Exploiting Chemistry for Better Packet Flow Management 3: Formal Analysis

Exploiting Chemistry for Better Packet Flow Management 3: Formal Analysis

The previous two posts introduced the ideas of Meyer and Tschudin [11] involving the application and exploitation of chemical kinetic theory to flow management in computer networking. The first part introduced the ideas and gave an overview of the entire work, and the second part took a deeper look into the formal model of a packet chemistry. This section discusses the analysis options available once a packet chemistry model has been created. 

This section can also be skipped for those less interested in the formal mathematics. Suffice it to say that there are a multitude of already created methods now available for the elegant analysis of computer networks when modeled by an artificial packet chemistry.

Formal Analysis of Artificial Packet Chemistry

By representing packet flow in a computer network as an artificial chemistry, a multitude of analyses are available, from high to low granularity. The authors give a heavily brief survey (and a good bibliography) of works that can be utilized to analyze these networks pulled from the physics and chemistry literature. A particular advantage of this method is the ability to study the transient states of the network rather than just steady states. The authors also claim the ability to determine the stability of the network flow based only on topology, a heavy advantage in design.

Stochastic Analysis at the Microscopic Level

The stochastic behavior of chemical reaction networks is described by the chemical master equation[10] which takes the form
\frac{\text{d}\mathbf{P}}{\text{d}t} = \mathbf{A}\mathbf{P}

which is a differential equation describing the evolution of state probabilities for a system. Here the states are discrete, and time is continuous. The matrix \mathbf{A} describes the transition rates (which can also be kinetic or reaction rates), and the stochastic process described is a Markov jump-process Since we’re on a network, the Markov jump process exists in an \mathcal{S}-dimensional integer lattice. Some work has been done to analyze several classes of chemical reaction networks to find the steady-state probability distribution of the state space. For example, if the total number of packets in the network has a bound, and the network contains only first order (unimolecular to unimolecular) reactions, the steady state probability distribution for the lengths of the queues in the network is a multinomial distribution[3]. On the other hand, if the network is open (we allow packets to exit the network completely), then the steady state probability distribution of the lengths of the queues follows a product of Poisson distributions (which is also Poisson)[3]. (This is an extremely desirable property, called a product-form.)

Deterministic Approximations

This is the most common approach utilized in computer network analysis today, simply because networks are so large and complex that stochastic modeling becomes too cumbersome. Here, the average trajectory is represented by a system of ordinary differential equations, building a fluid model. One downside to this in the networking space is that the analysis of protocols by this method requires manual extraction from source code and accuracy is uncertain.

In the chemistry sector (and now in the packet chemistry model), obtaining a fluid approximation is not only easier, but shown to be accurate. There are links between the stochastic master equation to several approximations[5,6] including a deterministic ODE model. Gillespie[5] showed that the ODE model accurately predicts the network flow trajectory in many cases.

One thing the authors note here is that the ODE model can be directly and automatically generated from the network topology. For example, a single server with a single queue (M/M/1) is simply modeled as one chemical species X. The arrival rate (inflow) is \lambda, and the service rate is proportional to the queue length, so \mu = kx, where x is the queue length. Then we get a simple differential equation
\dot{x} = \lambda-kx describing the change in queue length as the difference of inflow and outflow. In the steady state, \dot{x} = 0, which lets us look for a fixed point \hat{x} = \frac{\lambda}{k}. This is the steady-state queue length, which allows us to derive the expected waiting time T = \frac{1}{k}, showing that the latency of a packet under this model is independent of the arrival rate and fill level. This model when implemented automatically adjusts the service rate such that in the steady state, every packet sees the same latency.

It’s also important to determine just how stable this steady state is by analyzing the sensitivity of the network and states to perturbations. The authors list several citations to show that no new approaches are needed to do this; one can look to signal and control theory literature. In particular, a network designer would desire to predict the stability of a complex network by studying the topology as opposed to an analysis of the system of ODEs. Fortunately, modeling a network this way allows for the use of the Deficiency Zero Theorem for complex chemical networks that gives conditions for stability of steady-state[2,7].

The authors give a formal convergence proof that the example network above converges to a stable fixed point and is asymptotically stable, comparing it to the proof of a similar protocol Push-Sum (a gossip protocol in computer networks).


The next post in this series will discuss Meyer and Tschudin’s implementation of a scheduler based on the principles discussed thus far. 


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