Delay%20models%20in%20Data%20Networks

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Chapter 3

• Delay models in Data Networks

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Section 3.2

• Littles Theorem

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3.2 Littles Theorem

• average number of customers in system
• ? mean arrival rate
• Tmean time a customer spends in system

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Littles Theorem

• Proof
• N(t) number of customers in system at time t
• ?(t) number of customers who arrived in
  interval 0,t
• Ti time spent in system by the i-th customer

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Littles Theorem
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Littles Theorem
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3.2.3 Application of Littles Theorem

• Ex3.1
• ? arrival rate in a transmission line
• NQ average number of packets waiting in queue
• W average waiting time spent by a packet in
  queue
• NQ ?W

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Application of Littles Theorem

• If average Tx time
• ? ? ?
• ? Average number of packets under Tx
• I.e. fraction of time that s busy utilization
  factor

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Application of Littles Theorem

• Ex3.2
• N average number packets in network
• T average delay per packet
• also
• Ti average delay of packets arriving at node i

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3.3 M/M/1 Queuing System

• M/M/1
• First M arrival , Poisson
• Second M service , Exponential
• 1 server number

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M/M/1 Queuing System

• Arrival Poisson process
• A(t) number of arrivals from 0 to time t
• Number of arrivals that occur in disjoint
  intervals are independent
• Number of arrivals in any interval of length ? is
  Poisson distributed with parameter ? ?,

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M/M/1 Queuing System

• Properties of Poisson process
• Inter arrival times are independent and
  exponentially distributed with parameter ?
• tn time of the n-th arrival

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M/M/1 Queuing System

• For every t?0,? ?0

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M/M/1 Queuing System

• A A1A2?AK is also Poisson with
• rate ? ?1 ?2? ?K

Poisson
A1
merge
A2
..
AK
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M/M/1 Queuing System

P
Also Poisson with ?P
split
Poisson
?
Poisson with ?(1-P)
1-P
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M/M/1 Queuing System

• Service time Exponential distribution with
  parameter ?
• Sn service time of n-th customer

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M/M/1 Queuing System

• Properties of Exponential memoryless

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Markov chain formulation

• Let's focus at the times,0,?,2?,,k?,
• Nk number of customers in system at time k?
  N(k?)
• Where N(t) is continuous-time Markov Chain
• Nk is discrete-time
• Let Pij transition probabilities
  PNk1jNki

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Markov chain formulation
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Markov chain formulation

• Note
• During any time interval, the total number of
  transitions from state n to n1 must differ from
  the total number of transitions from n1 to n by
  at most 1
• I.e. frequency of transitions from n1 to n
  frequency of transitions from n to n1

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Markov chain formulation
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Markov chain formulation

• Take ?-gt0 ? Pn?Pn1?
• ?Pn1?Pn, n0, 1, ????
• where ? ?/ ? utilization
• ? Pn1 ? n1P0, n0,1,
• Since ?lt1, and

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Markov chain formulation
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M/G/1 System

• Let Ci customer I
• Wi waiting time of Ci
• Xi service time of Ci
• Ni of customers found waiting in
  queue when Ci arrives
• Ri residual service time of the
  customer in service when Ci arrives

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M/G/1 System
Xi- Ni
Ri
Xi-1
Ci start service
Ni
Ci arrives
In steady-state,
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M/G/1 System

• To calculate R, by graphical approach

Residual service time r(?)
M(t) of service completion in 0, t
X2
X1
XM(t)
Time ?
X2
X1
XM(t)
Ci starts service
t
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M/G/1 System

• Time avg of r(?) in 0, t

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M/G/1 System
P-K Formula (3.53)
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Ex3.15

• Consider a go back n ARQ

1
2
3

n-1
n
1
sender
time
Timeout (n-1) frames
1
2
3
time
receiver
Prop. delay

• Assume that error in the forward channel is p,
  return channel is error-free
• Packet arrives as a Poisson process with rate ?
  packets/frame

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Ex3.15

• Service time X from when a packet
  transmitted until it is successfully
  received
• 1 , if 1st tx is successful ?(1-p)
• X 1n, if 1st tx is un- successful 2nd is
  successful ? p(1-p)
• 1kn, if 1st k is un- successful(k1)th
  successful ? Pk(1-p)

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Ex3.15
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Ex3.15
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3.5.1 M/G/1 Queue with vacations

1. When the server has served all customers, it goes
   on vacation
2. If the system is still idle after a vacation
   interval, go on another vacation interval
3. If a customer arrives during a vacation, customer
   waits until the end of vacation. Chapter 1
   section 1.3.1 page 34in Network or Transport Layer

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M/G/1 Queue with vacations
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M/G/1 Queue with vacations

• Assume vacation intervals v1, v2 are iid and are
  independent of customers arrival service times.
• ?A customer must wait for the completion of the
  current service or vacation interval, and then
  the service of all customers waiting before it.

