D. R. McDonald | Searcharxiv

arXiv2017

Yaglom limits can depend on the starting state

We construct a simple example, surely known to Harry Kesten, of an R-transient Markov chain on a countable state space S with cemetery state delta. The transition matrix K on S is irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given non-absorption. Each starting state x in S results in a different Yaglom limit. Each Yaglom limit is an rho-invariant quasi-stationary distribution where rho=1/R and R is the convergence parameter of $K$. Yaglom limits that depend on the starting state are related to a nontrivial rho-Martin entrance boundary.

math.PR↗

arXiv2006

Mean field convergence of a model of multiple TCP connections through a buffer implementing RED

RED (Random Early Detection) has been suggested when multiple TCP sessions are multiplexed through a bottleneck buffer. The idea is to detect congestion before the buffer overflows by dropping or marking packets with a probability that increases with the queue length. The objectives are reduced packet loss, higher throughput, reduced delay and reduced delay variation achieved through an equitable distribution of packet loss and reduced synchronization. Baccelli, McDonald and Reynier [Performance Evaluation 11 (2002) 77--97] have proposed a fluid model for multiple TCP connections in the congestion avoidance regime multiplexed through a bottleneck buffer implementing RED. The window sizes of each TCP session evolve like independent dynamical systems coupled by the queue length at the buffer. The key idea in [Performance Evaluation 11 (2002) 77--97] is to consider the histogram of window sizes as a random measure coupled with the queue. Here we prove the conjecture made in [Performance Evaluation 11 (2002) 77--97] that, as the number of connections tends to infinity, this system converges to a deterministic mean-field limit comprising the window size density coupled with a deterministic queue.

math.PR↗