Page 402 - A First Course In Stochastic Models

MULTI-SERVER QUEUES WITH POISSON INPUT 397

holds with
σ[β(τ) − 1]
γ = 2 c−1 ,
(τ − 1) τ β
where τ and σ are given by (9.6.42) and (9.6.43). This result can be derived in a
similar way as expansion (9.6.11) for the M/D/c queue was obtained.

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The M /G/c queue
X
An exact and tractable solution for the M /G/c queue is in general not possible
except for the special cases of deterministic services and exponential services.
Using the solutions for these special cases, we can give useful approximations for
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the general M /G/c queue. A practically useful approximation to the average
delay in queue per customer is
app 2 2
W q = (1 − c )W q (det) + c W q (exp),
S S
2
2
provided that c is not too large (say, 0 ≤ c ≤ 2) and the traffic load is not
S S
very small. It was pointed out in Section 9.3 that the first-order approximation
1 2
S
2 (1 + c )W q (exp) is not applicable in the batch-arrival queue. A two-moment
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Table 9.6.3 The percentiles η(p) for the M /E 2 /c queue
Constant batch size Geometric batch size
c ρ p 0.80 0.90 0.95 0.99 0.80 0.90 0.95 0.99
1 0.2 exa 2.927 3.945 4.995 7.458 5.756 8.122 10.49 15.98
app 2.836 3.901 4.967 7.440 5.745 8.116 10.49 15.99
1 0.5 exa 5.107 7.170 9.231 14.02 9.044 12.84 16.64 25.45
app 5.089 7.154 9.219 14.01 9.040 12.84 16.64 25.47
2 0.2 exa 1.369 1.897 2.431 3.661 2.989 4.172 5.355 8.101
app 1.354 1.887 2.419 3.656 2.982 4.167 5.353 8.106

2 0.5 exa 2.531 3.561 4.592 6.985 4.600 6.498 8.395 12.80
app 2.535 3.567 4.599 6.996 4.601 6.501 8.401 12.81

5 0.2 exa 0.621 0.845 1.063 1.560 1.298 1.773 2.246 3.345
app 0.640 0.853 1.066 1.560 1.305 1.779 2.253 3.354
5 0.5 exa 1.063 1.476 1.889 2.846 1.898 2.657 3.417 5.179
app 1.069 1.482 1.895 2.853 1.905 2.665 3.425 5.190
10 0.5 exa 0.553 0.764 0.971 1.451 0.980 1.360 1.740 2.622
app 0.566 0.772 0.979 1.458 0.991 1.371 1.751 2.634

10 0.7 exa 0.923 1.295 1.667 2.530 1.547 2.181 2.815 4.287
app 0.930 1.302 1.673 2.536 1.556 2.190 2.824 4.297