Telecommunication Network

1
TelecommunicationNetwork

• Telecommunication Traffics
• Semester 1, 2006

2
Traffic Engineering
3
Teletraffic

• How many lines / switches do I need?
• Why cant I get through sometimes?
• What happens to the call?
• Erlangs formula
• Blocked?
• Delayed?
• Demand
• Dimensioning
• Grade of Service

4
Traffic Terminology

• Call Attempt
• any attempt on the part of the traffic source
  (subscriber) to obtain service.
• Call
• a series of dialling attempts to the same number
  where the last attempt is successful. call
  attempts gt calls.
• Call Attempt Factor (CAF)
• not all calls successful, called party busy,
  equipment busy, misdialledtypical (voice) CAF
  1.4 / 70 success  but they still use signalling
  equipment.

5
Typical Call Attempt Analysis

• Category
• Call was completed 70.7
• Called subscriber did not answer 12.7
• Called subscriber line was busy 10.1
• Call abandoned without system response 2.6
• Equipment blockage or failure 1.9
• Customer dialling error 1.6
• Called number change or disconnected 0.4

6
Call Holding Time

• Call holding time is the length of time during
  which during which a traffic source engages a
  traffic path or channel. 1 3 minutes typical,
  gt10 minutes infrequent for voice. packet short
  and computers long, have different
  distributions.

H average holding time, 3 minutes. Negative
exponential
7
Busy Hour

• Busy hour is that continuous 60 minutes time span
  of the day during which the highest usage occurs.

8
Busy Hour

• Note
• may not occur at the same time every day
• weekly variation
• week day /weekend variation
• seasonal variation
• Mathematical formulas assume the busy hour
  traffic intensity is the average of an infinite
  number of busy hours.

9
Busy Hour

• ADPH (Average Daily Peak Hour)
• - one determines the busiest hour separately for
  each day (different time for different days), and
  then averages over e.g. 10 days
• - the resolution of the start time of the busy
  hour may be either a full hour (ADPH-F) or a
  quarter of an hour (ADPH-Q)
• TCBH (Time Consistent Busy Hour)
• - a period of one hour, the same for each day,
  which gives the greatest average traffic over
  e.g. 10 days
• FDMH (Fixed Daily Measurement Hour)
• - a predetermined, fixed measurement hour (e.g.
  9.30-10.30) the measured traffic is averaged
  over e.g. 10 days
• aFDMH aTCBH aADPH

10
Quality of Services (QoS)

• A network cannot be dimensioned for the worst
  case peaks. Then, occasionally the requsted
  service is not available or the quality of the
  service is reduced.
• The dimensioning has to made according to the
  stated (statistical) criteria for the quality of
  service
• - grade of service (GoS) quality at the call
  level (e.g. telephone network)
• - quality of service (QoS) quality during a
  connection or session (e.g. ATM network)
• In a telephone network a call that cannot be
  immediately carried
• - may be blocked loss system
• - may have to wait (ringing tone) waiting system
• The GoS requirement
• - loss system P(call is blocked) lt x
• - waiting system P(waiting time gt z seconds) lt x
  

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QoS

• Loss system
• - typically a blocking may occur during the busy
  hour
• - this happens with a certain probability, which
  depends on the traffic intensity during the busy
  hour and the dimensioning of the network as
  described by Erlang's formula(so called B
  formula)
• - the blocking probabilties in different parts of
  the network can summed to aprroximately estimate
  the end-to-end blocking
• Waiting system
• - if connecting the call is not immediately
  possible, the call may be put in a waiting state
• - a small waiting time does not matter, a user
  may not notice it at all
• - long waiting times are unacceptable for the
  users
• - one sets an upper limit to the waiting time,
  after which the call is blocked
• - the behaviour of a waiting system is desribed
  by so called Erlang's C formula

12
QoS

• There may be reattempts after unsuccesful calls
• It is not reasonable to dimension the network for
  a very small blocking probability, since the call
  may be unseccessful due to other reasons with a
  much higher probability
• - B subscriber does not answer
• - B subscriber is busy
• - one has dialled a wrong number
• Often the set limit for the blocking probability
  is 1
• In other networks than the traditional POTS, the
  quality of service is described by many other
  quantities, in place of or in addition to the
  blocking probability
• In ATM networks and in packet networks, e.g. the
  Internet, the following may be important
• - packet / cell delays
• - delay variation (jitter)
• - the proportion of lost packets / cells
• - the proportion of erroneous packets / cells
• - throughput

