Page 217 - A First Course In Stochastic Models
P. 217
210 MARKOV CHAINS AND QUEUES
Proof For fixed , x > 0, let U ,x be a Poisson distributed random variable with
(x/ ) k
−x/
P {U ,x = k } = e , k = 0, 1, . . . .
k!
2
It is immediately verified that E(U ,x ) = x and σ (U ,x ) = x . Let g(t) be any
bounded function. We now prove that
lim E[g(U ,x )] = g(x) (5.5.2)
→0
for each continuity point x of g(t). To see this, fix ε > 0 and a continuity point x
of g(t). Then there exists a number δ > 0 such that |g(t) − g(x)| ≤ ε/2 for all t
with |t − x| ≤ δ. Also, let M > 0 be such that |g(t)| ≤ M/2 for all t. Then
∞
|E[g(U ,x )] − g(x)| ≤ |g(k ) − g(x)| P {U ,x = k }
k=0
ε
≤ + M P {U ,x = k }
2
k:|k −x|>δ
ε
= + MP {|U ,x − E(U ,x )| > δ}.
2
2
By Chebyshev’s inequality, P {|U ,x − E(U ,x )| > δ} ≤ x /δ . For small
1
2
enough, we have Mx /δ ≤ ε. This proves the relation (5.5.2). Next, we apply
2
(5.5.2) with g(t) = F(t). Hence, for any continuity point x of F(t),
∞ k
−x/ (x/ )
F(x) = lim E[F(U ,x )] = lim F(k )e
→0 →0 k!
k=0
∞ k
−x/ (x/ ) k
= lim e p j ( ),
→0 k!
k=0 j=1
where the latter equality uses that F(0) = 0. Interchanging the order of summation,
we next obtain
∞ ∞ k
−x/ (x/ )
F(x) = lim p j ( ) e ,
→0 k!
j=1 k=j
yielding the desired result.
The proof of Theorem 5.5.1 shows that the result also holds when F(t) has a
positive mass at t = 0. We should then add the term F(0) to the right-hand side
of (5.5.1). Roughly stated, Theorem 5.5.1 tells us that the probability distribution
of any positive random variable can be arbitrarily closely approximated by a mix-
ture of Erlangian distributions with the same scale parameters. The fact that the
Erlangian distributions have identical scale parameters simplifies the construction