Inventory model I: A retailer sells headphones one at a time according to demand which

forms a Poisson process at rate $\backslash$lambda : At Poisson arrival time tn $($n

th demand request$),$ the

inventory drops by $1$ if the inventory is non$-$empty. If the inventory is empty at a request

time, then nothing happens, that demand request is $$lost$.$ The amount in inventory

starts off as B $>=2.$ As soon as the Inventory drops down to $0,$ it will be re$-$stocked up to

B after an exponential amount of time L $($lead time$)$ at rate $\backslash$gamma $,$ independent of the past.

Again: during those L time units, all demand is lost. Let X$($t$)$ denote the inventory level

at time t$.$ The state space is thus S $=\{0,1,...,$ B$\}.$

$($a$)$ Argue that $\{$X$($t$)\}$ forms a CTMC$,$ and find both the holding time rates aj and the

embedded MC transition matrix P $=($Pi$,$j $).$

$($b$)$ Explain why $\{$X$($t$)\}$ is not a birth and death process meaning that X$($t$)$ can sometimes

make jumps $($change of state$)$ larger than size $1.$

$($c$)$ Solve the balance equations for the limiting distribution.

$($d$)$ Find the long$-$run average inventory level;

lim

t$->\backslash$infty

$1$

t

Z t

$0$

X$($s$)$ds$.$

$($e$)$ What proportion of demand is lost?

$($f$)$ What proportion of demand causes a re$-$stock