Page 205 - A First Course In Stochastic Models

198 MARKOV CHAINS AND QUEUES

the equilibrium distribution of the state at an arbitrary epoch in the system with
one source less. In particular, we find

M − 1 c M−1−c
p (1 − p)
c
the long-run fraction of lost service requests = .
c
M − 1 k M−1−k
p (1 − p)
k
k=0
(5.2.5)
The formulas (5.2.3) to (5.2.5) have been derived under the assumption of expo-
nentially distributed think times and exponentially distributed service times. This
assumption is not needed. The Engset model has the insensitivity property that the
formulas (5.2.3) to (5.2.5) remain valid when the think time has a general prob-
ability distribution with mean 1/α and the service time has a general distribution
with mean 1/µ. This insensitivity result requires the technical condition that either
of these two distributions has a positive density on some interval. We come back
to this insensitivity result in the next section. By letting M → ∞ and α → 0 such
that Mα remains equal to the constant λ, it follows from the Poisson approximation
to the binomial probability that the right-hand side of (5.2.3) converges to
i
e −λ/µ (λ/µ) /i!
, i = 0, 1, . . . , c
c
−λ/µ k
e (λ/µ) /k!
k=0
in agreement with (5.2.1). In other words, the Erlang loss model is a limiting case
of the Engset model. This is not surprising, since the arrival process of service
requests becomes a Poisson process with rate λ when we let M → ∞ and α → 0
such that Mα = λ.

5.3 SERVICE-SYSTEM DESIGN

The Erlang delay model has many practical applications. In particular, it can be
used to analyse capacity and staffing problems such as those arising in the area of
telemarketing and call centre design and in the area of healthcare facilities planning.
In this section it will be shown that a normal approximation to Erlang’s delay
formula is very helpful in analysing such problems. The normal approximation
enables us to derive an insightful square-root staffing rule.
The mathematical analysis of the M/M/c queue was given in Section 5.1.2. In
the M/M/c queue customers arrive according to a Poisson process with rate λ,
the service times of the customers are exponentially distributed with mean 1/µ
and there are c identical servers. It is convenient to denote the offered load to the
system by
λ
R = .
µ