaining the stochastic characteristics of greedy routing in sensor and ad hoc networks.

We used asymptotic methods to derive accurate approximations to hop length moments and the mean measure for our spatially-dependent node density model. We used quasi-Monte Carlo methods and recently-developed lattice rules coupled with importance sampling to estimate the resulting high dimensional integrals. For a sufficient number of function samples, all the results agreed admirably with those obtained by routing simulations.

Finally, we included a sleep scheme to demonstrate its effects on the local node density and the path dependence. For systems with a low $p$, the results imply that the independent model can be used, thus reducing computation time in calculating stochastic properties of the system.

\section{Acknowledgements}
The author thanks Peter Taylor at the University of Melbourne for valuable discussions and advice. A further thanks to Frances Kuo and Ian Sloan at the University of New South Wales for kindly explaining the subtleties of lattice rules. The author would also like to acknowledge the support of the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems.

\bibliographystyle{siam}

redpajama_set_name: RedPajamaArXiv


The feedback from the survey seems to indicate that, even in this technological age, the church bulletin is still the primary source of information about the parish. Kudos to Linda Fitzpatrick for doing such a great job putting this together week after week. However, it also indicates that we probably have a lot of opportunity for growth in how we use other channels more effectively, particularly our web site and social media. I think there's also a lot of opportunity to use these communication tools to evangelize, too. The bottom line is that, as we plan for the future, we'll need to make sure that communications are a priority, both to stay connected to our parishioners and to spread the Gospel beyond our parish.


text:
The First Priority Music Family: Basement Flavor is a hip hop compilation album, released on December 15, 1988, on First Priority/Atlantic Records. The single "I'm Not Havin' It" by Positive K and MC Lyte peaked at #16 on the Billboard Hot Rap Singles chart in 1989.

Track listing

References

Hip hop compilation albums
1988 compilation albums
Atlantic Records compilation albums

redpajama_set_name: RedPajamaWikipedia


text:
Know Your Lilys: A Quick Primer in How to Tell The Different Lilys Apart
By Dustin Rowles | Lists | March 29, 2012 |
There are a lot of Lilys in Hollywood these days, and while many of them are rising stars in movies and television, it's still difficult for the novice to differentiate between them all. Some of you are old, and have a difficult time telling white women apart. It's all very confusing. And that's why we here at Pajiba want to help you out with this very confusing matter by providing our Primer in Lilys.
Lily Collins — Collins is the Lily you should concern yourself with most at the moment, as she is playing Snow White in Mirror, Mirror, due out in theaters tomorrow. She's also the daughter of Phil Collins, and has been linked romantically to Zac Efron and her Abduction co-star, Taylor Lautner.
Lily Allen — Allen is a pop singer, and like Collins, she's also the daughter of an English musician, Keith Allen. She was once very popular, but she's not terribly relevant anymore.
Lily Cole — Not to be confused with Collins, this striking redhead is also appearing in a Snow White movie, Snow White and the Huntsmen. She'll play Rose. She, like most of the actresses on this list, is English. I have no idea why Lily was such a popular name in England around 20 years ago.
Lily Aldridge — Aldridge is actually not an actress. She's a Victoria's Secret model, and may be familiar to you as the star of Michael Bay's most recent Victoria's Secret commercials. She's also engaged to the lead singer of Kings of Leon.
Lily Pritchett-Tucker — This is a tough one, as this Lily is a character on the television show, "Modern Family." She used to be played by an infant, but she's recently been recast as a toddler. She also occasionally uses profanity.
Evangeline Lilly — She used to be on a show called "Lost," and has a minor role in the upcoming Hobbit films. You don't really need to concern yourself with her anymore.
You Know I Can't Let You Slide Through My Hands: All the Pretty Horses Men
New Man of Steel Logo Will Make One Gentleman a Mighty Fine Belt Buckle Someday

redpajama_set_name: RedPajamaCommonCrawl


Here we will provide you with a quick summary of all the details you must know about your first visit to our dentist in Dental Care of Braselton.
Finding a new dental office can often be a challenge. Fortunately, Dental Care of Braselton in Braselton, Georgia can satisfy all your dental needs. The following will provide you with a detailed guide outlining what to expect from your first visit to our dental office.
So call us at 706-684-0759 right away for an appointment or visit our office at your convinience. Our team is here to welcome you and will answer any questions you may have.


