## 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 $\{p_t\}_{t=1}^\infty$ and his adversary decides $\{q_t\}_{t=1}^\infty$. At time $t$, a decision $(p_t,q_t)$ results in a vector payoff $A(p_t,q_t)\in {\mathbb R}^k$. Given $a_t$ is the average vector payoff at time $t$, 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 $\{a_t\}_{t=1}^\infty$ approach a convex set ${\mathcal A}$.

## Power-of-k Choices

We consider a model of a large number of single server queues. When a job arrives, we assume it chooses between $k$ queues. The job then chooses to join the shortest of these queues. We show that such choice can dramatically reduce queue sizes.

## Little’s Law Here $\bar{N}$ is the long run queue length; $\bar{W}$ is the expected waiting time; $\lambda$ is the arrival rate at the queue.