Modeling M/M/R-JSQ-PS Sojourn Time Distributions for URLLC Services

The future of networking promises to support time-sensitive applications that require ultra low latencies and reliabilities of 99.999%. Recent advances in cellular and WiFi connections enhance the network to meet high reliability and ultra low latencies. However, the aforementioned services require that the server processing time ensures low latencies with high reliability, otherwise the end-to-end performance is not met. To that end, in this paper we use queueing theory to model the sojourn time distribution for ultra reliable low latency constrained services of M/M/R-JSQ-PS systems: Markovian queues with R CPUs following a join-shortest-queue processor sharing discipline (for example Linux systems). We develop open-source simulation software, and develop and compare six analytical approximations for the sojourn time distribution. The proposed approximations yield Wasserstein distances below 2 time units, and upon medium loads incur into errors of less than 4.7 time units (e.g., milliseconds) for the $99.999{^{\text {th}}}$ percentile sojourn time. Moreover, the proposed approximations are stable regardless the number of CPUs and stay close to the simulations regardless the service time distribution. To show the applicability of our approximations, we leverage on a real world vehicular dataset to scale a 99.999% reliable vehicular service.