Consider two different internet routers denoted by A and B. Suppose that traffic arriving at each router follows a Poisson process with a rate of e packets per second. Let Tkdenote the k th
packet interarrival time and let Yk= T1+ T2+ ... + Tkdenote the time of the k th packet arrival.
(a)Determine the probability that Tk < c.
(b) Let Xk= 1 if Tk< c and Xk= 0 otherwise. What type of random process is the sequence of random variables X1,X2,X3,...?
(c) Due to a design flaw, when router A is first installed, it crashes if 6 or more of the first 10 packets have interarrival times smaller than c . Give an expression for the probability that a newly installed router crashes.
(d) Let K be a random variable that indicates the index of the first interarrival time that is smallerthancseconds.Forexample,if T1 ?c, T2 ?c, T3 <c, and T4 <c,then K=3. Determine E[K ], i.e., the expected number of arrivals until the first gap in traffic of less than c seconds.
(e) Due to a design flaw, router B automatically blocks all packet arrivals for d seconds after the first interarrival time that is smaller than c seconds. Therefore, using the definition of K in part (d), YKdenotes the time that router B starts to block packets and YK + d denotes the time that it stops blocking packets. Give an expression for E[YK +d|K].
(f) DetermineE[YK+d]=E?E[YK+d|K]? usingthelawofiteratedexpectation.In other
words, determine the expected time at which router B stops blocking incoming packets.
(g) After router B stops blocking incoming packets, what is the distribution of the amount of time until the next packet arrives? (Hint: what is the distribution of the time between
YK +d andYK+1).
(h) Determine the distribution of the number of blocked packets given that router B blocks packets for dseconds.
packet interarrival time and let Yk= T1+ T2+ ... + Tkdenote the time of the k th packet arrival.
(a)Determine the probability that Tk < c.
(b) Let Xk= 1 if Tk< c and Xk= 0 otherwise. What type of random process is the sequence of random variables X1,X2,X3,...?
(c) Due to a design flaw, when router A is first installed, it crashes if 6 or more of the first 10 packets have interarrival times smaller than c . Give an expression for the probability that a newly installed router crashes.
(d) Let K be a random variable that indicates the index of the first interarrival time that is smallerthancseconds.Forexample,if T1 ?c, T2 ?c, T3 <c, and T4 <c,then K=3. Determine E[K ], i.e., the expected number of arrivals until the first gap in traffic of less than c seconds.
(e) Due to a design flaw, router B automatically blocks all packet arrivals for d seconds after the first interarrival time that is smaller than c seconds. Therefore, using the definition of K in part (d), YKdenotes the time that router B starts to block packets and YK + d denotes the time that it stops blocking packets. Give an expression for E[YK +d|K].
(f) DetermineE[YK+d]=E?E[YK+d|K]? usingthelawofiteratedexpectation.In other
words, determine the expected time at which router B stops blocking incoming packets.
(g) After router B stops blocking incoming packets, what is the distribution of the amount of time until the next packet arrives? (Hint: what is the distribution of the time between
YK +d andYK+1).
(h) Determine the distribution of the number of blocked packets given that router B blocks packets for dseconds.