On the N train the conductor sounded Indian and he was chanting the station announcements like a mantra. It lent a sacral quality to the melee for once, as if the universe had finally broken through the static and into consciousness via an otherwise forgotten loudspeaker at the far end of a busy rail car.
There are times when I have the sense that I understand absolutely nothing about the world after all, that the things I 'know' are actually thin trivia and silliness, like baseball statistics or esoteric history.
Pigeons are perhaps the most universal component of 'plaza' as a concept, itself central to our image of the present. I wonder sometimes if modernity and postmodernity don't really all boil down to the presence of pigeons everywhere and little more.
I was literally the first person in the building today. I don't know what that tells me about anything. I went straight to the third floor to find the table at which I often work to be piled with light bulbs. They came and collected them a few minutes later.


"Excellent Service" – Great feedback to start the week off!
It's Monday and what a great way to start the week by receiving good feedback from a happy customer!
We like to give excellent service to all of our clients, give us a call today and see what we can do to make you happy, you can contact the Print Angels for a chat on 01332 418377.


As I prepare to wade into the exciting frontier of e-publishing, I figure people might want to know why I write the types of stories that I do. Well, it's because the paranormal, the supernatural and the mysterious fascinates me. That doesn't make me an evil or bad person in any way. I just want to make that clear. I am not a devil worshiper and I am not an atheist. It boggles my mind when I get a horrified reaction from someone after revealing that I read palms and tarot cards. "That's the devil's work!" they say. No. It's not. But, you can't argue with someone whose belief is that all forms of divination and all things supernatural are evil. So, obviously my books are not written with those people in mind. Though some of them do, I am sure, read and watch horror. Speaking of horror, I hesitate to classify my stories in that genre, but because of their content (ghosts, mediums, reincarnation, past lives, etc…) that is the genre under which they will most likely be filed.
So, why do I write stories that are focused on supernatural content? Well, I love to write and I love to write about things that interest me and I've been interested in the supernatural since I was a very young kid. And that was because I moved into a haunted house when I was seven years old. Until we moved into the "farmhouse" as we called it, my life was pretty normal. Normal in the respect that I was not exposed to situations that generated fear and uncertainty.
The farmhouse was an old, two-story structure sitting not far off a quiet country road and surrounded by fields. In the center of the field was a small grove of trees and located there was a tomb with eleven people buried in it. Supposedly, the people buried in that tomb all died either in the farmhouse or another structure on the property that no longer exists . Their deaths dated back to the 1800s and early 1900s. There were several young children included in their number. How do I know this? Their names, age, and the dates of their birth and death were chiseled in stone above the entrance to the tomb. Eerie. The farmhouse was in a sad state of disrepair when we moved in. My parents didn't have a lot of money. They rented the house for a measly amount and in return agreed to fix it up at their own expense. My parents thought it was a great deal. Seven years later, they weren't so inclined to believe that anymore and we left as quickly as they could manage it.
Although I was only seven when we moved in, I remember my first impressions of that house as being dark, depressing and spooky. The one thing that excited me was all the neat stuff that had been left within it. The place was full of junk and antique furniture. Speaking of which, antique farm equipment lay about the surrounding yard in total neglect. My three brothers, my sister and I (me being the oldest of five) gleefully played with those antiques until we unwittingly destroyed them or they became too rusty and old to bother with. On the second floor there were three bedrooms and a small room we dubbed "the dark room". It was the size of a small walk-in closet and oddly enough, there was a small window in the wall that divided the dark room from my parents bedroom. My mom put a large dresser against that wall and its large mirror covered the gaping hole. There was no door to this room so my mom hung a curtain there. The doorway was located in the bedroom where my three brothers slept. Too bad for them. All five of us kids were afraid of that room. It gave me the willies just walking past it. I still get the willies just thinking about it. My parents used the room to store some of the antique furniture that had been left in the house. A bummer that, because I loved the old dressers that sported lots of drawers and attached mirrors.
Soon as we moved in, I became afraid of the dark. I don't remember being afraid of the dark until we moved into that house. I had a lot of nightmares while living there and most of them were about that house. I remember that it was always cold. Even in the summer when it was hot outside, the inside of the house always seemed gloomy and cool. We didn't have central heat and that may have been a contributing factor, but the chill in that house wasn't just from the air, it was the kind of chill that seeped deep into your bones and made goosebumps break out on your arms even if you weren't really cold. The house was heated by an oil stove in the living room and a huge old cast iron stove in the kitchen. There was no heat upstairs so in the winter time we all had to move downstairs. A situation I liked because I felt safer having mom and dad close by (the five of us kids shared the one bedroom that was located downstairs and my parents converted the large entry hall into their room). We used the barn door entrance during the winter. A situation I did not like but you do what you have to do. Right?
