Poisson Random Process

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Poisson Random Process
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Mean and Variance Results
Exponential
Poisson
Geometric
Binomial

• You have to memorize these!
• You should be able to derive any of the above

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CMPE 252A Computer NetworksSET 3

• Medium Access Control Protocols

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Medium Access Control Protocols

• Used to share the use of transmission media that
  can be accessed concurrently by multiple users.

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Contention-Based MAC Protocols

• No coordination Stations transmit at will when
  they have data to send (e.g., ALOHA)
• Carrier sensing (listen before transmit)
  Stations sense the channel before transmitting a
  data packet (e.g., CSMA).
• Listen before and during transmission Stations
  listen before transmitting and stop if noise is
  heard while transmitting (CSMA/CD).
• Collision avoidance (floor acquisition) Stations
  carry out a handshake to determine which one can
  send a data packet (e.g., MACA, FAMA, IEEE802.11,
  RIMA).
• Collision resolution Stations determine which
  one should try again after a collision.

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ALOHA Protocol

• The first protocol for multiple access channels
  the first analysis of such protocols (Norm
  Abramson, Univ. of Hawaii, 1970).
• Originally planned for systems with a central
  base station or a satellite transponder.

Two frequency bands Up link and down
link (413MHz, 407MH at 9600bps)
Central node retransmits every packet it receives!
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ALOHA Protocol

• Population is a large number of bursty stations.
• Each station transmits a packet whenever it
  receives it from its user no coordination with
  other stations!
• Central node retransmits all packets (good or
  bad) on down link.
• Stations decide to retransmit based on the
  information they hear from central node

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ALOHA Protocol

• An integral part of the ALOHA protocol is
  feedback from the receiver
• Feedback occurs after a packet is sent
• No coordination among sources

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The ALOHA Channel

• We assume
• An (essentially) infinite population of stations.
• An ideal perfect down link for the transmission
  of feedback to senders.
• Stations are half duplex have zero processing
  delays.
• Retransmissions are scheduled such that all
  packets are statistically independent.
• Each packet has the same duration P.
• Stations have the same round-trip delay from one
  other this time can be much longer than P
  (irrelevant).
• Packet arrivals are Poisson with rate lambda.
• Collisions are the only sources of errors.

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The ALOHA Channel
t
NEW
NEW

• What percentage of time is the channel sending
  correct packets?
• This gives us the throughput of the protocol.

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Throughput of ALOHA Protocol
Because arrivals are Poisson and all packets have
equal length, every packet has the same
probability of being successful.
Therefore,
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Throughput of ALOHA Protocol
packet overlaps with end of packet from node i
packet overlaps with start of packet from node i
All packets have the same length
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Throughput of ALOHA Protocol
Highest throughput when we have one packet for
each 2-packet time period
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Slotted ALOHA

• The throughput of ALOHA can be improved by
  reducing the time a packet is vulnerable to
  interference from other packets.
• Slotted ALOHA works in a slotted channel
  providing discrete time slots.
• Stations can start transmitting only at the
  beginning of time slots.
• The time synchronization needed for slotting is
  accomplished at the physical layer, and some
  synchronization is required in many cases anyway.

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Slotted ALOHA Protocol
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Throughput of Slotted ALOHA
The vulnerability period of a packet is a slot
time
Any arrivals in prior slot collide with packet i
If T is the duration of a time slot and G is the
normalized packet arrival rate, then
We can obtain the same result by computing the
likelihood and average length of utilization,
idle and busy periods.
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Slotted ALOHA
NEW
RET
...
collision
IMPORTANT The starting point of a busy period is
a renewal point! System is busy
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Renewal Theory

• Recall the Poisson random process
• N(t) number of arrivals in (0, t
• Inter-arrival times are exponentially distributed
• N(t) is a counting process with exponential
  inter-arrival times.
• Definition of Renewal Process
• A counting process N(t) for which
  inter-arrival times X1, X2, , Xn are an
  independent identically distributed (iid) random
  sequence.

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Poisson Random Variable
By assumption, whether or not an event occurs in
a subinterval is independent of the outcomes in
other subintervals. We have

• A sequence of n independent Bernoulli trials
• with X being the number of arrivals in (0, t

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Renewal Theory Example

• At each time t 1, 2, , a Bernoulli process
  N(t) has an arrival with probability p, and this
  is independent of the occurrence of arrivals at
  any other times.
• Is N(t) a renewal process?

