3
$\begingroup$

I am reading the famous paper by Halfin and Whitt, [1]. I'd like to prove remark (1) on page 575. The authors state

\begin{align} \frac{\beta \alpha}{(1-\alpha)} = \frac{\phi(\beta)}{\Phi(\beta)} \Rightarrow \alpha = \frac{\phi(\beta)}{\beta \Phi(\beta) + \phi(\beta)}, \end{align}

where $\phi(x)$ is the standard normal pdf and $\Phi(x)$ the standard normal cdf. The authors want to have bounds on $\alpha$, stating

\begin{align} 1 - \Phi(\beta) \le \alpha \le \frac{1-\Phi(\beta)}{1-\beta^{-2}\Phi(\beta)}, \quad \beta \ge 1. \qquad \qquad (1) \end{align}

The authors state that they use (1.8) on page 175 of Feller [2]. This equation reads

\begin{align} (x^{-1} - x^{-3})\phi(x) < 1 - \Phi(x) < x^{-1} \phi(x), \quad x > 0. \end{align}

By manipulating this equation I obtain

\begin{align} \frac{\phi(\beta)}{\beta^{-2}\phi(\beta) + \beta} < \alpha < \beta^{-1} \phi(\beta), \quad \beta > 0. \qquad \quad \quad (2) \end{align}

Both bounds $(1)$ and $(2)$ are indeed valid for their respective domains. Clearly, the difference for these bounds has to do with the fact that $\beta \ge 1$ in the first, and $\beta > 0$ in the second.

I do not know how to obtain $(1)$, does anyone else have an idea?

[1] S. Halfin, and W. Whitt, Heavy-traffic limits for queues with many exponential servers. PDF

[2] W. Feller, An introduction to probability theory and its applications, Vol. I, Ed. 3.

$\endgroup$
1
$\begingroup$

For every $x\gt0$, define $\varphi_-(x)$ and $\varphi_+(x)$ by $$ 1 - \Phi(x)=(x^{-1} - x^{-3})\varphi_+(x),\qquad 1 - \Phi(x) = x^{-1} \varphi_-(x). $$ For every $x\gt1$, $\varphi_-(x)\gt0$ and $\varphi_+(x)\gt0$, and Feller's (1.8) implies that $\varphi_-(x)\lt\varphi(x)\lt\varphi_+(x)$ (but if $0\lt x\lt1$ the factor $x^{-1}-x^{-3}$ reverts the inequality sign related to $\varphi_+(x)$). The definition of $\alpha$ and the fact that, for every $\beta\gt0$, the function $$u\mapsto\frac{u}{\beta\Phi(\beta)+u} $$ is nondecreasing on $u\gt0$ (while if $\beta\lt0$, the function is not defined on the whole interval $u\gt0$), yield $$ \frac{\varphi_-(\beta)}{\beta\Phi(\beta)+\varphi_-(\beta)}\lt\alpha\lt\frac{\varphi_+(\beta)}{\beta\Phi(\beta)+\varphi_+(\beta)}. $$ The LHS and the RHS of this inequality coincide with the LHS and the RHS in (1).

$\endgroup$
  • $\begingroup$ Thank you for your answer. Just one question. How does it follow from Feller (1.8) that $\varphi_-(x) < \varphi(x) < \varphi_+(x)$? $\endgroup$ – user60307 Jan 10 '14 at 8:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy