Asymptotic Behavior of a Generalized TCP Congestion Avoidance Algorithm

Abstract

The transmission control protocol (TCP) is a transport protocol used in the Internet. In Ott (2005), a more general class of candidate transport protocols called ‘protocols in the TCP paradigm’ was introduced. The long-term objective of studying this class is to find protocols with promising performance characteristics. In this paper we study Markov chain models derived from protocols in the TCP paradigm. Protocols in the TCP paradigm, as TCP, protect the network from congestion by decreasing the ‘congestion window’ (i.e. the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probabilitypof loss is assumed to be constant, the protocol gives rise to a Markov chain {Wn}, whereWnis the size of the congestion window after the transmission of thenth packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, asp→ 0. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein-Uhlenbeck process or can be driven by a Poisson process.

TCP and iso-stationary transformations

“…Several research papers [11,15,17,18,20,21] deal with establishing scaling limits under a "low loss" scenario for such Markov Chains. Challenges were to establish weak convergence of processes to a limiting process, and to establish weak convergence of stationary distributions to the limiting stationary distribution.…”

Steady state and scaling limit for a traffic congestion model

“…2.1), proving a full hydrodynamic limit for the empirical measure (5.1) in Theorem 5.1. The result relates partially to the limiting process from equation (2.1) in [19] (α = constant, n = 1, non-interactive dynamics) and the convergence to the process given by equation (2.7) quoted in [14] from [8] (α(x) = x, n = 1, non-interactive dynamics). Recurrence issues and the ergodic properties of the one-particle scaling limit process with rate α(x) are analyzed in greater detail in [13].…”

“…In a general model used in internet congestion control [2,9,10,14,17,19], related to classical autoregressive models [7,18], the time dependent data flow x(t) undergoes a biased random walk with linear steps in one direction (x moves to x + a, a > 0) and on a logarithmic scale in the other (x moves to γx, where 0 < γ < 1), belonging to a class of dynamics known in the literature [14] as AIMD (additive increase multiplicative decrease). In the present paper, we concentrate on one standard model, defined rigorously as the solution to the martingale problem given by (2.3), which we shall call the γ-process.…”

“…Various orders of magnitude of the jump size are considered in recent papers [14,17,19] under constant ζ but this is the first instance to our knowledge where the rates themselves are analyzed and the natural scale and criticality are identified. This approach is particularly important and the natural one since in the main application of the model, in which ζ represents the probability of loss of packets of information x, one expects non-increasing behavior in each component of x (or, in other models, the "average" level of congestionx).…”

“…It is useful to note in the discrete setting, where a = 1, the left side jump is x i → [γx i ] and a lower bound away from zero on x is imposed, the irreducibility of the process is immediate and Theorem 2.1 implies the existence of an unique invariant measure -see [19] for related models.…”