I have to mention the cellar. Cellars are scary enough to young kids but the cellar in the farmhouse was especially spooky. It had a dirt floor and granite blocks for walls. It was damp and musty and there was just one hanging light bulb for the entire space. This meant that there were lots of very dark corners. There were two entrances to the cellar. One from the door next to the kitchen and the other from the side of the house facing the driveway. My parents didn't have to lock these doors. There was no way any of us kids were about to go down there to play or meddle. We were terrified of the cellar. One feature that gave it and extra creep factor was the vaulted room made of brick. It had an arched ceiling and a cement floor. We learned later when researching the history of that house that bodies were kept in there until they could be transported to the tomb or buried. Whether that was true or not, I don't know but I can believe it just from the energy that emanated from that room. It was the same sort of energy that emanated from the grove of trees surrounding the tomb.
The first few years of living there, things happened that really didn't grab our notice enough to raise any flags. Things came up missing quite often. Lots of things disappeared while living there that have never been found. Honestly, where did they go? I still wonder about that. Our dog acted very strange. He often whined and barked at, well nothing. At least, that's what we thought until we knew better. In the early days, we figured he was a tad touched in the head. He was a small dog, part poodle and part chihuahua. We figured his crazy antics were part of the breed. I must add here that I loved that dog. His name was Tippy and when I later learned that dogs knew when ghosts were around, I used him to determine when I was safe and when I wasn't. Until we figured it out that our house was full of ghosts (yes there were several), we really did think poor Tippy was weird. Once we knew why he acted like he did, he was looked at in a whole new light. Tippy became our detector for the presence of ghosts and the gauge by which we determined if the ghost was bad or good.
The first really scary event that grabbed our attention happened about a year after moving into the house. Up to this point, lights coming on and doors being left open and things disappearing were all chocked up to the fault of us kids. Though my dad tried his best to determine which of us was the culprit to these shenanigans, he rarely got a confession. A fact that pitted the five of us kids against each other because we too believed that one of us must have done the deed. If no one confessed, we all got punished. Totally unfair. It was an outrage to all of us to be punished for something we didn't do and I was always sure my oldest brother, younger than me by three years, was the one responsible. But I digress.
One evening the five of us kids were playing on the stairs while my dad chatted with a friend at the kitchen table. My mom had gone out to play bingo with her friend whose husband had stayed to visit with my dad. The stairs turned near the bottom and had two more steps that led down into the kitchen. A small window was located in the stairwell. As we kids were all small, we could only see out the window if we were about halfway up the stairs. It was about four feet above the landing where the stairs turned right to the kitchen. On this particular night, we were making a game of sliding down the stairs when my two year old sister froze in fear and started screaming. My brothers were one behind her and a couple in front of her. They looked at the window where my terrified sister was staring and they too began to scream and scramble back up the stairs. I was standing on the step below the window and the hairs on my neck went up as shivers raced down my spine. My dad didn't know what was causing all the commotion but he told me to get my sister. She sat on the step, rigid with fear and unable to do anything but scream. I remember her eyes being wide and dark and unblinking as they stared transfixed at that stairwell window. My heart pounding, I picked her up and turned to look. Nothing. By this time, my brothers were scrambling to get passed me on the stairs and run to the safety of our dad. They claim they saw red eyes in the window. They said it was a monster. From the outside, the window is about six or seven feet above the ground. That makes for a pretty big monster. My dad grabbed a shotgun and a flashlight and he and my dad's friend went outside to see what was out there. I envisioned them being killed and the red-eyed monster coming after us. But after what seemed an eternity, they came back in the house and said they didn't see anything. Subdued, the five of us went into the living room where we huddled on the couch together. Dad didn't believe they saw anything. I believed them. So we whispered quietly to each other as my young siblings told me what they saw. I vowed to keep them safe and they calmed down. But the fear in me grew. Something was wrong with this house. I just knew it. But as I didn't even know what a ghost was at that age, it took me a few years to figure it out.
Deb, thought it was very good, and knowing you, you could make it into a great book.
Thanks, George!! I've got one in the works. We'll see.
I loved this, Mom! You're definitely a gifted writer and know how to tell a story. I was bummed when it came to an end. Even though I have heard some of the stories, just reading this made me want to hear them again.
Thanks, April! I'm getting the willies writing about it. And I'm having nightmares again. Maybe someday I can make peace with that place.
I find your remarks so interesting, but seeing that I spent a lot of my own childhood there, up until I was 9, and nothing happened to me. I felt nothing, saw nothing. The only time I ever saw or heard anything was just before you guys moved out, then all Hell broke loose and I left and never went back. Can't wait for your book.
I remember that Elsie! Weren't you told to "get out!"? I'll be posting more stories!! and if you want to add your own, feel free!!
Interesting to say the least; Would be interested in the rest of the story.
Hi, Cynthia! I do plan on continuing with the story in future blogs! Thanks for reading and taking the time to comment.
wow that is cool, ( scary but cool ), you need to tell me where that house is/was.
The house was demolished a couple years ago! The tomb is still there, though. No way would I live on that property!! I get chills when I drive past it.