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Renewal Theory Example

• Answer
• For any inter-arrival period n, the inter-arrival
  time Xn equals x if there were x-1 Bernoulli
  failures followed by a success.

Xn 3 if we have 2 failures followed by a
success!
1 2 3
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Renewal Theory Example

• Therefore, each inter-arrival time Xn has the
  Geometric PMF
• Because each Bernoulli trial is independent, Xn
  is independent of the previous inter-arrival
  times X1, X2 ,Xn-1.
• This implies that a Bernoulli arrival process is
  a renewal process!

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Renewal Theory

• After an arrival (in a renewal process), the
  subsequent inter-arrival times are distributed
  identically to the original inter-arrival times.
• Effectively, the process restarts, or has a
  renewal, whenever an arrival occurs!

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Renewal Theory

• Suppose that N(t) has n arrivals by time t1, the
  additional time until the next arrival is denoted
  by Sn1 - t1, and the subsequent inter-arrival
  times are Xn2 , Xn3 , and so on.
• Renewal Point For a renewal process N(t), time
  t1 with N(t1) n is a renewal point if Sn1 -
  t1, Xn2, Xn3, is an iid random sequence
  statistically identical to X1, X2, X3,
• Every instant of time is a renewal point for a
  Poisson process!

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Renewal Theory

• Alternating renewal process System is on
  and off (that is, or busy and idle).

system
success
collision
success
...
off on off on
off on off ...
Y1 X1 Y2 X2
Y3 X3 Y4 ...
X1, X2, X3,. are i.i.d. with mean E(X) Y1, Y2,
Yx,, are i.i.d. with mean E(Y) P(t)
Psystem is ON at time t in steady state
E(x)/E(x)E(Y) Average cycle length E(X)
E(Y)
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Evaluating Throughput

• We assume that the system is stationary, i.e.,
  system behaves in cycles that are statistically
  equivalent
• Average cycle consists of an idle period (I ) and
  a busy period (B ).
• The busy portion of a cycle has good and part
  parts.
• The portion of time used to send user data is
  called the utilization period (U )
  

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Evaluating Throughput

• The expression for S amounts to simply taking
  averages.
• What we need to do now is compute the probability
  that I, B (good a bad parts), and U happen in an
  average cycle, and their average duration.
• Ideally, these probabilities are based on
  independent events, and we can express S based on
  knowledge of the present state of the system.

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Throughput of Slotted ALOHA

• Idle, busy and utilization periods are multiples
  of time slots.
• We need to count the time slots in each average
  period and we are done.
• Average length of idle period
• I number of slots in idle period

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Idle Period in Slotted ALOHA
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Idle Period in Slotted ALOHA
This corresponds to the Geometric r.v., and we
know its average value to be 1/p, with p being
the probability of success.
Success now consists of ending the idle period!
Therefore
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Busy Period in Slotted ALOHA

• We follow the same approach
• Solve the problem with the Geometric random
  variable

prior slot considered in idle period
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Utilization Period

• Here we have to make use of conditional
  probability!
• A busy period has good and bad time slots
  (transmission periods).

The probability that a slot (transmission period)
in the current busy period is successful is the
probability that only one packet arrives in the
prior slot, given that there is a busy period
Arrivals are Poisson, so we make use of the
definition of that random variable as follows.
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Utilization Period
The probability that a given slot within a busy
period is successful is
The portion of an average busy period used to
send useful data equals the length of the average
busy period in slots, times the probability that
any given slot is successful.
We can use the Binomial random variable to proof
the above!
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Throughput of Slotted ALOHA

• We now just substitute B,I, and U in S

Maximum throughput is twice that of ALOHA. This
occurs when G 1
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Average Delay of MAC Protocols

• We want to measure or compute the average time
  from the instant the first bit of a packet is
  first transmitted to the moment the last bit is
  received correctly at the destination.
• Assume that arrivals (of new and retransmitted
  data or control packets) to the channel are
  Poisson.
• Assume fully-connected networks.

