Predrag R. Jelenkovic

The effect of multiple time scales and subexponentiality in MPEG video streams on queueing behavior

Guided by the empirical observation that real-time MPEG video streams exhibit both multiple time scale and subexponential characteristics, we construct a video model that captures both of these characteristics and is amenable to queueing analysis. We investigate two fundamental approaches for extracting the model parameters: using sample path and second-order statistics-based methods. The model exhibits the following two canonical queueing behaviors. When strict stability conditions are satisfied, i.e., the conditional mean of each scene is smaller than the capacity of the server, precise modeling of the interscene dynamics (long-term dependency) is not essential for the accurate prediction of small to moderately large queue sizes. In this case, the queue length distribution is determined using quasistationary (perturbation theory) analysis. When weak stability conditions are satisfied, i.e., the conditional mean of at least one scene type is greater than the capacity of the server, the dominant effect for building a large queue size is the subexponential (long-tailed) scene length distribution. In this case, precise modeling of intrascene statistics is of secondary importance for predicting the large queueing behavior. A fluid model, whose arrival process is obtained from the video data by replacing scene statistics with their means, is shown to asymptotically converge to the exact queue distribution. Using the transition scenario of moving from one stability region to the other by a change in the value of the server capacity, we synthesize recent queueing theoretic advances and ad hoc results in video modeling, and unify a broad range of seemingly contradictory experimental observations found in the literature. As a word of caution for the widespread usage of second-order statistics modeling methods, we construct two processes with the same second-order statistics that produce distinctly different queueing behaviors.

Least-recently-used caching with dependent requests

We investigate a widely popular least-recently-used (LRU) cache replacement algorithm with semi-Markov modulated requests. Semi-Markov processes provide the flexibility for modeling strong statistical correlation, including the widely reported long-range dependence in the World Wide Web page request patterns. When the frequency of requesting a page n is equal to the generalized Zipfs law c/nα, α > 1, our main result shows that the cache fault probability is asymptotically, for large cache sizes, the same as in the corresponding LRU system with i.i.d. requests. The result is asymptotically explicit and appears to be the first computationally tractable average-case analysis of LRU caching with statistically dependent request sequences. The surprising insensitivity of LRU caching performance demonstrates its robustness to changes in document popularity. Furthermore, we show that the derived asymptotic result and simulation experiments are in excellent agreement, even for relatively small cache sizes.

Reduced-Load Equivalence and Induced Burstiness in GPS Queues with Long-Tailed Traffic Flows

We analyze the queueing behavior of long-tailed traffic flows under the Generalized Processor Sharing (GPS) discipline. We show a sharp dichotomy in qualitative behavior, depending on the relative values of the weight parameters. For certain weight combinations, an individual flow with long-tailed traffic characteristics is effectively served at a constant rate. The effective service rate may be interpreted as the maximum average traffic rate for the flow to be stable, which is only influenced by the traffic characteristics of the other flows through their average rates. In particular, the flow is essentially immune from excessive activity of flows with ‘heavier’-tailed traffic characteristics. In many situations, the effective service rate is simply the link rate reduced by the aggregate average rate of the other flows. This confirms that GPS-based scheduling algorithms provide a potential mechanism for extracting significant multiplexing gains, while isolating individual flows. For other weight combinations however, a flow may be strongly affected by the activity of ‘heavier’-tailed flows, and may inherit their traffic characteristics, causing induced burstiness. The stark contrast in qualitative behavior illustrates the crucial importance of the weight parameters.

Dynamic packet fragmentation for wireless channels with failures

It was shown recently [7-9], under quite general conditions, that retransmission-based protocols may result in power-law delays and possibly zero throughput even if the distribution of packets (data units) is very concentrated, e.g., exponential or Gaussian. This phenomenon occurs irrespective of whether the cause of retransmissions is due to channel failures in the data link layer [7] or collisions in ALOHA-type protocols in the MAC layer [9]. These theoretical findings are in agreement with empirical measurements in [18], showing that the utilization of the 802.11 protocol is only 40%, basically due to retransmissions. In order to alleviate this problem, we propose a new dynamic packet fragmentation algorithm that can adaptively match channel failure characteristics. This algorithm is based on the mathematical insights developed in [7, 8]. As a first order approximation to the channel dynamics, we assume that the channel is either available for a period of time or unavailable. Then, our fragmentation algorithm divides the original packets into smaller ones whose size is bounded by the kth largest value among the last k+m channel availability periods. We also discuss mechanisms for aggregating smaller packets into larger ones, which, in combination with fragmentation, can further improve the performance. Under the renewal assumptions on the channel dynamics, we prove that our fragmentation method results in k additional moments for the total transmission time until all the fragments are successfully transmitted, i.e., the transmission time has a much more concentrated distribution and, in particular, the channel will always have a positive throughput. In addition, we argue that by tuning the parameter m, the number of introduced new packets can be kept reasonably small as well. Furthermore, we demonstrate through simulations that the superior performance of our fragmentation algorithm extends beyond the renewal assumptions used in the analysis to time varying and/or correlated channels. For practical implementation of the algorithm, we also discuss approaches to measuring the channel availability periods, especially in situations when the channel dynamics may not be directly observable. It is worth mentioning that our algorithm can be used for designing efficient checkpointing schemes in other systems that are prone to failures, e.g., distributed computing systems.

Can Retransmissions of Superexponential Documents Cause Subexponential Delays

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 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.

Large Deviation Analysis of Subexponential Waiting Times in a Processor-Sharing Queue

We investigate the distribution of the waiting timeV in a stable M/G/1 processor-sharing queue with traffic intensityp x] = P[ B > (1 -p) x](1 +o(1)) as x ? 8. Furthermore, we demonstrate that the preceding relationship does not hold if the service distribution has a lighter tail thane-vx.

Subexponential loss rates in a GI/GI/1 queue with applications

AbstractConsider a single server queue with i.i.d. arrival and service processes,

Asymptotic insensitivity of least-recently-used caching to statistical dependency

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