text:
\section{Introduction}
This paper fleshes out a remark made in \cite{helm:17}.  That paper establishes under very weak conditions the existence of an optimal $(s_*,S_*)$ ordering policy for inventory management in one-dimensional diffusion models.  The problem under consideration involves minimizing the long-term average expected cost of an inventory process $X$ taking values in some interval ${\cal E} \subseteq \mathbb R$ satisfying
\begin{equation} \label{controlled-dyn}
X(t) = x_0 + \int_0^t \mu(X(s))\, ds + \int_0^t \sigma(X(s))\, dW(s) + \sum_{k=1}^\infty I_{\{\tau_k \leq t\}} Y_k, \qquad t\geq 0,
\end{equation}
in which $x_0 \in {\cal E}$ denotes the initial inventory level, $(\tau,Y) = \{(\tau_k,Y_k): k \in \mathbb N\}$ is an ordering policy in which $\tau_k$ denotes the time of the $k^{\mbox{\footnotesize th}}$ order and $Y_k \geq 0$ is the size of the order.  The drift rate given by $\mu$ and the diffusion coefficient $\sigma$ are assumed to be continuous, with $\sigma$ non-degenerate; $W$ is a standard Brownian motion process.  Note the condition $Y_k \geq 0$ means that this model only allows the manager to increase the inventory through ordering; it does not provide him or her the ability to decrease inventory.  Define the set ${\cal R} = \{(y,z) \in {\cal E}^2: y < z\}$ in which $y$ represents the pre-order inventory level and $z$ is the post-order level and set $\overline{\cal R} = \{(y,z) \in {\cal E}^2: y \leq z\}$.  The difference between ${\cal R}$ and $\overline{\cal R}$ is that $\overline{\cal R}$ includes orders of size $0$ whereas ${\cal R}$ does not.

The long-term average criterion is
\begin{equation} \label{lta-cost}
J_0(\tau,Y) = \limsup_{t\rightarrow \infty} \mbox{$\frac{1}{t}$} \mathbb E\left[\int_0^t c_0(X(s))\, ds + \sum_{k=1}^\infty I_{\{\tau_k \leq t\}} c_1(X(\tau_k-),X(\tau_k))\right]
\end{equation}
in which $c_0\geq 0$ is the holding/back-order cost rate function and $c_1 \geq k_1 > 0$ denotes the cost for ordering.  Here, $k_1$ represents the fixed cost per order while $c_1(y,z)$ gives the total ordering cost for raising the inventory level from $y$ to $z$.  Notice, in particular, the assumption $c_1 \geq k_1$ means that actively ordering nothing incurs a cost of at least $k_1$ so differs from not ordering which has no cost.  The paper \cite{helm:17} also imposes minor additional conditions such as continuity on $c_0$ and $c_1$ in Condition 2.3.

The approach taken in \cite{helm:17} is to capture the expected behaviour of the inventory process $X$ associated with an ordering policy $(\tau,Y)$ and its costs through average expected occupation and average expected ordering measures.  These are defined for each $t > 0$ by
\begin{equation} \label{mus-t-def}
\begin{array}{rcll}
\mu_{0,t}(\Gamma_0) &=& \displaystyle \mbox{$\frac{1}{t}$} \mathbb E\left[\int_0^t I_{\Gamma_0}(X(s))\, ds\right], & \quad \Gamma_0 \in {\cal B}({\cal E}), \rule[-15pt]{0pt}{15pt} \\
\mu_{1,t}(\Gamma_1)  &=& \displaystyle \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{k=1}^\infty I_{\{\tau_k \leq t\}} I_{\Gamma_1}(X(\tau_k-),X(\tau_k))\right], & \quad \Gamma_1 \in {\cal B}(\overline{\cal R}).
\end{array}
\end{equation}
The long-term average cost is expressed using $\mu_{0,t}$ and $\mu_{1,t}$ as
$$J_0(\tau,Y) = \limsup_{t\rightarrow \infty} \left(\int_{\cal E} c_0(x)\, \mu_{0,t}(dx) + \int_{\overline{\cal R}} c_1(y,z)\, \mu_{1,t}(dy\times dz)\right).$$
The paper then employs weak convergence techniques to more easily show optimality of an $(s,S)$ policy in the general class of admissible policies.

One of the benefits of this approach is the simplicity of establishing tightness of $\{\mu_{0,t}\}$ as $t\rightarrow \infty$.  For example, when ${\cal E}$ is a bounded interval, viewing each $\mu_{0,t}$ as a measure on the closure $\overline{\cal E}$ of ${\cal E}$ immediately implies that $\{\mu_{0,t}\}$ is tight so there will exist limiting measures as $t\rightarrow \infty$.  When ${\cal E}$ is unbounded, it is natural to assume that $c_0$ converges to $\infty$ at each infinite boundary; this means that ``an infinite holding cost rate occurs when an infinite amount of inventory is either present or back-ordered.''  Under this assumption of inf-compactness on $c_0$, Proposition~3.3 of \cite{helm:17} shows that $\{\mu_{0,t}\}$ is tight whenever $J_0(\tau,Y) < \infty$; in this paper, \propref{mu0t-tight} establishes the same result for our particular model and ordering policy.

Imposing inf-compactness on $c_1$ would similarly imply $\{\mu_{1,t}\}$ is tight whenever $J_0(\tau,Y) < \infty$.  However, the assumption of inf-compactness is not natural for the ordering cost function $c_1$.  For example, the most commonly studied ordering cost function is $c_1(y,z) = k_1 + k_2(z-y)$ for $(y,z) \in {\cal R}$ in which the fixed cost of $k_1$ is charged along with a cost that is proportional to the size of the order.  When ${\cal E} = \mathbb R$, for example, the set $\{(y,z) \in {\cal R}: c_1(y,z) \leq K\}$ is not compact for any $K < \infty$.  In addition, if one were to allow efficiencies of scale by considering a concave function $H: \mathbb R_+ \rightarrow \mathbb R_+$ and taking $c_1(y,z) = k_1 + H(z-y)$, again the set $\{(y,z): c_1(y,z) \leq K\}$ is not compact for any $K < \infty$.

It would seem that a sufficient condition to ensure $\{\mu_{1,t}\}$ is tight ought to be that $J_0(\tau,Y) < \infty$.  Intuitively, placing many orders resulting in unbounded inventory levels means that $X$ ought to spend large amounts of time in regions where the function $c_0$ imposes a heavy penalty leading to an unbounded expected holding cost and hence $J_0(\tau,Y) = \infty$.

The purpose of this paper is to illustrate that this intuition is incorrect.  To illustrate the idea simply, we begin by considering a deterministic inventory model having constant demand rate and examine a particular ordering policy.  The analysis is fairly straightforward.  The second example is very similar but perturbs the deterministic demand by a Brownian motion process.  The ensuing drifted Brownian motion process requires more sophisticated analysis.

\section{Deterministic Demand Rate Example}
The aim of this example is to illustrate that the tightness of the average ordering measures cannot be easily characterized for the most commonly studied ordering cost function. The example consists of a deterministic inventory model so to distinguish this model from the stochastic models in the rest of the paper, lower case notation will be used.

An ordering policy $(\mathbf{t},\mathbf{y})$ consists of a sequence of times $\mathbf{t}=\{t_k: k\in \mathbb N\}$ and corresponding sequence of order sizes $\mathbf{y}=\{y_k: k\in \mathbb N\}$.  The inventory process $x$ taking values in $\mathbb R$ satisfies
\begin{equation} \label{determ-dyn}
x(t) = -t + \sum_{k=1}^\infty I_{\{t_k \leq t\}} y_k;
\end{equation}
notice that the initial inventory is $x(0)=0$ and demand is at a constant rate of $1$ unit per unit of time.  The cost structure consists of the holding/back-order rate function $c_0(x) = 2|x|$, the ordering cost function $c_1(y,z) = k_1 + (z-y)$ and the long-term average cost is
\begin{equation} \label{determ-lta-cost}
j_0(\mathbf{t},\mathbf{y}) = \limsup_{t\rightarrow \infty} \mbox{$\frac{1}{t}$} \left(\int_0^t c_0(x(s))\, ds + \sum_{k=1}^\infty I_{\{t_k \leq t\}} c_1(x(t_k-),x(t_k))\right).
\end{equation}
For each $t > 0$, the average occupation and and average ordering measures, $m_{0,t}$ and $m_{1,t}$ respectively, are defined by
\begin{equation} \label{determ-mus-t-def}
\begin{array}{rcll}
m_{0,t}(\Gamma_0) &=& \displaystyle \mbox{$\frac{1}{t}$} \int_0^t I_{\Gamma_0}(x(s))\, ds, & \quad \Gamma_0 \in {\cal B}({\cal E}), \rule[-15pt]{0pt}{15pt} \\
m_{1,t}(\Gamma_1) &=& \displaystyle \mbox{$\frac{1}{t}$} \sum_{k=1}^\infty I_{\{t_k \leq t\}} I_{\Gamma_1}(x(t_k-),x(t_k)), & \quad \Gamma_1 \in {\cal B}(\overline{\cal R}).
\end{array}
\end{equation}

The goal of this example is to identify a particular ordering policy $(\mathbf{t},\mathbf{y})$ with $j_0(\mathbf{t},\mathbf{y}) < \infty$, $\{m_{0,t}\}$ tight as $t\rightarrow \infty$, such that  the collection $\{m_{1,t}\}$ is not tight; that is, show that for some $\epsilon > 0$ and for any compact set $\Gamma \subset {\cal R}$, there is some $t$ such that $m_{1,t}(\Gamma^c) > \epsilon$.

We begin with a description of the ordering policy $(\mathbf{t},\mathbf{y})$ which runs in cycles, each of which is composed of two phases.  For cycle $i=1,2,3,\ldots$, Phase 1 consists of using the $(0,1)$-ordering policy a total of $2^{i-1}$ times.  Phase 2 is quite non-standard involving  a single $(0,2^{(i-1)/2})$-ordering policy followed immediately by using the $(2^{(i-1)/2},2^{(i-1)/2})$-ordering policy $2^{i-1}$ times; this phase consists of placing many orders of size $0$ in a row that incur the fixed cost but no additional cost in addition to the single order of higher amount.

Rather than use a single index for the order times and amounts, we employ a double index $(i,j)$ in which $i$ denotes the cycle and $j$ gives the number of the order within cycle $i$.

\begin{defn}[{\em The Policy} $(\mathbf{t},\mathbf{y})$] \label{determ-ty-def}
For the first cycle, define
$$\left\{\begin{array}{rcl} t_{1,1}&=&0, \\ y_{1,1}&=&1, \end{array}\right. \quad \mbox{and} \quad \left\{\begin{array}{rcl} t_{1,2}&=&1, \\ y_{1,2}&=&1, \end{array} \right. \quad \left\{\begin{array}{rcl} t_{1,3}&=&t_{1,2}=1 \\ y_{1,3}&=&0.\end{array} \right.$$
Observe that the first order occurs immediately, making $x(0+) = 1$, the inventory level then declines at rate 1 until it hits $\{0\}$ at time $1$ completing the first phase.  Phase 2 begins with an order up to level $2^{(1-1)/2}=1$ at time $1$, immediately followed at time $1$ by an order of size $0$; the process $x$ then decreases and hits $0$ at time $2$, completing Phase 2 of the cycle 1.

For cycle $i=2,3,4,\ldots$, define
$$\left\{\begin{array}{rcl} t_{i,1}&=&t_{i-1,2^i-1} + 2^{(i-1)/2}, \\ y_{i,1}&=&1, \end{array}\right. \quad \mbox{and}\quad \left\{\begin{array}{rcl} t_{i,j}&=& t_{i,j-1}+1,\\ y_{i,j}&=& 1, \end{array} \right. \mbox{ for } j=2, \ldots, 2^{i-1},$$
completing Phase 1 of cycle $i$.  Then define the orders during Phase 2 by
$$\left\{\begin{array}{rcl} t_{i,2^{i-1}+1}&=& t_{i,2^{i-1}}+1, \\ y_{i,2^{i-1}+1}&=&2^{(i-1)/2}, \end{array} \right. \quad \mbox{and} \quad \left\{\begin{array}{rcl} t_{i,2^{i-1}+1+j}&=&t_{i,2^{i-1}+1}, \\ y_{i,2^{i-1}+1+j}&=&0, \end{array} \right. \mbox{ for } j=1,\ldots,2^{i-1};$$
cycle $i$ ends when $x$ hits $\{0\}$ at time $t_{i+1,1}$.  Notice that as $i$ increases, the {\em number}\/ of $(0,1)$-ordering sub-cycles in Phase 1 increases exponentially by a factor of 2, followed in Phase 2 by a single order of (base $\sqrt{2}$) exponentially increasing {\em size}\/ which is immediately followed by a (base $2$) exponentially increasing {\em number}\/ of orders of size $0$.
\end{defn}

\begin{rem} \label{no-0-size-order}
When formulating the ordering costs, traditionally no distinction is made between not ordering and ordering nothing, with no cost incurred in either case.  In contrast, the formulation in this paper has orders of size $0$ incur the fixed cost $k_1$.  The policy $(\mathbf{t},\mathbf{y})$ uses $0$-size orders to create non-trivial masses in the average ordering measure $m_{1,t}$ at arbitrarily large values on the diagonal $z=y$ without affecting the length of Phase 2, provided $t$ is sufficiently large.  Under the traditional formulation, it is possible to place many orders of suitably small sizes, resulting in similar masses in neighbourhoods near the diagonal at arbitrary distances from the origin (with $t$ large), such that the length of Phase 2 is barely increased.  The analysis is essentially the same as in this manuscript but requires more careful bookkeeping without affecting the limiting results.  The costly $0$-size orders considerably simplify the computations.
\end{rem}

