Abstract
Multiclass queueing networks (McQNs) extend the classical concept of the Jackson network by allowing jobs of different classes to visit the same server. Although such a generalization seems rather natural, from a structural perspective, there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product–form equilibrium distributions, and stochastic monotonicity, do not immediately carry over to the multiclass framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this end, we introduce and study a class of Markov processes, which we call Q-processes, modeling the time evolution of the network configuration of any open, work-conservative McQN having exponential service times and Poisson input. We define a new monotonicity notion tailored for this class of processes. Our main result is that we show monotonicity for a large class of McQN models, covering virtually all instances of practical interest. This leads to interesting properties that are commonly encountered for “traditional” queueing processes, such as (i) monotonicity with respect to external arrival rates and (ii) star-convexity of the stability region (with respect to the external arrival rates); such properties are well known for Jackson networks but had not been established at this level of generality. This research was partly motivated by the recent development of a simulation-based method that allows one to numerically determine the stability region of a McQN parameterized in terms of the arrival-rates vector.
Highlights
Multiclass queueing networks (McQNs) arise as natural generalizations of conventional Jackson networks: in Jackson networks each station acts as a ·/M/1 single-class queue, in McQNs each network station is a multiclass queue
This leads to interesting properties that are commonly encountered for “traditional” queueing processes, such as (i) monotonicity with respect to external arrival rates and (ii) starconvexity of the stability region; such properties are well known for Jackson networks but had not been established at this level of generality
This research was partly motivated by the recent development of a simulationbased method that allows one to numerically determine the stability region of a McQN parameterized in terms of the arrival-rates vector
Summary
Multiclass queueing networks (McQNs) arise as natural generalizations of conventional Jackson networks: in Jackson networks each station (server) acts as a ·/M/1 single-class queue, in McQNs each network station is a multiclass queue. Validity of monotonicity properties is expected to imply that stability of a queueing network is a monotone property This is the case for Jackson networks as stability (to be thought of as subcriticality) is a monotone property with respect to (external) arrival, service and traffic rates. For McQNs, some results in the literature indicate that this is not necessarily the case when parameterized with respect to service rates (Bramson 1994b, Dai et al 1999) or traffic rates (Dumas 1997) This naturally raises the question (and casts some doubts on) whether stability of a McQN is monotone with respect to (external) arrival rates. Extensive simulation experiments indicated that a certain form of (stochastic) monotonicity with respect to external arrival rates could still be expected even for McQNs for which no stability conditions are known. Such a property would be enough to guarantee, for instance, that stability is a monotone property with respect to arrival rates
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