Engset and Bernoulli distribution got from condition as follows:
- Calling source (total customer) is limited.
- Total server (line) that processing is limited.
- Incoming call rate depend on not busy total customer.
- Customer that made a success of occupation server doesn't do calling again.
If S declares full scale customer and N declare total server, so when:
S > N: got Engset distribution (operative Engset traffic or Purechance traffic (type 2)
S £ N: got Bernoulli or Binominal distribution, (operative Bernoulli traffic)
Because incoming call rate or offered traffic to system is proportional with not busy total customer, so for n condition can be declared with equation:
Cn = (S – n) l
Where:
l : Incoming call rate per customer and is call intensities free calling source.
(S – n) : is total calling source (costumer) that still free.
Cn can be assumed as birth coefficient in n(=bn) condition, so that condition diagram for Engset and Bernoulli distribution can be described as follows:
Figure 4.3 Engset and Bernoulli distribution condition diagram.
Balance equation:
(S-n)l.P(n) = (n+1)s.P(n+1) n = 0, 1, 2, ….. , (N-1 atau S-1)
So that completion for:
n = 0: Sl.P(0) = s.P(1)
® P(1) = S(l/s).P(0)
n = 1: (S-1)l.P(1) = 2s.P(2)
® P(2) = S(S-1)(l/s)2.(1/2).P(0)
n = 2: (S-2)l.P(2) = 3s.P(3)
® P(3) = S(S-1)(S-2)(l/s)3.(1/3.2.1).P(0)
So on.
Got:
Pn = { S ! / n ! (S-n) ! }.(l/s)n.P(0)
Or:
Pn = æ S ö . (l/s)n.P (0)
è n ø
Where:
æ S ö is binomial coefficient = S!
è n ø n!(S – n)
Equation is demoted from diagram condition, and good operative for Engset also Bernoulli:
If Engset : end condition = N (S>N)
Binomial : end condition = S (S£N)
Completion for Engset Distribution : (S>N)
Blocking probability (n=N)
æ S öæ l öN
PN = è Nøè s ø
N
∑ æ S öæ l öj
j=0 è Nøè sø
PN = all busy/occupied line probability
= Time Congestion
Average offered rate:
N
C = l (S - S n Pn)
N=0
Value S n Pn in equation above represent average total busy server. Traffic that proceed by network is total average accepted calling during average service time period. This is equal to total average busy server in a certain time that is given. Thereby equation above can be made to be:
C = l (S – Y)
So offered traffic to be:
A = C tr = l tr (S – Y)
When system on N condition, offered traffic rate is (S-N)l, and all incoming call on that condition will refused, so that lost traffic is:
A – Y = (S – N) l tr PN
Thereby, now the GOS to be:
GOS = S – N . PN
S – Y
Indicating that is for Engset traffic, blocking probability and unequal grade of service, in this case time congestion and call congestion has different value.
Completion for Bernouli distribution: (S £ N)
With using equation:
Pn = æ S öæ l ön
è n øè s ø
where :
(l/s)n = {(l+s)/s}S.{1-l/(l+s)}S-n.{ l/(l+s)}n
Then got:
Pn = æ l+s öS . P0æ S ö.æ l ön æ1 - l öS-n
è s ø è n ø èl+sø è l+sø
Bernoulli formula:
Pn = æ S ö.pn æ1 - pöS-n
è n ø
From both above equation got:
{(l+s)/s}S.P0 = 1 and {l /(l+s)} = p
So:
P0 =. 1 = æ s öS
æl+söS èl+sø
è s ø
Equation above known as Bernoulli or Binomial equation, that explain servers not interdependent one with another.