\begin{prop}
The long-term average cost of the policy $(\mathbf{t},\mathbf{y})$ of \defref{determ-ty-def} satisfies
$$j_0(\mathbf{t},\mathbf{y}) < \infty.$$
\end{prop}

\begin{proof}
It is helpful to determine both the length of each cycle $i$ and the associated cost of cycle $i$.  Note that Phase 1 consists of $2^{i-1}$ $(0,1)$-sub-cycles, each of duration $1$ unit of time.  The length of Phase 2 is determined by the single $(0,2^{(i-1)/2})$-policy lasting $2^{(i-1)/2}$ units of time.  Thus, the length of cycle $i$ is
$$t_{i+1,1} -t_{i,1} = 2^{i-1} + 2^{(i-1)/2}.$$
In examining the costs incurred during cycle $i$, consider first the holding costs.  During Phase 1 by construction, $x(t) = 1-t+t_{i,j}$ for $t$ in the unit time interval $(t_{i,j},t_{i,j+1})$ for $j=1,\ldots,2^{i-1}$.  Phase 2 occupies the interval $(t_{i,2^{i-1}+1},t_{i+1,1})=(t_{i,2^{i-1}+1},t_{i,2^{i-1}+1}+2^{(i-1)/2})$ with $x(t) = 2^{(i-1)/2} - t + t_{i,2^{i-1}+1}$.  Thus, the holding cost during cycle $i$ is
\begin{eqnarray} \label{determ-cycle-i-holding-cost} \nonumber
\int_{t_{i,1}}^{t_{i+1,1}} c_0(x(t))\, dt &=& \displaystyle \sum_{j=1}^{2^{i-1}} \int_{t_{i,j}}^{t_{i,j+1}} 2(1-t+t_{i,j})\, dt + \int_{t_{i,2^{i-1}+1}}^{t_{i+1,1}} 2(2^{(i-1)/2}-t+t_{i,2^{i-1}+1})\, dt \\
&=& 2^{i-1} + 2^{i-1} = 2^i. \rule{0pt}{18pt}
\end{eqnarray}
During cycle $i$, there are $2^{i-1}$ orders of size 1, one order of size $2^{(i-1)/2}$ and $2^{i-1}$ orders of size $0$, resulting in a total ordering cost of
\begin{equation} \label{determ-cycle-i-order-cost}
\begin{array}{rcl} \displaystyle
\sum_{j=1}^{2^i+1} c_1(x(t_{i,j}-),x(t_{i,j})) &=& (k_1+1) 2^{i-1} + (k_1 + 2^{(i-1)/2}) + k_1 2^{i-1} \\
&=& (2k_1+1) 2^{i-1} + 2^{(i-1)/2} + k_1.
\end{array}
\end{equation}
Obviously, the total cost during cycle $i$ is the sum of \eqref{determ-cycle-i-holding-cost} and \eqref{determ-cycle-i-order-cost}.

It will be helpful to determine the total elapsed time during the first $n$ cycles for $n=1,2,3,\ldots$ as well as the total cost during these cycles.  The total time is
$$t_{n+1,1} = \sum_{i=1}^n (2^{i-1} + 2^{(i-1)/2}) = 2^n + \frac{2^{n/2}-1}{2^{1/2}-1} - 1$$
while the total cost of the first $n$ cycles is
\begin{eqnarray*}
\lefteqn{\int_0^{t_{n+1,1}} c_0(x(t))\, dt + \sum_{i=1}^n \sum_{j=1}^{2^i+1} c_1(x(t_{i,j}-),x(t_{i,j}))} \\
&\qquad =& \sum_{i=1}^n (2^i + (2k_1+1) 2^{i-1} + 2^{(i-1)/2} + k_1) \\
&\qquad =& (k_1+1)2^{n+1} + 2^n + \frac{2^{n/2}-1}{2^{1/2}-1} + k_1n - 2k_1 - 3.
\end{eqnarray*}

We now turn to an examination of the average cost over the interval $[0,t]$.  For each $t \geq 0$, define $i(t)$ to satisfy $t_{i(t),1} \leq t < t_{i(t)+1,1}$; that is, the time $t$ occurs during the $i(t)^{\mbox{\footnotesize th}}$ cycle.  Then
\begin{eqnarray} \label{determ-avg-cost-bd} \nonumber
\lefteqn{\frac{1}{t} \left(\int_0^t c_0(x(s))\, ds + \sum_{i=1}^{\infty}\sum_{j=1}^{2^i+2} I_{\{t_{i,j} \leq t\}} c_1(x(t_{i,j}-),x(t_{i,j}))\right)} \\  \nonumber
&\qquad \leq& \frac{1}{t} \left(\int_0^{t_{i(t)+1,1}} c_0(x(s))\, ds + \sum_{i=1}^{\infty}\sum_{j=1}^{2^i+2} I_{\{t_{i,j} \leq t_{i(t)+1,1}\}} c_1(x(t_{i,j}-),x(t_{i,j}))\right) \\
&\qquad =& \frac{t_{i(t)+1,1}}{t} \cdot \frac{1}{t_{i(t)+1,1}} \sum_{i=1}^{i(t)+1} (2^i + (2k_1+1) 2^{i-1} + 2^{(i-1)/2} + k_1) \\ \nonumber
&\qquad \leq& \frac{t_{i(t)+1,1}}{t_{i(t),1}} \cdot \frac{(k_1+1)2^{i(t)+2} + 2^{i(t)+1} + \frac{2^{(i(t)+1)/2}-1}{2^{1/2}-1} + k_1 (i(t)+1) - 2k_1 - 3}{2^{i(t)+1} + \frac{2^{(i(t)+1)/2}-1}{2^{1/2}-1} - 1} \\ \nonumber
&\qquad \leq& \frac{t_{i(t)+1,1}}{t_{i(t),1}} \cdot \frac{2k_1+3 + \frac{2^{-(i(t)+1)/2}-2^{-(i(t)+1)}}{2^{1/2}-1} + (k_1i(t) - k_1 - 3)2^{-(i(t)+1)}}{1 + \frac{2^{-(i(t)+1)/2}-2^{-(i(t)+1)}}{2^{1/2}-1} - 2^{-(i(t)+1)}} .
\end{eqnarray}
Using the expression for $t_{n,1}$ yields
$$\frac{t_{i(t)+1,1}}{t_{i(t),1}} = \frac{2^{i(t)+1} + \frac{2^{(i(t)+1)/2}-1}{2^{1/2}-1} - 1}{2^{i(t)} + \frac{2^{i(t)/2}-1}{2^{1/2}-1} - 1} = \frac{2 + \frac{2^{(1-i(t))/2}-2^{-i(t)}}{2^{1/2}-1} - 2^{-i(t)}}{1 + \frac{2^{-i(t)/2}-1}{2^{1/2}-1} - 2^{-i(t)}}$$
so $\lim_{t\rightarrow \infty} \frac{t_{i(t),1}}{t_{i(t)-1,1}} = 2$.  As a result, there exists some $T < \infty$ such that for all $t \geq T$, $\frac{t_{i(t)+1,1}}{t_{i(t),1}} \leq 3$ and therefore
$$\limsup_{t\rightarrow \infty} \frac{1}{t} \left(\int_0^t c_0(x(s))\, ds + \sum_{i=1}^\infty \sum_{j=1}^{2^i+2} I_{\{t_{i,j} \leq t\}} c_1(x(t_{i,j}-),x(t_{i,j}))\right) \leq 6k_1 + 9 < \infty,$$
completing the proof.
\end{proof}

Since the holding cost rate function $c_0$ is inf-compact and $j_0(\mathbf{t},\mathbf{y}) < \infty$, the next result is straightforward; we provide the proof of the result for the stochastic example in \propref{mu0t-tight}.

\begin{cor}
For the policy $(\mathbf{t},\mathbf{y})$ of \defref{determ-ty-def}, define the occupation measures $\{m_{0,t}\}$ by \eqref{mus-t-def}.  Then $\{m_{0.t}\}$ is tight as $t\rightarrow \infty$; that is, for each $\epsilon > 0$, there exists some compact set $\Gamma \subset \mathbb R_+$ and $T < \infty$ such that $m_{0,t}(\Gamma^c) < \epsilon$ for all $t \geq T$.
\end{cor}

Now we get to the point of the example.

\begin{prop}
For the policy $(\mathbf{t},\mathbf{y})$ of \defref{determ-ty-def}, the collection of ordering measures $\{m_{1,t}: t>0\}$ defined by \eqref{determ-mus-t-def} is not tight.
\end{prop}

\begin{proof}
Consider the measure $m_{1,t_{n+1,1}}$ for each $n\in \mathbb N$.  We note that there are $2^{n-1}$ orders of size $0$ at time $t_{n,2^{n-1}+1}<t_{n+1,1}$ when the process $x$ takes value $2^{(n-1)/2}$.  Thus,
$$m_{1,t_n}(\{(2^{(n-1)/2},2^{(n-1)/2})\}) = \frac{2^{n-1}}{2^n + \frac{2^{n/2}-1}{2^{1/2}-1} - 1} = \frac{1}{2}\cdot \frac{1}{1+\frac{2^{-(n/2)-1}-2^{-n}}{2^{1/2}-1} - 2^{-n}}.$$
Thus for any $\epsilon < \frac{1}{2}$, for any compact set $\Gamma \subset \overline{\cal R} = \{(y,z)\in \mathbb R^2: y\leq z\}$, there is an $n$ sufficiently large so that $(2^{(n-1)/2},2^{(n-1)/2}) \in \Gamma^c$ and $m_{1,t_n}(\{(2^{(n-1)/2},2^{(n-1)/2})\}) > \epsilon$.
\end{proof}

\section{Drifted Brownian Motion Example}
The second example considers a stochastic variation on the deterministic demand model.  The inventory level $X_0$ {\em without orders}\/ is given by the drifted Brownian motion process with
\begin{equation} \label{dbm-dyn}
X_0(t) = W(t) - t, \qquad t\geq 0;
\end{equation}
note that the initial inventory level is $x_0=0$ and the drift and diffusion coefficients are $\mu=-1$ and $\sigma=1$.  The cost functions remain as
$$c_0(x) = 2|x|, \quad \forall x \in \mathbb R, \qquad \mbox{and}\qquad c_1(y,z) = k_1 + (z-y), \quad \forall (y,z) \in \overline{\cal R}.$$

The ordering policy $(\tau,Y)$ is essentially the same as in the deterministic example, though adapted to the stochatic environment.  The policy runs in cycles, each of which is composed of two phases.  For cycle $i=1,2,3,\ldots$, Phase 1 consists of using the $(0,1)$-ordering policy a total of $2^{i-1}$ times; the length of each sub-cycle is now a random variable having mean $1$.  Phase 2 again involves a single $(0,2^{(i-1)/2})$-ordering policy followed immediately by using the $(2^{(i-1)/2},2^{(i-1)/2})$-ordering policy $2^{i-1}$ times.

The formal description of the ordering policy is now given; as in the deterministic example, we employ a double subscript notation.

\begin{defn}[{\em The Policy}\/ $(\tau,Y)$] \label{stoch-tauY-def}
For cycle $i=1$, define
$$\left\{\begin{array}{rcl}
\tau_{1,1} &=& 0, \\ Y_{1,1} &=& 1, \end{array} \right. \qquad \mbox{and} \qquad
\left\{\begin{array}{rcl}
\tau_{1,2} &=& \inf\{t\geq \tau_{1,1}: X(t) = 0\}, \\ Y_{1,2} &=& 1, \end{array} \right. \qquad
\left\{\begin{array}{rcl}
\tau_{1,3} &=& \tau_{1,2}, \\ Y_{1,3} &=& 0, \end{array} \right.$$
and for cycle $i = 2,3,4,\ldots$, define the orders in Phase 1 to be
$$\begin{array}{ll}
\left\{\begin{array}{rcl}
\tau_{i,1} &=& \inf\{t\geq \tau_{i-1,2^i+1}: X(t) = 0\}, \\ Y_{i,1} &=& 1, \end{array} \right. & \rule[-18pt]{0pt}{12pt} \\
\left\{\begin{array}{rcl}
\tau_{i,j} &=& \inf\{t\geq \tau_{i,j-1}: X(t) = 0\}, \\ Y_{i,j} &=& 1, \end{array} \right. & \quad j=2, \ldots, {2^{i-1}}, \rule[-18pt]{0pt}{12pt}
\end{array} $$
and the orders in Phase 2 by
$$\begin{array}{ll}
\left\{\begin{array}{rcl}
\tau_{i,2^{i-1}+1} &=& \inf\{t\geq \tau_{i,2^{i-1}}: X(t) = 0\}, \\ Y_{i,2^{i-1}+1} &=& 2^{(i-1)/2}, \end{array} \right. &  \rule[-18pt]{0pt}{12pt} \\
\left\{\begin{array}{rcl}
\tau_{i,j} &=& \tau_{i,2^{i-1}+1} \\
Y_{i,j} &=& 0, \end{array}\right. & j = 2^{i-1}+2, \ldots, 2^i+1.
\end{array} $$
\end{defn}

The same observation about $0$-size orders as in \remref{no-0-size-order} holds for the stochastic model as well.  The main result of the paper can now be stated.

\begin{thm}
For the drifted Brownian motion inventory model, let $(\tau,Y)$ be the ordering policy of \defref{stoch-tauY-def}, $X$ be the resulting inventory process, and $\{\mu_{0,t}\}$ and $\{\mu_{1,t}\}$, respectively, be the corresponding average expected occupation and ordering measures defined in \eqref{mus-t-def}.  Then
\begin{description}
\item[(a)] $J_0(\tau,Y) < \infty$;
\item[(b)] $\{\mu_{0,t}: t > 0\}$ is tight as $t\rightarrow \infty$; and
\item[(c)] $\{\mu_{1,t}: t > 0\}$ is not tight.
\end{description}
\end{thm}

\begin{proof}
This theorem is proven in pieces.  \propref{lta-holding-costs} shows that the long-term average holding costs are bounded.  In light of this proposition, \propref{mu0t-tight} then establishes the tightness of the average expected occupation measures $\{\mu_{0,t}\}$ as $t\rightarrow \infty$.  Finally, \propref{ordering-measure-results} shows both that the long-term average ordering costs are finite and that the average expected ordering measures $\{\mu_{1,t}\}$ are not tight.
\end{proof}

Our analysis depends on a careful construction of the inventory process $X$ under the ordering policy $(\tau,Y)$ of \defref{stoch-tauY-def}. Independent copies of the diffusion $X_0$ of \eqref{dbm-dyn} are pieced together at the jump times (see, e.g., the appendix of \cite{chri:14} for such a construction) with the implication that the various ordering sub-cycles are independent.

