1
Purdue University
ECE
ECE
547
547
Introduction to Computer
Introduction to Computer
Communication Networks
Communication Networks
Instructor:
Instructor:
Xiaojun
Xiaojun
Lin
Lin
Lecture 11
Lecture 11
Purdue University
M/M/1 system
M/M/1 system
¾
P
n
= lim
t
→
∞
P
n
(t).
¾
P
n
(t+
Δ
t) = f(P
j
(t))
Question:
If at time t+
Δ
t the queue is in state n, then what
are the possible states the queue can be in at time t?
Probability of n packets in system at time t
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2
Purdue University
M/M/1 system
M/M/1 system
¾
Let n
≥
1
a)
At time t, Queue is in state
≥
n+2 or
≤
n2 with probability
o(
Δ
T)
b)
Queue is in state n at time t
a)
No arrivals or departure in (t, t+
Δ
T]
b)
1 arrival and 1 departure in (t, t+
Δ
T]
c)
Any other scenario is o(
Δ
T)
c)
Queue is in state n1 at time t
a)
One arrival and no departure
b)
o(
Δ
T)
d)
Queue is in state n+1 at time t
a)
No arrival and 1 departure
b)
Other scenario is o(
Δ
T)
Purdue University
M/M/1 system
M/M/1 system
¾
P
n
(t +
Δ
t) =
o(
Δ
T) + P
n
(t)[
μΔ
t
λΔ
t + (1
λΔ
t)(1
μΔ
t) +o(
Δ
t)]
+ P
n1
(t)[(1
μΔ
t)
λΔ
t + o(
Δ
t)]
+ P
n+1
(t)[(
μΔ
t)(1
λΔ
t) + o(
Δ
t)]
¾
Simplifying the above and incorporating all (
Δ
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 Spring '09
 XIAOJUNLIN
 pn, Purdue University

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