13
Traffic Density / Intensity

• Traffic Density is defined as the number of
  simultaneous calls at a given moment.
• Traffic Intensity represents the average traffic
  density (occupancy) during any one hour period.
• Occupancy is any use of of a traffic resource
  regardless of whether or not a connection (call)
  is completed.
• Occupancy is the probability of finding the trunk
  busy is equal to the proportion of time for which
  the trunk is busy

14
Loss and Delay Systems

• A Loss System is one in which a call attempt is
  rejected when there is no idle resource to serve
  the call. (BCC Blocked Call Cleared)
• Blocked calls
• A Delay System is one in which call attempts are
  held in a waiting queue until resource are
  available to serve the calls.
• Delayed calls

15
Offered, Carried and Blocked Traffic

• Offered traffic is the traffic intensity that
  would occur if all traffic submitted to a group
  of circuits could be processed.
• Carried traffic is the traffic intensity actually
  handled by the group.
• Blocked traffic is that portion of traffic that
  cannot be processed by the group of circuits
  (I.e. offered traffic minus carried traffic).
• Blocked traffic may be rejected, retried or
  offered to another group of circuits (overflow).

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Traffic Units

• Telephone traffic may be defined as the occupancy
  of the transmission and switching equipment that
  comprise the network during the process of
  establishing the connection and while the call is
  in progress.
• Traffic Flow (no. of calls)(mean call holding
  time)
• If 100 calls are generated in 1 hour of 3 minutes
  average duration we have 3100 300 call minutes
  or 300/60 5 call hours.

17
Erlangs

• The international dimensionless unit of telephone
  traffic is called the Erlang after A. K. Erlang
  (1878 1929) a Danish scientist.
• Defined as one circuit occupied for one hour.
• 1 Erlang 1 Callhour / hour
• Busy hour traffic
• Erlangs (Calls/busy hour)(mean call holding
  time)
• (careful with units, all times in hours)

18
Example

• Call established at 2 am between a central
  computer and a data terminal. Assuming a
  continuous connection and data transferred at 34
  kbit/s what is the traffic if the call is
  terminated at 2-45am?
• Traffic (1 call)(45 min)(1 hour /60 min)
• or 0.75 Erlangs. Its nothing to do with the data
  rate of communication, only the call holding time.

19
More examples

• A group of 20 subscribers generate 50 calls with
  an average holding time of 3 minutes, what is the
  average traffic per subscriber?
• Traffic (50 calls)(3min)(1 hour/60 min)
• 2.5 Erlangs
• 2.5 / 20 or 0.125 Erlangs per subscriber.
• Individual (residential) calling rates are quite
  low and may be expressed in milli-Erlangs, i.e.
  0.125 Erlangs 125 milli-Erlangs.

20
Grade of Service

• Grade of Service is a measure of the probability
  that a percentage of the offered traffic will be
  blocked or delayed.
• the ability to interconnect users
• the rapidity with which that connection is made
• Commonly expressed as the fraction of calls or
  demand that
• fails to receive immediate service (blocked
  calls)
• is forced to wait longer than a given time
  (delayed calls)

21
Erlang Formula

• Agner Karup Erlang Copenhagen Telephone Company
  1908 addressed problem of how many (trunk) lines
  to install between the telephone exchange of one
  village and the next. No right answer! Cost /
  Quality trade off!
• one line cheap but people wait for call...
• one per villager expensive but people never wait!
• Derived a formula to calculate the probability of
  a call being blocked

22
Erlang B

• Simplest assumption that any blocked call is lost

where A Offered Traffic N Number of Servers
(Lines) Pb Probability of Blocking
23
Erlang B Sample Calculation

• A 3 Erlangs
• N 6 Lines
• Pb given by

Note 0! 1 A0 1
24
Traffic Tables

• Usual approach is to calculate the carried
  traffic A for a given number of lines N and
  probability of blocking Pb.
• P005 a blocking probability of 0.5 (1 in 200)
• P02 a blocking probability of 2 (1 in 50)

25
Dimensioning your System

• How many lines (radio channels) to carry how many
  voice calls?
• Collect traffic data - measure, find or guess
  minutes of traffic in each hour. Divide by 60
  Erlangs.
• Determine average busy hour.
• Choose a target grade of service, say P02.
• Use Erlang B Tables to determine number of lines
  needed.

