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22
Stochastically recursive sequences and their generalizations
 Siberian Adv. Math
, 1992
"... The paper deals with the stochastically recursive sequences { X ( n) } defined as the solutions of equations X ( n + 1) = f ( X ( n) , ξn) (where ξn is a given random sequence), and with random sequences of a more general nature, named recursive chains. For those the theorems of existence, ergodici ..."
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Cited by 40 (11 self)
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The paper deals with the stochastically recursive sequences { X ( n) } defined as the solutions of equations X ( n + 1) = f ( X ( n) , ξn) (where ξn is a given random sequence), and with random sequences of a more general nature, named recursive chains. For those the theorems of existence, ergodicity, stability are established, the stationary majorants are constructed. Continuoustime processes associated with ones studied here are considered as well. Key words and phrases: stochastically recursive sequence; recursive chain; generalized Markov chain; renovating event; couplingconvergence; ergodicity; stability; rate of convergence; stationary majorants; boundedness in probability; processes admitting embedded stochastically recursive sequences. CHAPTER 1.
Effective Bandwidths of Departure Processes from Queues with Time Varying Capacities
 In Proceedings of IEEE INFOCOM'95
, 1995
"... In this paper, we consider a queue with a time varying capacity and identify the effective bandwidth of the stationary departure process from such a queue. Two important observations are made: (i) the effective bandwidths for the transient departure process and the stationary departure process from ..."
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Cited by 29 (1 self)
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In this paper, we consider a queue with a time varying capacity and identify the effective bandwidth of the stationary departure process from such a queue. Two important observations are made: (i) the effective bandwidths for the transient departure process and the stationary departure process from such a queue are in general different, and (ii) sometimes it is necessary to build up the queue first in order to have a large excursion of the stationary departure process. The new result on the effective bandwidth of the stationary departure process is applied to intree networks with time varying capacities and priority tandem queues. Algorithms for approximating the tail distributions of queue lengths in such networks are derived.
DiscreteTime Analysis of Adaptive Rate Control Mechanisms
 In Proc. 5th Int. Conference on Data and Communications
, 1994
"... We analyze the performance of a generic feedback flow control mechanism which captures the properties of several such mechanisms recently proposed in the literature. These mechanisms dynamically regulate the rate of data flow into a network based on feedback information about the network state. They ..."
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Cited by 25 (4 self)
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We analyze the performance of a generic feedback flow control mechanism which captures the properties of several such mechanisms recently proposed in the literature. These mechanisms dynamically regulate the rate of data flow into a network based on feedback information about the network state. They are used in a variety of networks and they have been advocated for upcoming highspeed networks. However, they are difficult to model realistically. In this paper, we present a stochastic discretetime approach that yields models which are realistic and yet tractable and computationally easy to solve. For our generic mechanism, the feedback consists of an exponentially averaged estimate of the bottleneck service rate and queue size. We obtain a model described by nonlinear stochastic difference equations. We find the conditions under which these equations converge to a steadystate and we characterize the speed of convergence to steadystate. We then consider a linearized version of the mo...
Computable Exponential Bounds for Intree Networks with Routing
 Proc. INFOCOM'95
, 1995
"... In this paper, we re ne the calculus proposed in [5, 8, 9]. The new calculus, including network operations for multiplexing, inputoutput relation, and routing, allows us to compute tighter exponential bounds for the tail distributions of queue lengths in intree networks with routing. In particular ..."
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Cited by 9 (1 self)
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In this paper, we re ne the calculus proposed in [5, 8, 9]. The new calculus, including network operations for multiplexing, inputoutput relation, and routing, allows us to compute tighter exponential bounds for the tail distributions of queue lengths in intree networks with routing. In particular, if external arrival processes and routing processes are either Markov arrival processes or autoregressive processes, the stationary queue length at a local node is stochastically bounded above by the sum of a constant and an Erlang random variable. The decay rate of the Erlang random variable is not greater than ( in some cases equal to) the decay rate of the tail distribution of the stationary queue length. The number of stages of the Erlang random variable is the number of external arrival processes and routing processes contributing to its queue length. For the single queue case, both the lower and upper bounds are derived.
Analysis of Two Competing TCP/IP Connections
, 2002
"... Many mathematical models exist for describing the behavior of TCP/IP under an exogenous loss process that does not depend on the window size. The goal of this paper is to present a mathematical analysis of two asymmetric competing TCP connections where loss probabilities are directly related to th ..."
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Cited by 8 (0 self)
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Many mathematical models exist for describing the behavior of TCP/IP under an exogenous loss process that does not depend on the window size. The goal of this paper is to present a mathematical analysis of two asymmetric competing TCP connections where loss probabilities are directly related to their instantaneous window size, and occur when the sum of throughputs attains a given level. We obtain bounds for the stationary throughput of each connection, as well as an exact expression for symmetric connections. This allows us to further study the fairness as a function of the different round trip times. We avoid the simplifying artificial synchronization assumption that has frequently been used in the past to study similar problems, according to which whenever one connection looses a packet, the other one looses a packet as well.
Effect on network performance of common versus private acquisition sequences for impulse radio uwb networks
 in ICUWB 06
, 2006
"... Packet detection and timing acquisition for IRUWB networks such as 802.15.4a relies on the presence of an acquisition sequence (or preamble) at the beginning of each packet. A simple network design choice is to use a common acquisition sequence for the whole network. A second design choice is to us ..."
