Page 207 - A First Course In Stochastic Models

200 MARKOV CHAINS AND QUEUES

there are sufficiently many servers. The larger the number of servers, the higher the
server utilization before the service to the customers seriously degrades. A relatively
large value of P W does not necessarily imply bad service to the customers. For
example, take c = 100 and ρ = 0.95. Then on average 50.6% of the customers must
wait, but the average wait of a delayed customer is only 1 of its mean service time.
5
Moreover, on average, only 5% of the delayed customers have to wait more than 3
5
of the mean service time. The situation of many servers is encountered particularly
in the telephone call centre industry. Service level is a key performance metric
of a call centre. In practice it is often defined as ‘80% of the calls answered in
20 seconds’.

Square-root staffing rule
In the remainder of this section we take the delay probability as service mea-
sure. What is the least number c of servers such that the delay probability P W is
∗
below a prespecified level α, e.g. α = 0.20? From a numerical point of view
∗
it is of course no problem at all to find the exact value of c by searching
over c in formula (5.1.11) for a given value of R (= cρ). However, for prac-
titioners it is helpful to have an insightful approximation formula. Such a for-
mula can be given by using the normal distribution. The formula is called the
square-root staffing rule. This simple rule of thumb for staffing large call cen-
tres provides very useful information to the management. In its simplest form
the square-root formula is obtained by approximating the M/M/c queue with
many servers by the M/M/∞ queue. This approach was used in Example 1.1.3.
However, this first-order approximation can considerably be improved by using
a relation between Erlang’s delay probability in the M/M/c delay system and
Erlang’s loss probability in the M/M/c/c loss system. The improved approx-
∗
imation to the least number c of servers such that P W ≤ α is given by the
square-root formula
√
∗
c ≈ R + k α R, (5.3.1)
where the safety factor k α is the solution of the equation

k (k) 1 − α
= (5.3.2)
ϕ(k) α
with (x) denoting the standard normal probability distribution function and ϕ(x)
√ − x
1 2
= (1/ 2π)e 2 denoting its density. It is important to note that the safety fac-
tor k α does not depend on R. Also, it is interesting to point out the similarity
of the square-root staffing rule with the famous rule for the reorder point s in
the (s, Q)-inventory model with a service-level constraint. The factor k α can be
found by solving (5.3.2) by bisection. For example, for α = 0.8, 0.5, 0.2 and 0.1
the safety factor k α has the respective values 0.1728, 0.5061, 1.062 and 1.420.
The approximation (5.3.1) clarifies the interplay of the process parameters and