states, $1$ and $2,$ because the only time that ordering will take place

is when $Z_n=0,-1,-2,$ or $-3,$ in which case $2,2,4,$ and $4$ units

are ordered, respectively, leaving $x_n=2,1,2,1,$ respectively.

$($a$)$ Construct the $($one$-$step$)$ transition matrix.

$($b$)$ Use the steady$-$state equations to solve manually for the steady$-$

state probabilities.

$($c$)$ Now use the result given in Prob. $16\mathrm{.}5-2$ to find the steady$-$

state probabilities.

$($d$)$ Suppose that the ordering cost is given by $(2+2m)$ if an order

is placed and zero otherwise. The holding cost per period is $Z_n$

if $Z_{n}0$ and zero otherwise. The shortage cost per

period is $-4Z_n$ if $Z_n<mn\; id="EditorElement\_2434237">0</mn>$ and zero otherwise. Find the $($long$-$

run$)$ expected average cost per unit time.