Cellular Systems Cellular Concepts The cellular concept was a ...

EE5401 Cellular Mobile Communications
EE5401 Cellular Mobile Communications
P
0
=
C
∑ − 1
n=
0
1 ⎛ λ ⎞
⎜ ⎟
n!
⎝ µ ⎠
n
1
1 ⎛ λ ⎞
+ ⎜ ⎟
C!
⎝ µ ⎠
C
1
⎛ λ ⎞
⎜1
− ⎟
⎝ µ C ⎠
The probability that the call will not have
immediate access to a channel (i.e. having nonzero
delay)
Pr( delay
> 0) = ∑ ∞ P
=
k
k C
C
A 1
= P
C!
A
1−
C
where A = λ / µ . This is the Erlang C formula.
- This probability of non-zero delay that can be
tolerated is also known to be the GoS parameter of
the Erlang C system.
- If no channels are immediately available, the call
is delayed, and the probability that the delayed
call is forced to wait more than t seconds is given
by the probability that a call is delayed, multiplied
by the conditional probability that the delay is
greater than t seconds (the delay threshold).
Pr(delay > t)
= Pr(delay > 0) Pr(delay > t delay > 0)
= Pr(delay > 0) exp[ −(
C − A)
t / H ]
- The average delay D for all calls in a queued
system is given by
∞
H
D = ∫ Pr(delay > t)
dt = Pr(delay > 0)
0
C − A
or the average delay for those calls which are
queue is given by H /( C − A)
.
0
Note: Proof for
Pr( delay > t delay > 0) = exp[ −(
C − A)
t / H ]
Consider C=1, this corresponding to the M/M/1 queue
model. The waiting time W consists of the service times of
the existing N customers in the system, ie.,
W = S1 + S2
+ L+
S N where S i follows exponential
distribution fS ( t)
= µ exp( − µ t)
and W is the sum of these N
i
independent random variables. Note that implicity we
assume that N > 0 .
λ λ n
Also can easily show that P n = (1 − )( ) , i.e. n follows a
µ µ
geometric distribution (memoryless property).
The pdf of W is then given by
Erlang(
n+
1, µ )
64748 4
∞
∞ n
( µ t)
−µ
t λ λ n
fW
( t)
= ∑ f
W | n
( t,
n)
Pn
= ∑ µ e ⋅ (1 − )( )
n=
0
n=
0 n!
µ µ
−(
µ −λ)
t
= ( µ − λ)
e
Note that µ = 1/ H , A = λH
, integrate this from t to ∞ will
obtain Pr( delay > t delay > 0)
For M/M/C queue, the system behaves as an M/M/1 queue
with higher service rate C µ (rather than µ ).
Institute for Infocomm Research 69 National University of Singapore
Institute for Infocomm Research 70 National University of Singapore