The initial analysis examines the long-term average cost of the $(\tau,Y)$ policy.  It begins by focusing on the holding costs.  Notice that the only orders which affect the length of cycle $i$ are the $2^{i-1}$ times that the $(0,1)$ policy is used and the one time that the $(0,2^{(i-1)/2})$ policy occurs; the $2^{i-1}$ times that orders of size $0$ are placed do not change the state of the inventory or lengthen the sub-cycles so have no affect on the holding costs.  Thus it is sufficient to restrict the analysis solely to the non-zero orders.

Let ${\cal S}:=\{\sigma_1, \sigma_2,\sigma_3, \ldots\}$ denote the times of the non-zero orders.  Let $i\in \mathbb N$ denote the cycle and $j\in \{1,2,\ldots,2^{i-1}+1\}$ be the number of the non-zero order within cycle $i$.  Then observe that $\sigma_n = \tau_{i,j}$ where $n=2^{i-1}+i+j-2$.  The ensuing computations are simplified by a shift in the index for $j$.  For $i\geq 2$, define $\tau_{i,0} = \tau_{i-1,2^{i-1}+1}$ so that order number ``zero'' of the $i^{\mbox{\footnotesize th}}$ cycle is the last non-zero order, in fact the large order, of cycle $(i-1)$.

Our first result gives a strong law of large numbers result for the cycle lengths.

\begin{prop} \label{slln}
Let $(\tau,Y)$ be the ordering policy of \defref{stoch-tauY-def} and ${\cal S}$ be the times of the non-zero orders.  Then
\begin{equation} \label{slln-sigma-n}
\lim_{n\rightarrow \infty} \frac{\sigma_n}{n} = 1 \; (a.s.).
\end{equation}
\end{prop}

\begin{proof}
Define the independent (but not identically distributed) random variables
$$\beta_0 = \sigma_1 = 0, \quad \mbox{and} \quad \beta_k = \sigma_{k+1} - \sigma_k, \qquad k\in \mathbb N;$$
thus $\beta_k$ gives the random length of time of the $k^{\mbox{\footnotesize th}}$ inter-order interval.  Then
$$\sigma_n = \sum_{k=0}^{n-1} \beta_k, \qquad n \in \mathbb N.$$
Thus \eqref{slln-sigma-n} has the representation
\begin{equation} \label{representation}
\lim_{n\rightarrow \infty} \mbox{$\frac{1}{n}$} \sum_{k=0}^{n-1} \beta_k = 1,\; (a.s.).
\end{equation}

Notice that, apart from $\beta_0=0$, $\beta_n$ is either the length of a $(0,1)$ sub-cycle or a $(0,2^{(i-1)/2})$ sub-cycle.  More precisely, for each $i\ge 1$, $\beta_{2^{i}+i-1} $ is the length of the cycle arising from the $(0,2^{(i-1)/2})$ sub-cycle and for $n\neq 2^i-1+i$, $\beta_n$ is the length of a $(0,1)$ sub-cycle.  Using the Laplace transform of the hitting time of a drifted Brownian motion process (see Formula 2.0.1 (p.\;295) of \cite{boro:02}), one can determine that
$$\begin{array}{lclclcll}
\mathbb E[\beta_{2^{i-1}+i+j-2}]&=& 1, & \mbox{and} & \mbox{Var}(\beta_{2^{i-1}+i+j-2})&=&1, & \; j=1,2,\ldots, 2^{i-1},\/ i\ge 1, \\
\mathbb E[\beta_{2^{i}+i-1}]&=& 2^{(i-1)/2}, & \mbox{and} & \mbox{Var}(\beta_{2^{i}+i-1})&=& 2^{(i-1)/2}, & \;  i\geq 1.
\end{array}$$
Next observe
\begin{eqnarray*}
\sum_{n=2}^\infty \frac{\mbox{Var}(\beta_n)}{n^2} &=&
\sum_{i=2}^\infty \sum_{j=0}^{2^{i-1}}\frac{\mbox{Var}(\beta_{2^{i-1}+i+j-2})}{(2^{i-1}+i+j-2)^2} \\
&\leq& \sum_{i=1}^\infty \sum_{j=1}^{2^{i-1}} \frac{1}{(2^{i-1}+i+j-2)^2} + \sum_{i=1}^\infty \frac{2^{(i-1)/2}}{(2^{i-1}+i-1)^2} \\
&\leq& \sum_{n=1}^\infty \frac{1}{n^2} + \sum_{i=1}^\infty \frac{1}{2^{3(i-1)/2}} < \infty.
\end{eqnarray*}
By Kolmogorov's Strong Law of Large Numbers (cf.\ Theorem 2 (p.\;389) of \cite{shir:96}), it follows that
$$\lim_{n\rightarrow \infty} \left(\mbox{$\frac{1}{n}$} \sum_{k=1}^n \beta_k - \overline{\mu}_n\right) = 0, \;(a.s.),$$
in which $\overline{\mu}_n = \frac{1}{n} \sum_{k=1}^n \mathbb E[\beta_k]$.  Since $\beta_0=0$, notice that
$$\mbox{$\frac{1}{n}$} \sum_{k=1}^n \beta_k = \mbox{$\frac{n+1}{n}$} \cdot \left(\mbox{$\frac{1}{n+1}$} \sum_{k=0}^n \beta_k\right)$$
so \eqref{representation} holds if $\overline{\mu}_n \rightarrow 1$ as $n\rightarrow \infty$.

We now analyze the convergence of $\overline{\mu}_n$.  Again for $n\geq 2$, write $n=2^{i-1}+i+j-2$ with $i \ge 2$ and $0\leq j \leq 2^{i-1}$.  Notice that for $1 \leq k \leq i-1$, cycle $k$ contains $2^{k-1}$ sub-cycles generated by $(0,1)$ ordering policies having a mean length of $1$ and a single $(0,2^{(k-1)/2})$ sub-cycle with mean length $2^{(k-1)/2}$ while the partial cycle $i$ has $j$ sub-cycles from $(0,1)$,
Thus for $n \geq 2$,
\begin{align*}
\mbox{$\frac{1}{n}$} \sum_{k=1}^n \mathbb E[\beta_k] &

= \frac{1}{2^{i-1}+i+j-2} \left(\sum_{k=1}^{i-1} (2^{k-1} + 2^{(k-1)/2}) + j\right) \\
&= \frac{1}{2^{i-1}+i+j-2} \left(2^{i-1} - 1 +\frac{2^{(i-1)/2}-1}{2^{1/2}-1} + j\right) \\
& = \frac{1 -\frac{1}{2^{i-1}+j} + \frac{1}{2^{1/2}-1} \cdot\frac{2^{(i-1)/2}-1}{2^{i-1}+j} }{1 + \frac{i-2}{2^{i-1}+j}}.
\end{align*}
Obviously we have $$\lim_{i\to\infty}\frac{1}{2^{i-1}+j}=\lim_{i\to\infty}\frac{2^{(i-1)/2}-1}{2^{i-1}+j} =\lim_{i\to\infty} \frac{i-2}{2^{i-1}+j} =0 \text{ for each }j =0, 1, \dots, 2^{i-1}.$$
Therefore it follows that as $n\to\infty$ (and hence   $i\rightarrow \infty$),  $\frac{1}{n} \sum_{k=1}^n \mathbb E[\beta_k] $ converges to 1.
\end{proof}

We now establish a variant of the elementary renewal theorem; the fact that the cycles are not identically distributed means that we cannot simply apply the theorem.

\begin{prop} \label{elementary-renewal}
For $t \geq 0$, let $\displaystyle = \sum_{i=1}^\infty I_{\{\sigma_i \leq t\}} = \max\{n: \sigma_n \leq t\}$ be  the number of orders of positive size by time $t$.  Then
\begin{description}
\item[(a)] $\displaystyle \lim_{t\rightarrow \infty} \frac{N(t)}{t} = 1, \; (a.s.)$; and
\item[(b)] $\displaystyle \lim_{t\rightarrow \infty} \frac{\mathbb E[N(t)]}{t} = 1$.
\end{description}
\end{prop}

\begin{proof}
First by the definition of $N(t)$, $\sigma_{N(t)} \leq t < \sigma_{N(t)+1}$ for each $t \geq 0$.  Thus for each $t \geq 0$,
$$\frac{\sigma_{N(t)}}{N(t)} \leq \frac{t}{N(t)} < \frac{\sigma_{N(t)+1}}{N(t)} = \frac{N(t)+1}{N(t)} \cdot \frac{\sigma_{N(t)+1}}{N(t)+1}.$$
As $t\rightarrow \infty$, \propref{slln} implies first that $N(t) \rightarrow \infty\; (a.s.)$ and then establishes (a).

In order to prove (b), it is necessary to show that the collection $\{\frac{N(t)}{t}: t\geq 1\}$ is uniformly integrable.  By Lemma~3 (p.\ 190) of \cite{shir:96}, it suffices to show that $\sup_{t\geq 1} \mathbb E[(\frac{N(t)}{t})^2] < \infty$.  To establish this result, we compare the counting process $N$ with a similar renewal process $\widetilde N$ connected to a different policy.

Define the ordering policy $(\widetilde\tau,\widetilde Y)$ which always uses the $(0,1)$ policy and let $\{\widetilde{\sigma}_n: n\in\mathbb N\}$ denote the ordering times.  Define the renewal process $\widetilde N$ by
$$\widetilde N(t) = \max\{n: \widetilde\sigma_n \leq t\} = \sum_{i=1}^\infty I_{\{\widetilde\sigma_i \leq t\}}.$$
It then follows that $N(t) \leq \widetilde N(t)$ for each $t \geq 0$ and hence
$$\mathbb E\left[\left(\frac{N(t)}{t}\right)^2\right] \leq \mathbb E\left[\left(\frac{\widetilde N(t)}{t}\right)^2\right].$$
Using a standard renewal argument (see, e.g., the proof of Theorem~5.5.2 (pp.\;143,144) of \cite{chun:01}), it follows that the right-hand side is uniformly bounded for $t \geq 1$, establishing the uniform integrability of $\{\frac{N(t)}{t}: t \geq 1\}$ and hence (b).
\end{proof}

The next step on the way to showing $J_0(\tau,Y) < \infty$ is to analyze the holding costs over a single cycle.  Observe that $X(\sigma_i)$ is either $1$ or $2^{(k-1)/2}$ for some $k$; to simplify notation, let $z$ represent either value.  Next, $X(t) = z - (t-\sigma_i) + W(t-\sigma_i)$ for $t \in [\sigma_i,\sigma_{i+1})$ since no orders are placed on the interval $(\sigma_i,\sigma_{i+1})$ and, by the definition of $\sigma_{i+1}$, $z - (\sigma_{i+1}-\sigma_i) + W(\sigma_{i+1}-\sigma_i) = 0$.  Again to simplify notation, make the change of time $s=t-\sigma_i$ and define $\tau=\sigma_{i+1}-\sigma_i$.  Thus, $X$ satisfies $X(s) = z - s + W(s)$ for $s\in [0,\tau)$.

Define
\begin{equation} \label{cycle-holding-cost}
L_\tau = \int_0^\tau c_0(X(s))\, ds.
\end{equation}
Then by Proposition~2.6 of \cite{helm:16}, it follows that $\mathbb E[L_\tau] = z^2 + z$.  In applying this result, one uses the fact that the function $g_0$ of that paper is computed to be $g_0(x) = x^2 + x$ for this specific example.

