Notes on "Metascheduling for Continuous Media"

Proposes a scheduling framework for continuous media (e.g. audio, video streams) which can guarantee performance across multiple stages. The paper approaches scheduling from a theoretical perspective and focuses mostly on guaranteeing latency, throughput, and bounding jitter. The scheduler is capable of running real-time applications; however, the restrictive assumptions make it incompatible with OSs like Unix and standard internet protocols. Professional audio, video-on-demand, and telephony are the targeted applications of the scheduler. I consider high-quality, professional audio with low latency requirements the only potential remaining application of the scheduler; non-real-time technology has progressed to adequately support the other applications.

// The scheduling API.
// Fails if it cannot provide sufficient resources.
reserve(input: maximum message size and rate 
            success/failure flag
            session ID
            maximum logical delay
            cost function 
            minimum actual delay
            minimum unbuffered actual delay)
relax(input: session ID, new maximum logical delay)

In the scheduling framework, applications provide the maximum message size and rate to request resources. If accepted, a session is created for the resource that provides bounds on delay and a cost function. These bounds can be relaxed to minimize cost. Minimizing cost across resources is proven to be an NP-hard problem, but can be solved more easily if cost functions are (1) piecewise linear with a finite number of vertices, (2) monotonically decreasing, and (3) convex. For multi-stage pipelines, the paper proposes an iterative algorithm that minimizes cost.


Reasoning about Jitter

The paper rigorously proves a mathematical framework for reasoning about jitter and delay, focusing on arrival times and workahead. It defines a linear bounded arrival process (LBAP) as a message arrival process which depends on maximum message size \(M\), maximum message rate \(R\), and workahead limit \(W\) (allows bursts that exceed the long-term data rate \(MR\)). The paper proceeds to prove some mathematical bounds which are useful for limiting jitter and delay:

\(N_I(t_0, t_1)\) is the number of messages arriving at interface \(I\) on the time interval \([t_0, t_1)\):

\[N_I(t_0, t_1) \leq R|t_1 - t_0| + W\]

The workahead \(w(t)\) of an LBAP where \(w(t) \leq W\) is a measure of bursts:

\[w(t) = max_{t_0 < t} \{0, N(t_0, t) - R|t - t_0| \}\]

Bounds on the arrivals during a time interval:

\[N(t_1, t_2) \leq w(t_2) - w(t_1) + R | t_2 - t_1 |\]

Logical arrival time \(l(m_i)\) is when message \(m_i\) with actual arrival time \(a_i\) would arrive if workahead were not allowed:

\[\begin{align*} l(m_i) &= a_i + w(a_i) / R \\ &= \max(a_i, l(m_{i-1}) + 1 / R) \quad : \quad l(m_0) = a_0. \end{align*}\]

Using these equations, the paper establishes proofs and bounds over sequences of sessions.

Graph of workahead and arrival times.

CPU scheduling

Deadline-workahead scheduling is the policy describing how processes are scheduled on a CPU. The term real-time indicates that the process has a session; if sessions are done properly, this could allow the scheduler to correctly run real-time processes. Processes are categorized as one of the following (ordered by priority):

  1. Critical processes: real-time processes with an unprocessed message \(m\) where \(m\)’s logical arrival time has passed \(l(m) \leq t\).
    • Preemptively scheduled by earliest deadline.
  2. Interactive processes: nonreal-time processes for which fast response time is important.
  3. Workahead processes: real-time processes that have pending work, but are not critical.
  4. Background processes.


The mathematical approach to reasoning about jitter and cost-function based approach to allocating resources across multi-stage applications is interesting. Specifically, protocols for negotiating data rates and resource allocation exist in GStreamer, although these are generally done online using backpressure. There are definite applications to real-time scheduling, but the restrictive assumptions mean that this scheduling technique will likely only be implemented in specialized real-time systems.