$3.$ A single product inventory system with the storage capacity of $2$ units is replenished weekly using the following

policy: if there are $0$ units in stock, order up to the storage capacity; else, do not order. Weekly demands D are iid

rvs with the following pmf$.$ Any unmet demand is lost. Assume zero order replenishment time. For any week, the

ordering cost is $4$n$,$ where n is the number of units ordered during the week. The weekly storage cost is $03$m$,$ where

mis the number of units on hand at the beginning of the week $($before any ordering takes place$).$

d

$0$

$1$

$2$

$3$

P$($D $=$d$)0\mathrm{.}20\mathrm{.}30\mathrm{.}40\mathrm{.}1$

a$)$ Find the long$-$run fraction of time that the system is out of stock $($zero inventory$).(15$