We now establish a result similar to the law of large numbers for the holding costs.

\begin{prop} \label{cycle-cost-asymptotics}
Let $(\tau,Y)$ be given by \defref{stoch-tauY-def} for the drifted Brownian motion process, $X$ be the inventory process and denote the non-zero ordering times by the collection ${\cal S}$.  Then
\begin{equation}
\label{eq-limsup-holding-cost}
 \limsup_{n\rightarrow \infty} \mbox{$\frac{1}{n}$} \sum_{k=1}^{n} \mathbb E\left[\int_{\sigma_{k}}^{\sigma_{k+1}} c_0(X(s))\, ds\right] \le  3.
\end{equation}
\end{prop}

\begin{proof} For simplicity of notation, denote $L_{k} : = \int_{\sigma_{k}}^{\sigma_{k+1}} c_0(X(s))\, ds $ for $k =1, 2, \dots$
As in the proof of \propref{slln}, write the index $n$ as $n=2^{i-1}+i+j-2$ in which $i=2,3,\ldots$ and $j=0,1,\ldots,2^{i-1}$.  Recall, the large orders have indices with $j=0$ and hence $n = 2^{i-1} + i -2 $ for $i \ge 2$ so the formula for $\mathbb E[L_\tau]$ above establishes that
$$\begin{array}{rccl}
\mathbb E[L_{2^{i-1}+i+j-2}] &=& 2, & \quad j=1,\ldots,2^{i-1},\; i \in \mathbb N, \\
\mathbb E[L_{2^{i}+i-1}] &=& 2^{i-1} + 2^{(i-1)/2}, & \quad  i\in \mathbb N.
\end{array}$$
Consequently, for $n\geq 2$,
\begin{align*}
 \mbox{$\frac1n$} \sum_{\ell=1}^{n} \mathbb E[L_{\ell}]
 &  =  \frac{1}{2^{i-1}+i+j-2}   \left(\sum_{k=1}^{i-1} (2^{k-1}\cdot 2 + 2^{k-1} + 2^{(k-1)/2}) +j \cdot 2\right)  \\
 & =  \frac{1}{2^{i-1}+i+j-2} \left(3(2^{i-1} -1) +  \frac{2^{(i-1)/2}-1}{2^{1/2} -1} + 2 j \right) \\
 & \le   \frac{3 ( 2^{i-1}  +j ) + \frac{2^{(i-1)/2}-1}{2^{1/2} -1}-3}{2^{i-1}+i+j-2} \\
 & = \frac{3 + \frac{1}{2^{1/2}-1} \cdot\frac{2^{(i-1)/2}-1}{2^{i-1}+ j} - \frac{3}{2^{i-1}+ j}} {1 + \frac{i-2}{ 2^{i-1} + j}}.
\end{align*} Using similar computations as those in the end of   the proof of Proposition \ref{slln}, we see immediately that this ratio converges to 3 as $n\to \infty $ and hence $i \to \infty$. This gives \eqref{eq-limsup-holding-cost} as desired.
\end{proof}

We now parlay the asymptotics relative to cycles to verify that the long-term average holding costs related to $(\tau,Y)$ are finite.
\begin{prop} \label{lta-holding-costs}
Let $(\tau,Y)$ be given by \defref{stoch-tauY-def} for the drifted Brownian motion process having underlying diffusion given by \eqref{dbm-dyn} and let $X$ satisfy \eqref{controlled-dyn} .  Then
$$\limsup_{t\rightarrow \infty} \mbox{$\frac{1}{t}$} \mathbb E\left[\int_0^t c_0(X(s))\, ds\right] \leq 3.$$
\end{prop}

\begin{proof}
Again, denote the non-zero ordering times by ${\cal S}$ and observe that the orders of size $0$ do not affect the cycle lengths.  Again, for each $t \geq 0$, recall $N(t) = \max\{n: \sigma_n \leq t\}$ so that $\sigma_{N(t)} \leq t < \sigma_{N(t)+1}$.  Thus for positive $t$,
\begin{eqnarray*}
\mbox{$\frac{1}{t}$} \int_0^t c_0(X(s))\, ds &\leq& \mbox{$\frac{1}{t}$} \int_0^{\sigma_{N(t)+1}} c_0(X(s))\, ds
 \leq  \mbox{$\frac{1}{t}$} \int_0^{\sigma_{N(t)+2}} c_0(X(s))\, ds =  \mbox{$\frac{1}{t}$} \sum_{j=1}^{N(t)+1} L_{j},
\end{eqnarray*}
where as in the proof of \propref{cycle-cost-asymptotics}, we employed the notation $L_j = \int_{\sigma_j}^{\sigma_{j+1}} c_0(X(s))\, ds$ for each $j \in \mathbb N$.  Noting that $\{N(t)=0\} = \emptyset$ for each $t$, consider the expectation of the right-hand term above:
\begin{eqnarray} \label{approx-wald} \nonumber
\mathbb E\left[\sum_{j=1}^{N(t)+1} L_j\right] &=& \mathbb E\left[\sum_{k=1}^\infty I_{\{N(t)=k\}}\sum_{j=1}^{k+1} L_j\right] \\
&=& \mathbb E\left[\sum_{j=1}^\infty L_j\sum_{k=j-1}^\infty I_{\{N(t)=k\}}\right] = \sum_{j=1}^\infty \mathbb E\left[L_j I_{\{N(t) \geq j-1\}}\right].
\end{eqnarray}
Observe that $\{N(t) \geq j-1\} = \{N(t)<j-1\}^c = \{\sigma_{j-1} > t\}^c$ is independent of the process over the interval $[\sigma_j,\sigma_{j+1})$.  Therefore the right-hand side of \eqref{approx-wald} yields
\begin{eqnarray*}
\sum_{j=1}^\infty \mathbb E\left[L_j I_{\{N(t) \geq j-1\}}\right] &=& \sum_{j=1}^\infty \mathbb E[L_j] \mathbb E[I_{\{N(t) \geq j-1\}}] \\
&=& \sum_{j=1}^\infty \mathbb E[L_j] \left(\sum_{k=j-1}^\infty \mathbb P(N(t)=k)\right) \\
&=& \sum_{k=1}^\infty \sum_{j=1}^{k+1} \mathbb E[L_j] \mathbb P(N(t)=k) \\
\end{eqnarray*}
By \propref{cycle-cost-asymptotics}, $\limsup_{k\rightarrow \infty} \frac{1}{k}\sum_{j=1}^{k} \mathbb E[L_j] \le 3$ so for any $\epsilon > 0$ there exists some $K_0 < \infty$ such that $\frac{1}{k+1} \sum_{j=1}^{k+1} \mathbb E[L_j] <   3+ \epsilon$ for all $k \geq K_0$.  Thus
 \begin{align*}
&\sum_{k=1}^\infty \sum_{j=1}^{k+1} \mathbb E[L_j] \mathbb P(N(t)=k)  \\
 &\ \ =  \sum_{k=1}^{K_0-1}   \sum_{j=1}^{k+1} \mathbb E[L_j] \mathbb P(N(t)=k) +  \sum_{k=K_0}^\infty \left(\mbox{$\frac{1}{k+1}$} \sum_{j=1}^{k+1} \mathbb E[L_j]\right) (k+1) \mathbb P(N(t)=k) \\
&\ \ \leq \sum_{k=1}^{K_0-1}  \sum_{j=1}^{k+1} \mathbb E[L_j]  \mathbb P(N(t)=k)  +\sum_{k=K_0}^\infty (3 + \epsilon) (k+1) \mathbb P(N(t)=k) \\
&\ \ \leq \sum_{k=1}^{K_0-1} \sum_{j=1}^{k+1} \mathbb E[L_j] + (3 + \epsilon)\mathbb E[N(t)] +3 + \epsilon.
\end{align*}Since the first and last summands are constant, combining these upper bounds, dividing by $t$ and using \propref{elementary-renewal} yields
$$\limsup_{t\rightarrow \infty} \mbox{$\frac{1}{t}$}\mathbb E\left[\int_0^t c_0(X(s))\,ds\right] \leq \lim_{t\rightarrow \infty} \frac{(3 + \epsilon)\mathbb E[N(t)]}{t} =3 + \epsilon.$$
The result now follows since $\epsilon> 0$ is arbitrary.\end{proof}

It is now easy to show that $\{\mu_{0,t}\}$ is tight as $t\rightarrow \infty$.

\begin{prop} \label{mu0t-tight}
Let $(\tau,Y)$ be given by \defref{stoch-tauY-def} for the drifted Brownian motion inventory model and for $t > 0$, define $\mu_{0,t}$ by \eqref{mus-t-def}.  Then $\{\mu_{0,t}: t > 1\}$ is tight as $t\rightarrow \infty$.
\end{prop}

\begin{proof}
Choose $\epsilon > 0$ arbitrarily and pick $M$ such that $M > \frac{3}{\epsilon}+1$.  Let $T$ be such that for all $t \geq T$,
$$\mbox{$\frac{1}{t}$} \mathbb E\left[\int_0^t c_0(X(s))\, ds\right] < 3 + \epsilon.$$
Since $c_0(x) = 2|x|$ is inf-compact, define the compact set $\Gamma = [-\frac{M}{2},\frac{M}{2}] = \{x: c_0(x) \leq M\}$.  Then for each $t>T$,
$$ \mu_{0,t}(\Gamma^c) \leq \int_{\Gamma^c} \frac{c_0(x)}{M}\, \mu_{0,t}(dx) \leq \frac{1}{M}\int c_0(x)\, \mu_{0,t}(dx)  \leq \frac{3 + \epsilon}{M}  <  \epsilon.$$
Since $\epsilon$ is arbitrary, it follows that $\{\mu_{0,t}\}$ is tight as $t\rightarrow \infty$.
\end{proof}

The final task is to verify that the long-term average ordering costs are finite and that $\{\mu_{1,t}\}$ is not tight as $t\rightarrow \infty$.  The next proposition addresses both of these concerns since the analysis is very similar.

\begin{prop} \label{ordering-measure-results}
Let $(\tau,Y)$ be given by \defref{stoch-tauY-def} for the drifted Brownian motion inventory model.  For $t > 0$, define $\mu_{1,t}$ by \eqref{mus-t-def}.  Then
\begin{equation}
\label{eq-limsup-order-cost}
 \limsup_{t\rightarrow \infty} \int c_1(y,z)\, \mu_{1,t}(dy\times dz) \le 3k_1+2
\end{equation}
and $\{\mu_{1,t}: t > 1\}$ is not tight.
\end{prop}

\begin{proof}
We first address the lack of tightness for $\{\mu_{1,t}\}$.  Let $\Gamma\subset \overline{\cal R}$ be any compact set.  Then there exists some $N_0$ such that for all $i \geq N_0$, $(2^{(i-1)/2},2^{(i-1)/2}) \in \Gamma^c$.

Again, denote the times of non-zero orders by ${\cal S} = \{\sigma_n: n\in \mathbb N\}$ and for $n\geq 2$, write $n=2^{i-1}+i+j-2$ with $i \ge 2$ and $j=0,\ldots,2^{i-1}$.  Recall, the ``large'' order of size $2^{(i-1)/2}$ occurs at time $\sigma_{2^i+i-1}$.  Under policy $(\tau,Y)$, there are a further $2^{i-1}$ orders of size $0$ at time $\sigma_{2^i+i-1}$; denote this common time of ordering by $\widetilde\sigma_{2^i+i-1,j}$ for $j=1,\ldots,2^{i-1}$ for each for the $0$-size orders.  Thus
\begin{eqnarray*}
\mu_{1,t}(\Gamma^c) &=& \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{n=1}^\infty I_{\{\sigma_n \leq t\}} I_{\Gamma^c}(X(\sigma_n-),X(\sigma_n)) \right. \\
& & \left. \qquad +\; \sum_{i=1}^\infty \sum_{j=1}^{2^{i-1}} I_{\{\widetilde\sigma_{2^i+i-1,j}\leq t\}} I_{\Gamma^c}(X(\widetilde\sigma_{2^i+i-1,j}-),X(\widetilde\sigma_{2^i+i-1,j})) \right] \\
&\geq& \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{i=N_0}^\infty \sum_{j=1}^{2^{i-1}} I_{\{\widetilde\sigma_{2^i+i-1,j}\leq t\}} I_{\Gamma^c}(2^{(i-1)/2},2^{(i-1)/2}) \right] \\
&=& \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{i=N_0}^\infty 2^{i-1} I_{\{\sigma_{2^i+i-1}\leq t\}} \right].
\end{eqnarray*}
For each $t \geq 0$, define the processes $I$ and $J$ such that $N(t) = 2^{I(t)-1}+I(t)+J(t)-2$ in which $I(t)$ denotes the cycle in which order $N(t)$ occurs and $0 \leq J(t) \leq 2^{I(t)-1}$.  Since $N(t)$ is the number of non-zero orders placed by time $t\geq 0$, order number $N(t)$ is the $J(t)^{\mbox{\footnotesize th}}$ order within cycle $I(t)$; again $J(t)=0$ corresponds to the large order of the previous cycle.  Using the processes $I$ and $J$, it follows that for $0 \leq J(t) \leq 2^{I(t)-1}$,
\begin{eqnarray} \label{mu1-lb} \nonumber
\mu_{1,t}(\Gamma^c) &\geq& \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{i=N_0}^\infty 2^{i-1} I_{\{\sigma_{2^i+i-1}\leq t\}} \right] \geq \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{\ell=2^{N_0}+N_0-1}^{I(t)-1} 2^{\ell-1}\right] \\ \nonumber
&=& \mbox{$\frac{1}{t}$} \mathbb E\left[\sum_{\ell=1}^{I(t)-1} 2^{\ell-1} - \sum_{\ell=1}^{2^{N_0}+N_0-2} 2^{\ell-1}\right] \\
&=& \mbox{$\frac{1}{t}$} \left(\mathbb E\left[2^{I(t)-1}\right] - 2^{2^{N_0}+N_0-2}\right).
\end{eqnarray}
 By \lemref{elementary-renewal}, $N(t) \rightarrow \infty\; (a.s.)$ as $t\rightarrow \infty$ so $I(t) \rightarrow \infty$ as well.  Thus the asymptotics of $\mu_{1,t}(\Gamma^c)$ is determined by the asymptotics of the first summand above.

