Page 395 - A First Course In Stochastic Models

390 ALGORITHMIC ANALYSIS OF QUEUEING MODELS

Table 9.6.1 Exact and approximate results
2
2
2
c = 0 c = 0.5 c = 2
S S S
P delay L q P delay L q P delay L q
c = 2 exac 0.3233 0.177 0.3308 0.256 0.3363 0.487
ρ = 0.5 app 0.3333 0.194 0.3333 0.260 0.3333 0.479
app2 — 0.176 — 0.255 — 0.491
c = 5 exa 0.1213 0.077 0.1279 0.104 0.1335 0.181
ρ = 0.5 app 0.1304 0.087 0.1304 0.107 0.1304 0.176
app2 — 0.076 — 0.103 — 0.185
c = 10 exa 0.0331 0.024 0.0352 0.030 0.0373 0.048
ρ = 0.5 app 0.0361 0.025 0.0361 0.030 0.0361 0.047
app2 — 0.023 — 0.030 — 0.049
c = 2 exa 0.7019 1.445 0.7087 2.148 0.7141 4.231
ρ = 0.8 app 0.7111 1.517 0.7111 2.169 0.7111 4.196
app2 — 1.442 — 2.143 — 4.247
c = 5 exa 0.5336 1.156 0.5484 1.693 0.5611 3.250
ρ = 0.8 app 0.5541 1.256 0.5541 1.723 0.5541 3.191
app2 — 1.155 — 1.686 — 3.277
c = 25 exact 0.1900 0.477 0.2033 0.661 0.2164 1.173
ρ = 0.8 approx 0.2091 0.495 0.2091 0.663 0.2091 1.178
approx2 — 0.477 — 0.657 — 1.196
c = 50 exa 0.0776 0.214 0.0840 0.282 0.0908 0.471
ρ = 0.8 app 0.0870 0.207 0.0870 0.277 0.0870 0.488
app2 — 0.211 — 0.279 — 0.485

This two-moment approximation can be found in Cosmetatos (1976) and Page
(1972). The useful special-purpose approximation

√
1 4 + 5c − 2
app
L q (det) = 1 + (1 − ρ)(c − 1) L q (exp)
2 16cρ
to L q (det) was proposed in Cosmetatos (1976). The results in Table 9.6.1 for the
approximation (9.6.24) use this approximation to L q (det).

Asymptotic expansions
It is assumed that the probability distribution function B c (t) = B(ct) satisfies
Assumption 9.2.1. In other words, the service-time distribution is not heavy-tailed.
∞ st

Let B = sup[s | e {1 − B(ct)} dt < ∞]. Then, using (9.6.22) and Theorem
0
C.1 in Appendix C, it is a routine matter to verify that
app −j
p ∼ σ app τ as j → ∞, (9.6.25)
j