Page 16 - A First Course In Stochastic Models

THE POISSON PROCESS 7

(a) It will be obvious that the process {N(t)} satisfies the properties (A) and (B).
To verify property (C) note that

P {one arrival in (t, t + t]}
2
one arrival of type i and no arrival
= P
of the other type in (t, t + t]
i=1
= [λ 1 t + o( t)][1 − λ 2 t + o( t)]
+ [λ 2 t + o( t)][1 − λ 1 t + o( t)]

= (λ 1 + λ 2 ) t + o( t) as t → 0.

Property (D) follows by noting that

P {no arrival in (t, t + t]} = [1 − λ 1 t + o( t)][1 − λ 2 t + o( t)]
= 1 − (λ 1 + λ 2 ) t + o( t) as t → 0.

This completes the proof that {N(t)} is a Poisson process with rate λ 1 + λ 2 .
To prove the other assertion in part (a), denote by the random variable Y i the
interarrival time in the process {N i (t)}. Then

P {Z k > t, I k = 1} = P {Y 2 > Y 1 > t}

∞
= P {Y 2 > Y 1 > t | Y 1 = x}λ 1 e −λ 1 x dx
t
∞ λ 1

= e −λ 2 x λ 1 e −λ 1 x dx = e −(λ 1 +λ 2 )t .
t λ 1 + λ 2
By taking t = 0, we find P {I k = 1} = λ 1 /(λ 1 + λ 2 ). Since {N(t)} is a Poisson
process with rate λ 1 + λ 2 , we have P {Z k > t} = exp [−(λ 1 + λ 2 )t]. Hence

P {I k = 1, Z k > t} = P {I k = 1}P {Z k > t},

showing that P {I k = 1 | Z k = t} = λ 1 /(λ 1 + λ 2 ) independently of t.
(b) Obviously, the process {N i (t)} satisfies the properties (A), (B) and (D). To
verify property (C), note that

P {one arrival of type i in (t, t + t]} = (λ t)p i + o( t)
= (λp i ) t + o( t).

It remains to prove that the processes {N 1 (t)} and {N 2 (t)} are independent. Fix
t > 0. Then, by conditioning,