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This paper introduces the $\mathrm{M^T/G/1}$ queue model to analyze packet transmission in communication networks where primary packets trigger secondary packets after random delays. By leveraging the memoryless property of exponential delays, the authors develop a tractable Markov description and derive a finite system of linear algebraic equations to compute performance metrics for both primary and secondary customers. The findings reveal that this approach effectively captures the dynamics of causally dependent packet streams, providing insights into system performance that classical queueing models fail to address.
Exponential triggering delays can significantly alter packet stream dynamics, revealing performance metrics that classical models overlook.
In many communication networks, the transmission of a packet may automatically trigger the transmission of a subsequent packet from the same source after a (possibly random) delay, without requiring acknowledgment or feedback. Such behavior arises in multi-stage status updating, proactive protocols, and other applications where users generate causally dependent packet streams. In this paper, in order to analyze these systems, we introduce the $\mathrm{M^T/G/1}$ queue. In this model, primary customers arrive according to a Poisson process, and each primary customer triggers a secondary customer to join the queue after an independent delay. This arrival mechanism falls outside the scope of classical queueing models with renewal arrival processes. When the triggering delays follow an exponential distribution, we exploit the memoryless property to set up a tractable Markov description. By truncating the number of pending secondary customers, we derive a finite system of linear algebraic equations in the Laplace--Stieltjes transform domain and solve them using matrix-analytic methods. Based on the resulting workload distribution, we compute class-specific performance metrics using PASTA for primary customers and Palm conditioning for secondary customers. Finally, we validate the accuracy of this truncation through numerical experiments.
