- Recall: Slotted Aloha:
- Time is divided into
slots
- A transmission
can only begin
at the start of
a slot:
- When a message msg
arrives:
the message msg must wait until the start of a slot to be transmitted
- Time is divided into
slots
- Fact:
- Normally, we use:
- time unit = second
- Normally, we use:
- However:
- Expressions can often be simplified using a different time unit
- Time unit used in
Aloha analysis:
- 1 time unit =
length of
1 message
= length of 1 slot
- 1 time unit =
length of
1 message
- Throughput S:
S = p × G
Where: S = throughput G = offered load (arrival rate) p = fraction of transmissions that are successful = probability that a transmission is successful
- Converting S and
G for changing
time unit:
- G =
# messages that arrives
per time unit
- We may need to convert the value when the time unit changes
- Example:
- Suppose G = 5 arrivals per sec
- Length of
message =
2 sec
Then:
# arrivals = 5 in 1 sec (5/sec) = 10 in 2 sec = 10 per "length of message" G = 10 / time unit (with message length as time unit)
- G =
# messages that arrives
per time unit
- When is a transmission
successful:
- Consider a transmission x
that arrives
prior to
a slot s:
- A tranmission y
that arrive
during the
slot "s−1"
will collide with
the transmission x:
- Transmissions that
do not
arrive
during the
slot "s−1"
will not
collide with
the transmission x:
Or:
- Consider a transmission x
that arrives
prior to
a slot s:
- Conclussion:
- A transmission x
in slot s
is successful
if and only if:
- There are no (zero) message arrivals during the slot s − 1
- A transmission x
in slot s
is successful
if and only if:
- Throughput of
the slotted Aloha:
Throughput S = #successful transmissions per time unit (= slot) = Total # transmissions × fraction successful = G × fraction successful = p × G
p = fraction of successful transmission = probabability that a transmission in slot is successful = probability that no (0) message arrives in previous slot = Prob[ 0 arrivals in 1 time unit (t.u.) ]
Recall Poisson arrival process: e-Gτ (Gτ)k Prob[ k arrivals in τ t.u. ] = ------------- (G = arrival rate) k! (G × 1)0 e(−G × 1) Prob[ 0 arrivals in 1 t.u. ] = ---------------- 0! = e−G
Hence: Throughput S = p × G = e−G × G = G e−G
- Summary: throughput of the
slotted Aloha protocol
S = G × e−G (Slotted Aloha)
- Graph of Slotted Aloha's throughput (S) vs. offered load (G):
- Previously:
Throughput S = G × e−G (Slotted Aloha)
- The maximum can be found through
differentiation:
S = G × e−G S' = e−G - G × e−G
Solve: S' = 0 ==> e−G - G × e−G = 0 <==> 1 - G = 0 <==> G = 1
- The maximum achievable
throughput in the
unslotted Aloha network is:
S(1.0) = 1.0 × e-1.0 = 1.0 × 0.368 = 0.37Maximum channel utilization with slotted Alpoha = 37% (not very good)