26
Cellular Radio Example

• A single GSM carrier supports 8 (TDM) speech
  channels.
• From the table on slide 18 we can see that for
  N8 we can carry 3.63 Erlangs of traffic at P02
  or 2.73 Erlangs at P005.
• How many 3 minutes calls does this represent?
• at P02, 3.63 T 3 / 60 or T 72 calls
• at P005, 2.73 T 3 / 60 or T 54 calls
• i.e. we can carry 72 calls where 1 (attempt!) in
  50 gets blocked or 54 where 1 in 200 gets blocked.

27
Queuing Theory
28
Queuing Theory 2

• Queuing Theory (more statistics) governs the
  performance of packet based systems.

Buffer
Arriving packets
Departing packets

• Data packets arrive and are buffered ready to be
  read out on a transmission link at a rate B bit/s
• If arrival rate l in a time t is greater than B a
  queue will form
• System will have alternate periods of idle no
  queue and server busy queue not empty

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Queuing Theory 3

• Queuing theory enables us to determine the
  statistics of the queue, from which such desired
  performance characteristics as the time spent
  waiting in the queue or the probability that a
  packet is blocked or lost may be found.
• Statistics depend on
• the packet arrival process (usually assumed to be
  random and described by the Poisson distribution)
• the packet length distribution (c.f. customer
  service time)
• no of servers and their discipline
• FIFO firstin, firstserve
• LIFO lastin, firstserve

30
Revision

• Know Terminology
• call attempt, call, call attempt factor, call
  holding time, busy hour
• loss delay systems
• offered, carried and blocked traffic - Grade of
  Service
• The Erlang
• Use of Erlang B Tables
• traffic capacity for a give blocking probability
  and number of lines / circuits
• Types of queue / server systems, single / multi

31
TCOM 501 Networking Theory Fundamentals

• Lectures 4 5
• February 5 and 12, 2003
• Prof. Yannis A. Korilis

32
Topics

• Markov Chains
• M/M/1 Queue
• Poisson Arrivals See Time Averages
• M/M/ Queues
• Introduction to Sojourn Times

33
The M/M/1 Queue

• Arrival process Poisson with rate ?
• Service times iid, exponential with parameter µ
• Service times and interarrival times independent
• Single server
• Infinite waiting room
• N(t) Number of customers in system at time t
  (state)

34
Exponential Random Variables

• X exponential RV with parameter ?
• Y exponential RV with parameter µ
• X, Y independent
• Then
• minX, Y exponential RV with parameter ?µ
• PXltY ?/(?µ)
• Exercise 3.12

• Proof

35
M/M/1 Queue Markov Chain Formulation

• Jumps of N(t) t 0 triggered by arrivals and
  departures
• N(t) t 0 can jump only between neighboring
  states
• Assume process at time t is in state i N(t) i
  1
• Xi time until the next arrival exponential
  with parameter ?
• Yi time until the next departure exponential
  with parameter µ
• Ti minXi,Yi time process spends at state i
• Ti exponential with parameter ?i ?µ
• Pi,i1PXi lt Yi ?/(?µ), Pi,i-1PYi lt Xi
  µ/(?µ)
• P011, and T0 is exponential with parameter ?
• N(t) t 0 is a continuous-time Markov chain
  with

36
M/M/1 Queue Stationary Distribution

• Birth-death process ? DBE
• Normalization constant
• Stationary distribution

37
The M/M/1 Queue

• Average number of customers in system
• Littles Theorem average time in system
• Average waiting time and number of customers in
  the queue excluding service

38
The M/M/1 Queue

• ??/µ utilization factor
• Long term proportion of time that server is busy
• ?1-p0 holds for any M/G/1 queue
• Stability condition ?lt1
• Arrival rate should be less than the service rate

39
M/M/1 Queue Discrete-Time Approach

• Focus on times 0, d, 2d, (d arbitrarily small)
• Study discrete time process Nk N(dk)
• Show that transition probabilities are
• Discrete time Markov chain, omitting o(d)

40
M/M/1 Queue Discrete-Time Approach

• Discrete-time birth-death process ? DBE
• Taking the limit d?0
• Done!

