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Can Retransmissions of Superexponential Documents Cause Subexponential Delays?
Jelenkovic, P.R.; Jian Tan;
INFOCOM 2007. 26th IEEE International Conference on Computer Communications. IEEE
6-12 May 2007
Page(s):892
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900
Abstract:
Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel dynamics is modeled as an on-off process {(Ai, E/j)}iles1 with alternating independent periods when channel is available Ai and unavailable Ui, respectively. During each period of time that the channel becomes available, say Ai, we attempt to transmit the data unit. If L les Ai, the transmission was considered successful; otherwise, we wait for the next period Ai+i when the channel is available and attempt to retransmit the data from the beginning. We study the asymptotic properties of the total transmission time T and number of retransmissions N until the data is successfully transmitted. In recent studies it was proved that the waiting time T follows a power law when the distributions of L and A1 are of an exponential type, e.g., Gamma distribution. In this paper, we show that the distributions of N and T follow power laws with exponent alpha as long as logP[L > x] apalphalogP[A1 > x] for large x. Hence, it may appear surprising that we obtain power law distributions irrespective of how heavy or light the distributions of L and A1 may be. In particular, both L and A1 can decay faster than any exponential, which we term superexponential. For example, if L and A1 are Gaussian with variances sigma2 L and sigma2 A, respectively, then N and T have power law distributions with exponent alpha = sigma2 A/sigma2 L; note that, if sigma2 A<sigma2 L, the transmission time has an infinite mean and, thus, the system is unstable. The preceding model, as recognized in (Fiorini et al., 2005), describes a variety of situations where failures require jobs to restart from the beginning. Here, we identify that this mo- del also provides a new mechanism for explaining
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