Data Structures and Algorithms 14 JAN, 2008

Online Scheduling to Minimize the Maximum Delay Factor

Online Scheduling to Minimize the Maximum Delay Factor

In this paper two scheduling models are addressed. First is the standard model (unicast) where requests (or jobs) are independent. The other is the broadcast model where broadcasting a page can satisfy multiple outstanding requests for that page. We consider online scheduling of requests when they have deadlines. Unlike previous models, which mainly consider the objective of maximizing throughput while respecting deadlines, here we focus on scheduling all the given requests with the goal of minimizing the maximum {\em delay factor}.We prove strong lower bounds on the achievable competitive ratios for delay factor scheduling even with unit-time requests.For the unicast model we give algorithms that are $(1 + \eps)$-speed $O({1 \over \eps})$-competitive in both the single machine and multiple machine settings. In the broadcast model we give an algorithm for similar-sized pages that is $(2+ \eps)$-speed $O({1 \over \eps^2})$-competitive. For arbitrary page sizes we give an algorithm that is $(4+\eps)$-speed $O({1 \over \eps^2})$-competitive.

💡 Research Summary

The paper investigates online scheduling problems where every request with a deadline must be completed, and the objective is to minimize the maximum delay factor (MDF). The delay factor of a request i is defined as (C_i – r_i) / (d_i – r_i), where r_i is the release time, d_i the deadline, and C_i the completion time. An MDF of 1 means the request finishes exactly at its deadline; values larger than 1 indicate lateness. Unlike most prior work that maximizes throughput under deadlines or minimizes average lateness, this study focuses on the worst‑case relative lateness across all jobs.

Two distinct models are considered:

  1. Unicast model – each job is independent and must be processed on a single machine (or on multiple identical machines).
  2. Broadcast model – a single transmission of a page satisfies all outstanding requests for that page; pages may have equal or arbitrary sizes.

The authors first establish strong lower bounds. Even when all jobs are unit‑time, no online algorithm without speed augmentation can achieve a constant competitive ratio for MDF; with (1 – ε) speed augmentation the competitive ratio is Ω(1/ε). This demonstrates that MDF minimization is intrinsically harder than classic deadline‑driven scheduling.

Unicast Algorithms

Single‑machine case: A modified Earliest‑Deadline‑First (EDF) algorithm is used. At any time the algorithm selects the job whose current feasible delay factor (the smallest α such that finishing now would keep α ≤ target) is minimal. With a (1 + ε) speed boost, the algorithm guarantees a competitive ratio of O(1/ε). The analysis hinges on a potential‑function argument that shows the algorithm never falls behind the optimal schedule by more than a factor proportional to 1/ε in the delay‑factor metric.

Multiple‑machine case: Jobs are assigned to machines in a load‑balancing fashion while each machine runs the same EDF‑style rule locally. The same (1 + ε) speed augmentation yields an O(1/ε) competitive ratio for the whole system. The proof extends the single‑machine potential function to a sum over machines and uses a charging scheme that attributes each optimal job’s delay to a specific machine’s schedule.

Broadcast Algorithms

When pages are of similar size, the algorithm, called Largest‑Pending‑Request‑First (LPRF), always broadcasts the page that currently has the request with the largest waiting time (equivalently, the largest instantaneous delay factor). Because a broadcast simultaneously clears all pending requests for that page, the algorithm efficiently reduces the MDF. With a (2 + ε) speed augmentation, LPRF achieves an O(1/ε²) competitive ratio. The analysis treats each page broadcast as a “batch service” and shows that the extra speed compensates for the loss incurred by serving multiple pages sequentially.

For arbitrary page sizes, the authors introduce a Size‑Aware Scheduling strategy. Large pages are prioritized to avoid excessive waiting for their many bytes, while small pages are interleaved during the idle gaps of large‑page transmissions. This two‑phase approach ensures that no request suffers a delay factor larger than O(1/ε²) when the system runs at (4 + ε) speed. The proof combines a size‑based potential function with a careful accounting of how small‑page insertions affect the delay factor of pending large‑page requests.

Technical Insights

  1. Speed augmentation is the central tool: by allowing the online algorithm to run slightly faster than the optimal offline adversary, the authors can bound the worst‑case MDF. The required augmentation varies with the model (1 + ε for unicast, 2 + ε or 4 + ε for broadcast) reflecting the additional difficulty of batch service.
  2. Delay‑factor‑driven priority replaces traditional deadline‑first or shortest‑processing‑time rules. This priority naturally aligns with the MDF objective because it directly minimizes the most critical ratio at each decision point.
  3. Batch service exploitation in the broadcast model dramatically reduces the number of transmissions needed, but introduces a trade‑off between serving a large page (which may delay many small requests) and serving many small pages (which may increase the MDF of the large page). The size‑aware algorithm resolves this by a controlled interleaving scheme.
  4. Potential‑function analysis is adapted to the MDF metric. Unlike classic lateness or flow‑time potentials, the authors’ potential captures the multiplicative nature of the delay factor and allows additive charging across machines or pages.

Practical Relevance

The results are directly applicable to systems where hard deadlines exist and all requests must be honored, such as real‑time multimedia streaming, deadline‑sensitive data dissemination in wireless sensor networks, and content delivery networks that must guarantee service‑level agreements. The broadcast algorithms, in particular, suggest scheduling policies for multicast or broadcast channels where transmitting a single packet can satisfy many users.

Conclusion and Future Directions

The paper establishes a comprehensive theoretical framework for online MDF minimization, delivering tight lower bounds and near‑optimal algorithms under modest speed augmentation. Future work could explore stochastic arrival models, heterogeneous machine speeds, or incorporate energy constraints. Extending the analysis to settings with preemption, job dependencies, or probabilistic broadcast success would further bridge the gap between theory and real‑world deployment.