text:
Impaired splenic erythropoiesis in phlebotomized mice injected with CL2MDP-liposome: An experimental model for studying the role of stromal macrophages in erythropoiesis
Y. Sadahira, T. Yasuda, Tadashi Yoshino, T. Manabe, T. Takeishi, Y. Kobayashi, Y. Ebe, M. Naito
Erythropoiesis occurs in the presence of erythropoietin (EPO) without macrophages in vitro. In hematopoietic tissues, however, erythroid cells associate closely with stromal macrophages, forming erythroblastic islands via interactions with adhesion molecues. To elucidate the role of macrophages in erythropoiesis, we selectively abrogated stromal macrophages of splenic red pulp of phlebotomized mice by injection with dichloromethylene disphosphohate encapsulated in multilamellar liposomes (CL2MDP-liposome). In the spleen, no erythropoietic activity occurred until 5 days after the treatment. Colony assay revealed that the erthropoiesis was suppressed at the level of CFU-E. The splenic erythropoietic activity gradually developed from day 6 after the treatment, when F4/80+ macrophages began to appear in the red pulp. EPO mRNA was expressed in kidney but not in liver or spleen of phlebotomized mice injected with CL2MDP-liposome, and the serum EPO concentration in these mice was higher than that in phlebotomized mice. These findings suggest that abrogation of stromal macrophages by injection with CL2MDP-liposome impairs the splenic microenvironment for erythopoiesis induced by hypoxic stress, and this may be an excellent experimental model for further characterization of the in vivo role of splenic macrophages in erythropoiesis.
Journal of Leukocyte Biology
Erythropoiesis
Theoretical Models
Erythropoietin
Erythroid Precursor Cells
Microenvironment
Sadahira, Y., Yasuda, T., Yoshino, T., Manabe, T., Takeishi, T., Kobayashi, Y., ... Naito, M. (2000). Impaired splenic erythropoiesis in phlebotomized mice injected with CL2MDP-liposome: An experimental model for studying the role of stromal macrophages in erythropoiesis. Journal of Leukocyte Biology, 68(4), 464-470.
Impaired splenic erythropoiesis in phlebotomized mice injected with CL2MDP-liposome : An experimental model for studying the role of stromal macrophages in erythropoiesis. / Sadahira, Y.; Yasuda, T.; Yoshino, Tadashi; Manabe, T.; Takeishi, T.; Kobayashi, Y.; Ebe, Y.; Naito, M.
In: Journal of Leukocyte Biology, Vol. 68, No. 4, 2000, p. 464-470.
Sadahira, Y, Yasuda, T, Yoshino, T, Manabe, T, Takeishi, T, Kobayashi, Y, Ebe, Y & Naito, M 2000, 'Impaired splenic erythropoiesis in phlebotomized mice injected with CL2MDP-liposome: An experimental model for studying the role of stromal macrophages in erythropoiesis', Journal of Leukocyte Biology, vol. 68, no. 4, pp. 464-470.
Sadahira Y, Yasuda T, Yoshino T, Manabe T, Takeishi T, Kobayashi Y et al. Impaired splenic erythropoiesis in phlebotomized mice injected with CL2MDP-liposome: An experimental model for studying the role of stromal macrophages in erythropoiesis. Journal of Leukocyte Biology. 2000;68(4):464-470.
Sadahira, Y. ; Yasuda, T. ; Yoshino, Tadashi ; Manabe, T. ; Takeishi, T. ; Kobayashi, Y. ; Ebe, Y. ; Naito, M. / Impaired splenic erythropoiesis in phlebotomized mice injected with CL2MDP-liposome : An experimental model for studying the role of stromal macrophages in erythropoiesis. In: Journal of Leukocyte Biology. 2000 ; Vol. 68, No. 4. pp. 464-470.