We next determine bounds on $I(t)$.  Since $J(t) \leq 2^{I(t)-1}$ and $N(t) = 2^{I(t)-1}+I(t)+J(t)-2$, we have $N(t) \leq 2^{I(t)}+I(t) -2$ and hence
\begin{equation} \label{bd1}
2^{I(t)-1} \geq \mbox{$\frac{1}{2}$} (N(t) - I(t) +2).
\end{equation}
Since $I(t)\ge 1 $ and  $J(t) \ge  0$, $N(t) \ge  2^{I(t)-1}-1$ so
\begin{equation} \label{I-minus-one-lb}
I(t)-1 < \log_2(N(t)+1) \leq \log_2(N(t)+N(t)) = \mbox{$\log_2(\frac{N(t)\cdot 2t}{t}) = \log_2(\frac{N(t)}{t}) + \log_2(2t)$}.
\end{equation}
Using this estimate in \eqref{bd1} yields
$$2^{I(t)-1} > \mbox{$\frac{1}{2}$} (N(t) - \log_2(\mbox{$\frac{N(t)}{t}$}) - \log_2(2t)+ 1).$$
Employing this lower bound in \eqref{mu1-lb} and Jensen's inequality on the second summand, we have
$$\mu_{1,t}(\Gamma^c) \geq \mbox{$\frac{1}{2}$} \cdot \left(\frac{\mathbb E[N(t)]}{t} - \frac{\log_2(\mathbb E[\frac{N(t)}{t}])}{t} - \frac{\log_2(2t)}{t}\right) \stackrel{t\rightarrow \infty}{\longrightarrow} \mbox{$\frac{1}{2}$};$$
note that we have also used \propref{elementary-renewal} on the second summand.  Therefore for any $\epsilon < \frac{1}{2}$, $\mu_{1,t}(
\Gamma^c) > \epsilon$ for all $t$ sufficiently large.  Hence $\{\mu_{1,t}\}$ is not tight as $t\rightarrow \infty$.

Consider now the total ordering costs by time $t > 0$.  First, denote $(\tau,Y) = \{(\tau_k,Y_k): k \in \mathbb N\}$ to capture all of the orders.  Next, denote the non-zero orders by ${\cal S}$ and as above, for each $i \in \mathbb N$ and $j=1,\ldots, 2^{i-1}$, let $\widetilde\sigma_{2^i+i-1,j} = \sigma_{2^i+i-1}$ be the common time of the $0$-size orders in cycle $i$.

Since $t$ is finite and $0 \leq J(t) \leq 2^{I(t)-1} $,
\begin{align*}
\lefteqn{\mathbb E\left[\sum_{k=1}^\infty I_{\{\tau_k\leq t\}} c_1(X(\tau_k-),X(\tau_k))\right]} \\

   & =  \mathbb E\left[\sum_{i=1}^{I(t)-1} \left[2^{i-1}(k_1+1)  + (k_1 + 2^{(i-1)/2}) + 2^{i-1} k_{1}\right] + J(t) (k_{1}+1)\right]\\

    & \le     \mathbb E\left[\left( (2k_1+1) (2^{I(t)-1}-1) + \frac{2^{(I(t)-1)/2}-1}{2^{1/2}-1} + k_1 (I(t)-1)\right)+ 2^{I(t) -1} (k_{1}+1)\right]\\

&=  \mathbb E\left[(3k_1+2) 2^{I(t)-1}  + \frac{2^{(I(t)-1)/2}-1}{2^{1/2}-1} + k_1 I(t)  -3 k_{1} -1 \right].
\end{align*}
Since $I(t) \ge 1$ and $ J(t) \geq 0$, it follows that $2^{I(t)-1}    \le N(t) + 1 $ and so $2^{(I(t)-1) /2} \le (N(t)+1)^{1/2}$.  Using \eqref{I-minus-one-lb}, we also have $I(t) <  \log_{2}(\frac{N(t)}{t}) + \log_{2}(2t) + 1$.
Then it follows from Jensen's inequality  that \begin{align*}
\mathbb E&\left[\sum_{k=1}^\infty I_{\{\tau_k\leq t\}} c_1(X(\tau_k-),X(\tau_k))\right] \\

      & \le  (3k_{1} +2) \mathbb E[N(t)]    + \frac{     \mathbb E[(N(t)+1)^{1/2}]-1}{2^{1/2}-1}  + k_{1} \mathbb E\left[ \log_{2}(N(t)/t ) + \log_{2}(2t) + 1\right] + 1 \\
      & \le   (3k_{1} +2) \mathbb E[N(t)]    + \frac{   (\mathbb E[N(t)+1])^{1/2} - 1} {2^{1/2}-1}    + k_{1}\big [ \log_{2} (\mathbb E[N(t)/t] ) + \log_{2}(2t) + 1\big] + 1.
\end{align*}
Now divide both sides by $t$, and then send $t\to \infty$, obtaining \eqref{eq-limsup-order-cost}  from Proposition \ref{elementary-renewal}.
\end{proof}

\begin{rem}[Final Comments]
Under further analysis, one can obtain more precise results about the costs related to the ordering policy $(\tau,Y)$ of \defref{stoch-tauY-def}.  As in the analysis of \propref{cycle-cost-asymptotics}, a lower bound on the limit inferior of the Ces\`{a}ro mean of the expected cycle costs can be shown to be $\frac{5}{2}$.  With more extensive calculations including the variance of the holding costs per cycle, these bounds can be shown to be tight and moreover that
$$\liminf_{t\rightarrow \infty} \mbox{$\frac{1}{t}$} \int_0^t c_0(X(s))\, ds = \mbox{$\frac{5}{2}$} \; (a.s.) \quad \mbox{and} \quad \limsup_{t\rightarrow \infty} \mbox{$\frac{1}{t}$} \int_0^t c_0(X(s))\, ds = 3 \; (a.s.).$$
We also note that \propref{mu0t-tight} is proven for general inventory models in \cite{helm:17}; for completeness of the paper, we have included \propref{mu0t-tight} for this example.
\end{rem}

redpajama_set_name: RedPajamaArXiv


Why can't you stop her from being contentious?
……That you are to act like a man (ie – do not wimp out or be a milquetoast) ……..
1 Corinthians 16:12-14 – 13 Be on the alert, stand firm in the faith, act like men, be strong. 14 Let all that you do be done in love.
Colossians 3:19 – Husbands, love your wives and do not be embittered against them.
Be wise and understanding on what is really going on inside your wife. That includes recognizing her narcissism / insolent pride for what it really is. Pretending its not there does not lead to solutions.
There is a hint in these verses of creating some space. This space will give you some peace, and provide your wife with a "smaller target". This does not mean leaving or abdicating your responsibilities (the "corner of the roof" referred to would be in the same house). But rather than fighting the contentious one, you avoid it the fight. This has some similarities to the "avoid them" philosophy discussed here. But it is not completely avoiding, but rather fulfilling your responsibilities, showing love, etc., while at the same time stepping away from fights. Practically speaking it may mean being silent in the immediate face of the contentiousness, and then when the time is right to create some space.
Proverbs 17:14 – The beginning of strife is like letting out water, So abandon the quarrel before it breaks out.
And he who reproves a wicked man gets [a]insults for himself.
Fourth, consider setting boundaries and consequences (see here).
Proverbs 19:29 – Judgments are prepared for scoffers, And blows for the back of fools.
This does not mean saying, "stop nagging me or I"m leaving you". But there could be some smaller boundary/consequence combinations that help you more peacefully manage the situation.
43 "You have heard that it was said, 'You shall love your neighbor and hate your enemy.' 44 But I say to you, love your enemies and pray for those who persecute you, 45 so that you may [a]be sons of your Father who is in heaven; for He causes His sun to rise on the evil and the good, and sends rain on the righteous and the unrighteous.
What should we pray for? If the source of her contentiousness is her insolent pride, then praying that God would graciously remove that from her is a good place to start. You are then praying not only for your own relief, but also for her eternal good.
Sixth, your wife could have a family history of narcissism / insolent pride / and contentiousness. Dealing with a family history of insolence will be explored in the future.
Although dealing with a contentious wife may seem hopeless at times, we can hope in God and at that same realize that while God may not get us out of our circumstance quickly and easily, nothing is impossible with God.


Emperor Akbar Cardamoms are graded based on the size of the diameter of the pod. This is a generally accepted norm of cardamom grading.Emperor Akbar Cardamoms have multiple uses. They can be used in the form of seeds, husks, powder or for extracting its oil.
However, not all buyers give equal importance to the size of the pod. They have other uses and so other parameters such as seed content, oil content, etc. are important for such buyers. Our team understands that.
Therefore, we can customize Emperor Akbar Green Cardamom to such specific requirements.
We can also offer Organic Cardamom to the discerning customers.
These are customized grades which are not commonly available in the markets. Hence, these are specially packed in Emperor Akbar Blue Packs of 5 Kgs.


text:
$restriction.code /landing/headlines-featured
Ohio Men's And Women's Cross Country's 2019 Schedule Announced
The Ohio athletic department announced Tuesday the upcoming slate for the 2019 Bobcats cross country teams.
Ohio Football's Rourke Named To 2019 Davey O'Brien Award Watch List
Senior quarterback Nathan Rourke (Oakville, Ontario, Canada) of the Ohio football team has been named to the 2019 Davey O'Brien National Quarterback Award Watch List, as announced by the Davey O'Brien Foundation today.
Ohio Football's Rourke Named To 2019 Maxwell Award Watch List
Senior quarterback Nathan Rourke (Oakville, Ontario, Canada) of the Ohio football team has been named to the 2019 Maxwell Award Watch List, as announced by the Maxwell Football Club today.
Ohio Men's Basketball Fan Feedback Follow-Up
Thank you for taking time this past spring to respond to our call for feedback regarding your game day experience at our men's basketball games. We appreciate your input and will use it in an effort to better serve all Bobcats fans.
Ohio Field Hockey Announces 2019 Schedule
The Ohio field hockey program revealed its 2019 schedule today, as announced by second-year head coach Ali Johnstone.
Ohio Men's Basketball Matchups Announced for 2019 Myrtle Beach Invitational
The matchups for the second annual Myrtle Beach Invitational have been released. The event, hosted by the American Athletic Conference, will be played November 21, 22 and 24 at the HTC Center in Conway, S. C.
Ohio Swimming & Diving Announces 2019-20 Schedule
Three home meets against Akron, Ball State and Marshall highlight the Ohio swimming and diving team's 2019-20 schedule, which was announced by second-year head coach Mason Norman.
Five Ohio Student-Athletes Named MAC Distinguished Scholar Athletes
Five Ohio student-athletes have been honored by the Mid-American Conference with Distinguished Scholar Athlete recognition for the 2019 spring semester, as announced by the league office today.
Grusczynski Named Ohio Men's Golf Head Coach
Neal Grusczynski has been named the fourth head coach in the history of Ohio men's golf program, as announced by Interim Director of Athletics Amy Dean today.
Ohio Football's Rodgers Selected To Teamworks Football Leadership Committee
Ohio Assistant Athletic Director for Football Operations Chris Rodgers has been selected to the Teamworks Football Leadership Committee.
Loftus, Porvasnik Earn WGCA All-American Scholar Recognition
Freshmen Catherine Loftus and Alicia Porvasnik of the Ohio women's golf team were named Women's Coach Coaches Association (WGCA) All-American Scholars, the league announced today (July 8).
Ohio Athletics Unveils Recruiting Microsite
Ohio Athletics is pleased to unveil a new recruiting microsite that will share the experience of being a student-athlete at Ohio.
Jackson Promoted to Associate Head Coach of Ohio Women's Basketball
ATHENS, Ohio – Tavares Jackson has been promoted to associate head coach of the Ohio Women's Basketball program, head coach Bob Boldon announced today.
Former Ohio Baseball Player Logan Cozart Called Up to Triple-A Albuquerque
Former Ohio baseball right-handed pitcher Logan Cozart was called up to Triple-A Albuquerque of the Pacific Coast League, the Colorado Rockies announced.
Ohio Swimming & Diving's Van Lanen Named Nominee For 2019 NCAA Woman of the Year Award
Corrin Van Lanen (Green Bay, Wis.) of the Ohio swimming & diving team has been named Ohio's nominee for the 2019 NCAA Woman of the Year award.
Ohio Football's Rourke To Attend 2019 Manning Passing Academy
ATHENS, Ohio – Senior quarterback Nathan Rourke (Oakville, Ontario, Canada) of the Ohio football team is set to attend the 2019 Manning Passing Academy as a camp counselor this week (June 27-30) at Nicholls State University in Thibodaux, La.
Ohio Soccer Announces 2019 Fall Schedule
The Ohio women's soccer team has released its 2019 fall schedule, seventh-year head coach Aaron Rodgers announced today (June 26). The schedule sees nine home games, including four non-conference matchups and five Mid-American Conference pairings. The Bobcats will also face defending Big East Champion, Georgetown, who finished the 2018 season in the Women's College Cup Semifinals.
Ohio Athletics Signs Long-Term Multimedia Rights Deal With Learfield IMG College
Ohio Athletics has signed an agreement with Learfield IMG College as the department's exclusive rights holder for both multimedia rights and licensing rights for the next 12 years, it was announced today.
Ohio Swimming and Diving Named to CSCCA Scholar All-American Team
The College Swimming and Diving Coaches Association (CSCAA) has named the Ohio women's swimming and diving team to its Scholar All-America team for the Spring 2019 Semester. A record 480 institutions were named to the list, highlighted by Ohio's GPA of 3.46.
Ohio Softball to Host Summer Camp in July
The Ohio Softball program will be hosting their annual summer camp on Tuesday, July 16th, 2019. This camp is for softball players between 7th through the 12th grade and will take place at Walter Fieldhouse/Ohio Softball Field located on Ohio University's campus.
Former Ohio Volleyball Player Jamie Kosiorek Signs Professional Volleyball Contract in Denmark
Brøndby, Denmark – Former Ohio Volleyball player Jamie Kosiorek has signed a professional contract with Brøndby Volleyball Club in Brøndby, Denmark, BVC announced on Thursday (June 18).
Six Members of Ohio Baseball Named Academic All-MAC
CLEVELAND – Six members of the Ohio Baseball team were named Academic All-Mid-American Conference, the league announced today (June 18). Sophomore Treyben Funderburg (Beavercreek, Ohio), redshirt senior Nick Kamrada (Grand Ledge, Mich.), sophomore Aaron Levy (Cincinnati, Ohio), redshirt freshman Jack Liberatore (Greensburg, Pa.), redshirt freshman Trevor Lukkes (Tustin, Calif.) and senior Rudy Rott (Holmen, Wis.).
Ohio Baseball's Rudy Rott Named ABCA/Rawlings NCAA Division I All-America Third Team
GREENSBORO, N.C. – Ohio Baseball senior first baseman Rudy Rott (Holmen, Wis.) has been named to the 2019 ABCA/Rawlings NCAA Division I All-America Third Team, the American Baseball Coaches Association announced on Saturday (June 15).
Ohio Baseball's Rudy Rott Named ABCA/Rawlings NCAA Division I Mideast All-Region First Team
GREENSBORO, N.C. – Ohio Baseball senior first baseman Rudy Rott (Holmen, Wis.) has been named to the 2019 ABCA/Rawlings NCAA Division I Mideast All-Region first team, the American Baseball Coaches Association announced on Wednesday (June 12).
Ohio Volleyball Europe Blog - Recap From Vera Giacomazzi
I left Europe to go to America a couple years ago and America brought me back to Europe in the best way possible - with the whole Ohio squad.
Six Members of Ohio Softball Named Academic All-MAC
Six members of the Ohio softball team were named Academic All-Mid-American Conference, the league announced today (June 11). Deanna Cole, Alex Day, Morgan Geno, Alexa Holland, Madi McCrady and Katie Yun all earned the honor.
Ohio Announces Fan Favorite Vote As Part Of College Football 150th Anniversary Celebration
As part of its College Football 150th Anniversay celebration, Ohio Athletics is giving fans of the Bobcats the opportunity to vote on their favorite football student-athlete in program history.
Ohio Volleyball Europe Blog - Abby Arend in Maribor, Slovenia
Ohio Men's Basketball Season Tickets Now On Sale
Season tickets for the 2019-20 Ohio men's basketball season are now sale.
Ohio Volleyball Europe Blog - Lizzie Stephens
Ciao linci! Greetings from Pordenone, Italy! The team is currently on the Italia Rail traveling from Pordenone to Venice for the day. Yesterday, we spent time in Vera's hometown, San Vendemiano. When we arrived, we were greeted by Vera's mom, dad, sister, aunt, and two cousins. Her cousin, Elena, showed us around town and taught us about its historical points, including the medieval castle, the cathedral, and the town square. Our team is so grateful and happy we were able to experience Vera's hometown and spend time with her family.

