Can Retransmissions of Superexponential Documents Cause Subexponential Delays?

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, Ui)} i≥1 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 ≤ Ai, the transmission was considered successful; otherwise, we wait for the next period Ai+1 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 [1], [2], it was proved that the waiting time T follows a power law when the distributions of L and A 1 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 α as long as log P[L > x] ≈ α log P[A 1 > 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 σ 2 L and σ 2 A , respectively, then N and T have power law distributions with exponent α = σ 2 A /σ 2 L ; note that, if σ 2 A < σ 2 L , the transmission time has an infinite mean and, thus, the system is unstable.The preceding model, as recognized in [1], describes a variety of situations where failures require jobs to restart from the beginning. Here, we identify that this model also provides a new mechanism for explaining the frequently observed power law phenomenon in data networks. Specifically, we argue that it may imply the power laws on both the application as well as the data link layer, where variable-sized documents and (IP) packets are transmitted, respectively. We discuss the engineering ramifications of our observations, especially in the context of wireless ad hoc and sensor networks where channel failures are frequent. Furthermore, our results provide an easily computable benchmark for measuring the matching between the data and channel characteristics that permits/prevents satisfactory transmission.

Distribution of the Number of Retransmissions of Bounded Documents

“…Now, by continuity of G(x), there exists x 0 such that G(x 0 ) = HḠ(b); we can choose x 0 larger than in (2.11) by picking b 0 large enough. Next, using the elementary inequality 1 − x ≤ e −x , x ≥ 0, we upper bound the preceding expression by 12) where…”

“…It was first recognized in [6,18] that such mechanisms may result in long-tailed (power law) delays even if the job sizes and failure rates are exponential. In [12], it was noted that the same mechanism is at the core of modern communication networks where retransmissions are used on all protocol layers to guarantee data delivery in the presence of channel failures. Furthermore, [12] shows that the power law number of retransmissions and delay occur whenever the hazard functions of the data and failure distributions are proportional.…”

“…In [12], it was noted that the same mechanism is at the core of modern communication networks where retransmissions are used on all protocol layers to guarantee data delivery in the presence of channel failures. Furthermore, [12] shows that the power law number of retransmissions and delay occur whenever the hazard functions of the data and failure distributions are proportional. Hence, power laws may arise even if the data and channel failure distributions are both Gaussian.…”

“…These types of mechanisms have been employed on all networking layers, including, for example, Automatic Repeat reQuest (ARQ) protocol (e.g., see Section 2.4 of [4]) in the data link layer where a packet is resent automatically in case of an error; contention based ALOHA type protocols in the medium access control (MAC) layer that use random backoff and retransmission mechanism to recover data from collisions; end-to-end acknowledgement for multi-hop transmissions in the transport layer; HTTP downloading scheme in the application layer, etc. It has been shown that several well-known retransmission based protocols in different layers of networking architecture can lead to power law delays, e.g., ALOHA type protocols in MAC layer [13,10] and end-to-end acknowledgements in transport layer [11,9] as well as in other layers [12]. For other (non-retransmission) mechanisms that can give rise to heavy tails see [15] and the references therein.…”

“…However, these two are often highly correlated in most communication systems, meaning that longer data units have higher probability of error, thus violating the independence assumption. Recent work [12,13,11,9] has shown that, when the data size distribution has infinite support, all retransmission-based protocols could cause heavy-tailed behavior and possibly result in zero throughput, regardless of how light-tailed the distributions of data sizes and channel failures are. Nevertheless, in reality, packet sizes are upper bounded by the maximum transmission unit.…”