Slotted
Aloha
- synchronous
system: time divided into
slots
- slot
siz equals fixed packet transmission
time
- when
Packet ready for transmission, wait until start of next slot
- packets
overlap completely or not
at all
Slotted
Aloha performance
S
= G*prob[no other transmissions overlap]
=
G*prob[0 other attempted transmissions]
=
G*prob[0 other arrivals in previous slot]
=
Ge**(-G)
·
Aloha
is inefficient (and rude!):
doesn't listen before talking!
·
Carrier
Sense Multiple Access: CSMA
In
our analysis the access methods
to control multichannel system
and the data multichannel system
is based on slotted ALOHA protocol.
Both control and data channels
use the same time reference
which we call cycle. We define as cycle, the time
interval that includes one time
unit for control packet transmission
followed by a data packet transmission
period. Thus the cycle time
duration is T=L+1 time units
as Figure 2 shows. The stations
are synchronized for the transmission
on the control and data packet
during a cycle. A station generating
or retransmitting a data packet,
waits the beginning of the next
cycle, selects randomly one
of the v wavelengths
_{c1},...,
and sends a control packet over
the
_{cm }control
channel at the first time unit
of the
cycle. The control packet, as
Figure 2 displays, is consisting
of the transmitter address,
the receiver address and the
wavelength,
_{dk}, of the data channel. Immediately
after transmission of the control
packet the data packet is ransmitted
over
_{dk} wavelength data channel.
Figure
2.
Structure of the control packet
and the control multislot
TOP
If
more than one data packets in
a cycle use the same
_{dk},
a collision occurs and all overlapping
data packets are destroyed.
In case of unsuccessful (re)transmissions
on a control or a data channel
of the system, the stations
participating in it defer their
retransmissions for a random
time until successful retransmission.
The random time delay introduced
between two consecutive retransmissions
is uniformly distributed from
1 to K cycle times. In the receiving
mode a station's fixed tuned
receivers monitor the control
multichannel system to listen
its address. When the destination
station recognizes its address
on a
control channel of the system,
it tunes its tuneable receiver
over wavelength
_{dk} of
data channel for data packet
reception. If two or more successfully
(re)transmitted data packets
have the same destination during
a cycle, a receiver collision
occurs which is ignored in our
analysis. We also assume that
the average propagation delay
is negligible.
Analysis
Let G = the mean
offered traffic over (v-channel)
control multichannel system
in a control slot time, that
obeys to Poisson statistics.
G_{v} =
G/v, the traffic to j-th control
channel j
{1,2,...,v} given that each control channel is chosen
with equal and constant probability
pj = 1/v
P_{c} =
the probability of one Poisson
arrival in a control channel
during a time unit. Then
P_{c} =
G/v e^{-G/v} (1)
A_{v} = random
variable representing the number
of successfully (re)transmitted
control
packets in a control
slot time. Thus the probability
of finding A_{v}=k control
channels everyone with one Poisson
arrival during a time unit,
obeys to binomial probability
low.
Pr[
A_{v}=k ] = [ v! / (v-k)!
k! ] ( P_{c} )^{k} [ 1 - P_{c} ]^{(
v-k )} (2)
S_{c} = the average successful (re)transmission
rate of control packets during
a control slot time in steady
state.
S_{c} = E( A_{v} = k) =
k Pr[ A_{v} = k] = Ge ^{-G/v} (3)
P_{s} = the probability of successful
(re)transmission of a control
packet over the control multichannel
system. From (3) we take
P_{s} =
S_{c} / G = exp[-G /
v ] (4)
Consider that
A_{v}=k data packets
are (re)transmitted over data
multichannel system in the data
slot time of the ith cycle.
The
random distribution of the k
data packets in N data channels
gives N^{k} arrangements
each with probability N^{-k}.
P_{d}(k)
= the conditional probability
that only one from k data packets
are destined to a given data
channel n, n{1,2,...,N} in
the data slot time of the i_{th} cycle.
Thus the remaining k-1 packets are destined to the remaining
(N-1) data channels in (N-1)^{k-1} different ways.
The
P_{d}(k) can
be expressed as follows.
P_{d}(k) = ( k/N ) [ 1 - 1/N ]^{k-1} (5)
Using the approximation (1 - x )^{y} ÷ exp(-xy) for small x in (5),
we take
P_{d}(k)
( k/N ) exp[-(k-1)/N] (6)
P_{d} = the probability that a data
packet is (re)transmitted without
collisions in a data slot of
a cycle time in steady state,
regardless of successful (re)transmission
of the corresponding control
packet on control multichannel
system. Then according (6) we
take
P_{d} =( G / N ) exp[-G / N] (7)
P_{suc} = the
probability of success of a
data packet over a data channel,
given that the
corresponding
control packet has successful
(re)transmitted.
P_{su}_{c} = P_{s} P_{d} = ( G / N ) exp[-G / v ] exp[-G
/ N] (8)
A_{N} = random variable representing
the number of successfully (re)transmitted
data packets in a data slot
time of a cycle. The
probability of finding A_{N} = r successfully (re)transmitted
data packets can be evaluated
as follows.
