Milan Vojnovic

Power law and exponential decay of inter contact times between mobile devices

We examine the fundamental properties that determine the basic performance metrics for opportunistic communications. We first consider the distribution of intercontact times between mobile devices. Using a diverse set of measured mobility traces, we find as an invariant property that there is a characteristic time, order of half a day, beyond which the distribution decays exponentially. Up to this value, the distribution in many cases follows a power law, as shown in recent work. This power law finding was previously used to support the hypothesis that intercontact time has a power law tail, and that common mobility models are not adequate. However, we observe that the timescale of interest for opportunistic forwarding may be of the same order as the characteristic time, and thus, the exponential tail is important. We further show that already simple models such as random walk and random waypoint can exhibit the same dichotomy in the distribution of intercontact time as in empirical traces. Finally, we perform an extensive analysis of several properties of human mobility patterns across several dimensions, and we present empirical evidence that the return time of a mobile device to its favorite location site may already explain the observed dichotomy. Our findings suggest that existing results on the performance of forwarding schemes based on power law tails might be overly pessimistic.

Power Law and Exponential Decay of Intercontact Times between Mobile Devices

We examine the fundamental properties that determine the basic performance metrics for opportunistic communications. We first consider the distribution of intercontact times between mobile devices. Using a diverse set of measured mobility traces, we find as an invariant property that there is a characteristic time, order of half a day, beyond which the distribution decays exponentially. Up to this value, the distribution in many cases follows a power law, as shown in recent work. This power law finding was previously used to support the hypothesis that intercontact time has a power law tail, and that common mobility models are not adequate. However, we observe that the timescale of interest for opportunistic forwarding may be of the same order as the characteristic time, and thus, the exponential tail is important. We further show that already simple models such as random walk and random waypoint can exhibit the same dichotomy in the distribution of intercontact time as in empirical traces. Finally, we perform an extensive analysis of several properties of human mobility patterns across several dimensions, and we present empirical evidence that the return time of a mobile device to its favorite location site may already explain the observed dichotomy. Our findings suggest that existing results on the performance of forwarding schemes based on power law tails might be overly pessimistic.

Coupon replication systems

Motivated by the study of peer-to-peer file swarming systems a la BitTorrent, we introduce a probabilistic model of coupon replication systems. These systems consist of users aiming to complete a collection of distinct coupons. Users enter the system with an initial coupon provided by a bootstrap server, acquire other coupons from other users, and leave once they complete their coupon collection. For open systems, with exogenous user arrivals, we derive stability condition for a layered scenario, where encounters are between users holding the same number of coupons. We also consider a system where encounters are between users chosen uniformly at random from the whole population. We show that sojourn time in both systems is asymptotically optimal as the number of coupon types becomes large. We also consider closed systems with no exogenous user arrivals. In a special scenario where users have only one missing coupon, we evaluate the size of the population ultimately remaining in the system, as the initial number of users N goes to infinity. We show that this size decreases geometrically with the number of coupons K. In particular, when the ratio K/ log(N) is above a critical threshold, we prove that this number of leftovers is of order log(log(N)). These results suggest that, under the assumption that the bootstrap server is not a bottleneck, the performance does not depend critically on either altruistic user behavior or on load-balancing strategies such as rarest first.

FENNEL: streaming graph partitioning for massive scale graphs

Balanced graph partitioning in the streaming setting is a key problem to enable scalable and efficient computations on massive graph data such as web graphs, knowledge graphs, and graphs arising in the context of online social networks. Two families of heuristics for graph partitioning in the streaming setting are in wide use: place the newly arrived vertex in the cluster with the largest number of neighbors or in the cluster with the least number of non-neighbors. In this work, we introduce a framework which unifies the two seemingly orthogonal heuristics and allows us to quantify the interpolation between them. More generally, the framework enables a well principled design of scalable, streaming graph partitioning algorithms that are amenable to distributed implementations. We derive a novel one-pass, streaming graph partitioning algorithm and show that it yields significant performance improvements over previous approaches using an extensive set of real-world and synthetic graphs. Surprisingly, despite the fact that our algorithm is a one-pass streaming algorithm, we found its performance to be in many cases comparable to the de-facto standard offline software METIS and in some cases even superiror. For instance, for the Twitter graph with more than 1.4 billion of edges, our method partitions the graph in about 40 minutes achieving a balanced partition that cuts as few as 6.8% of edges, whereas it took more than 81/2 hours by METIS to produce a balanced partition that cuts 11.98% of edges. We also demonstrate the performance gains by using our graph partitioner while solving standard PageRank computation in a graph processing platform with respect to the communication cost and runtime.

