Exercise 7
M/M/
· · ·
Queueing and LossSystems
1. A pure loss system with negativ exponentially distributed interarrival times and
service times has exactly
n
service units.
Determine the probability of exactly
k
= 0
, . . . ,n
service units being occupied.
What will be the traﬃc level and the loss probability for this station?
2. A processor is connected to the receiving end of a data transmission line.
The
processor is able to buﬀer
K
symbols from this line.
The arrival process on this line is a Poisson process with the rate
λ
, and the service
time needed for each symbol is negative exponetially distributed with the completion
rate
μ
.
Symbols are lost if they meet a full buﬀer at arrival.
Calculate the state probabilities
p
k
, the waiting probability
p
w
, the loss probability
p
b
and the traﬃc level
Y
with
K
= 2 buﬀer places and a traﬃc load of
ρ
= 1.
3. Consider a queueing loss system M/M/n–s/FCFS with
n
service units and
s
buﬀer
places.
(a) Show the state space with all state transitions and transition probability densi
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 Spring '09
 haiza
 Poisson Distribution, Probability theory, communication networks, service units, state probabilities, state probabilities pk

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