LINDLEY'S RECURSION AND GI =GI=1 QUEUE in .NET framework

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10.3 LINDLEY'S RECURSION AND GI =GI=1 QUEUE
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distribution and its integrated tail distribution for An, respectively, G1 x x 0 P A > u du=EA . Theorem 10.3.1 (Pakes). If G1 P s (or G P s*), and EAn < ECn, then I
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There are several natural avenues for extending this result. In Willekens and Teugels [58] and Abate et al. [1] asymptotic expansion re nements to Theorem 10.3.1 were investigated. For extensions of Theorem 10.3.1 to Markov-modulated M =G=1 queues see Asmussen et al. [4], and to Markov-modulated G=G=1 queues (equivalently random walks) see Jelenkovic and Lazar [36]. Further extension of these results to more general arrival processes was obtained in Asmussen et al. [6]. Recently, Asmussen et al. [5] established an asymptotic relationship between the number of customers in a GI =GI =1 queue and their waiting time distribution. In the rest of this section recent results are presented on a GI =GI =1 queue with a nite buffer and truncated heavy-tailed arrival sequences.
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Finite Buffer GI=GI=1 Queue
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In engineering network switches it is very common to design them as loss systems. The main performance measures for these systems are loss probabilities and loss rates. Unfortunately, there are no asymptotic results in literature that address this problem under the assumption of long-tailed arrivals. Recently, I investigated this problem [31, 33]. Here, in Theorem 10.3.2, I present the main result from my earlier work [31]. The theorem gives an explicit asymptotic characterization of the loss rate in a nite buffer queue with long-tailed arrivals. This result, in combination with results from Jelenkovic and Lazar [35], yields a straightforward asymptotic formula for the loss rate in a uid queue with long-tailed M =G=I arrivals (for more details see Jelenkovic [31, 33]). In addition, I [31, 33] derived an explicit asymptotic approximation of buffer occupancy probabilities. This approximation is uniformly accurate for buffer sizes that are away from the buffer boundaries (zero and the maximum buffer size). Furthermore, as the maximum buffer size increases, the length of the buffer around the boundaries where the approximation does not apply stays constant. This precise knowledge of the buffer probabilities allows computation of various other functionals of the nite buffer queue. The evolution of a nite buffer queue is de ned with the following recursion: QB min QB An 1 Cn 1 ; B ; n 1 n n ! 0;
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ASYMPTOTIC ANALYSIS OF QUEUES WITH SUBEXPONENTIAL ARRIVALS
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where B is the buffer size. We assume that the queueing process is in its stationary regime. The loss rate is de ned as lB E QB An 1 Cn 1 B : n loss Theorem 10.3.2. EA < EC, then Let G1 be the integrated tail distribution of A. If G1 P s and lB E A B 1 o 1 loss
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HEURISTIC 10.3.4. Following the general heuristics for subexponential distributions the large buffer over ow is due to one (isolated) large arrival An . At the moment when this happens (say, time n) the queue length process is, because of the stability condition EA < EC, typically very small in comparison to B. Similarly, Cn is much smaller than B. Hence, the amount that is lost at the time of over ow is approximately QB An 1 Cn 1 B % An 1 B . n Accuracy of Theorem 10.3.2 was demonstrated [31, 33] with many numerical and simulation experiments. Here, an example is presented. Example 10.3.5. Take Cn  2 and an arrival distribution P A 0 1, 2 P A i 0:461969=i4 ; i > 0, EA 0:5553. Then, we numerically compute the loss rates lB for the maximum buffer sizes B 100i; i 1; . . . ; 7. The results are loss presented with circles in Fig. 10.1. Note that for B 700 we needed to solve a
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LOG10 (loss rate)
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