Parallel TCP Sockets: Simple Model, Throughput and Validation

We consider a simple model of parallel TCP connections defined as follows. There are N connections competing for a bottleneck of fixed capacity. Each connection is assumed to increase its send rate linearly in time in absence of congestion indication and otherwise decreases its rate to a fraction β of the current send rate. Whenever aggregate send rate of the connections hits the link capacity, a single connection is signalled a congestion indication. Under the prevailing assumptions, and assuming only in addition a mild stability condition, we obtain that the throughput is the factor of the link capacity, 1−1/(1+ const N), with const = (1+β)/(1−β). This result appears to be previously unknown; despite simplicity of its final form, it is not immediate. The result is of practical importance as it elucidates the throughput of parallel TCP sockets, an approach used widely to improve throughput performance of bulk data transfers (e.g. GridFTP), in regimes when individual connections are none or weakly synchronized. We argue that it is important to distinguish two factors that contribute to TCP throughput deficiency (F1) TCP window synchronization and (F2) TCP window adaptation in congestion avoidance. Our result is a good news as it suggests that in regimes when (F1) does not hold, already a few sockets are enough to almost entirely eliminate the deficiency due to (F2). Specifically, the result suggests that already 3 TCP connections yield 90% link utilization and 95% is almost achieved by 6 connections. This analytically proven result should provide incentive to throughput-greedy users to limit the number of their parallel TCP sockets as a few connections already ensure effectively 100% utilization, and any additional connection would provide only a marginal throughput gain. Opening too many sockets is not desirable as such transfers may beat down other connections sharing a link on the path of this transfer. However, there still remains a throughput deficiency due to (F1), which may provide incentive to users to open more sockets. The result found in this paper suggests that throughput-deficiency of parallel TCP sockets would be largely attributed to the synchronization factor (F1) and not to window control (F2). This motivates intelligent queueing disciplines that help mitigating the synchronization. As a by-product, the result shows that emulation of parallel TCP connections by MultTCP protocol is a good approximation. The implication of the result is that aggregate throughput achieved by connections is insensitive to a choice of loss policy which connection is signalled a congestion indication at congestion events. This perhaps somewhat counterintuitive throughput insensitivity property is showed not to hold for steady-state distribution of the aggregate send rate. We provide validation of our results by simulations and internet measurements.

On the long-run behavior of equation-based rate control

We consider unicast equation-based rate control, where, at some points in time, a source adjusts its rate to f(p, r). Here p is an on-line estimate of the loss-event rate, r, of the mean round-trip time, both as observed by this source, and f is a TCP throughput formula. It was generally believed that such a source would be TCP-friendly, that is, under the same operating conditions, its long-run time-average send rate (throughput) would not be larger than that of a TCP source. Our goal is to identify whether, and how far, this is true. First, we identify factors that play a role in TCP friendliness and find that it is important to study them separately. Then we analyze the importance of individual factors. A first factor is conservativeness (= throughput not larger than f(p, r)). We show that conservativeness is influenced by some convexity properties of f(p, r) with respect to p, and the covariance of the loss process. We show that in many real life cases these conditions result in conservativeness and, sometimes, excessive conservativeness. This explains the previously observed phenomena of throughput-drop when losses are high and f is the so-called PFTK formula. The second factor is that the source may experience considerably different loss-event rate than a TCP source. We identify and analyze two limit cases where this may lead to either TCP-friendliness or, in contrast, non-TCP-friendliness. Other factors such as round trip time and obedience of TCP to its own formula are found to be less significant. Our claims are obtained by analysis, and verified by numerical examples, simulations, laboratory and Internet experiments. Our results suggest that TCP-friendliness is difficult to verify in practice, whereas conservativeness is easier.

Stochastic analysis of some Expedited Forwarding networks

We consider stochastic guarantees for networks with aggregate scheduling, in particular, Expedited Forwarding (EF). Our approach is based on the assumption that a node can be abstracted by a service curve, and the input flows are regulated individually at the network ingress. Both of these assumptions are in line with EF. For a service curve node, we derive bounds on the complementary distributions of the steady-state backlog and backlog as seen by packet arrivals. We also give a bound on the long-run loss ratio for a service curve node where the buffer size is too small to guarantee loss-free operation. For a packet scale rate guarantee node, we use the delay from the backlog bound to obtain a probabilistic bound on the delay. Our analysis is exact under the given assumptions. Our results should help us to understand the performance of networks with aggregate scheduling, and provide the basis for dimensioning such networks.

Towards mobile ad-hoc WANs: terminodes

Terminodes are personal devices that provide functionality of both the terminals and the nodes of the network. A network of terminodes is an autonomous, fully self-organized, wireless network, independent of any infrastructure. It must be able to scale up to millions of units, without any fixed backbone or server. In this paper we present the main challenges and discuss the main technical directions.

Using Three States for Binary Consensus on Complete Graphs

We consider the binary consensus problem where each node in the network initially observes one of two states and the goal for each node is to eventually decide which one of the two states was initially held by the majority of the nodes. Each node contacts other nodes and updates its current state based on the state communicated by the last contacted node. We assume that both signaling (the information exchanged at node contacts) and memory (computation state at each node) are limited and restrict our attention to systems where each node can contact any other node (i.e., complete graphs). It is well known that for systems with binary signaling and memory, the probability of reaching incorrect consensus is equal to the fraction of nodes that initially held the minority state. We show that extending both the signaling and memory by just one state dramatically improves the reliability and speed of reaching the correct consensus. Specifically, we show that the probability of error decays exponentially with the number of nodes N and the convergence time is logarithmic in N for large N. We also examine the case when the state is ternary and signaling is binary. The convergence of this system to consensus is again shown to be logarithmic in N for large N, and is therefore faster than purely binary systems. The type of distributed consensus problems that we study arises in the context of decentralized peer-to-peer networks, e.g. sensor networks and opinion formation in social networks - our results suggest that robust and efficient protocols can be built with rather limited signaling and memory.

Bounds for independent regulated inputs multiplexed in a service curve network element

We consider the problem of bounding the probability of buffer overflow in a network node fed with independent arrival processes that are each constrained by arrival curves, but that are served as an aggregate. Existing results assume that the node is a constant rate server. However, in practice, one finds complex network nodes that do not provide a constant service rate, and thus, to which the existing bounds do not apply. Now many nodes can be adequately abstracted by a service curve property. We extend previous results to such cases. As a by-product, we also provide a slight improvement to the bound in Chang et al. (see Proc. Sigmettics 2001, Cambridge, MA, May 2001, p.184-193). Our bounds are valid for both discrete and continuous time models.