redpajama_set_name: RedPajamaCommonCrawl


text:
Topics Labor Relations
Legal Alert: Fourth Circuit Finds NLRB Lacked Authority to Issue Notice Posting Rule
The Fourth Circuit has become the second federal appeals court to strike down the National Labor Relations Board's rule requiring employers to post a notice of employee rights.
Executive Summary: The Fourth Circuit has become the second federal appeals court to strike down the National Labor Relations Board's rule requiring employers to post a notice of employee rights. In Chamber of Commerce v. NLRB (June 14, 2013), the court held that the Board exceeded the authority granted to it by the National Labor Relations Act when it promulgated the rule.
The Notice Posting Requirement
As discussed in our August 30, 2011 Legal Alert, NLRB to Require Posting of Notice of Employee Rights, the Board's rule requires employers to post a Notice of Employee Rights under the National Labor Relations Act (the Act). The rule provides that an employer's failure to post the Notice may: (1) be considered an unfair labor practice (ULP); (2) toll the statute of limitations for filing a ULP charge; and (3) be used as evidence of anti-union animus in proceedings before the Board.
The rule was challenged in federal courts in both the Fourth Circuit and the D.C. Circuit. In May 2013 the D.C. Circuit Court of Appeals vacated the rule, holding that it violates employers' free speech rights. For more information on this decision, please see our May 8 Legal Alert. On June 14, in Chamber of Commerce, the Fourth Circuit affirmed the federal trial court's decision that the Board lacked authority under the Act to enact the notice-posting rule.
The Fourth Circuit's Decision
In finding the rule invalid, the Fourth Circuit held that the Act only empowers the Board to carry out its statutorily defined "reactive roles" in addressing unfair labor practice charges and conducting union elections upon request. The court held that the language of Section 6 of the Act, which grants the Board the authority to issue rules necessary to carry out the provisions of the Act, requires that some section of the Act give the Board either the explicit or implicit authority to issue a rule. Because nothing in the Act charges the Board with informing employees of their rights under the Act, the court found "no indication in the plain language of the Act that Congress intended to grant the Board the authority to promulgate such a requirement."
Further, the court found no provision of the Act that a notice-posting rule is necessary to carry out. Specifically, the court held that Section 7, which lists rights protected by the Act, does not provide authority to issue the notice-posting rule because these rights are not functions or provisions to be carried out. Additionally, although the court acknowledged that the Board has the authority to define what constitutes a ULP under Section 8(a)(1), it held that this provision does not give the Board authority to "enact the unprecedented rule at issue here."
Finally, the court examined the legislative history of the Act and held that the Board's responsibilities for conducting representation elections and adjudicating ULP charges under Sections 9 and 10 of the Act are reactive roles that are not set in motion until a party files a representation petition or ULP charge. According to the court, these provisions do not justify an expansion of the Board's role to include proactive regulation of employers' conduct.
Employers' Bottom Line: The Board has not indicated whether it will seek review of this decision with the Supreme Court. Currently, however, employers are not required to post the Notice of Employee Rights.
If you have any questions regarding this decision or other labor or employment law issues, please contact the FordHarrison attorney with whom you usually work.


text:
Enbridge's Mainline overhaul has outraged oil producers, but investors are cheering
Investors see monetizing existing infrastructure as a safer bet than trying to build new pipelines
Enbridge's Mainline is North America's largest pipeline network, transporting nearly 3 million barrels per day, or 70 per cent, of crude from western Canada to the U.S. Midwest.	Daniel Acker/Bloomberg
By Nia Williams and Rod Nickel
CALGAR/WINNIPEG — Enbridge Inc.'s plan to overhaul contracts on its Mainline pipeline system has outraged many Canadian shippers but is cheered by investors who see monetizing existing infrastructure as a safer bet than trying to build new pipelines.
The Mainline is North America's largest pipeline network, transporting nearly 3 million barrels per day, or 70 per cent, of crude from western Canada to the U.S. Midwest.
Many shippers opposed say Enbridge is abusing its market power and the changes will hurt Canadian producers by imposing unfair terms and tolls. The proposed revamp comes at a time when export pipeline capacity is so constrained the Alberta government has curtailed production to support crude prices.
Oil firms weigh in on Enbridge's proposed overhaul of Mainline pipeline contracts, and not all of them are complaining
Enbridge pipeline contracts will give Midwest refiners too much power, oilsands producers say
Canadian energy regulator may delay Enbridge Mainline plan after companies ask for intervention
Regulatory hurdles and well-organized environmental opposition are clogging up North American pipeline projects. Enbridge is facing U.S. court challenges on its aging Line 3 and Line 5 pipelines, and delays dog the proposed government-owned Trans Mountain expansion and TC Energy's Keystone XL.
Contracting out the Mainline would be a smart move to lock in revenue while the Trans Mountain (TMX) and Keystone XL (KXL) projects are still uncertain, investors said. If those projects go ahead and the Mainline remains a so-called "common carrier" system relying on monthly nominations, it could see volumes dip.
"Under scenarios in which KXL is going to be built and TMX, the Mainline is going to be the pipe of last resort," said Ryan Bushell, president of Newhaven Asset Management, which holds shares in Enbridge and Canadian Natural Resources Ltd.
"Given how hard it is getting (new pipelines) done, Enbridge should be able to contract the Mainline out as they see fit."
Enbridge proposes switching to long-term fixed-volume contracts instead of monthly nominations on 90 per cent of Mainline capacity in 2021 when its current tolling arrangement expires.
OPEN SEASON DISPUTE
The Mainline contributes 30 per cent of Enbridge's adjusted earnings before interest, tax, depreciation and amortization, according to the latest quarterly report.
"We are confident that Mainline contracting is supported by shippers representing a majority of our volumes," Enbridge spokesman Jesse Semko said, adding the company makes decisions in the best interest of customers and shareholders.
The company launched a two-month open season to solicit bids for capacity on Aug. 2. Since then, some of Canada's biggest producers, including Canadian Natural and Suncor Energy, have complained about the plan and dozens of shippers wrote to the Canada Energy Regulator (CER) last week about whether it should intervene in the open season.
Some shippers argue the Mainline should remain a common carrier pipeline open to all customers, as legislated in Canada's 1949 Pipelines Act. Other shippers and investors say it is Enbridge's asset to monetize as it wants.
A number of investors also own shares in Canadian oil producers.
"If you're Enbridge, why wouldn't you try to make this as profitable for you as possible? I'm rooting for (any compromise) to be bended toward Enbridge, just because it's relatively more important to them," said Bruce Campbell, chairman of Campbell, Lee & Ross, which holds Enbridge and Suncor.
Enbridge has already lowered oil-shipping requirements on the Mainline contracts by two-thirds. The move came after smaller producers complained they would struggle to meet the initial volume requirements.
Enbridge shares have gained 8 per cent year-to-date, and Newhaven's Bushell said they would be stronger if not for ongoing uncertainty about Line 3 and Line 5 legal challenges. Rival TC Energy is up 39% over the same period.
GMP FirstEnergy analysts said so far the dispute over the Mainline changes had not impacted Enbridge stocks.
"When you're investing in a utility, you're looking for contracted revenue and therefore contracted cash flow. As an investor from the Enbridge perspective, we like it," said Stephen Kallir, vice-president of investments at BlueSky Equities.
 Run the joules: Cold snap sees Sask. set natural gas records Husky ordered to pay $3.8 million following 225,000-litre pipeline spill near...
Latest Business Videos
Canadian home sales rise for fifth straight month as mortgage rates decline
Alberta heads to the polls today. The fate of carbon taxes for the entire country ...
Liberals leave themselves little wiggle room in the event of a recession
Meet the Canadian who invented the way to not die of AIDS
'Unprecedented' Supreme Court decision on Trans Mountain should be message for Quebec: Kenney
Supreme Court dismisses B.C. case against Trans Mountain pipeline
Run the joules: Cold snap sees Sask. set natural gas records


Is my heart open to God's surprises?
"The surprises of God put us on the path, they put us on our journey without waiting.
Having an open heart sounds easier said than done. Opening your heart is a scary thing to do because you don't know what other people will do to you. Openness requires trust – trusting others to treat us well and be worthy of our trust. Those things aren't guaranteed when you open your heart.
I think most of us live our lives guarding our hearts for various reasons. Maybe people have disappointed and hurt us before. Maybe it's easier to close your heart than to get hurt again. Maybe being vulnerable is just a tad too difficult to deal with.
But Pope Francis asked the right question. If our hearts are closed, then we wouldn't be about to hear what God is trying to tell us. If we can't hear what He's trying to tell us, then we can't even get started on our respective journeys. We're all basically puttering around in busy-ness that leads nowhere.


text:
Research Paper: Cultural Challenges Faced By Business
The face of business is changing rapidly and globalization is the most probable answer to some of the problems faced by businesses. With the world shrinking into a village, easy communications and efficient logistics it is no wonder that businesses think internationally.
By going global, a business caters to a larger market increasing the probability of higher revenue and thus more profit. But this greater exposure and a market does not come at a low price. Companies decide to globalize if their competencies are relevant, appropriable and transferable to the foreign markets, but it should also be ensured that these competencies can withstand the barrage of challenges that new markets pose.
Globalization and entering a foreign market comes with its pack of challenges. There are political, legal, social, financial, economic and environmental barriers which the company will have to cross to create its niche in a foreign country. Getting through them once does not guarantee success as every different region of the world and every different country will pose a barrier of a different kind and a separate analysis and planning will have to be done for each market. A study of US multinational corporations found out that poor intercultural management skills still constitute a major management problem. (Deresky,1996)
Even in countries where language does not present a barrier, slight cultural changes and differences in dialect and verbal representations, nonverbal gestures and assumptions can create challenges. This can be represented by the fact that the number one billion in the US is one thousand million in the UK. Non-verbal gestures also play an important role in business communication. Eye contact is a symbol of confidence in many Western countries but in Asian countries or in Mexico it is a sign of disrespect. Bowing is considered to be a sign of submission in most countries but again in countries like Japan it signifies initiation and ending of conversations. (Ferraro,2007) Thus, knowledge of the minute cultural, verbal and nonverbal details will ensure that cross-cultural communication does not lead to misunderstandings.