41
Transition Probabilities?

• Ak number of customers that arrive in Ik(kd,
  (k1)d
• Dk number of customers that depart in Ik(kd,
  (k1)d
• Transition probabilities Pij depend on
  conditional probabilities Q(a,d n) PAka,
  Dkd Nk-1n
• Calculate Q(a,d n) using arrival and departure
  statistics
• Use Taylor expansion e-ld1-ldo(d),
  e-md1-mdo(d), to express as a function of d
• Poisson arrivals PAk 2o(d)
• Probability there are more than 1 arrivals in Ik
  is o(d)
• Show probability of more than one event (arrival
  or departure) in Ik is o(d)
• See details in textbook

42
Example Slowing Down

• M/M/1 system slow down the arrival and service
  rates by the same factor m
• Utilization factors are the same ?stationary
  distributions the same, average number in the
  system the same
• Delay in the slower system is m times higher
• Average number in queue is the same, but in the
  1st system the customers move out faster

43
Example Statistical MUX-ing vs. TDM

• m identical Poisson streams with rate ?/m link
  with capacity 1 packet lengths iid, exponential
  with mean 1/µ
• Alternative split the link to m channels with
  capacity 1/m each, and dedicate one channel to
  each traffic stream
• Delay in each queue becomes m times higher
• Statistical multiplexing vs. TDM or FDM
• When is TDM or FDM preferred over statistical
  multiplexing?

44
PASTA Theorem

• Markov chain stationary or in steady-state
• Process started at the stationary distribution,
  or
• Process runs for an infinite time t?8
• Probability that at any time t, process is in
  state i is equal to the stationary probability
• Question For an M/M/1 queue given t is an
  arrival time, what is the probability that
  N(t)i?
• Answer Poisson Arrivals See Time Averages!

45
PASTA Theorem

• Steady-state probabilities
• Steady-state probabilities upon arrival
• Lack of Anticipation Assumption (LAA) Future
  inter-arrival times and service times of
  previously arrived customers are independent
• Theorem In a queueing system satisfying LAA
• If the arrival process is Poisson
• Poisson is the only process with this property
  (necessary and sufficient condition)

46
PASTA Theorem

• Doesnt PASTA apply for all arrival processes?
• Deterministic arrivals every 10 sec
• Deterministic service times 9 sec
• Upon arrival system is always empty a10
• Average time with one customer in system p10.9
• Customer averages need not be time averages
• Randomization does not help, unless Poisson!

1
47
PASTA Theorem Proof

• Define A(t,td), the event that an arrival occurs
  in t, t d)
• Given that a customer arrives at t, probability
  of finding the system in state n
• A(t,td) is independent of the state before time
  t, N(t-)
• N(t-) determined by arrival times ltt, and
  corresponding service times
• A(t,td) independent of arrivals ltt Poisson
• A(t,td) independent of service times of
  customers arrived ltt LAA

48
PASTA Theorem Intuitive Proof

• ta and tr randomly selected arrival and
  observation times, respectively
• The arrival processes prior to ta and tr
  respectively are stochastically identical
• The probability distributions of the time to the
  first arrival before ta and tr are both
  exponentially distributed with parameter ?
• Extending this to the 2nd, 3rd, etc. arrivals
  before ta and tr establishes the result
• State of the system at a given time t depends
  only on the arrivals (and associated service
  times) before t
• Since the arrival processes before arrival times
  and random times are identical, so is the state
  of the system they see

49
Arrivals that Do not See Time-Averages

• Example 1 Non-Poisson arrivals
• IID inter-arrival times, uniformly distributed
  between in 2 and 4 sec
• Service times deterministic 1 sec
• Upon arrival system is always empty
• ?1/3, T1 ? NT/?1/3 ? p11/3
• Example 2 LAA violated
• Poisson arrivals
• Service time of customer i Si ?Ti1, ? ? 1
• Upon arrival system is always empty
• Average time the system has 1 customer p1 ?

