TRUNCATED POISSION DISTRIBUTION

• Previously we assume all channels to be infinite. This conforms with the requirements of Offered Traffic where it cannot be calculated and the channels are infinite.

• In this topic we will look into the FINITE channels where there

will be blockings once all the channels are occupied. STATE PROBABILITIES

• We will get the similar cut equations as what we’ve discussed before in the infinite case, but in this case, the number of states (which represents the busy channels) is limited and the normalisation conditions now becomes:

( )1

0 !0

−

= ⎭⎬⎫

⎩⎨⎧

= ∑n

j

j

jAp

• So the distribution will be called as the Truncated Poisson

Distribution, or Erlang’s first formula, i.e. Erlang-B formula, given as…

( ) ni

iA

iA

ipn

j

i

i

≤≤=∑=

0,

!

!

0

TRAFFIC CHARACTERISTICS OF ERLANG’S B-FORMULA • By calculating the state probabilities, we can find the

performance measures defined by state probabilities.

o Time Congestion It’s the probabilities that all n channels are busy at a

random of time EQUAL to the proportion of time all channels are busy.

Thus we can assume that i = n and …

From

( ) ni

iA

iA

ipn

j

i

i

≤≤=∑=

0,

!

!

0

We derive it to…

( ) ( )

!...

!21

!2

nAAA

nA

npAE n

n

n

++++==

(Erlang’s Famous B-formula)

o Call Congestion:

The probability that the call will lost is EQUAL to the proportion of call attempts blocked. If we consider one time unit, we find ( )ABB n= :

( )( )

( ) ( )AEnpvp

npB nn

v

==⋅

⋅=∑=0λ

λ

o Carried Traffic:

If we use the cut equation between states [i – 1] and [i] we get: -

( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )ipiip

ipiipipiip

pppp

⋅=−⋅

⋅=−⋅−⋅−=−⋅

⋅=⋅⋅=⋅

1

1112

22110

μλ

μλμλ

μλμλ

L

Which translates to …

( ) ( ) ( ){ }

( ){ }AEAY

npAipiipY

n

n

i

n

i

−⋅=

−⋅=⋅=−⋅= ∑∑==

1

1111 μ

λ

Where A is the offered traffic. The carried traffic will be less than both A and n.

o Lost Traffic:

( )AEAYAA nl ⋅=−= o Traffic congestion:

( )AEA

YAC n=−

=

• We thus have E = B = C because the call intensity is independent

of the state (valid for all Poisson processes)

ERLANG-B FORMULA GRAPH

TRAFFIC CARRIED BY THE i'th TRUNK (THE UTILISATION ia ):

• Random Hunting:

o In this case all channels carry the same traffic on the average. The total carried traffic is independent of the hunting strategy and we find the utilisation:

( ){ }

nAEA

nYaa n

i

−===

1

o We observed that for a given congestion E we obtain the

highest utilisation for large channel groups (economy of scale)

• Ordered hunting = Sequential hunting: The traffic by channel i is the difference between the traffic lost from i – 1 channels:

( ) ( ){ }AEAEAa iii −= −1.

Note that also that the traffic carried by the i channels is independent of the total number of channels and thus no influence on the traffic carried by channel i.

IMPROVEMENT FUNCTION:

• The improvement function shows the increase in carried traffic, when the number of channels increased by one (1), from n to n+1: Mathematically we can equate it as …

( )

{ } { }( ) ( ) ( ){ }

1

1

1

1

11

+

+

+

+

=−=

−−−=−=

n

nnn

nn

nnn

aAEAEAAF

EAEAYYAF

And… ( ) 10 ≤≤ AFn PEAKEDNESS

• Defined as the ratio between the variance and the mean value of the distribution of the number of busy channels. Thus for Truncated Poisson Distribution,

( ) ( ){ } nnn aAEAEAm

Z −=−−== − 11 1

2σ

DURATION OF STATE [i] • Total intensity for leaving state [i] is constant and equal to

( )μλ i+ . Thus, the duration of the time in state [i] is exponentially distributed with density function:

( ) ( ) ( )

( ) ( ) ( ) nientfnieitf

tnn

tii

=⋅=

≤≤⋅+=−

+−

,0,

μ

μλ

μμλ

GENERAL PROCEDURE FOR STATE TRANSITION DIAGRAMS

• Very important tool in Teletraffic theory where it models the formulation and solution of the traffic problems.

• We can generalise the steps to produce state transition diagrams.

STEP 1: Construction of the state transition diagram

• Define the states of the system in a unique way. • Draw the states as circles. • Consider the states one at a time and draw all possible arrows

for transitions away from this state due to: -

o The arrival process (new arrival or phase shift in the arrival process)

o The departure (service) process (the service time

terminates or shifts phase).

• With this we may obtain the complete state transition diagram. STEP 2: Set up the equations describing the system.

• If the conditions for statistical equilibrium are fulfilled, the steady state equations can be obtained from

o Node equations (general) o Cut equations.

STEP 3: Solve the balance equations assuming statistical equilibrium.

• Express all state probabilities by for example the probability of state [0], p(0).

• Find p(0) by normalisation.

STEP 4

• Calculate the performance measures expressed by the state probabiltities.

NOTE:

• In practice, we let the non-normalised value of the state probability q(0) equal to one, and then calculate the relative values q (i), (i = 1, 2, …).

• By normalising we will find that …

( ) ( ) ,,...,1,0, niQ

iqipn

==

Where ( )∑

=

=n

vn vqQ

0

• Time congestion becomes: -

( ) ( )n

n

n QQ

Qnqnp 11 −−==

EVALUATION OF ERLANG’S B-FORMULA

• Earlier on, we obtained the Time Congestion formula as: -

( ) ( )

!...

!21

!2

nAAA

nA

npAE n

n

n

++++==

• However for numerical calculations, the formula above is not

appropriate since both n! and nA increase so quickly which results the occurrence of overflow in the computer.

• With the recursive formula, from the numerical point of view,

it’s more stable.

( ) ( )( ) 1, 0

1

1 =⋅+

⋅=

−

− EAEAn

AEAAEn

nn

Example: We consider an Erlang-B loss system with n = 6 channels, arrival rate

2=λ calls per time unit, and departure rate of 1=μ departure per time unit, so that the offered traffic is A = 2 Erlang. If we denote the non-normalised relative state probabilities by q(i)…

(a) Draw the state transition diagram and find the state probabilities when the system is in statistical equilibrium.

(b) Calculate the Time Congestion

(c) Calculate the Traffic Congestion

(d) Calculate the Call Congestion