Modelling M/M/R-JSQ-PS sojourn time distribution for Ultra-Reliable Low Latency Communication services

The future Internet promises to support time-sensitive services that require ultra low latencies and reliabilities of 99.99%. 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 queuing theory to model the sojourn time distribution for Ultra-Reliable Low Latency Communication services of M/M/R-JSQ-PS systems: Markovian queues with R CPU servers 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 1.78 time units (e.g., milliseconds) for the 99.99th percentile sojourn time. Moreover, the proposed sojourn time approximations are stable regardless the number of CPUs and stay close to the simulations.