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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