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Cited by 6 (5 self)
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Packet detection and timing acquisition for IRUWB networks such as 802.15.4a relies on the presence of an acquisition sequence (or preamble) at the beginning of each packet. A simple network design choice is to use a common acquisition sequence for the whole network. A second design choice is to use an acquisition sequence private to destinations. It potentially yields a larger network throughput, but requires additional complexity for sources to learn the acquisition sequence of their destination. In this paper, we evaluate the effect of a common or private acquisition sequence on the network throughput. Our analysis is based on analytical modeling and simulations. We show that a private acquisition sequence yields a substantial increase in throughput. The throughput difference grows with the number of concurrent transmitters and interferers. We also show the presence of a compounding effect similar to the exposed terminal issue in 802.11 networks. I.
Understanding the simulation of mobility models with Palm calculus
 IEEE TRANSACTIONS ON MOBILE COMPUTING ITMMC3
, 2005
"... The simulation of mobility models such as the random waypoint often cause subtle problems, for example the decay of average speed as the simulation progresses, a difference between the long term distribution of nodes and the initial one, and sometimes the instability of the model. All of this has to ..."
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Cited by 6 (2 self)
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The simulation of mobility models such as the random waypoint often cause subtle problems, for example the decay of average speed as the simulation progresses, a difference between the long term distribution of nodes and the initial one, and sometimes the instability of the model. All of this has to do with time averages versus event averages. This is a well understood, but little known topic, called Palm calculus. In this paper we first give a very short primer on Palm calculus. Then we apply it to the random waypoint model and variants (with pause time, random walk). We show how to simply obtain the stationary distribution of nodes and speeds, on a connected (possibly non convex) area. We derive a closed form for the density of node location on a square or a disk. We also show how to perform a perfect (i.e. transient free) simulation without computing complicated integrals. Last, we analyze decay and explain it as either convergence to steady state or lack of convergence.
The Random Trip Model: Stability, Stationary Regime, and Perfect Simulation
, 2006
"... We define "random trip", a generic mobility model for random, independent node motions, which contains as special cases: the random waypoint on convex or non convex domains, random walk on torus, billiards, city section, space graph, intercity and other models. We show that, for this model, a necess ..."
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Cited by 6 (0 self)
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We define "random trip", a generic mobility model for random, independent node motions, which contains as special cases: the random waypoint on convex or non convex domains, random walk on torus, billiards, city section, space graph, intercity and other models. We show that, for this model, a necessary and sufficient condition for a timestationary regime to exist is that the mean trip duration (sampled at trip endpoints) is finite. When this holds, we show that the distribution of node mobility state converges to the timestationary distribution, starting from origin of an arbitrary trip. For the special case of random waypoint, we provide for the first time a proof and a sufficient and necessary condition of the existence of a stationary regime, thus closing a long standing issue. We show that random walk on torus and billiards belong to the random trip class of models, and establish that the timelimit distribution of node location for these two models is uniform, for any initial distribution, even in cases where the speed vector does not have circular symmetry. Using Palm calculus, we establish properties of timestationary regime, when the condition for its existence holds. We provide an algorithm to sample the simulation state from a timestationary distribution at time 0 (“perfect simulation”), without computing geometric constants. For random waypoint on the sphere, random walk on torus and billiards, we show that, in the timestationary regime, the node location is uniform. Our perfect sampling algorithm is implemented to use with ns2, and is available to download from
Smoothing Point Processes As a Means to Increase Throughput
 TR RC 16866, IBM Research Division
, 1992
"... Unlike the leakybucket scheme which regulates the input rate, the lters we study in this paper reduce the variability of interarrival times (subject to a maximum delay constraint for each customer). These lters are called \smoothing" lters for point processes. By considering the output proces ..."
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Cited by 5 (0 self)
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Unlike the leakybucket scheme which regulates the input rate, the lters we study in this paper reduce the variability of interarrival times (subject to a maximum delay constraint for each customer). These lters are called \smoothing" lters for point processes. By considering the output processes of various queueing systems, we show that (i) an in niteserver queue acts as a smoothing lter for a doubly stochastic Poisson process and (ii) a singleserver queue with deterministic service times acts as a smoothing lter for a stationary and ergodic point process. Based on (ii), we provide a smoothing algorithm that satis es a maximum delay constraint. The algorithm is shown to be robust and consistent. We then consider two examples where these lters can be applied to increase throughput: one is the singleserver loss system with exponential service times, the other is the leakybucket scheme with token buer size 1. Numerical examples and simulations are also given for comparing the tradeo between delay and throughput. Our approach is based on the variability ordering and a cut criterion for the majorization ordering. The criterion appears to be new and to be of independent interest.
The entropy and delay of traffic processes in ATM networks
 in Proceedings of the Conference on Information Science and Systems (CISS
, 1995
"... Abstract: In this paper we discuss the application of some recent results obtained from studying the asymptotics and the entropy of flows in large queueing systems to the analysis of entropy variations of traffic processes in ATM networks. We argue that certain important queueing nodes/networks (e.g ..."
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Cited by 3 (0 self)
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Abstract: In this paper we discuss the application of some recent results obtained from studying the asymptotics and the entropy of flows in large queueing systems to the analysis of entropy variations of traffic processes in ATM networks. We argue that certain important queueing nodes/networks (e.g. ·/M/1, ·/GI/ ∞ nodes and Jackson networks) are entropy increasing in nature, in that the entropy of the arrival process to such a node/network is strictly less than the entropy of the departure process except when the arrival process is Poisson (which has maximum entropy). The entropy increasing nature of ·/GI/ ∞ queues is established for a class of input processes. Based on these entropy maximization results and with the aid of some simulations, we investigate the connection between the entropy and delay of ATM traffic processes, and examine the effectiveness of entropy as a measure of “burstiness ” of traffic streams in ATM virtual circuits. 1