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M/G/1 Queue with vacations

• Where R is the mean residual time for completion
  of service or vacation when the customer arrives.

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M/G/1 Queue with vacations

• Let L(t) of vacations completed by t
• M(t) of services completed by t

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Because Fraction of time occupied with vacation
1-?
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Ex3.16 FDM, SFDM, TDM

• m streams of traffic with rate ?/m(Poisson)
• FDM system Divide available bandwidth into m
  subchannels. Transmission time for a packet on
  each of these subchannels is m.

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FDM
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Slotted FDM System

• Packet trans starts only at time m, 2m,When the
  queue is idle, server takes a vacation of m. (if
  idle again after vacation, take another)

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TDM System

• Look at SFDM queue, -gtsame queue
• WTDMWSFDM

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Summary
Service time
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Reservations Polling
Satellite
Collision -gt solutionpolling or reservation

S1
D1
D1
S2
D2
S1
D1
D1
S2
D2
Cycle
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Reservation Polling

• M Poisson traffic streams with rate ?/m
• Gated System only those packets which arrive
  prior to the users preceding reservation period
  are transmitted.
• Exhaustive system all packets are transmitted
  including those that arrive during this data
  period
• Partially gated all packets that arrive up to
  the beginning of the data interval.

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Single-User

• Gated system

m1
Di starts
Di arrives
Wi
time
S
D
D
S
D
D

D
Di ends tx
Ri
Vl(I)
l(i)-th reservation interval Ni of packets
arrive in front of Di
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Single-User

• A reservation(vacation)starts when the system has
  served all packets which arrive prior to the
  preceding reservation interval.
• A vacation(M/G/1 queue with vacation) starts when
  the system has served all packets which have
  arrived.(corresponds to exhaustive system)

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Single-User
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Single-User
Single-user gated
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Multi-User
Packet i starts
Packet i arrives
Wi
time
S
D
D
D

S
D
D

S
D
D
Ri
Pakcet i ends
Ni
SumYi

• Ni is redefined as of packets which must be
  transmitted before packet i

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Multi-user

• Where Yi includes all reservation intervals
  packet I must want for.

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Multi-User

• If number the users 0, 1, 2,,m-1, the l-th
  reservation interval is used to make reservation
  for user l mod m

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Multi-user
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Multi-user

• For an exhaustive system
• Let ?ljE ( Yj packet i arrives in user ls
  reservation or data intervals and belongs
  to user (lj) mod m)

Packet i belongs to each user with same prob.
1/m
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Multi-user
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Multi-user

• All users have equal average data length in
  steady state.
• P(packet i arrives during user ls data interval)
• P(packet i arrives during user ls reservation
  interval)

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Multi-user
(Yipkt i arrives in user ls data or reser.
int.) X P(pkt i arrives in user ls reser. Or
data int. )
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Multi-user

• If Vls have same dist.

Exhaustive system (3.69)
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Multi-user

• - The partially gated system is the same as the
  exhaustive system except that if a packet arrives
  in its own users data interval (with prob. ?/m),
  it is delayed an extra cycle of reservation
  periods(mV)
• Y is increased by

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Multi-user

• The fully gated system is the same as partially
  gated system except if a pkt arrives during a
  users own reservation interval (prob. (1-?)/m)
• It is delayed by an additional mV
• Y is increased by

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Priority Queuing

• N classes of customers class i arrives a Poisson
  process with rate ?I
• service time
• Each class joins a separate queue

?1
Server
?2
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Priority Queuing

• Single server will server customers from the
  highest priority queue first
• Non-preemptive
• - a lower priority customer, once started, is
  allowed to finish, when a high priority customer
  arrives.
• Preemptive resume
• - Service for a low priority customer is
  interrupted when a high priority customer arrives
  and is resumed from the point of interruption
  when all higher priority customers have been
  served

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Non-preemptive

• Let NQkavg. in queue for priority k
• Wk avg. queueing time for priority k
• ?k ?k/?k system utilization for
  priority k
• R mean residual service time.

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Non-preemptive
Where ?1W2 is the avg. of higher priority
customers that arrives while you are waiting
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Non-preemptive
Similarly,
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Non-preemptive
Rthe residual time
Where 2nd moment of the service
time avg. over all priority
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Non-preemptive
??
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Preemptive
Note that Tk will not be affected by customers
from class k1 to n
Work due to class 1 to k-1 who arrives when this
customer is waiting (B)
Unfinished work of Class 1 to k (A)
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Preemptive