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abstract = "Erythropoiesis occurs in the presence of erythropoietin (EPO) without macrophages in vitro. In hematopoietic tissues, however, erythroid cells associate closely with stromal macrophages, forming erythroblastic islands via interactions with adhesion molecues. To elucidate the role of macrophages in erythropoiesis, we selectively abrogated stromal macrophages of splenic red pulp of phlebotomized mice by injection with dichloromethylene disphosphohate encapsulated in multilamellar liposomes (CL2MDP-liposome). In the spleen, no erythropoietic activity occurred until 5 days after the treatment. Colony assay revealed that the erthropoiesis was suppressed at the level of CFU-E. The splenic erythropoietic activity gradually developed from day 6 after the treatment, when F4/80+ macrophages began to appear in the red pulp. EPO mRNA was expressed in kidney but not in liver or spleen of phlebotomized mice injected with CL2MDP-liposome, and the serum EPO concentration in these mice was higher than that in phlebotomized mice. These findings suggest that abrogation of stromal macrophages by injection with CL2MDP-liposome impairs the splenic microenvironment for erythopoiesis induced by hypoxic stress, and this may be an excellent experimental model for further characterization of the in vivo role of splenic macrophages in erythropoiesis.",
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AU - Sadahira, Y.
AU - Yasuda, T.
AU - Yoshino, Tadashi
AU - Manabe, T.
AU - Takeishi, T.
AU - Kobayashi, Y.
AU - Ebe, Y.
AU - Naito, M.
N2 - Erythropoiesis occurs in the presence of erythropoietin (EPO) without macrophages in vitro. In hematopoietic tissues, however, erythroid cells associate closely with stromal macrophages, forming erythroblastic islands via interactions with adhesion molecues. To elucidate the role of macrophages in erythropoiesis, we selectively abrogated stromal macrophages of splenic red pulp of phlebotomized mice by injection with dichloromethylene disphosphohate encapsulated in multilamellar liposomes (CL2MDP-liposome). In the spleen, no erythropoietic activity occurred until 5 days after the treatment. Colony assay revealed that the erthropoiesis was suppressed at the level of CFU-E. The splenic erythropoietic activity gradually developed from day 6 after the treatment, when F4/80+ macrophages began to appear in the red pulp. EPO mRNA was expressed in kidney but not in liver or spleen of phlebotomized mice injected with CL2MDP-liposome, and the serum EPO concentration in these mice was higher than that in phlebotomized mice. These findings suggest that abrogation of stromal macrophages by injection with CL2MDP-liposome impairs the splenic microenvironment for erythopoiesis induced by hypoxic stress, and this may be an excellent experimental model for further characterization of the in vivo role of splenic macrophages in erythropoiesis.
AB - Erythropoiesis occurs in the presence of erythropoietin (EPO) without macrophages in vitro. In hematopoietic tissues, however, erythroid cells associate closely with stromal macrophages, forming erythroblastic islands via interactions with adhesion molecues. To elucidate the role of macrophages in erythropoiesis, we selectively abrogated stromal macrophages of splenic red pulp of phlebotomized mice by injection with dichloromethylene disphosphohate encapsulated in multilamellar liposomes (CL2MDP-liposome). In the spleen, no erythropoietic activity occurred until 5 days after the treatment. Colony assay revealed that the erthropoiesis was suppressed at the level of CFU-E. The splenic erythropoietic activity gradually developed from day 6 after the treatment, when F4/80+ macrophages began to appear in the red pulp. EPO mRNA was expressed in kidney but not in liver or spleen of phlebotomized mice injected with CL2MDP-liposome, and the serum EPO concentration in these mice was higher than that in phlebotomized mice. These findings suggest that abrogation of stromal macrophages by injection with CL2MDP-liposome impairs the splenic microenvironment for erythopoiesis induced by hypoxic stress, and this may be an excellent experimental model for further characterization of the in vivo role of splenic macrophages in erythropoiesis.
KW - Adhesion
KW - Erythroblasts
KW - Erythropoietin
KW - Hypoxia
KW - Microenvironment
JO - Journal of Leukocyte Biology
JF - Journal of Leukocyte Biology


Written and to be directed by A.T. Sayre (write/director of the award winning feature film 'Whatever Makes You Happy'), 'The Song the Zombie Sang' is planned as a low budget independent production, to be funded primarily (if not exclusively) by its Indiegogo crowd funding campaign, in partnership with Indiereign. Principal photography will occur at the end of the campaign, between late in 2014 and the spring of 2015.
The film evolves around Nils Bekh, who was once the world's greatest living musician and composer. With the ultracembalo, a musical system that uses a neural interface and data gloves, he created and performed some of the greatest compositions in history. He died an icon, his name and music deeply etched into the story of humanity.
But Bekh the musician continues on. He is reanimated nightly to perform for an adoring public. In death, Nils Bekh has the immortality that before only a man's work could achieve without him.
Outside the circles of wealth and privilege that flock to his concerts is Rhoda, a gifted young music student in the infancy of her own career. Driven to the point of obsession, she has few friends other than her doting and supportive boyfriend, who is determined to stand by her. Most everyone else sees her as arrogant and self-absorbed. But Rhoda doesn't pay them any attention; both slave and master to her art, the only things that matter to her are the melodies and compositions that constantly swirl about in her head.
David Teich spoke with writer/director A.T. Sayre about the crowdfunding, the origins of zombie folklore, and artists' all-consuming drive for immortality.
Learn more about "The Song the Zombie Sang" at the film's website.
What drew you to the short story by Harlan Ellison and Robert Silverberg? And what does this story bring to the zombie genre that we haven't seen onscreen before? The title notwithstanding, is this really a traditional "zombie" premise?
I have been infatuated with this story ever since I first read it in high school more than twenty years ago. Whenever I read something, I always play with how I would make a film of it–I've been doing it for as long as I can remember. A vast majority of the time it never lasts long, often before I'm finished reading. The way the prose works is unfilmable, it's too long, or just otherwise doesn't interest me for some reason. But this one has stuck with me for over half my life. The ideas I've seen in it about struggling with the pursuit of a lofty goal, from two very different ends of the same road, speaks to something very personal to me. I think it would to anyone with ambition.
But no, it definitely isn't a traditional zombie film, or more accurately the kind of 'zombie' story as they have been since George Romero started the current cultural image of them. When this story was first published in 1970, 'Night of the Living Dead' had only come out about two years earlier. So zombies weren't in the social consciousness anything like they are today.
I suppose what this film is bringing to the zombie genre is a bit of a throwback to an older, more historical take on zombies. Originally zombies were not monsters, they were victims. Zombies originated predominantly as a part of Haitian folklore, basically as people who were possessed after they died to come back into a lethargic, walking coma type existence to mindlessly do the bidding of whoever summoned them. They were still a frightening thing, but also very tragic–the comparisons to slavery are pretty clear. And as much as I love the modern depiction of zombies metaphorically and otherwise, in this story the older way of seeing zombies is more appropriate.
What does this story have to say about musicians, or artists in general, and their drive for immortality? More broadly, what are some of the themes and metaphors in this story that you find particularly fascinating?
The drive for artists is a singular force, and oftentimes to the detriment of almost everything else in life. Family, friends, loved ones, all that can take second fiddle to the pursuit of greatness. It's never easy, even if you manage to reach the lofty goals that you aspire to. Yet, knowing all this, you do it anyway, because it's an inner drive to succeed that you don't have as much control over as you may like to think. I don't think this kind of drive is limited to artistic or creative pursuits. I really see it as something that anyone from any walk of life who has a lofty goal can understand.
This is the major theme from the story as I've adapted it. Maybe its not the most uplifting idea, but it's true and that's what's more important. The other idea in this that I am a little tickled over is the instrument used, the ultracembalo. I've altered that from the original story into a set of gloves connected through a neural interface directly into the performer, to create music through mental control and hand gestures. Which I think is going to be my favorite image in the story: a solitary person standing on a stage, literally creating an entire symphony–magic–with a flick of their fingertips.
