Efficient analysis of the MMAP[K]/PH[K]/1 priority queue

Abstract

In this paper we consider the MMAP/PH/1 priority queue, both the case of preemptive resume and the case of non-preemptive service. The main idea of the presented analysis procedure is that the sojourn time of the low priority jobs in the preemptive case (and the waiting time distribution in the nonpreemptive case) can be represented by the duration of the busy period of a special Markovian fluid model. By making use of the recent results on the busy period analysis of Markovian fluid models it is possible to calculate several queueing performance measures in an efficient way including the sojourn time distribution (both in the time domain and in the Laplace transform domain), the moments of the sojourn time, the generating function of the queue length, the queue length moments and the queue length probabilities.

Refined large deviations asymptotics for Markov-modulated infinite-server systems

“…In the present paper the focus lies on the latter application, where the model with an infinite number of servers typically serves as a proxy for its counterpart with a large but finite number of servers. The Markov modulation of the arrival rates and service rates facilitates the modeling of some sort of 'burstiness'; although the concept of Markov modulation has been around for a few decades, it still spurs a considerable amount of research effort ( Horváth, 2015;O'Reilly, 2014 ). For instance, the model can be used to describe the fluctuations in the users' activity level (where each user alternates between transmitting data or being silent).…”

Differential Approximation and Sprinting for Multi-Priority Big Data Engines

“…This can be viewed as a single server queue serving K priority classes. We thus opt to employ the recent method proposed in [22], which is capable of obtaining the response time distribution and its moments for a fairly general priority queue with K priority classes under both preemptive and non-preemptive scheduling.…”

“…One key reason to choose [22] as the latency model for DiAS is its support for Phase-Type (PH) job processing times [28], which is a class of distributions that can capture fairly general behaviours. Further, PH distributions are closed under a number of operations, a feature that we exploit to model the detailed job processing times of concurrent tasks.…”

“…Background on the MMAP[K]/PH[K]/1 priority queue. Horváth [22] analytically derived the latency distribution for an MMAP[K]/PH[K]/1 priority queue, where processing times follow PH distributions, differentiated for the K job classes, and arrivals follow a Marked Markovian Arrival Process (MMAP) with K different streams [28], one for each priority class. This class of arrival processes can capture fairly general behaviors, including correlations among arrival streams or general inter-arrival times.…”

“…It is extremely challenging to optimize the performance of big data engines with priority scheduling as performance conflicts arise across disparate job priorities as well as across performance targets, i.e., latency vs. resource efficiency. Meeting the latency targets in priority systems is a long standing challenge from both system [14,36] and modelling [20,22] perspectives due to the complex dynamics and performance requirements across diverse priority classes. This is partly because the processing order of low-priority jobs highly depends on the high-priority jobs, especially when low-priority jobs are evicted during periods of resource shortage.…”