The Multi-Source Probabilistically Preemptive M/PH/1/1 Queue With Packet Errors

Abstract

Analytical modeling of Age of Information (AoI) and Peak AoI (PAoI) has recently drawn a lot of attention in the context of quantitative assessment of information freshness in status update systems. In this paper, we study a probabilistically preemptive bufferless M/PH/1/1 queue fed with information update packets from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> separate information sources for which a new information packet arrival from source- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is allowed to preempt a packet from source- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> in service, with a probability depending on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> . To make the model even more general than the existing ones, we assume a distinct phase-type (PH-type) service time distribution, a distinct packet error and retransmission probability, for each of the information sources. Using sample path arguments and the theory of Markov Fluid Queues (MFQ), the exact distributions of the AoI and PAoI are numerically obtained for each of the sources. Numerical examples are provided to demonstrate the impact of various system parameters on AoI performance. In the context of a two-source system, we present a methodology on how to optimally choose the preemption probabilities and packet generation rates so as to minimize certain AoI-oriented cost functions.