Pr[ A_{N} = r ] = [ N! / (N-r)! r! ] (
P_{suc} )^{r} [ 1
- P_{suc} ]^{(N-r)} (9)
S_{N} = the average number of successful
(re)transmissions of data packets
in a data slot time of a cycle
in steady state.
S_{N} = E( A_{N}=r
) =
r Pr[ A_{N}=r ] = G exp[-G / v ] exp[-G
/ N] (10)
From the above
equation results that the S_{N} is a function of G, v and N.
Thus for N=1 and v=1 we have
S_{N} = G exp(-2G). This
value coincides to the unslotted
ALOHA throughput. The explanation
is that a successful transmission
from the system requires successful
transmission from both control
and data channel.
S
= the throughput of the system
is defined as the average number
of successful (re)transmissions
of data packets during a cycle
time in steady state.
S = [ L / (L+1) ] S_{N} = [ L / (L+1) ] G
e^{-G/v} e^{-G/N} (11)
S_{nor} = S / N,
the throughput per data channel
in steady state.
D = the interval between the time a data packet is generated
and the moment that has been
successfully (re)transmitted
from a data channel. Data packet
delay D, is composed from three
parts as follows:
D = D_{w} + D_{r} + T (12)
D_{w} = the
interval between the time a data packet is generated till the beginning of
the next cycle.
D_{r} = the delay from the transmission
of a control packet, to the
successful reception of the
corresponding data packet from
a
data channel.
F_{r} = the probability of successful
(re)transmission of a data packet
over a data channel. From (10)
we take
F_{r} = S_{N} / G = e^{-G/v} e^{-G/N} (13)
Q_{r}(n)
= the probability of successful
retransmission of a data packet
after n trials.
Assuming that the
probability of success is the same on any try, n has a geometric
distribution, i.e.
Q_{r}(n) = F_{r} (1 - F_{r})^{n-1} (14)
R = the average number of trials for successful transmission of a data
packet. So
R
= E[ n ] = n
Q_{r}(n) = 1 / F_{r} = e^{G/v} e^{G/N} (15)
D_{b} = (K+1) / 2, t
he average delay between
successive retransmissions
The average packet
delay D, is
E [ D ] = E [ D_{w} ] + E [ D_{r} ] + T
= T / 2 + (R-1)(1 + D_{b})T
+ T = (L+1) [ 1.5 + [e^{G/v} e^{G/N}- 1 ][ (K+3)/2)]
(16)
TOP
Numerical
Results
In Figure 3,
we assume a multichannel system
with fix number of data channels,
N=10, and several control channels
v=5,10,15. The Figure illustrates
the average delay E[D] versus
the throughput per data channel
S_{no}_{r}.
It is evident from the figure
that the system performance
measures improves as v increases
for fix values of N. The performance
behaviour can be explained as
follows:
Figure
3The average delay E[D]
versus the throughput per data
channel S_{nor} for
v=5,10,15( control channel)
systems and N=10 data channels
with L=100 and K=10
If
we set the first derivative
of equation (11) with respect
of G equal to zero, we find
the optimal value of G_{opt} that maximize the throughput
of the system. After some calculations
we take
G_{opt} = v
N / ( v + N ) (17)
S_{max} = [ L / (L+1) ] 1/e v N / (
v + N ) (18)
The corresponding
S_{nor} is
S_{nor}(max)
= [ L / (L+1) ] 1/e [ 1 / (
1 + N/v ) ] (19)
From the above
results we observe that for
fix values of N as v increases
S_{nor}(max) increases
and for large values of v ,
S_{nor}(max)
approaches 1/e.
Figure
4 The average delay E[D]
versus the throughput per data
channel S_{nor} for N=5,10,15( data channel)
systems and v=10 control channels
with L=100 and K=10
Figure 4 present the average delay E[D] versus the throughput
per data channel S_{nor} fix number of control channels,
v=10,
and several data channels N=5,10,15.
It is obvious from the figure
that as N increases the performance
behaviour is getting
worse. Also equation (19) denotes
for fix value of v as N increases
S_{nor}(max)
deteriorates. Also Figures 3,4
show that the
system is unstable because there
are two different values of
average delay associated with
a given throughput.
Figure
5 shows the average delay E[D]
versus the throughput per data
channel S_{nor} for
v=10,N=5 (channel) system with
K=10,30,50. For fix value of
K, and in the lower part of
the curves the delay increases
very slowly with the throughput
showing the high throughput
and low delay desirable regions.
As traffic increases approaching
its optimum value G_{opt}=vN/(v+N)
which corresponds to S_{max}=[L/(L+1)]1/evN/(v+N),
and the average number of retransmissions,
R, becomes significant, then
the average delay, D_{b}=(K+1)/2,
between successive retransmissions
begins to make a noticeable
difference in the average delay
E[D] as Figure 5 shows.
Figure 5 The average delay E[D]
versus the throughput per data
channel S_{nor} for v=10,N=5 (channel)
system with K=10,30,50 |