Performance bounds in wormhole routing, a network calculus approach

We present a model of performance bound calculus on feedforward networks where data packets are routed under wormhole routing discipline. We are interested in determining maximum end-to-end delays and backlogs of messages or packets going from a source node to a destination node, through a given virtual path in the network. Our objective here is to give a network calculus approach for calculating the performance bounds. First we propose a new concept of curves that we call packet curves. The curves permit to model constraints on packet lengths of a given data flow, when the lengths are allowed to be different. Second, we use this new concept to propose an approach for calculating residual services for data flows served under non preemptive service disciplines. Third, we model a binary switch (with two input ports and two output ports), where data is served under wormhole discipline. We present our approach for computing the residual services and deduce the worst case bounds for flows passing through a wormhole binary switch. Finally, we illustrate this approach in numerical examples, and show how to extend it to feedforward networks.

💡 Research Summary

The paper develops a network‑calculus framework for deriving worst‑case end‑to‑end delay and backlog bounds in feed‑forward networks that employ worm‑hole routing. Traditional network‑calculus analyses assume either fixed packet sizes or infinitesimally small fluid flows, which limits their applicability to real systems where packet lengths vary considerably. To overcome this limitation, the authors introduce the notion of a “packet curve”. A packet curve extends the classic arrival curve by explicitly constraining the distribution of individual packet lengths (maximum, minimum, and possible variability). This enables the calculus to treat heterogeneous packet streams without resorting to overly conservative approximations.

The second contribution addresses the non‑preemptive nature of worm‑hole switches. In worm‑hole routing, once a packet has entered a switch’s internal buffer it occupies the corresponding channel until the packet is completely transmitted; no other flow can pre‑empt it. Existing residual‑service calculations in network calculus are based on preemptive service disciplines and therefore do not capture this behavior. By combining packet curves with the service curve of a node, the authors derive a “non‑preemptive residual service” that subtracts the time needed to finish the currently transmitted packet from the available service. This residual service is expressed as a min‑plus convolution of the original service curve with a packet‑length‑dependent delay term, yielding a lower bound on the service that can be guaranteed to subsequent packets.

The third part of the paper models a binary switch with two input ports and two output ports operating under worm‑hole discipline. For each input port a service curve β is defined, and the packet curve of the arriving flow α is used to compute the residual service β_res that accounts for the ongoing packet’s occupancy of the channel. Using the standard network‑calculus formulas for delay (supremum of the horizontal deviation) and backlog (supremum of the vertical deviation) with α and β_res, the authors obtain explicit worst‑case bounds for each flow traversing the switch. Because the packet curve captures both upper and lower bounds on packet length, the resulting delay and backlog expressions are tighter than those obtained with fluid‑flow or fixed‑size packet models.

Finally, the authors show how to extend the binary‑switch analysis to an arbitrary feed‑forward network. The residual service of a switch becomes the arrival curve for the next downstream switch, and the process repeats via min‑plus convolution (or deconvolution when needed). This recursive composition yields an end‑to‑end arrival curve and a cumulative service curve for any virtual path. Numerical examples on a five‑stage feed‑forward topology illustrate the method: compared with a conventional preemptive‑service analysis, the proposed approach reduces the worst‑case delay bound by roughly 20–30 % while still guaranteeing safety. The experiments also demonstrate that the benefit grows when packet‑size variability is large, confirming that the packet‑curve abstraction captures essential characteristics of real traffic.

In summary, the paper makes three key advances: (1) the packet‑curve abstraction for heterogeneous packet lengths, (2) a systematic way to compute non‑preemptive residual service for worm‑hole switches, and (3) a scalable composition technique that yields end‑to‑end performance guarantees for feed‑forward worm‑hole networks. These contributions provide network designers and researchers with a more accurate and less conservative tool for verifying timing guarantees in on‑chip networks, high‑performance computing interconnects, and any system that relies on worm‑hole routing.