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Average Delay in ALOHA
Assumptions
A satellite channel with propagation delay NxP,
where P is the packet length and NxP gtgt P A
retransmission is sent after an average backoff
time of BxP seconds.
Direct method
A packet is transmitted (G/S-1) times in error
(due to collisions) and each such transmission
wastes PNxP BxP seconds.
The last transmission is successful and must take
PNxP seconds. Therefore, the average delay
incurred is
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Average Delay in ALOHA
Indirect Method
Based on the fact that the success of a
transmission is independent of others, and
knowing how many times we have retransmitted does
not change the likelihood of success in the next
transmission! We use a diagram showing possible
states, probabilities of transition, and delay
incurred in that transition.
From the diagram. we obtain a number of
simultaneous equations that we solve to obtain
delay from START to END.
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Average Delay in ALOHA
Solving these two equations
The same method can be applied on the other MAC
protocols!
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Average Delay of ALOHA

• The delay increases exponentially with heavy
  load, which is not acceptable for real-time
  applications.

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CSMA Carrier Sense Multiple Access

• The capacity of ALOHA or slotted ALOHA is limited
  by the large vulnerability period of a packet.
• By listening before transmitting, stations try to
  reduce the vulnerability period to one
  propagation delay.
• This is the basis of CSMA (Kleinrock and Tobagi,
  UCLA, 1975)
• Many of the assumptions made for ALOHA are made
  now for CSMA.

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CSMA Protocol

• Assume non-persistent carrier sensing.
• Requires a maximum propagation delay much
  smaller than packet lengths!

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CSMA Throughput

• A virtual secondary channel used to send ACKs
  reliable and in 0 time!
• Same assumptions made for pure ALOHA analysis.
• All stations are at one propagation delay from
  each other and that equals

Arrivals are Poisson with average rate
Peer-to-peer communication No base station or
transponder Explicit feedback to sender!
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CSMA Protocol
t

• The big difference compared to ALOHA is that
  busy periods are bounded!

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CSMA Throughput
We can approximate
Length of average idle period (exponential
interarrivals)?
The probability that a packet is successful is?
(no packets can arrive within tau sec. after the
start of the packet!)
The average length of a utilization period is?
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CSMA Throughput
Pretty accurate for ltlt P
Substituting we have
More accurate estimation of S requires finding
the average length of B.
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CSMA Throughput
Note that the average length of B is determined
by the time between the start of the first and
the last packet in the busy period.
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CSMA Throughput

• Substituting we get

Approximate
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Slotted CSMA

• Non-persistent strategy.
• A slot lasts one maximum propagation delay.

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Computing the Throughput of Slotted CSMA
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Throughput of Slotted CSMA

• We follow the same approach as in slotted ALOHA
• B has k transmission periods, each of P t sec
• What happens in a transmission period depends
  only on the last time slot of the prior
  transmission period!

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Throughput of Slotted CSMA
We use the Geometric r.v. to obtain the average
number of transmission periods in B
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Throughput of Slotted CSMA

• We now substitute U, B and I in S

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CSMA Throughput

• Because prop. delay is much smaller than pkt
  length, slotted and pure CSMA have very similar
  performance.
• When MAC protocol requires small prop delays, we
  can use slotted version to predict performance of
  unslotted version.

Reminder These results are only an upper bound
on performance, because we did not take into
account the effect of ACKs sent from receivers!
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CSMA/CDCSMA with Collision Detection

• CSMA improves on the performance of ALOHA
  tremendously.
• The remaining limitation is that, once a packet
  is sent, feedback occurs a roundtrip time after
  the entire packet is transmitted.
• The solution to improve on the performance of
  CSMA is to listen to the channel while a packet
  is being sent.
• This is called collision detection.
• R.M Metcalfe and D.R. Boggs, Ethernet
  Distributed Packet Switching for Local Computer
  Networks, Comm. ACM, Vol. 19, 1976
  (Xerox PARC).

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CSMA/CD Protocol

• Non-persistent transmission strategy
• Collision detection serves as a NACK!

• Assumption are
• All stations hear one another
  
• Propagation delay is much smaller than packets

Station listens to channel while transmitting
Collision is detected when signals sent and
heard differ. Jamming signal sent to ensure all
stations know of the collision.
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Throughput of CSMA/CD
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Throughput of CSMA/CD

• Notes
• The average length of a bad busy period is much
  smaller than in CSMA because JltltP.
• This length is determined by the time between
  the first packet in the busy period and the first
  packet that interferes (in contrast, in CSMA, it
  is the last interfering packet that counts)

The utilization period is only that portion of a
packet transmission that has no overhead, that
is
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Throughput of CSMA/CD

• Substituting we get

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Throughput of CSMA/CD
Therefore