A Numerical Approach to Stability of Multiclass Queueing Networks

Abstract

The Multi-class Queueing Network (McQN) arises as a natural multi-class extension of the traditional (single-class) Jackson network. In a single-class network subcriticality (i.e. subunitary nominal workload at every station) entails stability, but this is no longer sufficient when jobs/customers of different classes (i.e. with different service requirements and/or routing scheme) visit the same server; therefore, analytical conditions for stability of McQNs are lacking, in general. In this note we design a numerical (simulation-based) method for determining the stability region of a McQN, in terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity properties enjoyed by the associated Markovian queue-configuration process. Stochastic monotonicity is a quite common feature of queueing models and can be easily established in the single-class framework (Jackson networks); recently, also for a wide class of McQNs, including first-come-first-serve (FCFS) networks, monotonicity properties have been established. Here, we provide a minimal set of conditions under which the method performs correctly. Eventually, we illustrate the use of our numerical method by presenting a set of numerical experiments, covering both single and multi-class networks.

Stochastic Monotonicity of Markovian Multiclass Queueing Networks

“…Stochastic monotonicity of (Markovian) McQNs became relevant recently when, motivated by the lack of closedform stability conditions, the authors investigated simulation-based methods for approximating stability regions (Leahu and Mandjes 2017). 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.…”

“…is finite, and we extend this notation to bounded linear operators U defined on (subspaces of) S ) , namely U : sup{|Ux| : x ≤ 1}. On S ) , we denote the natural (component-wise) ordering Leahu andMandjes: Stochastic Monotonicity of Multiclass Queueing Networks Stochastic Systems, 2019, vol. 9, no.…”

“…• For any (k, l) ∈ 7, h (k,l) (ξ) gives the transition rate corresponding to a (k, l)-transition from state ξ, and f (k,l) (ξ) denotes the state-space transform after performing a (k, l)-transition from ξ. Leahu andMandjes: Stochastic Monotonicity of Multiclass Queueing Networks Stochastic Systems, 2019, vol. 9, no.…”

“…It turns out that strong monotonicity does not carry over beyond the elementary framework, the main reason being that the same transition probabilities from a given pair of ordered states are not stochastically ordered (with respect to any sensible stochastic ordering that is compatible with ⊆ on X); hence, the results in Massey (1987) do not apply in this framework. Leahu andMandjes: Stochastic Monotonicity of Multiclass Queueing Networks Stochastic Systems, 2019, vol. 9, no.…”

Detecting Markov Chain Instability: A Monte Carlo Approach

“…A first paper to consider this approach is [24]. Then, concurrently to our work, [15] proposes a simulation based method for determining the stability region of a multiclass queueing network with respect to its arrival rate, when it is possible to verify that the stability region satisfies particular stochastic monotonicity properties. Given the variety of models and counter-examples discussed above, we place importance on the generality of settings to which our algorithm is suitable.…”