Queueing Delay Analysis of Mixed Traffic in Time Sensitive Networks

In many emerging Time Sensitive Networking (TSN) applications such as industrial control and automotive, periodic traffic with end-to-end latency constraints is buffered with nonmission-critical aperiodic traffic. As queueing delay is a central component of end-to-end latency, we study the effect of aperiodic traffic on the queueing delay of periodic traffic via characterizing the probability distribution of queue size and delay in an (M + D)/M/1 queue, a queue with Poisson and periodic inputs, infinite waiting capacity and a single memoryless server. Since obtaining the closed form distributions of the queue size and delay is intractable, we approximate the behavior of the (M + D)/M/1 queue when the service and arrival rates are close. We use a Markov chain with a quasitoeplitz matrix, enabling us to use an existing technique to study this class of Markov chains. We determine the transition matrix of the Markov chain, investigate the setting in which the approximation works well, and compute the stationary distribution. We plot and compare the stationary distributions of queue size and delay between the (M + D)/M/1 queue and our model; we observe that our model is a favorable match.