This distribution got from condition as follows:
- Incoming call is randomly, with incoming call rate = a (constant, doesn't depending total occupation) because total calling source is infinite (big).
- There's only birth process, there is no death process.
- Total server (channel) that accommodates (to cultivate) is infinite (big), so that that incoming call always can be served by servers.
For condition there is no death, so dn = 0 and equation condition birth-death process be
dPn(t) / dt = Pn-1(t).an-1 – an.Pn(t) for n ³ 1
dP0(t) / dt = - a0.P0(t) for n = 0
Observation assumption is begun in t = 0, that is when does system present in condition to zero where there is no birth occur. Got equation:
ì 1 for n = 0
Pn(0) = í
0 for n ¹ 0
With that condition, than got equation solution:
P0(t) = e-at
From equation above will got (for n = 1) :
dP1(t) / dt = ae-at – aP1(t)
the solving :
P1(t) = at.e-at
for n = 2,
P2(t) = (at)2 e-at / 2!
for n = 3,
P3(t) = (at)3 e-at / 3!
So on, ….. , so that with induction get general resolve as,
Pn(t) = (at)n e-at
n!
The above equation is known as Poisson Distribution Equation or Poisson Arrival Process Equation. This equation expresses probability system with total occupation as much as n on time t., this represent existence probability n arrival in time interval t. Equation also derivable with pay attention condition equation so that furthermore can see the condition diagram form, that is:
dPn(t) = - (bn + dn).Pn(t) + bn-1(t) + dn+1.Pn+1(t)
dt
Whit b1, b2, ….. , bn and d1, d2, ….. , dn, where:
bn : Birth Coefficient
For monstrous calling source, this mean that incoming call rate is constant and hasn't depending on how big calling that success occupy device (server)
so : b1 = b2 = ….. = bn = a (constant)
dn : Death Coefficient
In condition there is no death, so dn = 0, while bn = a, because constant calling rate (doesn't depending n), so that birth-death process equation condition be:
dPn(t) / dt = Pn-1(t).an-1 – an Pn(t) for n ³ 1
dPn(t) / dt = - an Pn(t) for n = 0
To finish this equation, observation assumption is begun in t = 0, that is when does system present in condition to zero where there is no birth that occur. The equation:
ì 1 for n = 0
Pn(0) = í
0 for n ¹ 0
With that condition, for n = 0, the solution is:
Pn(t) = e-at
for n = 1
dp1(t) / dt = ae-at – aP1(t)
The solution:
P1(t) =at e-at
for n = 2
P2(t) = (at)2 e-at / 2!
for n = 3
P3(t) = (at)3 e-at / 3!
So on, ….. , so that with induction is got general resolve as,
Pn(t) = (at)n e-at / n!
The result is Poisson distribution equation. In this case (at) be incoming call average rate multiply with long time occupation average rate = traffic = A, so that equation can be written as,
Pn = An e-A
n!
Condition diagram:
Figure 4.1 Condition diagram for Poisson distribution
Birth coefficient bn = a
Taken from incoming 1 calling probability during dt time = a. dt, (a: incoming call average rate in 1 hour (busy).
Death coefficient dn = n.s
Taken from the end probability just any 1 occupation during dt time = n. s.dt ( n. s : end occupation average rate in condition n in 1 hour (busy rate)).
