[Paper Review] Random walks in a queueing network environment

Mark Andrew Gannon, E. Pechersky+2 more|arXiv (Cornell University)|Oct 6, 2014

Advanced Queuing Theory Analysis16 references3 citations

TL;DR

This paper proposes exactly solvable models of continuous-time Markov processes where a random walk interacts with a queueing network environment, using detailed balance equations to derive stationary distributions. The key result is a product-form stationary distribution closely related to the well-known product-form formula in queueing theory, valid under reversibility and sub-criticality conditions.

ABSTRACT

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The tool is the detailed balance equations.

Motivation & Objective

• To develop exactly solvable models of a random walk interacting with a queueing network environment, where the walk's position influences and is influenced by the network's transition intensities.
• To establish conditions under which the joint process of the random walk and the queueing network admits a stationary distribution using detailed balance equations.
• To generalize existing product-form results in queueing theory by incorporating reversible dynamics between a distinguished particle (random walker) and a network of queues.
• To explore applications in communication networks, random trapping, localization, and condensation phenomena through a unified framework.
• To lay the groundwork for future extensions to non-reversible settings, processor-sharing rules, and infinite-volume configurations.

Proposed method

• Model the environment as a symmetric, homogeneous Jackson or Gordon–Newell (closed) queueing network with finite sites and symmetric transmission rates.
• Introduce a distinguished particle (random walker) whose movement is coupled to the queueing network’s state, with transition intensities depending on local queue sizes.
• Apply detailed balance equations (DBEs) to derive the stationary distribution of the joint process, ensuring reversibility.
• Construct the stationary distribution as a product-form expression involving geometric and generalized gamma-type terms, depending on the network’s parameters and the walker’s position.
• Use symmetry and irreducibility conditions on the transmission matrix B to ensure positive recurrence and existence of the stationary measure.
• Verify the stationary distribution by checking that the detailed balance equations hold for all allowed transitions, including arrivals, departures, jumps, and walker-induced state changes.

Experimental results

Research questions

• RQ1 Under what conditions does a random walk interacting with a queueing network environment admit a stationary distribution?
• RQ2 How does the interaction between the walker and the queueing network affect the form of the stationary distribution?
• RQ3 Can the stationary distribution be expressed in a product-form, and if so, what are the conditions on the transition intensities and network structure?
• RQ4 What role does reversibility play in enabling exact solvability of the joint process?
• RQ5 How do sub-criticality and symmetry in the network’s transmission matrix affect the existence and structure of the stationary distribution?

Key findings

• The stationary distribution for the open network model is given by a product of geometric distributions with parameter λ/μ, as in standard Jackson networks, when the sub-criticality condition λ/μ < 1 holds.
• For the closed network model with fixed numbers of walkers and tasks (|y| = M, |n| = N), the stationary distribution takes the form π(y,n) = 1/Ξ × ∏[γ̄_j(n_j)]^{y_j}, where Ξ is a normalizing constant.
• In the open-closed model with fixed total number of tasks N, the stationary distribution is π(y,n) = 1/Ξ_N,Λ × ∏[ξ_l/η_l × γ̄_l(n_l)]^{y_l}, with Ξ_N,Λ < ∞ ensuring integrability.
• The detailed balance equations are satisfied under the proposed transition rate structures, confirming the correctness of the derived stationary distributions.
• The stationary distributions are shown to be product-form, closely resembling classical product-form solutions in queueing networks, but extended to include the influence of a random walker.
• The models are proven positive recurrent (PRR) under the stated conditions, including irreducibility of the transmission matrix and convergence of the normalizing constants.