Page 394 - A First Course In Stochastic Models

MULTI-SERVER QUEUES WITH POISSON INPUT 389

It is a matter of simple algebra to derive from (9.6.13) that
α (z)
app
P q (z) = λp , (9.6.22)
c−1
1 − λβ (z)
where

∞

c−1 −λ(1−z)t
α(z) = {1 − B e (t)} {1 − B(t)}e dt,
0
∞ −λ(1−z)t

β(z) = {1 − B(ct)}e dt.
0
app
The discrete FFT method can be used to obtain the p for j ≥ c.
j
Also, the generating function P q (z) enables us to obtain an approximation to
∞ ′

the average queue size. Since L q = (j − c)p j , the derivative P (1) yields
j=c q
an approximation to L q . By differentiation of (9.6.22), we find after lengthy alge-
bra that

app c 1 2
L q = (1 − ρ)γ 1 + ρ (1 + c ) L q (exp), (9.6.23)
S
E(S) 2
2
2
2
where c = σ (S)/E (S) and
S

∞
c
γ 1 = {1 − B e (t)} dt.
0
2
The quantity L q (exp) denotes the average queue size in the M/M/c queue. If c ≤
S
2
1, the constant γ 1 is very well approximated by (c+1) −1 2 −1 (1−c )E(S).
c E(S)+c
S S
The approximation (9.6.23) has the term γ 1 in common with the approximation pro-
posed in Boxma et al. (1979). This approximation improves the first-order approx-
imation 1 $ 1 + c 2 % L q (exp) to L q through
2 S
1 2 2L q (exp)L q (det)
Box
L q = (1 + c ) ,
S
2 2αL q (det) + (1 − α)L q (exp)
& 2 '
1 E(S )
where α = − c − 1 and L q (det) denotes the average queue size in
c−1 γ 1 E(S)
the M/D/c queue.
Table 9.6.1 gives for several examples the exact and approximate values of
2
P delay and L q . We consider the cases of deterministic service (c = 0), E 2 service
S
2
2
(c = 0.5) and H 2 service with the gamma normalization (c = 2). In the table
S S
we also include the two-moment approximation
app2 2 2
L q = (1 − c )L q (det) + c L q (exp). (9.6.24)
S S