

\subsection{\bf Telephone Calls as a Birth Process---I}

\begin{slide}
\begin{itemize}
\item Consider example given on previous page.

\item Consider every second. 

\begin{description}
\item{$\bullet$} $\exists\ \mbox{probability}\ p = N/3600$
of receiving one call:  

\item{$\bullet$} $\exists$ essentially no probability of
receiving $>1$ call.
\end{description}

\item Thus, each second can be described by variable $y_i ( = 0\
\mbox{no,} = 1\ \mbox{one call})$ which has Poisson distribution with
parameter $\lambda=N/3600$.

\item The number of calls in time $t$ seconds is ${\displaystyle
\sum^t_{i=1}} y_i$ which from additivity is also Poisson with parameter
${\displaystyle \sum^t_{i=1}} \lambda_i = pt$.

\item So
\[ 
P_k(t) = e^{-pt} \frac{(pt)^k}{k!} 
\]
is probability of receiving $k$ calls in $t$ seconds.  
\end{itemize}
\end{slide}