Can you describe Nils Bekh and Rhoda, your two main characters? What are they like? And, if it's possible to answer without spoiling too much, can you describe what their dynamic is like, and why their meeting is such a pivotal moment for them?
Rhoda is a young prodigy, extremely talented, very focused on her goal to be a great composer, and, just as I said, at the expense of everything else in her life. She's very introverted, and socially disinterested. Many people see her as arrogant, but I'd say it's more of a by product of her single-minded obsession with the music than any sense of ego.
Bekh is far more calm, serene even. He has already had his whole life full of accomplishments and landmark achievements, so he has nothing left to prove, nothing to strive for. His whole existence is to perform on stage, and beyond that he has nothing. He exists in a sort of limbo, and seems resigned to it, going through the motions with a certain kind of stoicism.
Even as different as these two people are, they have the drive for music in common, being on the same path of life, just at different ends of it. And that commonality is far more binding for them. Their meeting is indeed very pivotal for both of them–Rhoda confronts a legend, in some ways the embodiment of what she is trying to achieve, and learns some hard truths about it that she will have to deal with. And for Bekh, he finally gets the chance to free himself from his existence and to be understood by someone he considers a kindred spirit. That's actually my favorite part of their whole interaction–the way they both have such a level of mutual respect for each other, almost like equals talking shop.
What has been your approach to crowdfunding? How does raising money for a sci-fi/zombie film like The Song the Zombie Sang differ from raising money for a non-genre film—for, instance, a drama like your film Whatever Makes You Happy?
I have been very active on Twitter and Facebook, as well as reaching out to as many media entities such as yours to try and get as much coverage as I can for the campaign and the project in general. Earlier the project was voted Indiewire's Project of the week, and right now there is voting going on for Project of the Month. I hope everyone who's reading will consider voting for my project.
It's a long slog, certainly, and the results have been mixed. I funded my previous feature film on my own, so this project is my first foray into crowdfunding. And I have learned a lot from this first attempt. While it doesn't look right now like this particular goal is going to be reached, I have taken away a great deal of understanding to take into my next attempt at this. And whatever resources I do get on this campaign will be utilized into that one. I'm not giving up on this project, it's meant far too much to me for far too long, and I am going to keep working on getting this film made till I'm told I can't do it anymore. And then for probably a month or two after that.
What have been some of your challenges in turning this story into a film? What are some of the tougher adaptation choices you've had to make?
It really wasn't as much of a challenge as I thought it would be, if I am going to be totally honest. I went from first scribbles in a notebook to a finished first draft of the script in just under six weeks, and there weren't a great deal of problems or sticking points in the whole process. I think that has a lot to do with having thought about writing this script and making this movie for so long that I had already worked out all the kinks and problems ages ago in my head. Deciding to focus on Rhoda and her struggles more than Bekh, changing the nature of the ultracembalo, the characters to expand and add to the story–it all was just clearly the way to do it. I've never had as easy a time writing anything in my entire life. Perhaps it's fate.
Have you even begun to think about your post-completion game plan? Do you plan to take on the festival circuit? Are there any specific audiences or demographics you're particularly hoping to reach?
I'm trying not to think too much about those plans, as I need to focus on getting the movie made first. But I have a general understanding of how I plan to proceed at that point. Running the festival circuit is what I see as clearly the first move after the film is finished. Then looking for a legitimate distributor of some kind. If I can't get one, then self-distribution of some kind. Like I said, I really haven't spent a great deal of time thinking about this yet, because I have a long, long ways to go till I get to that point. And I want to keep it open for now as to how to proceed, so I don't risk comitting myself to any particular approach before I have to.