The small town of Krefeld is just a stone's throw from Düsseldorf and then Rhine, with ease of access to both the Netherlands and Belgium as well. A Krefeld car service from Blacklane is just the ticket for getting to where you need to be in the town with the minimum of fuss. Whether you're headed to a business engagement within the town itself, or hoping for a stylish taxi alternative to one of the nearby cities of the Rhine-Ruhr region, Blacklane is your answer. You can quickly and easily secure your ground transportation in advance by using our app for Android and iOS - meaning you can sort your transport while you're on the move.
Krefeld, though a small town, is regionally notable for its historical contributions to the German textile industry. Its proximity to the cities composing of the Rhine-Ruhr region means Krefeld has never been out of the loop when it came to German industry and business. If you're on town for professional reasons, a Krefeld chauffeur service from Blacklane is just the ticket for getting from point A to point B. Our First Class service is a great way to make a fantastic first impression and is guaranteed to turn some heads with your arrival, while an hourly hire means you can zip from appointment to meeting around the city with a personal chauffeur at your beck and call.
Düsseldorf International Airport is your most likely point of entry is you're getting to Krefeld by air. If you can have arranged for an airport transfer to pick you up directly after your landing, you can be in town in twenty minutes. Your chauffeur will greet you in the airport, help you with your luggage and whisk you away to your Krefeld hotel or accommodation. If you're on your way out of town, your personal chauffeur is the safest pair of hands in town when it comes to getting to your outbound flight in good time. Professional, locally-knowledgeable and experienced, our chauffeurs are handpicked for their ability and efficiency.


To provide premium traffic management accompanied with the "best" resources, delivering outstanding service at a competitive rate.
M1 Traffic Control specialise in supplying the same quality of service to businesses of all sizes, offering quick turnaround on TMPs, outstanding levels of customer service and Professional Personnel on the ground.
M1 Traffic Control has established a strong relationship with many companies over the years in a diverse range of industries and business types, from Federal, state and local governments to construction and civil based companies. Liberty Hire is an equipment hire company, specialising in the leasing of construction equipment such as VMS boards, arrow boards and lighting towers. When M1 Traffic Control needs equipment, we hire from Liberty Hire, as we have established a strong and trusting relationship with the company.
M1 Traffic Control is in contact with VicRoads on a daily basis, thus we have created a great rapport with them and gained good relationships with many VicRoads contacts. We are a VicRoads prequalified company and therefore we can work on VicRoads' roads, given we have applied for the appropriate permits. Our permits team liaises with VicRoads daily regarding MOAs (Memorandum of Authority) and consent for works permits.
Our traffic controllers on site also communicate with VicRoads when activating permits prior to commencing works on site.
M1 Traffic Control have well recognised relationships with major event organisers such as Peter Jones for the Melbourne Cup Parade and the Docklands Australia Day Fireworks, Vision Australia with Carols by Candlelight and Frankston City Council for their annual Festival of Lights and Waterfront Festival.
M1 Traffic Control has also established workplace diversity as we have created an inclusive environment that accepts any individual differences.
M1 Traffic Control proudly provide equal opportunity by employing people of all genders, races, religions, cultures and employing people with disability.


text:
Michael Long, an undercover police officer, is shot while investigating a case and left for dead by his assailants. He is rescued by Wilton Knight, a wealthy, dying millionaire and inventor who arranges life-saving surgery, including a new face and a new identity–that of Michael Knight. Michael is then given a special computerized and indestructible car called the Knight Industries Two Thousand (nicknamed KITT), and a mission: apprehend criminals who are beyond the reach of the law. The series depicts Michael's exploits as he and KITT battle the forces of evil on behalf of the Foundation for Law and Government.
Genre: Action, Action & Adventure, Crime, Drama, Sci-Fi, Sci-Fi & Fantasy
St. Elsewhere is an American medical drama television series that originally ran on NBC from October 26, 1982 to May 25, 1988. The series starred Ed Flanders, Norman Lloyd and William Daniels as teaching doctors at a lightly-regarded Boston hospital who gave interns a promising future in making critical medical and life decisions. The series was produced by MTM Enterprises, which had success with a similar NBC series, the police drama Hill Street Blues, during that same time; both series were often compared to each other for their use of ensemble casts and overlapping serialized storylines. St. Elsewhere was filmed at CBS/MTM Studios, which was known as CBS/Fox Studios when the show began; coincidentally, 20th Century Fox wound up acquiring the rights to the series when it bought MTM Enterprises in the 1990s.
Known for its combination of gritty, realistic drama and moments of black comedy, St. Elsewhere gained a small yet loyal following over its 6-season, 137-episode run; the series also found a strong audience in Nielsen's 18-49 age demographic, a young demo later known for a young, affluent audience that TV advertisers are eager to reach. The series also earned critical acclaim during its run, earning 13 Emmy Awards for its writing, acting, and directing. St. Elsewhere was ranked #20 on TV Guide's 2002 list of "The 50 Greatest TV Shows of All Time.", with the magazine also selecting it as the best drama series of the 1980s in a 1993 issue.
Cheers is an American sitcom television series that ran for 11 seasons from 1982 to 1993. It was produced by Charles/Burrows/Charles Productions in association with Paramount Network Television for NBC and created by the team of James Burrows, Glen Charles, and Les Charles. The show is set in a bar named Cheers in Boston, Massachusetts, where a group of locals meet to drink, relax, and socialize. The show's theme song, written and performed by Gary Portnoy, and co-written with Judy Hart Angelo, lent its famous refrain, "Where Everybody Knows Your Name", as the show's tagline.
After premiering on September 30, 1982, it was nearly canceled during its first season when it ranked last in ratings for its premiere. Cheers, however, eventually became a highly rated television show in the United States, earning a top-ten rating during 8 of its 11 seasons, including one season at #1. The show spent most of its run on NBC's Thursday night "Must See TV" lineup. Its widely watched series finale was broadcast on May 20, 1993, and the show's 275 episodes have been successfully syndicated worldwide. Nominated for Outstanding Comedy Series for all eleven of its seasons on the air, it has earned 28 Emmy Awards from a then-record 117 nominations. The character Frasier Crane was featured in his eponymous spin-off show, which later aired up until 2004 and included guest appearances by virtually all of the major and minor Cheers characters.
Australian journalist Guy Hamilton travels to Indonesia to cover civil strife in 1965. There—on the eve of an attempted coup—he befriends a Chinese Australian photographer with a deep connection to and vast knowledge of the Indonesian people, and also falls in love with a British national.
Genre: Drama, Romance, War
Death Wish II
Paul Kersey is again a vigilante trying to find five punks who murdered his housekeeper and daughter in Los Angeles.
A young woman crash-lands her plane in Jamaica. A local named Countryman rescues her and leads her away from the authorities, who have fabricated a story about the plane, involving drug and arms smuggling by the CIA, in order to gain popularity in an upcoming election.
Country: Jamaica
The Final Option
When SAS Captain Peter Skellen is thrown out of the service for gross misconduct due to unnecessary violence and bullying, he is soon recruited by The People's Lobby, a fanatical group aiming to hold several US dignitaries hostage. But Skellen's dismissal is a front to enable him to get close to the terrorist group. Can he get close enough to stop the Lobby from creating an international incident?
Country: Switzerland, UK
An unfortunate and naive girl is set-up by her boyfriend and convicted of drug smuggling. She is sent to a women's correctional facility where she must constantly struggle to survive.
Burden of Dreams
German director Werner Herzog begins work on his 1982 epic "Fitzcarraldo" but soon runs into serious setbacks, from casting problems to his own stubborn refusal to use special effects. After having to reshoot much of the film because the lead actor was recast, his crew must then haul an old-fashioned steamboat over a mountain using manpower alone. With a resolve bordering on insanity, Herzog struggles to realize his vision, vowing to see the film completed — even if it leads to his undoing.
Turkey Shoot
In the near future, after an unspecified holocaust, survivors are herded into prison camps. There, they are hunted for sport by the leaders of the camp. Paul, one of the newest prisoners, is determined not to go down as quietly as the others.
Genre: Drama, Horror, Sci-Fi, Thriller
An orphan in a facility run by the mean Miss Hannigan, Annie believes that her parents left her there by mistake. When a rich man named Oliver "Daddy" Warbucks decides to let an orphan live at his home to promote his image, Annie is selected. While Annie gets accustomed to living in Warbucks' mansion, she still longs to meet her parents. So Warbucks announces a search for them and a reward, which brings out many frauds.
Genre: Comedy, Drama, Family, Musical
Steve Freeling lives with his wife, Diane, and their three children, Dana, Robbie, and Carol Anne, in Southern California where he sells houses for the company that built the neighborhood. It starts with just a few odd occurrences, such as broken dishes and furniture moving around by itself. However, when he realizes that something truly evil haunts his home, Steve calls in a team of parapsychologists led by Dr. Lesh to help before it's too late.
Scientists in the Antarctic are confronted by a shape-shifting alien that assumes the appearance of the people it kills.
Genre: Horror, Mystery, Sci-Fi
As Kevin Flynn searches for proof that he invented a hit video game, he is "digitalized" by a laser and finds himself inside The Grid, where programs suffer under the tyrannical rule of the Master Control Program. With the help of TRON, a security program, Flynn seeks to free The Grid from the MCP.
Based on the John Irving novel, this film chronicles the life of T S Garp, and his mother, Jenny. Whilst Garp sees himself as a "serious" writer, Jenny writes a feminist manifesto at an opportune time, and finds herself as a magnet for all manner of distressed women.
After a gentle alien becomes stranded on Earth, the being is discovered and befriended by a young boy named Elliott. Bringing the extraterrestrial into his suburban California house, Elliott introduces E.T., as the alien is dubbed, to his brother and his little sister, Gertie, and the children decide to keep its existence a secret. Soon, however, E.T. falls ill, resulting in government intervention and a dire situation for both Elliott and the alien.
Genre: Family, Sci-Fi
In the smog-choked dystopian Los Angeles of 2019, blade runner Rick Deckard is called out of retirement to terminate a quartet of replicants who have escaped to Earth seeking their creator for a way to extend their short life spans.
Country: Hong Kong, USA
Genre: Sci-Fi, Thriller
It's 1961, two years after the original Grease gang graduated, and there's a new crop of seniors and new members of the coolest cliques on campus, the Pink Ladies and T-Birds. Michael Carrington is the new kid in school – but he's been branded a brainiac. Can he fix up an old motorcycle, don a leather jacket, avoid a rumble with the leader of the T-Birds, and win the heart of Pink Lady Stephanie?
Genre: Comedy, Drama, Musical, Romance
When former Green Beret John Rambo is harassed by local law enforcement and arrested for vagrancy, the Vietnam vet snaps, runs for the hills and rat-a-tat-tats his way into the action-movie hall of fame. Hounded by a relentless sheriff, Rambo employs heavy-handed guerilla tactics to shake the cops off his tail.
A Christmas Truce
The Last American Virgin
The friendship of a group of young friends struggling with teen sex, drugs, and work is jeopardized by a romantic interest which may turn pals into bitter rivals.
Halloween III: Season of the Witch
Dr. Daniel Challis and Ellie Grimbridge stumble onto a gruesome murder scheme when Ellie's novelty-salesman father, Harry, is killed while in possession of a strange mask made by the Silver Shamrock mask company. The company's owner, Conal Cochran, wants to return Halloween to its darker roots using his masks — and his unspeakable scheme would unleash death and destruction across the country.
Rocky III
Now the world champion, Rocky Balboa is living in luxury and only fighting opponents who pose no threat to him in the ring, until Clubber Lang challenges him to a bout. After taking a pounding from Lang, the humbled champ turns to former bitter rival Apollo Creed for a rematch with Lang.
Genre: Drama, Sport
A trainer attempts to retrain a vicious dog that's been raised to kill black people.
A Midsummer Night's Sex Comedy
Centred around a weekend party at the home of inventor Andrew Hobbs and his wife Adrian, attended by randy doctor Maxwell Jordan, his nurse Dulcy, renowned philosopher Dr.Leopold Sturgis and his fiancée, this is a light comedy concerning their various emotional, intellectual and sexual entanglements, loosely based on Ingmar Bergman's 'Smiles of a Summer Night' .
Set in 1959, Diner shows how five young men resist their adulthood and seek refuge in their beloved Diner. The mundane, childish, and titillating details of their lives are shared. But the golden moments pass, and the men shoulder their responsibilities, leaving the Diner behind.
Stopping briefly in a small Texas town, an itinerant race car driver finds that his stock car, on a trailer behind his motor home, has just been quickly and expertly stripped. He chases down the miscreants, who turn out to be six orphan children. He has no recourse to the law, for the corrupt local sheriff takes most of the proceeds of their thievery in exchange for not putting them in an orphanage. They are charming rogues who are in turn charmed by him. Disliking their arrangement with the sheriff, they stow away with him, and he finds himself becoming a reluctant stepfather. Thanks to their enthusiasm and incredible mechanical know-how, he begins to make a name for himself on the racing circuit. But the sheriff doesn't take kindly to losing his extra income…
Andy is a new teacher at a inner city high school that is like nothing he has ever seen before. There is metal detectors at the front door and everything is basically run by a tough kid named Peter Stegman. Soon, Andy and Stegman become enemies and Stegman will stop at nothing to protect his turf and drug dealing business.
Victor Victoria
A struggling female soprano finds work playing a male female impersonator, but it complicates her personal life.
Genre: Comedy, Music, Musical, Romance
A nebbish of a morgue attendant gets shunted back to the night shift where he is shackled with an obnoxious neophyte partner who dreams of the "one great idea" for success. His life takes a bizarre turn when a prostitute neighbour complains about the loss of her pimp. His partner, upon hearing the situation, suggests that they fill that opening themselves using the morgue at night .
On another planet in the distant past, a Gelfling embarks on a quest to find the missing shard of a magical crystal and restore order to his world, before the grotesque race of Skeksis find and use the crystal for evil.
Barbara Hershey stars as Carla Moran, a hard-working single mother until the night she is raped in her bedroom by someone – or something – that she cannot see. Despite skeptical psychiatrists, she is repeatedly attacked in her car, in the bath, and in front of her children. Could this be a case of hysteria, a manifestation of childhood sexual trauma, or something even more horrific?
Genre: Biography, Drama, Horror
Dr. Alec Holland, hidden away in the depths of a murky swamp, is trying to create a new species – a combination of animal and plant capable of adapting and thriving in the harshest conditions. Unfortunately he becomes subject of his own creation and is transformed. Arcane, desperate for the formula, attempts to capture the Swamp Thing. An explosive chase ensues that ultimately ends with a confrontation between Holland and a changed Arcane…
Genre: Horror, Sci-Fi
After fifteen years of marriage, an affluent couple divorce and take up with new partners.
A film adaptation of the classic sword and sorcery hero, Conan the Barbarian. A horde of rampaging warriors massacre the parents of young Conan and enslave the young child for years on The Wheel of Pain. As the sole survivor of the childhood massacre, Conan is released from slavery and taught the ancient arts of fighting. Transforming himself into a killing machine, Conan travels into the wilderness to seek vengeance on Thulsa Doom, the man responsible for killing his family. In the wilderness, Conan takes up with the thieves Valeria and Subotai. The group comes upon King Osric, who wants the trio of warriors to help rescue his daughter who has joined Doom in the hills.
Based on the real-life experiences of Ed Horman. A conservative American businessman travels to a South American country to investigate the sudden disappearance of his son after a right-wing military takeover. Accompanied by his son's wife he uncovers a trail of cover-ups that implicate the US State department which supports the right-wing dictatorship.
Genre: Biography, Drama, History, Mystery, Thriller
Pink Floyd: The Wall
A troubled rock star descends into madness in the midst of his physical and social isolation from everyone.
Genre: Animation, Drama, Fantasy, Music, Musical
The Dorm That Dripped Blood
A crazed killer stalks college students who gave up their vacation to clean a deserted dormitory.
A Polish contractor, Nowak, leads a group of workmen to London so they can provide cheap labor for a government official based there. Nowak (Irons) has to manage the project and the men as they encounter the tempations of the West and loneliness and separation from their families. Nowak is the only one of the group who speaks English, and he uses this as a tool over his team. When the unrest in Poland leads to a military takeover, Nowak is faced with a much more difficult situation than he expected.
Zack Mayo is a young man who has signed up for Navy Flight School. He is a Navy brat who has a bad attitude problem. Sgt. Foley is there to train and evaluate him and will clearly find Zack wanting. Zack meets Paula, a girl who has little beyond family and must decide what it is he wants to do with his life.


