Page 208 - A First Course In Stochastic Models

SERVICE-SYSTEM DESIGN 201

Table 5.3.2 The exact and approximate values of c ∗
α = 0.5 α = 0.2 α = 0.1
exa app exa app exa app
R = 1 2 2 3 3 3 3
R = 5 7 7 8 8 9 9
R = 10 12 12 14 14 16 15
R = 50 54 54 58 58 61 61
R = 100 106 106 111 111 115 115
R = 250 259 259 268 267 274 273
R = 500 512 512 525 524 533 532
R = 1000 1017 1017 1034 1034 1046 1045

increases the manager’s intuitive understanding of the system. In particular, the
square-root staffing rule quantifies the economies of scale in staffing levels that
can be achieved by combining several call centres into a single call centre. To
illustrate this, consider two identical call centres each having an offered load
of R erlangs of work and each having the same service requirement P W ≤ α.
√
For two separate call centres a total of 2(R + k α R) agents is needed, whereas
√
for one combined call centre 2R + k α 2R agents are needed. A reduction of
√ √
(2 − 2)k α R agents.
The quality of the approximation (5.3.1) is excellent. Rounding up the approx-
imation for c to the nearest integer, numerical investigations indicate that the
∗
approximate value is equal to the exact value in most cases and is never off by
more than 1. Table 5.3.2 gives the exact and approximate values of c for several
∗
values of R and α.

Derivation of the square-root formula
The following relation holds between the delay probability P delay in the M/M/c
delay system and the loss probability P loss in the M/M/c/c loss system:

(1 − ρ)P delay
P loss = . (5.3.3)
1 − ρP delay
This relation can be directly verified from the explicit formulas (5.1.11) and (5.2.2)
for P delay and P loss . In Section 9.8 we establish the relation (5.3.3) in a more general
framework by showing that the state probabilities in a finite-capacity queue with
Poisson arrivals are often proportional to the state probabilities in the corresponding
infinite-capacity model. By formula (5.2.2),

c
e −R R /c!
P loss = .
c
−R k
e R /k!
k=0