Tracking in a spaghetti bowl: monitoring transactions using footprints

The problem of tracking end-to-end service-level transactions in the absence of instrumentation support is considered. The transaction instances progress through a state-transition model and generate time-stamped footprints on entering each state in the model. The goal is to track individual transactions using these footprints even when the footprints may not contain any tokens uniquely identifying the transaction instances that generated them. Assuming a semi-Markov process model for state transitions, the transaction instances are tracked probabilistically by matching them to the available footprints according to the maximum likelihood (ML) criterion. Under the ML-rule, for a two-state system, it is shown that the probability that all the instances are matched correctly is minimized when the transition times are i.i.d. exponentially distributed. When the transition times are i.i.d. distributed, the ML-rule reduces to a minimum weight bipartite matching and reduces further to a first-in first-out match for a special class of distributions. For a multi-state model with an acyclic state transition digraph, a constructive proof shows that the ML-rule reduces to splicing the results of independent matching of many bipartite systems.