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
