Page 209 - A First Course In Stochastic Models

202 MARKOV CHAINS AND QUEUES

For fixed R, let the random variable X R be Poisson distributed with mean R. Then
the above formula for P loss can be written as

P {X R = c}
P loss = .
P {X R ≤ c}
A Poisson distribution with mean R can be approximated by the normal distribution
with mean R and standard deviation R when R is large. Now take
√
c = R + k R
√
for some constant k. Then P {X R ≤ c} = P {(X R − R)/ R ≤ k} and so, by the
normal approximation to the Poisson distribution,
P {X R ≤ c} ≈ (k).
Writing P {X R = c} = P {c − 1 < X R ≤ c}, we also have that

1 X R − R 1 1

P {X R =c}=P k − √ < √ ≤ k ≈ (k) − k − √ ≈ √ ϕ(k).
R R R R
This gives
1 ϕ(k)
P loss ≈ √ . (5.3.4)
R (k)
By (5.3.3) and ρ = R/c, we have P delay = cP loss /(c − R + RP loss ). Substituting
√ √
c = R + k R in this formula, noting that k R << R for R large and using
(5.3.4), we find with the abbreviation z = ϕ(k)/ (k) that
√
−1 −1
(R + k R)z Rz k k (k)
P delay ≈ ≈ = 1 + = 1 + . (5.3.5)
kR + Rz kR + Rz z ϕ(k)
Equating the last term to α gives the relation (5.3.2). This completes the derivation
of the square-root formula (5.3.1).

5.4 INSENSITIVITY

In many stochastic service systems in which arriving customers never queue, it turns
out that the performance measures are insensitive to the form of the service-time
distribution and require only the mean of the service time. The most noteworthy
examples of such service systems are infinite-server systems and loss systems.
In the M/G/∞ queue with Poisson arrivals and infinitely many servers, rather
simple arguments enable us to prove that the limiting distribution of the num-
ber of busy servers is insensitive to the form of the service-time distribution; see
Section 1.1.3. The Erlang loss model with Poisson input and the Engset model with
finite-source input provide other examples of stochastic service systems possessing
the insensitivity property. Other examples of stochastic service systems having the
insensitivity property will be given in this section and in the exercises. Nowadays