50
Distribution after Departure

• Steady-state probabilities after departure
• Under very general assumptions
• N(t) changes in unit increments
• limits an and exist dn
• an dn, n0,1,
• In steady-state, system appears stochastically
  identical to an arriving and departing customer
• Poisson arrivals LAA an arriving and a
  departing customer see a system that is
  stochastically to the one seen by an observer
  looking at an arbitrary time

51
M/M/ Queues

• Poisson arrival process
• Interarrival times iid, exponential
• Service times iid, exponential
• Service times and interarrival times independent
• N(t) Number of customers in system at time t
  (state)
• N(t) t 0 can be modeled as a continuous-time
  Markov chain
• Transition rates depend on the characteristics of
  the system
• PASTA Theorem always holds

52
M/M/1/K Queue

• M/M/1 with finite waiting room
• At most K customers in the system
• Customer that upon arrival finds K customers in
  system is dropped
• Stationary distribution
• Stability condition always stable even if ?1
• Probability of loss using PASTA theorem

53
M/M/1/K Queue (proof)

• Exactly as in the M/M/1 queue
• Normalization constant
• Generalize Truncating a Markov chain

54
Truncating a Markov Chain

• X(t) t 0 continuous-time Markov chain with
  stationary distribution pi i0,1,
• S a subset of 0,1, set of states Observe
  process only in S
• Eliminate all states not in S
• Set
• Y(t) t 0 resulting truncated process If
  irreducible
• Continuous-time Markov chain
• Stationary distribution
• Under certain conditions need to verify
  depending on the system

55
Truncating a Markov Chain (cont.)

• Possible sufficient condition
• Verify that distribution of truncated process
• Satisfies the GBE
• Satisfies the probability conservation law
• Another even better sufficient condition
  DBE!
• Relates to reversibility
• Holds for multidimensional chains

56
M/M/1 Queue with State-Dependent Rates

• Interarrival times independent, exponential,
  with parameter ?n when at state n
• Service times independent, exponential, with
  parameter µn when at state n
• Service times and interarrival times independent
• N(t) t 0 is a birth-death process
• Stationary distribution

57
M/M/c Queue

• Poisson arrivals with rate ?
• Exponential service times with parameter µ
• c servers
• Arriving customer finds n customers in system
• n lt c it is routed to any idle server
• n c it joins the waiting queue all servers
  are busy
• Birth-death process with state-dependent death
  rates
• Time spent at state n before jumping to n -1 is
  the minimum of Bn minn,c exponentials with
  parameter µ

58
M/M/c Queue

• Detailed balance equations
• Normalizing

59
M/M/c Queue

• Probability of queueing arriving customer finds
  all servers busy
• Erlang-C Formula used in telephony and
  circuit-switching
• Call requests arrive with rate ? holding time of
  a call exponential with mean 1/µ
• c available circuits on a transmission line
• A call that finds all c circuits busy,
  continuously attempts to find a free circuit
  remains in queue
• M/M/c/c Queue c-server loss system
• A call that finds all c circuits busy is blocked
• Erlang-B Formula popular in telephony

60
M/M/c Queue

• Expected number of customers waiting in queue
  not in service
• Average waiting time (in queue)
• Average time in system (queued serviced)
• Expected number of customers in system

61
M/M/8 Queue Infinite-Server System

• Infinite number of servers no queueing
• Stationary distribution Poisson with rate ?/µ
• Average number of customers average delay
• The results hold for an M/G/8 queue

62
M/M/c/c Queue c-Server Loss System

• c servers, no waiting room
• An arriving customer that finds all servers busy
  is blocked
• Stationary distribution
• Probability of blocking (using PASTA)
• Erlang-B Formula used in telephony and
  circuit-switching
• Results hold for an M/G/c/c queue

