Problem H$2\mathrm{.}3[10$ points$]$ Consider a system with two servers, no queue, and two types of jobs. Type $i,i=$

$1,2,$ jobs arrive at the beginning of a slot with probability $p_i.$ Both types of jobs require two units of service, i$.$e$.,$

if a job starts service at the beginning of time slot $k,$ it leaves at the end of slot $k+1,$ and both types of jobs

are indistinguishable once they start service. However, if jobs of both types arrive at the start of a time slot and

only one server is available, type $1$ jobs have precedence over type $2$ jobs, i$.$e$.,$ the free server is assigned to the

type $1$ job, while the type $2$ job is discarded. If both servers are busy when jobs of either type arrive, the jobs are

discarded.

a$)[3$ points$]$ Identify a state representation that will allow you to represent the evolution of the system over

time as a discrete time Markov chain.Describe that Markov chain by providing its state transition proba$-$

bility matrix $P.$

Hint: As this is a discrete time system, there are multiple possible state representations depending on when

we sample the system. Some might result in a slightly more complex state space, but may make it easier

to directly compute $($from the state probabilities$)E[N],$ while others might provide a more direct view of

blocking probabilities $($by capturing the state of the system as seen by arrivals$).$

b$)[3$ points$]$ Using the Markov chain from the previous question, provide expressions, functions of the sta$-$

tionary probabilities of the chain, for the probabilities $P{b}_i$ that an arriving job of type $i,i=1,2,$ is

discarded.

c$)[4$ points$]$ Assuming $p_1=0\mathrm{.}7$ and $p_2=0\mathrm{.}2,$ compute the values of $P{b}_1$ and $P{b}_2$ as well as the average

number jobs in the system $E[N].$