TCP and iso-stationary transformations
J.S.H. van Leeuwaarden
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A.H. Lpker
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T.J. Ott
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T.J. Ott Ott Associates, Chester,
NJ, USA
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J.S.H. van Leeuwaarden ( )
We consider piecewise-deterministic Markov processes that occur as scaling limits of discrete-time Markov chains that describe the Transmission Control Protocol (TCP). The class of processes allows for general increase and decrease profiles. Our key observation is that stationary results for the general class follow directly from the stationary results for the idealized TCP process. The latter is a Markov process that increases linearly and experiences downward jumps at times governed by a Poisson process. To establish this connection, we apply space-time transformations that preserve the properties of the class of Markov processes.
1 Introduction
This paper deals with a class M(r, ; , ; FQ) of piecewise-deterministic Markov
processes (PDMPs) related to the Transmission Control Protocol (TCP) for data
transmission over the Internet. The process X(t ) M has the following behavior.
Between random collapse times (k)kN the evolution is deterministic according to
The random times k are governed by a state-dependent Poisson process with rate
X(t ) and > 1, and at k the process is multiplied by a random variable Qk in
[0, 1) with distribution FQ(x), i.e., X(k) = QkX(k), where the random variables
(Qk)kN are independent of {X(t ) : t k}. If = 0 and = 0, the PDMP X(t ) has
a linear increase profile and independent losses and is known as the additive-increase
multiplicative-decrease (AIMD) algorithm, also referred to as idealized TCP [21]. In
general, if X(0) = x, the process X(t ) increases deterministically as
(x1 + (1 )rt ) 11 , =1,
until the time 1 of the first jump. It is clear that a larger leads to a more aggressive
increase profile. Also, a larger will lead to a more aggressive decrease profile,
because jumps will occur more frequently. All possible combinations of and
together present a diverse pallet of increase-decrease profiles for the dynamics of
TCP.
The dynamical behavior of TCP was originally modeled as a discrete-time Markov
chain [21]. 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.
Reference [17] studies a very general class of TCP control mechanisms for which
it conjectures limiting behaviors. The limiting processes include what in this paper
is < 1, = 0, Q = c with c a constant. [18] proves weak convergence of
stationary distributions for < 1, = 0, Q = c without establishing weak convergence of
processes, but includes rate of convergence results. [20] proves all results conjectured
in [17]: weak convergence of processes as well as of stationary distributions. [15]
extends the results for Q = c to a random Q. The class M is the class of scaling limits
that occurs in [15].
In this paper we deal solely with the stationary behavior for processes in M.
Recent time-dependent results can be found in [14, 19]. The key parameters in the
description of M are and for which we assume that > 1. This is necessary
to guarantee that the process is stable and admits a stationary distribution, cf.
Theorem 1. Our contribution lies in the application of spacetime transformations. We
show that by a state-space transformation of the type Y (t ) = X(t ) and a time
transformation Z(u) = X(t (u)) with t (u) = 0u X(t (s)) ds, stationary results for the
process X(t ) M(r, ; , ; FQ) follow directly from the stationary results for the
idealized TCP process Y (t ) M(1, 0; , 0; FQ). In this way, all known stationary
results for idealized TCP can be transferred directly to results for any process in M.
For a discussion of general PDMPs we refer to [7]. The processes in this paper
belong to the special class of growth-collapse processes for which we refer to [6].
Growth-collapse processes are also referred to as stress release models [16, 22, 23].
There is also a close relationship to so-called repairable systems (see [13] and [12]).
In [5] and [4] a different class of stress release models with additive jumps is
discussed. The papers [24] and [12] give conditions for ergodicity of a very general
class of growth collapse models, including our setting (model 4 in [24] and model 1
in [12]); the latter paper also gives conditions for non-explosiveness.
Spacetime changes for general Markov processes can be found in the classical
literature [9, 10]. For the special class of PDMPs in this paper, such spacetime
changes allow us to unify several earlier results. Ott et al. [21] use a space
transformation to solve the idealized TCP case for packet time ( < 1, = 0, Q = c)
and use a time transformation to obtain the limiting stationary distribution for clock
time ( = 0, = 1, Q = c). Dumas et al. [8] consider (...truncated)