63
M/M/8 and M/M/c/c Queues (proof)

• DBE
• Normalizing

64
Sum of IID Exponential RVs

• X1, X2,, Xn iid, exponential with parameter ?
• T X1 X2 Xn
• The probability density function of T
  isGamma distribution with parameters (n, ?)
• If Xi is the time between arrivals i -1 and i of
  a certain type of events, then T is the time
  until the nth event occurs
• For arbitrarily small d
• Cummulative distribution function

65
Sum of IID Exponential RVs

• Example 1 Poisson arrivals with rate ?
• t1 time until arrival of 1st customer
• ti ith interarrival time
• t1, t2,, tn iid exponential with parameter ?
• tn t1 t2,tn arrival time of customer n
• tn follows Gamma with parameters (n, ?).
• For arbitrarily small d

66
Sojourn Times in a M/M/1 Queue

• M/M/1 Queue FCFS
• Ti time spent in system (queueing service) by
  customer i
• Ti exponentially distributed with parameter µ-?
• Example of a sojourn time of a customer
  describes the evolution of the queue together
  with the specific customer
• Proof
• Direct calculation of probability distribution
  function
• Moment generating functions
• Intuitive Exercise 3.11(b)

67
M/M/1 Queue Sojourn Times (proof)
68
M/M/1 Queue Sojourn Times (proof)
69
M/M/1 Queue Sojourn Times (proof)
70
M/M/1 Queue Sojourn Times (proof)
71
Moment Generating Function
72
Interference and Capacity of Cellular Network
Systems

• Transferring knowledge to future leaders

Presented by Professor Johnson I Agbinya
jagbinya_at_uwc.ac.za
73
Traffic Engineering

• Problems with Connecting Phones with Switches
• Many switches required - to connect n phones
  together, s (n-1)n/2 switches are required
• slow connection speeds
• too many regular faults
• high maintenance costs and cost of switches

74
Traffic Engineering - Considerations

• Design for flexibility and account for low and
  high traffic periods
• peak traffic period occur sometimes in the
  mornings and afternoons. Low traffic weekends
• high traffic usually 10 to 20 of total capacity,
  all users need not be directly connected
• cellular systems depend on trunking to connect a
  large number of users

75
Trunking

• In a trunked radio system, each user is allocated
  a channel on a per call basis, and on termination
  of call, previously occupied channel is
  immediately returned to the pool of available
  channels
• Therefore a large number of users share a small
  pool of channels in a cell on a per call basis
• Access is provided to each user on demand
• When all channels are in use, a new user or
  demand is (denined) blocked

76
Unit of Traffic - Erlang

• The unit of telephone traffic intensity is called
  the Erlang, in honour of a Danish mathematician
• Definition One Erlang is one channel occupied
  continuously for one hour. In data
  communications, an 1 E 64 kbps

77
Grade of Service (GoS)

• A measure of the performance of a telephone
  system
• GOS is a measure of the ability of a user to
  access a trunked system during the busiest hour
• Also an indication of the user not being able to
  secure a channel during the busiest hour
• Telephone networks are designed with specified
  GOS, usually for the busiest hour. If a
  subscriber is able to make a call during the
  busiest hour, he will be able to make a call at
  any other time

78
Grade of Service (1)

• Definition
• GOS is the probability of having a call blocked
  during the busiest hour. For example, if GOS
  0.05, one call in 20 will be blocked during the
  busiest hour because of insufficient capacity
• GOS is used to determine the number of channels
  required
• GOS could be determined by
• competition between operators (measure of good
  service)
• regulation - a national communication authority
  might decide to impose a grade of service on its
  operators

79
How To Estimate Telephone Traffic

• Definitions, let
• Au Erlangs be traffic intensity generated by each
  user
• h be average duration of a call (hour)
• l is the average number of call requests per
  hour. Then
• For a system containing U users, the total
  offered traffic intensity A is
• In a trunked system of C channels, the traffic
  intensity per channel is

80
Types of Trunked Systems

• Two types of trunked systems are used
• (a) blocked calls cleared (Erlang B, M/M/m queue)
• (b) blocked calls delayed (Erlang C formula)
• Characteristics of Blocked calls Cleared Model
• Call arrival rate Poisson (exponential)
  distribution
• Infinite number of users
• Memoryless, channel requests at any time
• infinite number of channels in pool

