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1 ln P qb > b : I ln b
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9:121
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We now compare these expressions with those obtained in Sections 9.10.2 and 9.11.2. Under the condition c rin < 1; we have lim 1 1 ln P qb > b 2 I ln b 2 2d 9:123 9:122
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in the lognormal case, and lim 1 ln P qb > b a 1 I ln b 9:124
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in the regularly varying case, while no such conclusion can be drawn in the Weibull case. We note that the bounds given by Likhanov in 8 in this volume hold under the weaker condition that c rin be noninteger, and automatically imply Eq. (9.124) (under (9.122)). In Case II with 0 < R < I, we have v* $ Rt so that t G inf and Proposition 9.12.1 yields R lim inf
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c rin y 1 1 y
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9:125
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9:126
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9.14 APPENDIX: THE BASIC UPPER BOUND
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Interestingly enough, this lower bound is not as good as the one obtained in Proposition 9.8.1 by applying the general buffer asymptotics based on large deviations arguments [Section 9.6].
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We have presented recent results on the large buffer asymptotics for an in nite capacity multiplexer with constant release rate and M =G=I inputs. When the distribution of session durations has an exponential tail, these asymptotics are completely identi ed by means of large deviations arguments. The results are not as complete in the nonexponential case. Even in the more restricted setup where session durations have a subexponential distribution, only lower and upper bounds are available in general; they are derived through a variety of techniques, including the asymptotics of Pakes for the GI =GI =1 queue. These bounds have been shown to be tight for the generalized Pareto case. The issue is still open for other subexponential distributions (e.g., Weibull and lognormal), although the bounds are known to be tight in the lognormal case under some conditions on the release rate.
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APPENDIX: THE BASIC UPPER BOUND
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We begin the derivation of the companion upper bound (9.5) by establishing a basic asymptotic upper bound. For each m 1; 2; . . . and b > 0, de ne the quantities  A m; b  m max P Sn > b and B m; b  P sup
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n 1;...;m
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The representation (9.12) and a union bound argument lead to P qI > b A m; b B m; b 2 max A m; b ; B m; b : 9:127
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We now x y > 0. Next, substitute b=y for m in Eq. (9.127), take the logarithm and divide by h b . Letting b go to in nity in the resulting inequality, we obtain the basic asymptotic upper bound lim sup
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1 ln P qI > b h b
max e y ; f y ;
9:128
where we have used the notation e y  lim sup
1 ln A b=y ; b h b
9:129
BUFFER ASYMPTOTICS FOR M=G=I INPUT PROCESSES
and f y  lim sup
1 ln B b=y ; b : h b
9:130
In the next two sections we show that e y a y and f y b y with a y and b y given by Eqs. (9.49) and (9.50), with the immediate consequence that Eq. (9.48) follows from Eq. (9.128).
APPENDIX: UPPERBOUNDING e y y > 0
Fix y > 0. It is plain that 1 1 1 ln A b=y ; b ln b=y h b h b h b max ln P Sn > b ; b > 0: 9:131 Under Condition (ii) and Eq. (9.36), it is a simple matter to check that lim sup
n 1;...; b=y
1 ln b=y Kg y : h b
9:132
For the second term of Eq. (9.131), we have 1 max ln P Sn > b h b n 1;...; b=y 1 sup ln P S b=x > b h b x>y   S b=x 1 b ln P > sup b=x b=x x>y h b   S b=x 1 ln P >x sup b=x x>y h b   v b=x S b=x 1 >x ; sup ln P b=x x>y h b v b=x
b > 0;
and invoking Eq. (9.36) again we conclude 1 lim sup max ln P Sn > b b3I h b n 1;...; b=y  ! vt 1 St ln P >x : lim sup sup t t3I x>y h tx vt 9:133