The number of packets received at a router input during an interval of duration t seconds is modeled by a Poisson random variable. The probability of arriving k packets in the interval is where parameter β is the packet arrival rate, measured in packets/second. a)Write the equation of the probability density function of the random variable of Poisson .

b)Show that the arrival time between packets T is in this case an exponential random variable and determine the value of the parameter.

c)If the packets arrive with an equal speed of 2048 kilobits/second and the packet size is equal to 1024 bits determine the value of β.

d)Determine the value of the duration t of the observation interval so that the probability of no packet arriving is greater than 0.9.

e)Determine the duration t value of the observation interval so that the probability of arriving at least one packet is greater than 0.9.