In P$2$P systems, user cooperation is a key element to achieve good system performance. In this

problem, we will study how incentive schemes improve system performance. Lets consider a P$2$P

file sharing system. Assume there is one seed, who has the complete file of interest, in the system

at the beginning with service rate $$s$.$ Leechers, who want to download the file, arrive the system

in accordance with a Poisson process of rate $\backslash$lambda $,$ and denote by $$l the service rate of leechers.

Further, assume resource is fair shared by all leechers. That is if there are K leechers in the

system, the seed will provide each leecher a service rate $$s

K $.$ Finally, assume the file download

time of a leecher is exponential distributed with mean $1$

$$a if the leecher receives an aggregate

service rate $$a$.$

$($i$)$ If the system doesn$$t adapt any incentive scheme, leechers will not provide service to system

$($to other leechers$)$ and they will leave the system as soon as they finish their downloads.

Model this system as a CTMC by defining states and sketch the corresponding state transition diagram.

$($ii$)$ Now we enforce an incentive scheme so that leechers will provide a portion, say $\backslash$alpha $,$ of their

service rates to the system $($fair shared by all other leechers$)$ but they still leave the system

after they download the file. Model this system as a CTMC by defining states and sketch

the corresponding state transition diagram.

$($iii$)$ Find the stability condition of the two systems $($with and without incentive scheme$).$

$($iv$)$ Let $$s $=15,$l $=10,\backslash$alpha $=0\mathrm{.}2.$ Assume now there are $5$ leechers in the system. Compute

the expected time until all leechers finish their downloads for both systems.

$($v$)$ Comment how incentive schemes improve system performance in terms of system stability

and file download delay.

$($iv$)$ Now we adapt a good incentive scheme so that leechers not only provide a portion of their

service rates to the system but also stay in the system $($become seeds$)$ for an exponential

time with mean $1$

$\backslash$beta after they download the file. Model this system as a CTMC by defining

states and sketch the corresponding state transition diagram.

$($Hint: Now the number of seeds in the system is not fixed. You need to define states that

incorporate seed dynamics.$)$

$3.$ Consider the following Markov Chain $($MC$),$ whose transition diagram is shown in Figure $1.$ For

state $3...8,$ each state will jump to any of its neighbors with equal probability. $($Note that state

$3$s neighbors are $4$ and $8.$ You cannot go from state $3$ to state $1.)$