THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC

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6.2 6.2.1

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WAITING TIME TAIL BEHAVIOR Introduction

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In Section 6.1 we have already mentioned a result of Cohen [13] that relates the (regularly varying) tail behavior of service and waiting times in the GI =G=1 queue with the FCFS discipline; it shows that the waiting time is ``one degree heavier'' than the service time tail, in the case of regular variation. In Section 6.2.2 this result, and an extension, will be discussed in some more detail for the M =G=1 FCFS queue. A similar result for the M =G=1 queue with the PS discipline (due to Zwart and Boxma [42]) will be discussed in Section 6.2.3. That result shows that, under processor sharing, the waiting time tail is just as heavy as the service time tail. In Section 6.2.4 we prove that the latter phenomenon also occurs in the M =G=1 queue with the LCFS-PR discipline. In the present section we rst introduce some notation, and we present a very useful lemma that relates the tail behavior of a regularly varying probability distribution and the behavior of its Laplace Stieltjes transform (LST) near the origin. Consider the M =G=1 queue. Customers arrive according to a Poisson process with rate l; their service times B1 ; B2 ; . . . are i.i.d. (independent, identically distributed) random variables with nite mean b and LST bfsg. A generic service time is denoted by B. By B* we denote a random variable of which the distribution is that of a residual service time: P B* > x 1 b I

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P B > u du;

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x ! 0:

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Its LST is given by b*fsg : 1 bfsg =bs. The traf c load r : lb of the M=G=1 queue is assumed to be less than one, so that the steady-state waiting time distribution exists. A very useful property of probability distributions with regularly varying tails is a characterization of the behavior of its LST near the origin. Let F be the distribution of a nonnegative random variable, with LST ffsg and nite rst n moments m1 ; . . . ; mn (and m0 1). De ne " fn fsg : 1

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# s j ffsg mj : j! j 0

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Lemma 6.2.1. lent:

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Let n < n < n 1, C ! 0. The following statements are equivafn fsg C o 1 sn L 1=s ; 1 F t C o 1 s 5 0; s real; t 3 I: 6:3 6:4

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1 n n t L t ; G 1 n

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6.2 WAITING TIME TAIL BEHAVIOR

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The case C > 0 is due to Bingham and Doney [4]. The case C 0 is treated in Boxma and Dumas [12, Lemma 2.2]. The case of an integer n is more complicated; see Bingham et al. [5, Theorem 8.1.6 and Chap. 3]. 6.2.2 The M=G=1 FCFS Queue

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We rst formulate the main result of Cohen [13] for the GI =G=1 queue with FCFS discipline in full generality. There is no need to specify the interarrival time distribution; the mean interarrival time and traf c load are denoted by 1=l and r lb (as before). In what follows, W denotes the steady-state waiting time. Theorem 6.2.2.

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For r < 1 and n > 1, n 1 n x r x x3I L x ( P W > x $ A L x : b 1 r b 6:5

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P B > x $ n 1

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Pakes [29] has extended this result to the larger class of subexponential distributions (i.i.d. stochastic variables X1 and X2 have a subexponential tail if t3I P X1 X2 > t =P X1 > t $ 2 . His result states that P W > P if and only if P B* > P , and if either is the case then P W > x $

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