Data Communications & Computer Networks. Careers Unlimited – Utah M/M/1 Questions I have problem to solve 4 question on queuing theory.for more details see

Data Communications & Computer Networks. Careers Unlimited – Utah M/M/1 Questions I have problem to solve 4 question on queuing theory.for more details see the attachment 1. Find the average buffer occupancy, in messages, at the outgoing link of a
concentrator for the following case (assume the infinite buffer, M/M/1 model is
valid) :
[1] 10 terminals, each generating on the average of one message every 4 sec,
are statistically multiplexed at the concentrator. The messages are 40 bits
long, on the average. The capacity of the outgoing link is C = 1000 bps.
2. A concentrator buffer holds a maximum of N messages. Modeling it as an M/M/1
queue with finite buffer, find the probability that messages are blocked in each of
the following cases. Compare results.
[1] N = 2, ρ=0.1
[2] N = 4, ρ=0.1
[3] N = 4, ρ=0.8
What is the probability the buffer is empty in each of these cases?
3. Consider a queue with state-dependent Poisson arrival rate λn and departure rate
µn. Show that the equation governing the state probabilities after statistical
equilibrium has set in is given by
(λ + µ ) = µ +1 +1 + λ −1 −1 ≥ 1
Show the solution to this equation is given by
=
λ0 λ1 λ2 … λ −1

µ1 µ2 µ3 … µ
4. The “queue with discouragement” is one in which the arrival rates decrease with
increasing queue size. Specifically, let λ = λ⁄( + 1) and µn = µ. Show that the
probability of state of the queue is given by
=
ρ −ρ
!
ρ =
λ
µ

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