Condition Equation In Telecommunication Traffic Engineer

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

Incoming Call Pattern and Occupation In Telecommunication Traffic Engineer

Incoming call to device switching equipment to changeable and occupation dynamics, has random. The pattern can be described with time interval probability distribution between incoming call and go on it occupation.

Explanation that given only be emphasized in incoming call pattern, occupation pattern explanation is being assumed equivalent so that will not be discussed.

For states has randomly, like in incoming call pattern, so operative exponential distribution, that is:

p.d.f : f(t) = a.e-at
P.D.F : F(t) = 1 – e-at = P( tx £ t )
a : constant
tx : time interval between call (random variable)

Average time interval value between incoming call counted based on equation:

~
tr = ∫ t. f(t).dt = 1/a
t=0

So average incoming call rate value (total call/hour) = a and P(> t) = 1 – P(£ t) = 1 – F(t).
[P(> t) be probability that time interval between incoming call is bigger then t (is complement from P(£ T))].





Figure 3.3 Exponential distribution curve p.d.f and P.D.F






Figure 3.4 Incoming call in interval (t, t + Dt).

Incoming call probability has range interval ( t, t + Dt ):

P ( t <> t and tx £ t + Dt
t+Dt t
( t < 0 =" P" also =" {"> t + Dt ) } – { ( 1 – P ( > t ) }

With condition that a incoming call in interval ( t, t + Dt ), after known that during time t there is no call






Figure 3.5 incoming call in interval ( t, t + Dt ).

Because time t there is no incoming call, so existing incident only incoming call with time interval is bigger than t. On condition that, so the probability is:

( tx £ t + Dt / tx > t ) = P ( t <> t )
= e-at – e-a ( t + Dt ) e-at = 1 - e-a Dt
= a. Dt – ( a2.(Dt)2 / 2! ) – ( a3.(Dt)3 / 3! ) - …..
= P(Dt)

For limit Dt ® 0, then :

P(Dt) = a. Dt – 0 (Dt)

Where ‘0 (Dt)’ is a Dt function that quicker will be 0 from its own Dt if, Dt ® 0 ( a = incoming call rate)

So:
P(Dt) doesn't depending from t (how long incoming call has goes on), only depending from Dt.
Only may be 1 (one) event that occur, because when more than 1 event, harga proportionate tribe value with (Dt)2 will be 0.

Analogy between incoming call time interval with occupation time:

|Dt| |Dt|
0 t t + Dt 0 t t + Dt
Incoming call incoming call

f(t) = e-at --- p.d.f --- f(t) = 1/h. e-t/h

Average time interval value Average occupation time value
Between call = 1/a occupation = h

F(t) = 1 - e-at --- P.D.F --- F(t) = 1 - e-t/h

P( tx £ t ) P( tx - t )

Time interval probability Time occupation probability
Between incoming call go on during same or
Is same or smaller than t smaller than t

State Diagram Birth-Death Process Utilization In Telecommunication Traffic Engineering

Terminology of condition (state) used to representation one condition about total line from busy bundle (occupied) in one time interval. So that contemplated with state probability is state duration probability to go on in one time interval (1 busy hour).

To make easiest representation about state and the change, can be used State Diagram Form.

If to exist incoming call or to exist occupation ended, then will occur state change (transition), from one state to the other state.