81
Trunking Efficiency

• Trunking efficiency is the measure of the number
  of users that can be offered a specified grade of
  service with a configuration of fixed channels
• With a GOS of 0.1 and 10 trunked channels, 10
  Erlangs can be supported
• Dividing the channels into 2 groups of 5 at the
  same GOS will support 1.36 Erlangs per group or a
  total of 2.72 Erlangs
• Lesson Effective grouping of trunked channels is
  essential for efficient capacity provision

82
Traffic Intensity Models

• Three traffic intensity model tables are used in
  practice
• Erlang B tables (blocked calls cleared) can over
  estimate
• Engset formula (probability of blocking in low
  density areas) used where Erlang B model fails
• Erlang C tables (blocked calls delayed or held in
  queue indefinitely)
• Poisson tables (blocked calls held in queue for a
  limited time only)

83
System Utilization

• Let
• A be the offered traffic
• C be the carried traffic,
• B is blocking probability
• Lost traffic (A- C) BA
• Probability of blocking
• System utilization

84
State Transition

• How does the state of a telecommunication network
  change from time to time as calls are received?
• The system state is shown
• At any point in time, the system could be in any
  state from 0 to N, where N is the number of lines
  in the system

85
State Transition (1)

• l is the forward state transition rate
• Let vk Prsystem is in state k, then, the
  state transition probabilities are given by the
  expression
• and

86
Erlang B Formula

• Determines the probability that a call is blocked
• Is a measure of the GOS for trunked systems with
  blocked calls cleared
• Erlang B formula

87
Erlang B - Markov Process

• Erlang B process is modelled with a Markov state
  transition model
• Let the offered traffic A lh Erlang h as the
  holding time and the weights wk as un-normalised
  probabilities
• The probability for being in state N is the ratio
  of the weight for state N, to the sum of the
  weights for all states

88
Erlang B - Markov Process (1)

• The Erlang blocking probability for N channels is
• Erlang B formula provides data for calculating
  the number of channels required based on a
  blocking level () and the offered traffic
  (Erlang value)

89
Using the Erlang B Table

• The objective is to determine the number of
  trunks required for a given Erlang value and a
  blockage level. Three steps are required
• Locate the column with the desired blockage
  level
• While staying in the same column, find the row
  with the desired Erlang value (round off the
  Erlang value as necessary)
• Find the number of trunks in the selected row (at
  the intersection)

90
Example Computations

• Example 1 An average of one call for every 40
  seconds is offered in a system of five lines.
  What is the blocking probability when the average
  holding time of a call is 100 seconds?
• Solution N5 l1/400.025 offered traffic A
  lh0.025x100 2.5 erlang. The un-normalised
  probabilities are shown in table
• Blocking probability E5(2.5) (0.813802 /
  11.670573) 6.973.

91
Example Computations

• Example 2 Find the blocking probability for a
  cell when the offered traffic A1.75 erlangs for
  N5 channels.
• Solution Using the Erlang B table, the offered
  traffic is given by

92
Planning for Cell Capacity

• Assume that in a telephone network the call
  arrival rate is l calls per hour and the mean
  holding time for a call is tn (hours per call).
• Example 3 There are 100 subscribers with the
  following telephone traffic profile 20 make 1
  call/hour for 6 minutes 20 make 3 calls/hour for
  half a minute 60 make 1 call/hour for 1 minute.
  The traffic they generate is
• 20x1x (6/60) 2 E
• 20x3x(0.5/60) 0.5 E
• 60x1x(1/60) 1 E
• ie., a total of 3.5 E. On average, each
  subscriber generates 35 mE.
• In practice on average telephone subscribers
  generate between 25 to 35 mE during the busiest
  hour

93
Planning for Cell Capacity

• Example 4 Use the Erlang B table to compute the
  number of channels required for a cell when the
  expected number of calls per hour is 3000,
  blocking probability of 2 and the average length
  of a call is 1.8 minutes.
• Solution The offered traffic for this case is A
  qxT/60 3000x1.8/60
• 90 Erlangs. Erlang B table indicates that
  103 channels
• are required.