Bhatt U, Walker D, Raynolds M, Bieniek P, Epstein H, Comiso J, Pinzon J, Tucker C, Steele M, Ermold W, et al. Changing seasonality of panarctic tundra vegetation in relationship to climatic variables . ENVIRON RES LETT. 2017;12 (5):17.


text:
THE city may have missed out on its traditional New Year's Eve fireworks, but the organisers of the Fun Fair Summer Splash plan to light up the sky this month. Organisers also say they are well-versed in running events in the time of COVID and know "what's needed to keep everyone safe". It will be the third Fun Fair in Bathurst in the past four months. The first Fun Fair - featuring showbags, carnival food, rides from Sydney, local market stalls and food vans - was held over two weekends last September. There was some criticism within Bathurst about how wise it was to hold the event, but Jade Evans, one of the Fun Fair owner/operators, said after the first weekend that the protocols had worked well. "The mayor came down on Saturday and we had the police come down to check everything was COVID safe and they were happy with it. We've passed with flying colours," she said. The second Fun Fair was held over two weekends last November and December. What is being called the Fun Fair Summer Splash, featuring fireworks, water slides, rides and food trucks, will be held at Bathurst Showground from January 23 to 26. IN OTHER NEWS: Bathurst Regional Council urged to build a new dam "We'll be the first fireworks in Bathurst since COVID, and we've worked with council on a plan to ensure we can bring the town a display that is not only spectacular but also COVID safe," Ms Evans said. She said the fireworks will be held on Saturday, January 23 and Tuesday, January 26. Entry to the event can be pre-purchased online (for four hours of unlimited rides - and including the firework displays) or for $35 at the gates. Entry to just the fireworks will be $5 and will include a voucher for a free ride or food item. "For those who haven't purchased a four-hour unlimited rides pass during the day of the 23rd or 26th, the $5 entry fee to see the fireworks is a nominal cost, which includes a free ride or food voucher, that allows us to restrict entry to the best viewing area so that we can monitor seating in a COVID safe way," Ms Evans said. "All seating spots will be mapped out in accordance with NSW Health and local council requirements." Ms Evans said the Fun Fair operators - made up of a group of families who usually work with large events like the Royal Easter Show, The Ekka and Melbourne Shows - are now well versed in running COVID safe events "and are already known to local authorities who have commended us on going above and beyond what's required". Organisers say the Fun Fair Summer Splash COVID safety plan includes maintaining 1.5m social distancing, wiping down rides frequently, staff wearing masks, allowance of four square metres per person for the fireworks display, customer numbers being monitored by three COVID marshalls and the event team, sanitiser stations on-site and available at entry, exit and at every ride and attraction, and contactless digital sign-in with Service NSW. They are also recommending the use of contactless payment options where possible. They say cash will be accepted, but EFTPOS and online facilities are preferred. Visit https://www.ticketebo.com.au/jubilee-amusements-au
/images/transform/v1/crop/frm/7PapGKjYPrPEgYfvAPt3Wq/5f38464d-2e50-4bcc-bc82-f0f592f9b0b1.jpg/r0_64_1280_787_w1200_h678_fmax.jpg
Fun Fair back for third time - and fireworks are planned
RETURNING: Bec Blundell and Abby Barnes at the Bathurst Fun Fair during its first visit to the Showground last September. Photo: PHIL BLATCH
THE city may have missed out on its traditional New Year's Eve fireworks, but the organisers of the Fun Fair Summer Splash plan to light up the sky this month.
Organisers also say they are well-versed in running events in the time of COVID and know "what's needed to keep everyone safe".
It will be the third Fun Fair in Bathurst in the past four months.
The first Fun Fair - featuring showbags, carnival food, rides from Sydney, local market stalls and food vans - was held over two weekends last September.
FUN TIME: Willow and Faith Evans with Lauren Foster at the Bathurst Fun Fair last September. The event will return to Bathurst Showground.
There was some criticism within Bathurst about how wise it was to hold the event, but Jade Evans, one of the Fun Fair owner/operators, said after the first weekend that the protocols had worked well.
"The mayor came down on Saturday and we had the police come down to check everything was COVID safe and they were happy with it. We've passed with flying colours," she said.
The second Fun Fair was held over two weekends last November and December.
What is being called the Fun Fair Summer Splash, featuring fireworks, water slides, rides and food trucks, will be held at Bathurst Showground from January 23 to 26.
IN OTHER NEWS:Bathurst Regional Council urged to build a new dam
"We'll be the first fireworks in Bathurst since COVID, and we've worked with council on a plan to ensure we can bring the town a display that is not only spectacular but also COVID safe," Ms Evans said.
She said the fireworks will be held on Saturday, January 23 and Tuesday, January 26.
Entry to the event can be pre-purchased online (for four hours of unlimited rides - and including the firework displays) or for $35 at the gates.
Entry to just the fireworks will be $5 and will include a voucher for a free ride or food item.
"For those who haven't purchased a four-hour unlimited rides pass during the day of the 23rd or 26th, the $5 entry fee to see the fireworks is a nominal cost, which includes a free ride or food voucher, that allows us to restrict entry to the best viewing area so that we can monitor seating in a COVID safe way," Ms Evans said.
"All seating spots will be mapped out in accordance with NSW Health and local council requirements."
Ms Evans said the Fun Fair operators - made up of a group of families who usually work with large events like the Royal Easter Show, The Ekka and Melbourne Shows - are now well versed in running COVID safe events "and are already known to local authorities who have commended us on going above and beyond what's required".
Organisers say the Fun Fair Summer Splash COVID safety plan includes maintaining 1.5m social distancing, wiping down rides frequently, staff wearing masks, allowance of four square metres per person for the fireworks display, customer numbers being monitored by three COVID marshalls and the event team, sanitiser stations on-site and available at entry, exit and at every ride and attraction, and contactless digital sign-in with Service NSW.
They are also recommending the use of contactless payment options where possible.
They say cash will be accepted, but EFTPOS and online facilities are preferred.
Visit https://www.ticketebo.com.au/jubilee-amusements-au


text:
Bellaspira hannyae é uma espécie de gastrópode do gênero Bellaspira, pertencente a família Drilliidae.

Drilliidae
Espécies descritas em 1988

redpajama_set_name: RedPajamaWikipedia


text:
Shazam! Wrath of the Gods – Trailer for a new movie. We were expecting such a great ad!
Dimitri Pavuchka is dead. The Russian was burned in the apartment
The match of the Polish national team with a great opponent is approaching. We got to know the plans!
Małgorzata Kożuchowska stopped the car wrongly. The star can get a fine and penalty points
CNN: Russia has, on some sections of the front in Ukraine, reduced artillery fire by about 75 percent
Sigmund Shortle January 10, 2023 2 min read
The US news channel CNN reported, citing its sources in Washington and Kiev, that the Russian army has reduced the intensity of its artillery fire by about 75 percent in some areas of the front line in Ukraine. According to CNN's interlocutors, this is further evidence of Russia's increasingly weak position on the battlefield.
CNN notes that US and Ukrainian officials are unable to explain this clearly. The TV station reported it Russia The rationing of artillery shells could be due to a lack of supplies, but this could also be part of a broader reassessment of tactics in the face of the successes of Ukrainian forces at the front.
tvn24.pl coverage: Russia's attack on Ukraine
In any case, the stark decrease in the intensity of artillery fire is further evidence of Russia's increasingly vulnerable position on the battlefield almost a year after the start of the regime's armed invasion. Ukraine CNN reported, citing its American and Ukrainian interlocutors. In their opinion, this is also associated with the growing military support for Ukraine from its Western allies.
CNN reported that United State And Germany announced last week that they would supply Ukrainian forces with armored fighting vehicles for the first time, in addition to another battery of Patriot missiles.
Kramatorsk after Russian bombing Reuters
Access to 40-year-old ammo
So far, questions about Russia's arsenal have mostly focused on precision-guided munitions such as cruise missiles and ballistic missiles. However, according to US officials, the very low rate of artillery fire may indicate that the long and brutal fighting has also had a significant impact on Russian conventional arms shipments.
See also Botox, steroids? This is how Putin's face changed during the war [WIDEO]
In December last year, a US military official, who asked not to be named, told Reuters that the Russians were seeking 40-year-old missiles because of the rapid depletion of their stocks.
"Our city was very beautiful." Interview in Bakhmut was interrupted by Russian bombingReuters
Various estimates
The decline in Russian artillery fire rate is not uniform, a US Defense Department official told CNN, adding that there are days when the Russians are still using more artillery shells, especially around and near eastern Ukrainian cities like Bakhmut and Krymina. Kherson in the south. Country.
American and Ukrainian officials, interviewed on American television, gave very different assessments of the bombing. The first claims that the number of missiles Russia uses in Ukraine has dropped from 20,000 to 5,000 a day. Ukrainians, in turn, estimate that this number has decreased from 60,000 to 20,000 per day.
Main image source: mil.ru
Sigmund Shortle
"Coffee enthusiast. Troublemaker. Incurable introvert. Subtly charming twitter scholar. Award-winning social mediaholic. Internet buff."
Previous United States of America. Missing Anna Walsh. Bloody knife found, husband accused of misrepresentation
Next China: Foreign Ministry spokesman Zhao Lijian dismissed. The end of "wolf diplomacy"?
Russia. Media: Dmitry Pavuchka, a high-ranking manager linked to Roscosmos, died in the fire. Another mysterious death in the Russian elite
January 26, 2023 Sigmund Shortle
The New York Times: Europe's Center of Gravity Shifts East
The "black alien" complains about his fate: sometimes they don't want to serve me in a restaurant
January 26, 2023 Rowena Robinson