We argue, in a slightly informal manner, that queueing networks implicitly optimize a utility function subject to constraints on network capacity. We start with the simple example of a closed queueing network and, as we shall discuss, a motivating example is the Transmission Control Protocol which controls the number of packets in transfer on an Internet connection.

# Author: appliedprobability

## Gale-Eisenberg Market

The Gale-Eisenberg is a nice example were the distributed decisions of buyers and sellers have an equilibrium which solves an optimization problem.

## Sanov’s Theorem

Sanov’s asks how *likely* is it that the empirical distribution some IIDRV’s is *far* from the distribution. And shows that the relative entropy determines the likelihood of being far.

## Entropy and Boltzmann’s Distribution

Entropy and Relative Entropy occur sufficiently often in these notes to justify a (somewhat) self-contained section. We cover the discrete case which is the most intuitive.

## Distributed Random Access Scheduling

We consider a network of wireless routers. The routers that are close together can interfere if they transmit simultaneously. So schedules need to avoid such collisions. We want each each wireless node to achieve a transmission rates that equals its arrival rate. One might want to implement a policy like MaxWeight or simply estimate the vector of arrival rates and accordingly choose the correct transmission rate. However, this is complicated by the fact that the routers do not know the arrival and transmission rates of their neighbors; all they can do is sense if their neighbors are transmitting or not.

## Max-Weight Scheduling

We define a discrete-time queueing network where there are restrictions on which queues can be served simultaneously. We give a policy for serving queues which is stable whenever it is possible to stabilize the queueing network.

## Blackwell Approachability

Sequentially a player decides to play and his adversary decides . At time , a decision results in a vector payoff . Given is the average vector payoff at time , Blackwell’s Approachability Theorem is a necessary and sufficient condition so that, regardless of the adversary’s decisions, the player makes the sequence of vectors approach a convex set .