94
Frequency Planning-Sectored Antenna

• Assume we have 666 channels to be allocated and a
  frequency reuse value of 7, we can draw up the
  frequency plan which indicates how channels are
  allocated to cells. We assume the cells are
  sectored at 120 degrees.
• Channel separation for each cell sector 3x7
  21, and the frequency plan to use is shown

95
Engset Formula

• In low population density areas, Erlang B formula
  estimates the blocking probability too highly.
• Engset formula estimator is better in such
  situations. The formula is
• s is the number of sources (population),
• n is the number of lines or channels,
• a is the arrival intensity per free source and
• m is the reciprocal of the hold time (1/h)

96
Markov Model for Engset Formula

• As in Erlang B, there is a Markov model for the
  Engset process.
• It models how the state of the system change
  with time
• The probability of call blocking is not equal to
  the probability of being in a state n
• Engset Distribution State Transition

97
Erlang C Formula

• Erlang C formula is used to model the second
  type of trunked systems when blocked calls are
  delayed instead of cleared
• when a channel is not available immediately, the
  call is queued (delayed instead of being thrown
  away) until a channel becomes available
• It provides the probability that the call is
  blocked after waiting a specific length of time
  in the queue - this is the measure of GOS
• The Erlang C formula is

98
Probability of delayed call waiting More than t
seconds

• If no channel is available, it waits for one to
  be available. What then is the probability of the
  call being delayed for more than t seconds?
• The probability of waiting more than t seconds
  (GOS) is the product of the probability that the
  call is delayed multiplied by the conditional
  probability that the delay is more than t
  seconds
• The average delay D for all calls in a queued
  system is

99
Kendal Notation A/B/n/p/k
100
M/M/1 Queue

• One server, one queue, FIFO service
• exponentially distributed interarrival and
  service times
• infinite population, infinite capacity
• Can model as a birth-death process

Notation pn steady-state probability of being
in state n
101
M/M/1 Queue

• Using stochastic flow balance equations
• pn (?/?)n p0 ?np0, n0,1,,?
• ? is traffic intensity (lt1 for stability)
• Probabilities sum to one. Therefore,
• ? pn 1, n 0, 1, , ?
• p0(?0 ?1 ?2 ) 1
• p0 (1-?)
• pn ?n(1-?)
• Important performance measures follow

102
M/M/1 Queue

• Utilization prob. of one or more jobs in system
• U 1- p0 ?
• Mean jobs in System
• En ?npn , n 0, 1, , ?
• En ?/(1-?)
• Mean response time (Littles law)
• number in system arrival rate x response time
• En ?Er
• Er (1/?)(?/(1-?)) (1/?)(1/(1-?))

103
M/M/1 Queue

• Mean of jobs in queue
• Enq ?(n-1)pn , n 1, , ?
• Enq (?2)/(1-?)
• Can also be obtained using En Enq Ens
• Mean waiting time in queue (Littles Law)
• Number in queue arrival rate x mean waiting
  time
• Enq ?Ew
• Ew (1/?)((?2)/(1-?)) ?((1/µ)/(1-?))

104
M/M/1 Queue

• Prob. of finding n or more jobs in system
• P( in system n) ?pj , j n,n1, , ?
• ?(1-?)?j ?n
• Waiting time and response time distributions
• Waiting times in queue exponentially distributed
• P0 lt w t 1 - ?e-?t(1-?)
• Response times exponentially distributed
• P0 lt r t 1 - e-?t(1-?)

105
M/M/1 Queue Example

• Packets arrive at 100 packets/second at a router.
  The router takes 1 ms to transmit the incoming
  packets to an outgoing link. Using an M/M/1
  model, answer the following
• What is utilization?
• Probability of n packets in router?
• Mean time spent in the router?
• Probability of buffer overflow if router could
  buffer only 5 packets?
• Buffer requirement to limit packet loss to 10-6?

106
M/M/1 Queue - Example

• Arrival rate
• ? 100 pps
• Service rate
• ? 1/.001 1000pps
• Traffic intensity
• ? 0.1
• Mean packet residence time at router
• r (1/?)(1/(1-?))
• 1.01 ms
• Prob. of buffer overflow
• P( 6) ?6 10-12
• To limit loss to less than 10-6
• ?n 10-6
• n gt log(10-6)/log(0.1) gt 3