In dt: n condition ® n + 1 condition (found 1 incoming call)
P(1 incoming call on n condition during dt time) = bn . dt + 0 (dt)
In dt: n condition ® n – 1 condition (found 1 end occupation)
P(1 end occupation on n condition in dt time) = dn . dt + 0 (dt)
dn = death coefficient
p(more than 1 come/end event occur during dt time) = 0 (dt)"little"
0 (dt) = a function from dt that the value is bitterer will be quicker become 0 from dt its self when dt ® 0
Table 3-1:
Several possibilities (in) a condition presents in n, moment t + dt
Condition
on
t Condition
on
(t + dt)
Transition P
(transition on dt/ condition on t)
n n nothing that come and or end (1 - bn.dt).(1 - dn.dt) =
1 - bn.dt – dn.dt + 0(dt)
n – 1 n 1 incoming call and nothing that end bn-1.dt(1 – dn-1.dt) + 0(dt) =
bn-1.dt + 0(dt)
n + 1 n Nothing that come and 1 end occupation (1-bn+1.dt).dn-1.dt + 0 (dt) =
dn+1.dt + 0(dt)
The other condition n More than 1 transition 0(dt)
Completion with using addition and multiplication theorem is produce equation:
Pn(t + dt) = Pn(t) (1 – bn.dt – dn.dt) + Pn-1(t).bn-1.dt – Pn+1(t).dn+1.dt + 0 (dt)
{Pn(t + dt) – Pn(t)}/dt = - (bn + dn).Pn(t) + bn-1.Pn-1(t) + dn-1.Pn-1(t)
For dt limit ® 0:
dPn(t)/dt = - (bn + dn).Pn(t) + bn-1.Pn-1(t) + dn+1.Pn+1(t)
With n = 1, 2, 3, …..
Equation above called as Condition Equation. For completion equation is necessary to pay attention a balance condition from network operation. In a condition balance (steady state), condition probabilities don't change towards time (not time function), so that:
dP/dt = 0 (Pn(t) ¹ f(t))
For:
n = 0 : 0 = -b0.P0 + d1.P1
b0.P0 = d1.P1
n = 1 : (b1 + d1).P1 = - b0.P0 + d2.P2
n = 2 : (b2 + d2).P2 = - b1.P1 + d3.P3
n = 3, 4, ….. , s/d n : (bn + dn).Pn = - bn-1.Pn-1 + dn+1.Pn+1
Equation Substitution:
b1.P1 = d2.P2
b2.P2 = d3.P3
b3.P3 = d4.P4
bn.Pn = dn+1.Pn+1
As balanced equation, has explanation how many times change from condition -n to n+1 equal to how many times change from condition n+1 to n, the condition diagram, showed in picture 3.6.
Figure 3.6 Condition diagram
