- Definitions:
- Arrival rate λ
=
number of
new messages
that arrives
per time unit
- Offered load (rate) G
=
number of
(old and new) messages
that contend for the channel
per time unit
- Throughput S =
number of messages
that are
successfully transmitted
per time unit
A message is successfully transmitted if its transmission does not collide with any other message transmission.
- Arrival rate λ
=
number of
new messages
that arrives
per time unit
- Simplifying assumptions:
- All packets have the
same length
(So the transmission time for any packet is the same duration)
- Use the
packet transmission time
as time unit
(So a packet is transmitted in 1 time unit)
- The packet arrival process
is a Poisson process with
rate λ
The offered load is also Poisson distributed with rate G
(We will see that the offered load G is dependent on the arrival rate λ)
- All packets have the
same length
- A comment on the assumption:
- It is acceptable to
assume that the arrival process
is Poisson distributed
(Arrival processes with infinite population can usually be approximated with a Poisson process
- However, the
offered load is
not Poisson distributed
The number of retransmissions is not a Poisson process
It is very complicated
The assumption that offered load G is Poisson distributed is an important simplifying assumption that allows us to keep the analysis simple....
- Result of this simplifying assumption:
- Probab[k transmissions in t time units] = (G × t)k e(−G × t)/(k!)
where:
- G = arrival rate (offered load)
- e = 2.71828... (base of natural log)
(This is the Poison probability distribution)
- It is acceptable to
assume that the arrival process
is Poisson distributed
- Offered load =
all packets that
contend
for the transmission medium
- Packets that
contend for the
transmission medium:
- Newly arriving packets
- "Old" (existing) packets
that were
unsuccessful in their
transmission
(A packet is unsuccessful when it collided with some other packet)
- Newly arriving packets
- Situation:
The offered load G consists of:
- Newly arriving packets
λ
- The fraction of unsuccessful transmissions of the offered load G: (1 - Ҏ[success]) × G
- Newly arriving packets
λ
- Therefore:
G = λ + (1 - Ҏ[success]) × G <==> G × Ҏ[success] = λ λ <==> G = ----------- Ҏ[success]
- When is a transmission successful:
- Consider a transmission x:
Observe that:
- A transmission that begins
between (t-1, t] will
will collide with x
when x begins its transmission at
time t
- A transmission that begins between (t, t+1] will will collide with x while x is still transmitting
- A transmission that begins
between (t-1, t] will
will collide with x
when x begins its transmission at
time t
- The transmission x
is successful
if and only if:
- There are no transaction attempts that begins (= arrives) during the time interval (t-1, t+1]
- Consider a transmission x:
- Throughput S achieved by the Aloha protocol:
- A transmission attempt is successful is there are 0 arrivals during the time interval (t-1,t+1] !
Given: # of transmission attempts = G / t.u. (offered load)
Throughput: # of packets transmitted = #successful transmission attemps = #transmission attempt × fraction that are successful = #transmission attempts × Probab[ a transmission attempt is successful ]
Therefore: Probab[ a transmission attempt is successful ] = Probab[ 0 arrivals in the period (t-1,t+1] ] = Prabab[ 0 arrivals in 2 time units ] (G × 2)0 e(−G × 2) = ---------------- 0! = e−2G
Hence: #packets transmitted = #transmission attempts × e−2G = G × e−2G
Result: Throughput achieved by AlohaS = G × e−2G
- Graph of Aloha throughput (S) vs. offered load (G):
Maximum achieved throughput:
- Maximum achieved for G = 0.5
- S(0.5) = 0.5 × e-1 = 0.5/2.71828 ~= 0.18
Maximum Channel utilization:
- The highest achievable throughput
is only 0.18 transmissions per time unit
Since 1 transmission takes 1 time unit, the maximum channel utilization is 0.18 = 18%
- Maximum achieved for G = 0.5
- Slotted transmission:
- Every transmission must begin at the start of a slot
- When is a transmission successful:
- Consider these
transmission attempts:
- The transmission x
is successful
if and only if:
- There are
no (other) transaction attempts
that begins
in the same slot
as the transmission x
- A transmission attempt will begins in the same slot as the transmission x if it arrived during the previous slot
In other words:
- A transmission in slot (t, t+1] is successful if there are 0 (other) arrivals in slot (t-1, t]
- There are
no (other) transaction attempts
that begins
in the same slot
as the transmission x
- Consider these
transmission attempts:
- Throughput S achieved by the Slotted Aloha protocol:
- A (slotted) transmission attempt is successful is there are 0 arrivals during the time interval (t-1,t] !
Given: #transmission attempts = G / t.u. (offered load)
Throughput: #packets transmitted = #successful transmission attemps = #transmission attempt × fraction that are successful = #transmission attempts × Probab[ a transmission attempt is successful ]
Therefore: Probab[ a transmission attempt is successful ] = Probab[ 0 arrivals the previous slot ] = Prabab[ 0 arrivals in 1 time units ] (G × 1)0 e(−G × 1) = ---------------- 0! = e−G
Hence: #packets transmitted = #transmission attempts × e−G = G × e−G
Result: Throughput achieved by AlohaS = G × e−G
- Graph of Slotted Aloha's throughput (S) vs. offered load (G):
Maximum achieved throughput:
- Maximum achieved for G = 1
- S(1) = 1 × e-1 = 1/2.71828 ~= 0.37
Maximum Channel utilization:
- The highest achievable throughput
is only 0.37 transmissions per time unit
Since 1 transmission takes 1 time unit, the maximum channel utilization is 0.37 = 37%
- Maximum achieved for G = 1