6.5 HEAVY TRAFFIC LIMIT THEOREMS FOR WAITING TIME

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(ii) wf sg is regular for Re s < 0, continuous and uniformly bounded by one for Re s 0, with wf g its boundary value at Re s 0; s (iii) wf0g of0g 1. These conditions constitute a Riemann boundary value problem for the functions ofsg and wfsg. In Cohen [16] conditions on af g and bf g are provided, under which the solution of the boundary value problem is obtained. af g and bf g as speci ed in Eqs. (6.48) and (6.49) satisfy these conditions. The solution reads [16] ofsg eH s ; wf sg eH s ; where 1 H s 2pi iI

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x iI

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Re s > 0; Re s < 0;

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with the integral de ned as a principal value integral at in nity and as a principal value singular Cauchy integral at s if Re s 0 (cf. Cohen [16]). The above representation for ofsg has been exploited in Cohen [17] (see also Cohen [19]) to obtain the following four heavy traf c limit theorems (the rst two were already obtained in Boxma and Cohen [10], but under slightly stronger conditions). To present them, we need the following nomenclature. De nition 6.5.1. The tail of A t is said to be heavier than that of B t whenever one of the following cases occurs: (i) na < nb ; (ii) na nb , ba I, bb < I; (iii) na nb , ba bb I, and f 0. Analogously, the tail of B t is called heavier than that of A t if above the a and b indices are interchanged and f 0 is replaced by f I. A t and B t are said to have similar tails when the tail of A t is not heavier than that of B t and the tail of B t is not heavier than that of A t . Theorem 6.5.2. Consider the stable GI =G=I FCFS queue with A t and B t speci ed by Eqs. (6.48) and (6.49). If the tail of B t is heavier than that of A t , then the ``contracted'' waiting time DB r W =a converges in distribution for r 4 1, the limiting distribution Rnb 1 t is speci ed by its LST 1= 1 rnb 1 , and the coef cient of contraction DB r is that root of the equation Cb rxnb 1 Lb x 1 r; x > 0; 0 < 1 r ( 1; 6:55

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with the property that DB r 5 0 for r 4 1.

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THE SINGLE SERVER QUEUE: HEAVY TAILS AND HEAVY TRAFFIC

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Theorem 6.5.3. Consider the stable GI =G=I FCFS queue with A t and B t speci ed by Eqs. (6.48) and (6.49). If the tail of A(t) is heavier than that of B(t), then the ``contracted'' waiting time DA r W =a converges in distribution for r 4 1, the limiting distribution is the negative exponential distribution with unit mean and LST 1= 1 r , and the coef cient of contraction DA r is that root of the equation Ca xna 1 La x 1 r; x > 0; 0 < 1 r << 1; 6:56

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with the property that DA r 5 0 for r 4 1. Theorem 6.5.4. Consider the stable GI =G=1 FCFS queue with A t and B t speci ed by Eqs. (6.48) and (6.49). If A t and B t have similar tails and na nb 2, then the ``contracted'' waiting time DAB r W =a converges in distribution for r 4 1, the limiting distribution is the negative exponential distribution with unit mean and LST 1= 1 r , and the coef cient of contraction DAB r is that root of the equation Ca xLa x Cb rxLb x 1 r; with the property that DAB r 5 0 for r 4 1. The previous three theorems give conditions under which DW converges in distribution. D appears to be of order (roughly) 1 r 1= min na ;nb 1 . The limiting distribution is exponential when the interarrival time distribution is heavier, and is a heavy-tailed distribution Rn 1 t (which is related to the Mittag Lef er function) when the service time distribution is heavier. It should be noted that R1=2 t p 1 et erfc t . The complementary error function erfc x is a special function that has been investigated in much detail; this has resulted in considerable insight into the behavior of the limiting distribution in this special case (cf. Boxma and Cohen [10]). The most complicated case occurs when r 4 1 while A t and B t have similar tails, with 1 < n : na nb < 2. We introduce the following notation. Let da : with D : Let F r a : 1 2pi iI

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x > 0; 0 < 1 r ( 1;

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