Problem

Consider a discrete time queuing system to which messages arrive in accordance
with a Poisson point process at intensity [messages/sec]. The n-th arriving message,
n >= 1, requests the system to provide a service that is equal to Sn [sec/message].
Thus, it wishes to occupy the processor of the system for Sn seconds. Assume the
process {Sn, n = 1, 2, 3, . . . } to be a sequence of independent identically distributed
(i.i.d.) random variables, that is also independent of the arrival process. Assume
the service time distribution to follow an exponential distribution with mean service
time that is equal to 5 seconds. Thus, B(t) = P{Sn = 0).

(a) Calculate the average amount of cumulative service (also called work) requested
from the system over the period (0, t).

(b) Use the result in (a) to calculate the long term average amount of work per unit
time requested from the system (this is also known as the offered load rate).

(c) Calculate the variance of the amount of cumulative service (work) requested
from the system over the period (0, t).

(d) Assume that it is known that the message arrival rate is equal to
??= 1 [message/sec], and that at least 10 messages have arrived to the system over the
period (0,10) sec. Calculate the probability that exactly 10 messages have arrived over this period.

(e) Given the information provided in (d), compute the probability that at least 10
messages will arrive also in the interval (10, 20) sec.

em (this is also known as the offered load rate).

(c) Calculate the variance of the amount of cumulative service (work) requested
from the system over the period (0, t).

(d) Assume that it is known that the message arrival rate is equal to
??= 1 [message/sec], and that at least 10 messages have arrived to the system over the
period (0,10) sec. Calculate the…